Liquid Crystal on Silicon Displays characterization for...

Liquid Crystal on Silicon Displays characterization for Diffractive Applications and for Holographic Data Storage

Francisco Javier Martínez Guardiola

http://www.eltallerdigital.com/

http://www.ua.es/

INSTITUTO DE FÍSICA APLICADA A LAS CIENCIAS Y LAS TECNOLOGÍAS

LIQUID CRYSTAL ON SILICON

DISPLAYS CHARACTERIZATION FOR

DIFFRACTIVE APPLICATIONS AND

FOR HOLOGRAPHIC DATA STORAGE

IN PHOTOPOLYMERS“Caracterización de pantallas LCoS para aplicaciones en óptica difractiva y

almacenamiento holográfico de información en fotopolímeros.”

Fco. Javier Martínez Guardiola

Memoria presentada para aspirar al grado de Doctorpor la Universidad de Alicante

"Doctorado en Física Aplicada a las Ciencias y las Tecnologías"Dirigida por Andrés Márquez Ruiz

y Sergi Gallego Rico

INSTITUTO DE FÍSICA APLICADA A LAS CIENCIAS Y LAS TECNOLOGÍAS

LIQUID CRYSTAL ON SILICON

DISPLAYS CHARACTERIZATION FOR

DIFFRACTIVE APPLICATIONS AND

FOR HOLOGRAPHIC DATA STORAGE

IN PHOTOPOLYMERS“Caracterización de pantallas LCoS para aplicaciones en óptica difractiva y

almacenamiento holográfico de información en fotopolímeros.”


Memoria presentada para aspirar al grado de Doctorpor la Universidad de Alicante

"Doctorado en Física Aplicada a las Ciencias y las Tecnologías"Dirigida por Andrés Márquez Ruiz

y Sergi Gallego Rico

Ph.D. Candidate Director Director

Alicante, June 2015

Liquid Crystal on Silicon Displays Characterization for Diffractive applica-tions and for Holographic Data Storage in PhotopolymersCaracterización de pantallas LCoS para aplicaciones en óptica difractiva y alma-

cenamiento holográfico de información en fotopolímeros.

Author: Fco. Javier Martínez GuardiolaAdvisor: Andrés Márquez RuizAdvisor: Sergi Gallego Rico

Text printed in AlicanteFirst edition, June 2015

Abstract

In this PhD Thesis I present some methods for characterizing PA-LCoSmicrodisplays. It is useful to fully characterize this type of devices forevaluating its performance required in different applications. We havetested its validity in different applications such as diffractive opticselements (DOEs). Finally we apply these microdisplays in a full holo-graphic data storage scheme using a photopolymer as holographic re-cording medium. We evaluate the capability of PVA/AA photopolymerfor this holographic data storage system that incorporates as a noveltya convergent correlator geometry.

Resumen

En esta Tesis presentamos diferentes métodos, parcial y completo, pa-ra la caracterización de pantallas PA-LCoS. La caracterización de estosdispositivos es útil para evaluar su rendimiento en las diferentes apli-caciones en las que pueda ser útil. También se evalúa la validez delmodelado en diferentes aplicaciones como elementos ópticos difracti-vos. Finalmente, aplicamos estos displays en un montaje completo parael almacenamiento holográfico de información donde se usa un foto-polímero como material de registro. También estudiamos la capacidady comportamiento de un fotopolímero en este tipo de sistema de alma-cenamiento de información, donde se utiliza una geometría novedosaen este tipo de aplicaciones conocida como correlador convergente.

Acknowledgements

This thesis is the result of the research work carried out during the lastfour years, supported by the Spanish Ministerio de Trabajo y Compet-itividad of Spain under projects FIS2011-29803-C02-01 and FIS2011-29803-C02-02 and by the Generalitat Valenciana under project PRO-METEO/2011/021.

I want to thank all the people that surrounds me, specialy to my ad-visors.

de verdad, gracias.


June 2015

Contents

List of Figures xi

Acronyms xiii

1 Resumen de la Tesis 11.1 Contexto de la Tesis Doctoral . . . . . . . . . . . . . . . . . . . . 1

1.2 Moduladores Espaciales de Luz y pantallas LCoS . . . . . . . . . 4

1.2.1 Física de los Cristales Líquidos . . . . . . . . . . . . . . 4

1.2.2 Birrefrigencia y Propiedades Ópticas . . . . . . . . . . . 7

1.2.3 Dispositivos de Cristal Líquido . . . . . . . . . . . . . . . 9

1.3 LCoS de Alineación Paralela . . . . . . . . . . . . . . . . . . . . 14

1.4 Aplicación en Óptica Difractiva y en Almacenamiento Holográfico

de Información . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 Objetivos Principales de la Tesis . . . . . . . . . . . . . . . . . . 20

1.6 Resultados presentados en las publicaciones . . . . . . . . . . . . 22

1.7 Conclusiones y desarrollos futuros . . . . . . . . . . . . . . . . . 26

2 Summary of the Thesis 292.1 Ph. D. Thesis Context . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 SLMs and Liquid Crystal on Silicon microdisplays . . . . . . . . 32

2.2.1 Liquid Crystal Physics . . . . . . . . . . . . . . . . . . . 32

2.2.2 Birefringence and Optical Properties . . . . . . . . . . . . 35

2.2.3 Liquid Crystal devices . . . . . . . . . . . . . . . . . . . 37

2.3 Parallel Aligned LCoS . . . . . . . . . . . . . . . . . . . . . . . 41

ix

CONTENTS

2.4 Application in Diffractive Optics and in Holographic Data Storage 432.5 Main Goals of this Thesis . . . . . . . . . . . . . . . . . . . . . . 472.6 Results presented in the publications . . . . . . . . . . . . . . . . 492.7 Conclusions and future developments . . . . . . . . . . . . . . . 52

3 Papers of this Thesis 553.1 Extended linear polarimeter to measure retardance and flicker: ap-

plication to liquid crystal on silicon devices in two working geo-metries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Electrical dependencies of optical modulation capabilities in digit-ally addressed parallel aligned liquid crystal on silicon devices . . 67

3.3 Retardance and flicker modeling and characterization of electro-optic linear retarders by averaged Stokes polarimetry . . . . . . . 79

3.4 Averaged Stokes polarimetry applied to evaluate retardance andflicker in PA-LCoS devices . . . . . . . . . . . . . . . . . . . . . 85

3.5 Predictive capability of average Stokes polarimetry for simulationof phase multilevel elements onto LCoS devices . . . . . . . . . . 99

3.6 [Unpublished] Exploring binary and ternary modulations on a PA-LCoS device for holographic data storage in a PVA/AA photopolymer109

4 Complete list of publications related with the Ph.D. Thesis 1294.1 Papers in Journals . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.2 Papers in Conferences . . . . . . . . . . . . . . . . . . . . . . . . 130

Bibliography 133

x

List of Figures

1.1 Transición desde el estado Cristalino hasta el de Líquido Isotrópico. 5

1.2 Vector Director mostrado por un Cristal Líquido (a) Cristal Líquidoformado por moléculas alargadas (b) LC formado por moléculas enforma de disco. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Mesofases de los Cristales Líquidos. . . . . . . . . . . . . . . . . 6

1.4 Deformaciones elásticas presentes en los LC (a) “Splay” (b) “Twist”(c) “Bend”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Propagación de la luz en un medio anisótropo. Las curvas , i e ii,están sujetas a diferentes índices de refracción. . . . . . . . . . . 9

1.6 Representación del la rotación helicoidal de 90o en las celdas Twis-ted Nematic (TN). . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7 Vista esquemática de los tres tipos de celdas con alineación paralela. 12

1.8 Estructura de un dispositivo LCoS de solo-fase. . . . . . . . . . . 13

1.9 Dominios típicos de modulación de amplitud compleja. (a) Modu-lación completa ideal (b) Solo Amplitud (c) Modulación Solo-Fasecon una profundidad de 2π (d) Modulación Solo-Fase con una pro-fundidad de π (e) Binario de Amplitud (f) Binario de Fase (g) Tern-ario (h) Amplitud y Fase acoplada . . . . . . . . . . . . . . . . . 18

2.1 Transition from Crystalline state to Isotropic liquid. . . . . . . . . 33

2.2 Director exhibited by liquid crystal (a) Liquid Crystal formed byrod-like molecules (b) LCs formed by disk-like molecules. . . . . 33

2.3 Phases of liquid crystals. . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Elastic deformations showed in LCs (a) Splay (b) Twist (c) Bend. . 35

xi

LIST OF FIGURES

2.5 Light propagation in an anisotropic medium. The curves, i and ii,are subject to different refractive index. . . . . . . . . . . . . . . 36

2.6 Representation of the 90o helical rotation of LC molecules in Twis-ted Nematic cells. . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7 Schematic view of the three different PA cells . . . . . . . . . . . 392.8 Structure of phase-only LCoS device. . . . . . . . . . . . . . . . 402.9 Typical complex amplitude modulation domains. (a) Ideal full

range of modulation. (b) Amplitude only. (c) Phase-only modula-tion with 2π depth (d) Phase-only modulation with π depth (e) Bin-ary amplitude (f) Binary phase (g) Ternary (h) Coupled amplitude-phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

xii

Acronyms

AM Active Matrix

BA Bend-Aligned

BER Bit Error Rate

BIM Binary Intensity Modulated

CCD Charge-Coupled Device

CMOS Complementary Metal Oxide Semiconductor

DOE Diffractive Optical Element

DoP Degree of Polarization

DVI Digital Visual Interface

ECB Electrically Controlled Birefringence

FPTV Front-Projection Television

HDSS Holographic Data Storage System

HPDLC Holographic Polymer Dispersed Liquid Crystal

HTM Hybrid Ternary Modulation

ITO Indium Tin Oxide

LC Liquid Crystal

xiii

ACRONYMS

LCD Liquid Crystal Display

LCoS Liquid Crystal On Silicon

LUT Look-Up Table

NPBS Non Polarizing Beam Splitter

PA Parallel Aligned

PA-LCoS Parallel Aligned Liquid Crystal on Silicon

PVA Polyvinyl Alcohol

PVA/AA Polyvinyl Alcohol-Acrylamide

PWM Pulse Width Modulation

RPTV Rear-Projection Television

SLM Spatial Light Modulator

SOP State of Polarization

SRAM Static Random Access Memory

STN Super Twisted Nematic

TFT Thin Film Transistor

TN Twisted Nematic

TV Television

VA Vertical Aligned

xiv

CHAPTER

1Resumen de la Tesis

1.1 Contexto de la Tesis DoctoralEn esta tesis se presenta un análisis sobre la utilización de los novedosas pantallas

de Cristal líquido de alineación paralela (PA-LCoS) en aplicaciones de óptica di-

fractiva y de almacenamiento holográfico de información (HDS). Se muestra cómo

se puede modelar, caracterizar y optimizar el dispositivo para su uso en las posibles

aplicaciones.

En esta tesis el dispositivo PA-LCoS actúa como Modulador Espacial de Luz

(SLM) con tres propósitos diferentes, cada uno de ellos de un gran interés en dis-

tintos campos de investigación en Óptica y Fotónica, así como en otros campos

de la Física y la Ingeniería. Por ejemplo, se muestra cómo puede actuar de con-

versor de estados de polarización (SOP), como un modulador de solo-amplitud, y

como modulador de solo-fase. Concretamente, los dispositivos PA-LCoS se están

usando ampliamente en Óptica difractiva y puede que tengan una gran repercusión

en el almacenamiento holográfico si la tendencia es utilizar páginas de datos con

una codificación multinaria de solo-fase. Ambas áreas de la Fotónica son el ob-

jeto de estudio de esta tesis, de esta forma se utiliza el conocimiento adquirido con

el modelado y la caracterización de los dispositivos PA-LCoS en la primera parte

1

1. RESUMEN DE LA TESIS

de la tesis, para aplicarlo en una segunda parte a un sistema de almacenamiento

holográfico de datos (HDSS).

Los dispositivos PA-LCoS son un buen ejemplo de la madurez alcanzada en la

tecnología de Cristal Líquido (LC), por ello es interesante presentar los aspectos

más relevantes de la física de los dispositivos basados en LC, centrándonos en el

manejo de la luz, que es el contexto en el que se desarrolla esta tesis. La tecnología

LC es conocida por el púbico en general debido a la gran cantidad de pantallas

existentes en el mercado de consumo. El desarrollo en el área comercial ha produ-

cido un crecimiento en las capacidades y la especialización de los dispositivos LC

para aplicaciones de Óptica y Fotónica. Para optimizar su uso en un campo más

sofisticado y exigente serán requisitos ineludibles: el adecuado modelado, carac-

terización y optimización de la pantalla. En el caso de la óptica difractiva, se está

experimentando un incremento continuo en el número de aplicaciones relevantes,

más si cabe desde que el campo se introdujo en la nano-escala.

En nuestro caso, en primer lugar estamos interesados en la utilización de la

pantalla PA-LCoS para mostrar elementos ópticos difractivos (DOE) programables.

Estos dispositivos ópticos posibilitan la realización de las mismas funciones que los

elementos ópticos convencionales, pero aplicando el fenómeno de la difracción en

vez de la leyes de la refracción y la reflexión. En realidad, estos habilitan más

posibilidades: por ejemplo pueden realizar simultáneamente varias acciones, como

la deflexión y enfoque del haz, algo que no es posible con los elementos ópticos

convencionales. La programación proviene del hecho de poder cambiar la función

óptica en tiempo real cuando el DOE se implementa en un SLM como es un PA-

LCoS.

En este breve repaso de los campos de investigación relacionados con este tra-

bajo, me gustaría destacar las grandes posibilidades que ofrece la Holografía, y

su integración total en las propiedades de amplitud y fase de los campos electro-

magnéticos. En particular, el almacenamiento holográfico de información (HDS)

es una de la aplicaciones más prometedoras, los sistemas de almacenamiento holo-

gráfico de información (HDSS) se han propuesto desde la invención del láser en

los años ’60. La competencia sufrida por otras tecnologías de almacenamiento,

normalmente más baratas y robustas, han relegado, hasta ahora, a los HDSS a los

2

1.1 Contexto de la Tesis Doctoral

laboratorios de investigación. Sin embargo estos presentan una densidad de alma-

cenamiento extrema, y la posibilidad de lectura en paralelo de una gran cantidad

de información. Esta capacidad de lectura en paralelo es una ventaja única en com-

paración con las tecnologías de almacenamiento óptico que se basan en una lectura

y escritura secuencial. Los elementos centrales de cualquier HDSS son el SLM,

que se encarga de introducir la página de datos, y el material de almacenamiento

holográfico, que es donde se almacena la información, estos dos aspectos son de

especial relevancia para nuestro grupo de investigación.

En el transcurso de la tesis se han desarrollado dos métodos para caracteri-

zar la pantalla PA-LCoS, y se han aplicado para resolver algunos problemas: el

primer método está basado en el polarímetro lineal clásico, y el segundo en po-

larimetría de Stokes. El primero, quizá más sencillo de implementar, ya que no

usa ningún ninguna instrumentación especial, ofrece una primera aproximación a

la caracterización. Se puede usar para obtener configuraciones personalizadas, así

como para seleccionar los formatos de direccionamiento más estables u optimizar

la rangos dinámicos de amplitud y/o fase para una aplicación dada. El segundo

método permite la caracterización completa, con los datos obtenidos y calibra-

dos se puede simular el rendimiento de la pantalla en una gran variedad de situa-

ciones. En particular, se ha probado su capacidad de predicción en la realización

de DOE multinarios de fase. Finalmente, se ha empleado la pantalla PA-LCoS en

un montaje completo de HDS, realizado en la última etapa de la tesis. Este montaje

utiliza la geometría de correlador convergente en el haz objeto, y un fotopolímero

que podemos fabricar nosotros mismos, y que se puede utilizar en varias aplica-

ciones óptica y fotónicas, una de ellas es utilizarlo como memoria holográfica. Con

este esquema en el laboratorio se compararán la modulación Binaria de Intensidad

(BIM) y una modulación pseudo-Híbrida Ternaria.

Esta tesis está sustentanda en cinco artículos que han sido publicados en una

serie de revistas internacionales en el “Journal Citation Report” del “ISI Web of

Knowledge”, todas con factor de impacto. Este es el contenido central que com-

pone esta Tesis Doctoral que presento como compendio de artículos. En la ac-

tualidad también tenemos un artículo relevante para la tesis, que trata sobre al-

macenamiento holográfico de información, en proceso de revisión por pares. Se

3


mostrarán los puntos más importantes tratados en dicho artículo, ya que corres-

ponde con la continuación natural de las cinco publicaciones anteriores y conecta

con el principal objetivo de interés de nuestro grupo de la Universidad de Alicante,

que tiene un importante bagaje en fotopolímeros como materiales de registro y en

memorias holográficas.

En las siguientes secciones, describiré la física implicada y la utilización de

pantallas de cristal líquido (LCD) como SLM, haciendo hincapié en la descripción

y características de las pantallas PA-LCoS. También se presentará el contexto para

su aplicación en óptica difractiva y en HDS. Después, se presentarán los objetivos

de la tesis, la descripción de los resultados conseguidos en las distintas public-

aciones. Y finalmente, se presentará un resumen de las conclusiones y posibles

desarrollos futuros.

1.2 Moduladores Espaciales de Luz y pantallas LCoS

1.2.1 Física de los Cristales Líquidos

Los materiales de cristal líquido se corresponden con una estado de la materia

que comparte las propiedades típicas tanto de los líquidos isotrópicos como de los

sólidos cristalinos [1]. Esto significa que los LC pueden fluir como un líquido

pero también muestran cierto orden estructural en la distribución de sus molécu-

las, algo análogo a la estructura cristalina que normalmente encontramos en los

sólidos. Los LC se pueden clasificar por la forma de sus moléculas constituyentes

en dos grandes grupos: Calamíticos y Discóticos. Los calamíticos están formados

por moléculas de una forma alargada mientras que los discóticos están formados

por moléculas en forma de disco. La mayoría de los LC interesantes para aplica-

ciones electro-ópticas son los formados por moléculas alargadas [2], estos muestran

unas propiedades anisotrópicas uniaxiales. Para un grupo de moléculas de LC, la

orientación media de sus ejes largos define una magnitud electro-óptica relevante

conocida como eje director, normalmente indicado por el vector ~n, que en estos

materiales uniaxiales se corresponde con la orientación media de todos los ejes

largos. Los LC calamíticos pertenecen a los LC de tipo termotrópico, lo cuales

presentan diferentes fases dependiendo de la temperatura: Son sólidos cristalinos a

4


bajas temperaturas y líquidos isotrópicos a altas temperaturas. En la figura 1.1 serepresenta la evolución con la temperatura de los LC termotrópicos. En la figura1.2 se muestra como se define el eje director en un LC de material calamítico ydiscótico.

Figure 1.1: Transición desde el estado Cristalino hasta el de Líquido Isotrópico.

Figure 1.2: Vector Director mostrado por un Cristal Líquido (a) Cristal Líquido form-ado por moléculas alargadas (b) LC formado por moléculas en forma de disco.

Los materiales calamíticos presentan varias fases intermedias (mesofases) [3]

5


conocidas como: fase esméctica, fase nemática y fase colestérica. Una mesofases

se define como un estado intermedio entre el estado cristalino y estado líquido. La

mayoría de los materiales utilizados en Óptica y Fotónica son compuestos nemáti-

cos, como en el caso estudiado en esta tesis.

Figure 1.3: Mesofases de los Cristales Líquidos.

En la figura 1.3 se ilustran las diferentes mesofases. En la fase nemática existe

un amplio rango de orden en la orientación de las moléculas pero no hay orden

en la posición de éstas, por lo tanto el eje director, ~n, mantiene la orientación en

un amplio volumen dentro del material. En la fase esméctica los ejes largos de

las moléculas son paralelos, pero las moléculas se ordenan en planos, con lo que

muestran tanto orden traslacional como orientacional. En la fase colestérica las

moléculas están agrupadas en capas, cada una de las capas está rotada con respecto

a las adyacentes, esto provoca que el vector director describa una hélice. La fase

colestérica se produce en los LC nemáticos pero formados por moléculas quirales

[3]. La helicidad del vector director puede ser dextrógiro o levógiro dependiendo

de la quiralidad de las moléculas.

Las propiedades ópticas más relevantes de los LC provienen de la anisotropía

que presentan las moléculas constituyentes. La constante dieléctrica del material

depende del eje considerado, donde el eje director indica la orientación del eje

extraordinario. Esta anisotropía óptica, o birrefrigencia, es lo que hace a los LC tan

interesante para las aplicaciones ópticas. La birrefrigencia se definirá y tratará con

más detalle en la sección 1.2.2.

6


Además de la anisotropía a frecuencias ópticas del campo electromagnético,

los LC también tienen otra propiedad interesante. Cuando se les aplica un campo

eléctrico o un voltaje, éste induce un momento eléctrico dipolar en las moléculas

nemáticas, y éstas se reorientan para alinearse con este momento dipolar inducido.

A esto se le conoce como reorientación inducida por campo eléctrico [4]. Esto sig-

nifica que el eje director cambia su orientación con el campo eléctrico aplicado, por

lo tanto un haz que atraviese la capa de cristal líquido verá un indice extraordinario

efectivo diferente, lo que modifica la diferencia de fase entre las componentes ordi-

naria y extraordinaria del campo eléctrico. Esto nos lleva a un cambio en el estado

de polarización conforme se atraviesa el LC. Utilizando esta propiedad podemos

fabricar dispositivos que, por ejemplo, controlen la respuesta óptica de una matriz

de píxeles controladas por voltaje. El primer dispositivo comercial que utilizó esta

tecnología fue una calculadora electrónica presentada por Sharp en mayo de 1973

[5].

Debido a este comportamiento tan especial, que comparte propiedades de lí-

quidos y cristales, en los LC se puede hablar de elasticidad y viscosidad. Si se

pretende cambiar la orientación de los LC aparecerán unas fuerzas que se opondrán

a la deformación. Estas fuerzas son muy débiles comparadas con las de los sólidos.

Normalmente para reorientar las moléculas se aplica un campo eléctrico con lo que

la orientación final estará definida por el equilibrio entre este campo eléctrico y

los momentos de fuerzas debido a las constantes elásticas. Cualquier deformación

se puede dividir en tres deformaciones básicas, denominadas: “splay”, “twist”and “bend”. Estas deformaciones se ilustran en la figura 1.4. Estas propiedades

viscoelásticas son muy importantes para determinar el comportamiento de los LC

cuando se les aplica un voltaje. El tiempo de respuesta del LC y el voltaje de control

necesarios estarán definidos por estas constantes [6].

1.2.2 Birrefrigencia y Propiedades Ópticas

Dado que los LC nemáticos son uniaxiales y anisótropos, se pueden definir dos

índices de refracción, uno a lo largo del eje director, n‖, y otro perpendicular a

dicha dirección, n⊥, con lo que la birrefrigencia óptica vendrá dada por:

7


Figure 1.4: Deformaciones elásticas presentes en los LC (a) “Splay” (b) “Twist” (c)“Bend”.

∆n = n‖ − n⊥ (1.1)

El eje paralelo al eje director es el eje extraordinario. Los LC presentan dife-rente índice de refracción dependiendo tanto del ángulo entre el eje director y ladirección del haz incidente como de la dirección de polarización del haz. Debido aesto, la birrefrigencia es la característica más evidente que hace a los LC útiles enÓptica y Fotónica. Esta propiedad permite rotar el plano de oscilación de la luz, esdecir, cambiar el estado de polarización (SOP) del haz incidente mientras atraviesael medio.

El vector campo eléctrico del haz de luz que interacciona con el LC puededescomponerse en dos componentes linealmente polarizadas: una a lo largo deleje director, y otra perpendicular a éste [5]. Si existe una diferencia de velocida-des entre las dos orientaciones, las dos componentes se verán afectadas de maneradistinta. Éste comportamiento está representado en la figura 1.5. Debido a la reo-rientación que se produce en las moléculas cuando se aplica un campo eléctrico, lainteracción de la luz con el material puede ser controlada por voltaje, con lo que elSOP a la salida variará según dicho voltaje.

Esta propiedad que nos permite seleccionar la birrefrigencia variando el voltaje,hace a los materiales de LC muy útiles para ser utilizados en pantallas, ya que lavariación del SOP se puede convertir en una modulación de intensidad con soloponer un polarizador lineal a la salida. Esta birrefrigencia seleccionable tam-bién es muy útil en Óptica y Fotónica, ya que nos permite cambiar localmente

8


Figure 1.5: Propagación de la luz en un medio anisótropo. Las curvas , i e ii, estánsujetas a diferentes índices de refracción.

las propiedades de un frente de onda, actuando así como Modulador Espacial de

Luz (SLM).

La gran demanda de dispositivos de LC en el mercado de las pantallas ha pro-

vocado un gran desarrollo en la tecnología de los cristales líquidos. Otro de los

grandes avances, que hay que destacar, ha sido el acceso a las técnicas litográficas

y de la tecnología de semiconductores, que han permitido la miniaturización de los

dispositivos [7].

Estos avances son los responsables de que las pantallas de cristal líquido (LCD)

sean unos dispositivos muy asequibles. En los ’80 y principios de los ’90 el interés

por utilizar estas pantallas de LC como SLM se amplió enormemente. Hay muchos

artículos importantes de esos años en los que se analiza el uso de pantallas de LC

comerciales como SLM [8–12].

En la siguiente sección se hará un breve repaso de la tecnología y su evolu-

ción, desde los primeros desarrollos hasta los más recientes dispositivos de Cristal

Líquido sobre Silicio (LCoS).

1.2.3 Dispositivos de Cristal Líquido

Hay diferentes tipos de pantallas y estructuras para los dispositivos de LC. Desde

que el primer aparato fuera introducido al mercado en 1970, por Martin Schadt en

Suiza [13], la evolución ha avanzado rápidamente.

Si centramos nuestra atención en el control electrónico, el primer LCD fue uno

con direccionamiento directo, es decir, un dispositivo con varios displays de 8 seg-

9


mentos, con lo que se aplicaba el voltaje de manera independiente a cada segmento.

Justo después de la aparición del primer dispositivo, en 1971, se presentó el con-

cepto de matriz direccionada para televisión (TV) que utilizaba LC [14]. Lechner et

al. presentaron una pantalla de 2x18 píxeles [14], y Takata et al., justo un año des-

pués, una pantalla de 120x120 píxeles [15]. En la pantalla de Takata se utilizó un

grabado de las caras internas de la celda de cristal líquido para alinear las molécu-

las, los autores aseguran que esta técnica de alineamiento reduce la interferencia

eléctrica (“crosstalk”) entre píxeles.

Es importante remarcar que para poder mostrar una imagen, es necesario tener

la posibilidad de direccionar una matrix completa de celdas de LC. Obviamente

el direccionamiento directo no puede ser la solución cuando el número de píxeles

excede de cierto valor, con lo que se utiliza un direccionamiento multiplexado:

la imagen se muestra línea a línea utilizando unas señales de sincronismo tanto

verticales como horizontales. En este sentido, se ha desarrollado desde el principio

el concepto de control por Matriz Activa (AM), utilizado para mandar la imagen a

la pantalla mediante un control electrónico.

Ya en 1975, T. Peter Brody presentó la primera matriz que utilizaba transistores

de capa delgada (“Thin Film Transistors”, TFT) y técnicas de la microelectrónica

[16]. Esta pantalla no utilizaba LC pero fue un primer paso hacia la miniaturización

de las pantallas, y representó una ruptura tecnológica.

Otro punto de interés son las distintas geometrías que se pueden encontrar en

las celdas de LC. Es interesante conocer cómo se ordenan y alinean las moléculas

dentro de la celda de LC. En su estado nemático natural las moléculas están ali-

neadas con sus ejes longitudinales prácticamente paralelos los unos a los otros. Si

grabamos unos microsurcos en las capas de alineación de las caras interiores de la

celda, las moléculas se alinearán en la dirección de estos microsurcos. Las capas

siguientes del LC se alinearán en esa dirección de pulido por la interacción con las

moléculas anteriores. Pero si encerramos el material de LC entre dos vidrios donde

los microsurcos se han realizado en direcciones ortogonales, forzaremos a los LC

a rotar su dirección esos 90o, formando una hélice. En la figura 1.6 podemos ver

una representación esquemática de este modo llamado “Twisted Nematic” (TN). Es

importante señalar que el momento eléctrico dipolar inducido no tiene un sentido

10


predeterminado, con lo que se debe dar una dirección preferente para dicha reo-

rientación. Esto se realiza generando un pequeño ángulo de inclinación (Θ) en la

capa de alineación con respecto al plano del vidrio.

Figure 1.6: Representación del la rotación helicoidal de 90o en las celdas TwistedNematic (TN).

Este modo TN fue el usado en el primer dispositivo LCD de Schadt [13]. Otro

modo de operación utilizado en pantallas es el llamado “Super Twisted Nematic”

(STN). En el modo STN las moléculas se giran más de 90o, normalmente 180o,

240o o 270o. Este modo STN fue introducido para mejorar el rendimiento de los

LCD sin usar TFT, y también mejora el ángulo de visionado [3]. Para conseguir

estos ángulos de giro el LC nemático tiene que tener un ángulo intrínseco de rota-

ción entre las capas, por lo tanto se utilizan normalmente LC colestéricos, además

del pulido en las ventanas de vidrio.

El principal problema de los modos TN y STN es que la fase y la amplitud están

acopladas, por lo que no es posible cambiar de manera independiente la amplitud

11


y la fase. Para las aplicaciones de pantallas esto no es importante ya que el ojo noes sensible a la fase, pero para su uso como SLM esto es algo tedioso de manejar[17].

Otra configuración de los LCD es la de alineación paralela. En este tipo de dis-positivo las moléculas de LC están alineadas en la misma dirección en las dos caras.Con esta configuración podemos hacer modulación de solo-fase si el haz incidenteestá polarizado linealmente y a lo largo del eje director. Son normalmente cono-cidas como Alineación Paralela (PA) o dispositivos de Birrefrigencia ControladaEléctricamente (ECB).

Se pueden encontrar distintas configuraciones en este método como con lasde alineación paralela, las de Alineación Vertical (VA), y de Alineación Combada(Bend, BA).

Figure 1.7: Vista esquemática de los tres tipos de celdas con alineación paralela.

En la figura 1.7 se pueden ver las diferencias entre las tres configuraciones. Enla PA y la BA la única diferencia es el ángulo de pre-inclinación alterno entre lasdos caras de la celda.

Los primeros intentos de utilizar los dispositivos LC como SLM se realizaroncon los dispositivos TN desponibles en las pantallas de TV comerciales [8–12,17–19]. En la actualidad han aparecido nuevos dispositivos que mezclan los dosprincipales avances introducidos en la sección anterior: Estos son el uso de losLC para la modulación del frente de onda, y las técnicas utilizadas en la industriamicroelectrónica para producir dispositivos más sofisticados, que recordemos fueintroducido en 1975 [16].

Gracias a la combinación de los importantes avances en estos dos campos, sepudieron desarrollar los dispositivos de Cristal Líquido sobre Silicio (LCoS). És-tos son unos dispositivos reflectivos que utilizan la potente y extendida tecnología

12


microelectrónica para reducir el tamaño del píxel e integrar el control de la matriz

en el mismo substrato de Silicio. El Silicio puede ser pulido y metalizado para

obtener un muy buen espejo. A este substrato de Silicio se le añade una capa de LC

que puede ser usado como SLM. La tecnología LCoS fue inicialmente desarrollada

para sistemas de televisión de proyección trasera o delantera (FPTV, RPTV) ya que

se podía conseguir una muy alta resolución [20]. A pesar de la caída del mercado

de este tipo de pantallas, los dispositivos PA-LCoS son unos componentes muy ver-

sátiles y pueden ser usados en muchas aplicaciones como modulador de solo-fase

[20, 21].

Figure 1.8: Estructura de un dispositivo LCoS de solo-fase.

En la figura 1.8 se puede ver la estructura básica de un dispositivo LCoS. La

estructura de un SLM formado por una capa de LC sobre una capa metalizada de

aluminio que forma la estructura de la matriz de píxeles. La tensión en la capa

de aluminio está controlada por la electrónica contenida en el substrato de Silicio.

El substrato utiliza tecnología CMOS (CMOS es el acrónimo de “Complementary

Metal-Oxide Semiconductor”). El dispositivo mostrado en la figura 1.8 es un dis-

positivo de solo-fase ya que la capa de LC tiene alineación paralela, por lo tanto

podremos utilizarlo como un dispositivo ECB.

La integración de una circuitería de direccionamiento de alto rendimiento per-

mite un control más preciso del voltaje aplicado a cada píxel, así como se puede

optimizar el factor de llenado (“fill factor”). Al ser reflectivo se puede conseguir

una mayor profundidad de modulación de fase con el mismo grosor de LC [21].

En resumen, las ventajas de los LCoS reflectivos provienen de la capacidad

de integración de la tecnología CMOS, que es capaz de producir también unos

13


espejos de mucha calidad. Permiten un tamaño de píxel pequeño, y resaltar que lacircuitería de direccionamiento puede ser integrada en el propio chip de Silicio.

En óptica difractiva, una buena característica para un SLM es que actué soloen régimen de fase, ya que modular en amplitud siempre implica una pérdida deenergía, y también hay aplicaciones que necesitan modular solo la fase como enmetrología óptica [22], interconexiones reconfigurables [23, 24], sensado de frentesde onda de haces de luz estructurada [25], trampas ópticas holográficas [26], y unlargo etcétera. Estos dispositivos PA-LCoS son el último avance disponible en elmercado, por lo que será útil conocer sus problemas y rendimiento en diferentesaplicaciones.

Uno de los objetivos de esta tesis será utilizar un dispositivo comercial PA-LCoS como SLM, por lo que estos nuevos dispositivos necesitan ser caracteriza-dos de la manera más extensa posible. Por esta razón, en la siguiente sección sepresentan algunos aspectos de éstos.

1.3 LCoS de Alineación ParalelaEn esta tesis se utiliza un dispositivo PA-LCoS. Es importante presentar los requis-itos necesarios para un dispositivo de solo-fase, así como los problemas o limita-ciones que puedan tener. Por ejemplo, para usarse como dispositivo de solo-fasedebe ser capaz de modular el retardo al menos desde 0 a 2π. Para conseguir estoentran en juego tanto el grosor de la celda como los índices de refracción del LC.

Si obviamos el ángulo de pre-inclinación, y suponemos incidencia perpendic-ular, se puede encontrar que la diferencia en el retardo, entre las dos componenteslinealmente polarizadas del haz incidente (a lo largo del eje director y a lo largo deleje ordinario), vendrá dado por:

δ = 2π∆nd/λ (1.2)

done d es el grosor de la celda de LC, y ∆n es la birrefrigencia tal y como sedefinió en la ecuación 1.1. λ es la longitud de onda del haz incidente. Como elLCoS trabaja en reflexión, el retardo total será doble:

δ = 2π∆n2d/λ (1.3)

14

1.3 LCoS de Alineación Paralela

Por lo tanto, si la modulación del retardo se necesita que sea 2π, se puede

obtener que el grosor necesario será:

d = λ/2∆n (1.4)

Se concluye que en reflexión se puede usar una celda de LC más delgada. Esto

será una buena opción si necesitamos una respuesta en el cambio de fase rápida, ya

que sabemos que la respuesta en el tiempo depende directamente de “d2” [27, 28].

Y aún mejor, si solo necesitamos una modulación en fase de π, el grosor puede

reducirse a la mitad.

Dado que la tecnología utilizada en la fabricación de los LCoS puede hacer que

el pixel sea más pequeño, se podrán conseguir mayores ángulos de difracción [21].

Con esta técnica el factor de llenado puede ser optimizado gracias a la resolución

que es capaz de alcanzar la tecnología microelectrónica.

La tendencia en la industria de las pantallas ha sido producir dispositivos di-

reccionados digitalmente, ya que estos ofrecen un control más preciso, repetible y

robusto que las señales analógicas. Un problema común que presenta este direc-

cionamiento es la fluctuación, o “flicker”, en la modulación del haz de salida. Esto

es debido a la naturaleza pulsante de las señales digitales. En los dispositivos con

direccionamiento digital el voltaje se define mediante una señal de Modulación de

Anchura de Pulso (PWM). La viscosidad del LC limita las fluctuaciones, pero las

moléculas siguen apreciando una variación residual en la señal de tensión aplicada.

Esto produce una fluctuación en la orientación de las moléculas de LC del orden de

milisegundos, y afecta a la respuesta óptica produciendo, por ejemplo, una reduc-

ción en la eficiencia [29, 30]. Otros autores han reducido esta fluctuación enfriando

el dispositivo ya que una menor temperatura incrementa la viscosidad [31].

Ahora sabemos que las pantallas con direccionamiento digital presentan algún

nivel de “flicker”. Debido a esto, será necesaria una medida precisa del “flicker”.

Por lo tanto, a lo largo de esta tesis se han desarrollado algunos métodos para eval-

uar dicho “flicker”. La caracterización de estas fluctuaciones es importante para

conocer la utilidad del dispositivo en las distintas aplicaciones. Se verá cómo en

los DOEs más complejos el efecto del “flicker” es menor. Tanto las pantallas analó-

gicas como las digitales presentan otros problemas como los efectos anamórficos,

15


esto es, que la respuesta es diferente dependiendo de la orientación (filas o colum-

nas) de la variaciones en la señal de voltaje que hay que enviar a la pantalla para

formar la imagen [32]. Esto es debido a la estructura de la señal multiplexada

para filas y columnas (necesaria para mostrar la imagen): el ancho de banda es

diferente para filas que para columnas. Otro efecto no deseado es conocido como

efecto “fringe field”, que está producido por la interferencia entre píxeles vecinos,

y es debido a la distribución del campo eléctrico causado por las distintas tensiones

aplicadas en el LC [33].

Los LCoS digitales pueden utilizar distintos esquemas de direccionamiento

[34]. Estos esquemas mandan la información (o voltaje) con una señal de tipo

PWM o similar. Por lo tanto las fluctuaciones serán distintas para cada uno de

estos esquemas. Algunas pantallas, como la utilizada en esta tesis, puede confi-

gurarse para diferentes aplicaciones, y longitudes de onda, solamente con cambiar

este esquema de direccionamiento, y cada uno de los distintos esquemas presentará

fluctuaciones distintas.

Es importante, para las distintas aplicaciones, medir la amplitud de las fluc-

tuaciones causadas por el circuito activo utilizado [35], ya que, por ejemplo, el

“flicker” causa inestabilidades en la posición del haz en pinzas ópticas [26]. Por

esto, la medida de este “flicker” es uno de los principales objetivos de esta tesis.

Dependiendo de la aplicación, hay muchos regímenes de modulación interesan-

tes, y muchas opciones que pueden ser configuradas. Por lo tanto, se necesita un

profundo análisis de todas estas características, y de cómo afectan al rendimiento y

al “flicker” las distintas opciones de configuración.

El SOP del frente de onda es importante para algunas aplicaciones específicas

[36], por este motivo es necesario buscar una caracterización de la pantalla basada

en el SOP, y medir cómo el “flicker” afecta a éstas. Con la ayuda de un polarímetro

se ha desarrollado un método que permite predecir el SOP a la salida, así como la

depolarización producida por el “flicker”.

16

1.4 Aplicación en Óptica Difractiva y en Almacenamiento Holográfico deInformación

1.4 Aplicación en Óptica Difractiva y en Almacena-miento Holográfico de InformaciónLas micropantallas basadas en LCoS se han hecho muy populares en óptica di-

fractiva. Hay muchos grupos de investigación trabajando con estos dispositivos en

multitud de aplicaciones.

Una de estas interesantes aplicaciones es un sensor holográfico. Messer y otros

[37] desarrollaron un sistema que puede adaptar la onda de referencia para distintas

posiciones de un objeto sin utilizar lentes, la función de la lente es realizada por un

PA-LCoS. Esto permite cambiar dinámicamente y adaptarse a distintas distancias

del objeto sin necesidad de partes mecánicas.

Otra aplicación muy interesante es el uso del SLM en litografía holográfica,

donde se utilizan para fabricar microestructuras 3D en un sustrato [38]. En esta

aplicación el SLM se utiliza para generar digitalmente el holograma que corres-

ponde con una micropirámide. El SLM permite utilizar múltiples puntos de láser

focalizados a distintas distancias. Jenness y su equipo presentan una herramienta

efectiva para realizar microfabricación tridimensional de cualquier geometría y en

paralelo [38].

Además, la capacidad de los LCoS de manejar grandes intensidades de luz

permite que sean utilizados en aplicaciones con láseres de alta potencia, como es el

caso de la ablación de microestructuras en paralelo con matrices de varios puntos

[39].

Desde la introducción en 1996 por el equipo de Hayasaki de la posibilidad de

utilizar el SLM como pinzas ópticas se ha intensificado el uso de los SLM basados

en LC [20]. De hecho, ya en 1998 el grupo de Dufresne presentó un método para

crear múltiples pinzas ópticas a partir de un solo haz láser, utilizando elementos de

óptica difractiva [40].

Se pueden encontrar muchas más aplicaciones, tales como óptica adaptativa,

proyección holográfica y más, hay un breve resumen muy interesante realizado por

Colling et al. [41].

Finalmente, queremos destacar el uso de SLM y especialmente los basados

en LC para almacenamiento holográfico de información. En esta aplicación el

SLM tiene el papel de servir como punto de entrada de datos, es decir, se utiliza

17


para modular la señal óptica en varios regímenes, como por ejemplo: Binario deAmplitud [42], Binario de Fase, Modulación Ternaria, etc. [43].

Si queremos usar estos dispositivos de LC como SLM tendremos que tener encuenta la capacidades de modulación del dispositivo, en la figura 1.9 se presentanlos dominios complejos de amplitud que pueden usarse para distintas aplicaciones.

Figure 1.9: Dominios típicos de modulación de amplitud compleja. (a) Modulacióncompleta ideal (b) Solo Amplitud (c) Modulación Solo-Fase con una profundidad de2π (d) Modulación Solo-Fase con una profundidad de π (e) Binario de Amplitud (f)Binario de Fase (g) Ternario (h) Amplitud y Fase acoplada

La figura 1.9 muestra los diferentes esquemas que pueden realizarse o que serí-an útiles. Algunos de estos estarán disponibles con distintos dispositivos LC, peroes importante conocer la aplicación para seleccionar el dispositivo adecuado.

La figura 1.9(a) muestra el caso ideal donde la amplitud y la fase son totalmenteconfigurable, este caso es imposible de conseguir ya que solo existe un parámetrode control, el voltaje. Los otros dominios podrán ser aplicados o seleccionadosdependiendo de la aplicación.

La utilización de PA-LCoS como SLM para reproducir DOEs, que se puedanusar como página de datos, juega un papel muy importante en los HDSS y todoel desarrollo realizado por la industria en los LCoS abre nuevas posibilidades dereducir el tamaño del píxel, junto con el incremento del número de éstos, comoconsecuencia se abre una vía para incrementar la densidad de datos que se puedealmacenar. El HDS han sido una tecnología muy prometedora desde hace varios

18

1.4 Aplicación en Óptica Difractiva y en Almacenamiento Holográfico deInformación

años [44, 45], y se han realizado muchos esfuerzos para controlar y mejorar los

HDSS como se puede ver en los manuales de Coufal et al. [43] y más reciente-

mente por Curtis et al. [46]. Debido a los grandes avances en este área se han

desarrollado algunos prototipos comerciales [47], incluso un buen grupo de com-

pañías multinacionales [48, 49] están interesadas actualmente en un sistema de

archivo de larga duración [50]. Desde el surgimiento de la idea del almacenami-

ento holográfico ha habido varios problemas que han ralentizado el proceso de

expansión de esta tecnología, como son las fuentes de luz de bajo rendimiento, los

generadores de páginas de datos o SLM, las cámaras, el sistema óptico, el material

de registro o la arquitectura de multiplexado [43, 46]. Algunos de estos problemas

han sido parcial o completamente solucionados. El progreso continuo producido en

la tecnología de los SLM, donde los microdispositivos PA-LCoS han sustituido a la

tecnología LCD anterior en la mayoría de las aplicaciones de óptica y fotónica, nos

hacen explorar la posibilidad de implementar un HDSS utilizando este dispositivo.

En este sentido, el sistema completo utilizado en esta tesis ha sido diseñado

utilizando un PA-LCoS como entrada de página de datos. Con este dispositivo se

han implementado diferentes esquemas de modulación, o formas de codificar la

información. De los presentados en la figura 1.9, en nuestros experimentos hemos

implementados la modulación en binario de intensidad (BIM) (donde solo se ne-

cesitan dos niveles de intensidad), figura 1.9(e), y una modulación híbrida ternaria

(HTM), figura 1.9(g). En el HTM se necesitan tres valores de nivel de gris, dos de

ellos con un alto e igual nivel de intensidad transmitida y con un desfase relativo de

180o, los niveles ON, el tercer nivel será el nivel OFF que tendrá un nivel de trans-

misión de intensidad bajo. Otros esquemas como el binario de fase, mostrado en

la figura 1.9(f), tiene la ventaja de tener más intensidad disponible, pero el proceso

de detección es más complicado, ya que no consiste solo en recuperar un plano de

intensidad en una cámara CCD.

Siguiendo con el diseño y la caracterización de este HDSS, es importante notar

que hemos elegido un correlador convergente como alternativa al procesador 4-

f, que es el más usado en plataformas de memorias holográficas. El correlador

convergente permite un escalado sencillo de las dimensiones de la transformada de

Fourier. Adicionalmente se ha comparado este sistema con el 4-f clásico [51, 52].

19


En último lugar, pero no por ello menos importante, el material de registro

utilizado en el HDSS es un elemento clave, como se puede ver en la referencia

[46]. Existen algunos requisitos especiales para los materiales de registro holo-

gráfico para que se pueden usar como memorias holográficas, como un encogi-

miento inferior al 0.5% [53], un elevado grosor, alta resolución, bajo nivel de

ruido, etc. En este sentido los fotopolímeros son unos materiales muy atractivos

para esta aplicación, muestran unas buenas cualidades ópticas, escasas perdidas,

bajo “scattering”, y una alta modulación de índice de refracción. En el campo

de los fotopolímeros, nuestro grupo de investigación tiene una amplia experiencia

en la preparación de fotopolímeros gruesos para almacenamiento holográfico de

información [54, 55]. En concreto, en esta tesis un polímero basado en polivinil-

alcohol-acrilamida (PVA/AA) se ha utilizado como material de registro. Sobre este

material el grupo tiene algunos estudios sobre multiplexación de hologramas y con

distintos tipos de modulación [56, 57].

1.5 Objetivos Principales de la TesisDebido a la novedad que suponen los dispositivos LCoS, el primer objetivo será

la caracterización del PA-LCoS utilizado. Como se ha visto esto debe realizarse

para poder controlar la configuración del dispositivo de entrada de datos de nuestro

HDSS. Este HDS comienza una nueva línea de investigación en memorias holo-

gráficas utilizando un dispositivo novedoso y por reflexión. Otro punto novedoso

es la utilización de la geometría de correlador convergente.

Necesitamos un método fácil y rápido de conocer la posibilidades de cada una

de las configuraciones que el fabricante pone a nuestra disposición. El dispositivo

tiene muchas opciones, pero ninguna de ellas están caracterizadas ópticamente. En

esta tesis se ha desarrollado un método de caracterización que puede ayudarnos

a configurar el dispositivo para una aplicación específica, o incluso un rango de

modulación específica.

La caracterización del SOP a la salida es muy útil para aplicaciones de óptica

difractiva. Sería interesante tener un modelo lo suficientemente preciso y con capa-

cidad de predicción. En esta tesis se ha realizado, con ayuda de un polarímetro, un

20

1.5 Objetivos Principales de la Tesis

completa caracterización basada en el SOP. Se ha encontrado una forma de cono-

cer el “flicker” y el retardo para cada nivel de gris. Se muestra cómo se puede

conseguir esto y cómo se usa para predecir el comportamiento del dispositivo en

aplicaciones de óptica difractiva.

El método propuesto trata de ser lo más general posible, de hecho, puede ser ap-

licado para cualquier dispositivo retardador que presente algún tipo de fluctuación

o variación del retardo introducido.

Todos los métodos propuestos han sido validados y probados, y se ha realizado

de distintas formas.

Después de que el PA-LCoS haya sido modelado y caracterizado, se ha aplicado

en un completo HDSS, en el que hace la función de entrada de información, y el

material de registro utilizado el bien conocido y caracterizado por nuestro grupo

PVA/AA.

Los artículos que componen esta tésis pueden encontrarse en el Capítulo 3. Los

principales objetivo cubiertos por cada artículo son:

• En la Sección 3.1 y 3.2 se presenta un método de caracterización basado en

el polarímetro clásico, tradicionalmente utilizado para retardadores lineales.

Se ve cómo este método falla con los retardadores lineales que presentan

fluctuaciones, y cómo se puede corregir para obtener información sobre la

amplitud de la fluctuación. El método se presenta y valida mediante las me-

didas en tiempo real de las fluctuaciones en el retardo.

• En el artículo presentado en la Seccción 3.2 se utiliza el método extendido

para generar configuraciones digitales personalizadas que pueden ser útiles

en aplicaciones concretas.

• En la Sección 3.3 presentamos un método de caracterización basado en la

polarimetría de Stokes. Mostraremos cómo se pueden predecir los estados

de polarización a la salida y cómo caracteriza completamente el dispositivo.

Podemos conocer el retardo medio y la amplitud de las fluctuaciones para

todos los niveles de gris. Este método se valida utilizando su capacidad de

predicción: Puede predecir el SOP a la salida y el grado de despolarización

(DoP).

21


• La Sección 3.4 muestra un artículo donde se aplica el método de Mueller-

Stokes para diferentes esquemas de direccionamiento y diferentes geometrías

del montaje óptico, mostrando que el método sigue siendo válido y útil en

una gran variedad de situaciones.

• En la Sección 3.5 se aplica el método presentado en la sección 3.3 a un e-

lemento difractivo como son las redes “Blazed”. Se muestra que tiene una

buena capacidad de predicción incluso con elementos difractivos más com-

plejos.

• En la Seccción 3.6 utilizamos el PA-LCoS como entrada de datos en un com-

pleto HDSS. En este artículo, que ha sido enviado a Optics Express y que

sigue bajo revisión en estos momentos, se mezclan tres elementos novedosos

como son: el PA-LCoS utilizado en HDSS, el correlador convergente en el

haz objeto, y el fotopolímero PVA/AA.

Una breve descripción de los resultados obtenidos se presenta en la sección 1.6.

Esta descripción tiene la intención de servir como guía de lectura y mostrar los

principales puntos de intereses tratados en esta tesis.

Finalmente en la sección 1.7 se discutirán los objetivos conseguidos y algunos

desarrollos futuros que se podrían afrontar.

1.6 Resultados presentados en las publicacionesPara conseguir los objetivos propuestos en esta tesis el primer paso será carac-

terizar la pantalla PA-LCoS. Ésta actúa como un retardador lineal sintonizable,

por lo que la primera aproximación será aplicar el método clásico utilizado para

retardadores lineales fijos. Se ha observado que este método tiene sus limitaciones

cuando se aplica a retardadores que presentan fluctuaciones, pero demostramos

que esas limitaciones, bajo ciertas condiciones, pueden servirnos para medir la

amplitud de dicha fluctuación sin utilizar ningún tipo de equipamiento especial que

no pueda ser encontrado en una laboratorio de óptica.

El método presentado en la referencia [58] se aprovecha de los puntos extremos

de la característica retardo en función del nivel de gris. En estos puntos extremos

22

1.6 Resultados presentados en las publicaciones

el retardo es conocido (será múltiplo de 180o). Sabiendo esto encontramos una

relación entre la diferencia entre el retardo real en estos puntos (180o o 0o) y el

valor de retardo medido. La relación encontrada es una relación lineal en un gran

rango de amplitudes de fluctuación, por lo que se hace sencillo corregir los datos

de retardo en función del nivel de gris medido. Y además de esto, tenemos una

estimación de la amplitud del “flicker” para la configuración digital utilizada, lo

que nos permite comparar el rendimiento de las distintas configuraciones que nos

proporciona el fabricante.

Adicionalmente al trabajo anterior, aprendimos cómo sacar provecho del “soft-

ware” provisto por el fabricante para generar nuestras propias configuraciones.

Para esto es necesario un profundo estudio y conocimiento del dispositivo. En-

tender los parámetros que el fabricante nos permite configurar es algo crucial, por

esta razón presentamos los parámetros Vbright y Vdark. Estos son, respectivamente,

los valores alto y bajo del voltaje aplicado a la celda de LC para una determinada

secuencia digital. Debido a la naturaleza pulsante de la señal digital es fácil pensar

que una diferencia menor entre Vbright y Vdark implicará una menor amplitud de

la fluctuación. Estos voltajes definen también el rango de voltaje aplicado en una

configuración, y dado que hay una relación entre el voltaje aplicado y el nivel de

gris, estos voltajes definirán el rango de retardo disponible para una configuración

digital concreta. La relación entre el voltaje aplicado y el nivel de gris se modula

por medio de una curva de gamma, que también es configurable en estos disposi-

tivos. Las curvas de gamma definen la relación entre el nivel de gris y el voltaje

aplicado a la celda, ésto lo hacen asignando la secuencia digital que debe ser envi-

ada a la celda para cada uno de los niveles de gris. La forma de las curva de gamma

definirá la forma del retardo introducido por el dispositivo en función del nivel de

gris.

Con el método del polarímetro extendido y los conceptos anteriores en mente,

se ha desarrollado un método para linealizar la respuesta del dispositivo, o incluso

crear una nueva configuración que se adapte mejor a una aplicación concreta [59].

El método utiliza la corrección presentada en el artículo anterior [58] y genera

la curva de gamma apropiada para la linealización necesaria. Esta linealización

se puede realizar en intensidad o en retardo. También se presenta cómo elegir

23


los valores de Vbright y Vdark apropiados y cómo afectan estos al rango de retardo

disponible.

Como se presentó en las secciones anteriores es útil conocer como afecta el dis-

positivo al SOP. En este sentido se prueba un proceso de caracterización utilizando

la polarimetría Mueller-Stokes. El polarímetro utilizado en esta parte de la tesis es

uno basado en lámina rotante. En los anteriores trabajos vimos cómo la frecuen-

cia de las fluctuaciones están en el rango de los kHz, por lo que es imposible que

nuestro polarímetro sea capaz de seguir la evolución temporal de las fluctuaciones

en el SOP. Pero hemos encontrado que usando unas medidas promediadas se puede

calcular la amplitud del “flicker” y el retardo medio introducido para cada nivel de

gris. Esto nos permite una caracterización completa del dispositivo. Este método

se presenta en la referencia [60].

Dada la importancia de este método de caracterización completa, se necesita un

estudio más profundo. Queríamos saber si el método es aplicable a otras config-

uraciones eléctricas provistas por el fabricante, y si se ve afectado por la geometría

del sistema óptico montado en el laboratorio. En nuestro caso se utilizan varias

configuraciones: incidencia perpendicular a la pantalla (con la ayuda de una cubo

separador no polarizante, NPBS), con incidencia cuasi-perpendicular para separar

el haz incidente del reflejado, y con una incidencia a 45o, para poder conocer lo

límites y posibilidades del montaje y del PA-LCoS en las distintas geometrías de

trabajo. Encontramos que el método de promediado de los parámetros de Stokes es

robusto, presenta una buena repetitividad y reproducibilidad, por lo que podemos

concluir que el método es útil en muchas situaciones y diferentes configuraciones

de montaje óptico [61].

De las medidas obtenidas para caracterizar el dispositivo se puede predecir cual-

quier SOP a la salida, siempre que se conozca el SOP del haz incidente podremos

calcuar el SOP y el grado de despolarización (DoP). Estos resultados se presentan

en la referencia [60]. De todas formas, se ha ido más lejos en la capacidad de

predicción y se ha comprobado con DOE más complejos como las redes “blazed”.

En la referencia [62] mostramos cómo el modelo predice la eficiencia en difrac-

ción para redes blazed multinivel, y no solo predice la perdida de eficiencia, si no

que también predice la amplitud de la fluctuación en el primer orden de difracción.

24

1.6 Resultados presentados en las publicaciones

Por lo tanto tenemos una predicción tanto estática como dinámica para DOEs com-

plejos. Gracias a este trabajo se ha observado que: conforme se eleva el número de

niveles de cuantización la degradación debida al “flicker” se reduce. De hecho se

reduce, hasta el punto de que la eficiencia es similar a la ideal. Este trabajo también

muestra que hay que tener en cuenta otras fuentes de degradación de la señal como

son el “fringing field”, que se observa claramente cuando son pocos los niveles de

cuantización.

Todo el conocimiento acumulado sobre los PA-LCoS durante la realización de

esta tesis ha sido muy útil a la hora de diseñar el HDSS. Con todas las técnicas

aprendidas, podemos configurar el HDSS para utilizar distintas codificaciones en

las páginas de datos. El último artículo que se ha usado en esta tesis esta enviado a

Optics Express, pero está en proceso de revisión en estos momentos. Se ha incluido

para ilustrar mejor la utilidad de esta tesis.

En nuestro HDSS se ha probado la modulación binaria de intensidad (BIM) y la

modulación HTM. Está última está analizada y demostrada para LCD de tipo TN,

pero no había estudios con los dispositivos PA-LCoS, por lo que analizamos los

retos a los que hay que hacer frente. Detectamos problemas tanto teóricos como

experimentales cuando se trata de utilizar la codificación HTM. Mostramos que

una modulación puramente HTM no es posible con este tipo de dispositivo, pero

se ha desarrollado una modulación pseudo-HTM que sí se puede implementar con

nuestro dispositivo. Se ha diseñado un nuevo esquema que reduce el orden cero

de intensidad y mantiene un buen rendimiento en la reconstrucción de la página de

datos.

En este último artículo también se prueba la geometría del correlador conver-

gente, o correlador VanderLugt, que proporciona más flexibilidad para escalar la

transformada de fourier en el plano del material de registro [63], y permite la intro-

ducción de un desenfoque que incrementará la densidad de páginas almacenadas.

Se utiliza también un fotopolímero muy versátil como material de registro. Se ha

elegido este material debido a que tiene muchas propiedades que podemos cambiar,

lo que nos ofrece un gran potencial para futuros desarrollos.

25


1.7 Conclusiones y desarrollos futurosPara resumir y después de todo el trabajo realizado podemos extraer algunas con-

clusiones de esta tesis:

• Es importante conocer las características y el funcionamiento de los PA-

LCoS. Por esta razón, se ha desarrollado un método de caracterización fácil

de aplicar y sin necesidad de ningún equipo especial, que permite comparar

las diferentes configuraciones de direccionamiento. Obtenemos también el

rango de retardo disponible y tenemos una estimación de la amplitud de las

fluctuaciones, lo que nos permite seleccionar la configuración digital más a-

decuada a nuestra aplicación. Dado que hemos medido el “flicker”, el rango

de retardo y cómo varía éste en función del nivel de gris, podemos con-

cluir que las secuencias 5_5 tienen menos fluctuaciones, la linealidad y la

respuesta de retardo en función del nivel de gris dependen de la curva de

gamma, y el rango de retardo está relacionado con los valores de Vbright y

Vdark, la curva de gamma y la longitud de onda.

• El método ha sido validado y prueba que los valores estimados están muy

cercanos a los valores reales, no solo del retardo medio sino también de la

amplitud del “flicker”.

• Es obligatorio saber cómo ajustar el dispositivo para una aplicación específica.

En este sentido, se ha presentado un método para diseñar configuraciones

personalizadas. También presentamos una explicación más profunda de al-

gunos parámetros que pueden ser configurados por el “software” del fabric-

ante. Y como se ha dicho, este diseño se puede hacer sin necesidad de ningún

equipo que podría no estar disponible en un laboratorio de óptica.

• El método de caracterización basado en la extensión del método clásico es

muy potente pero incompleto. Pensando en la importancia de conocer el SOP

a la salida del sistema, se ha desarrollado un sistema de caracterización po-

larimétrico completo. El método de promediado de los parámetros de Stokes

es un método de caracterización completo, predice no solo el SOP a la salida,

sino también el DoP, y lo hace para todos los niveles de gris.

26

1.7 Conclusiones y desarrollos futuros

• La capacidad de predicción de la caracterización de Stokes ha probado quepuede predecir resultados incluso cuando se aplica a un DOE más complejocomo una red Blazed. Se ha observado una buen ajuste entre los predichoy lo medido, permitiendo una evaluación tanto dinámica como estática de laeficiencia en difracción. También se han observado otras fuentes de errorescomo el anamórfico o el “fringing field”.

• Se han analizado dos esquemas distintos de modulación en nuestro HDSS.Para este propósito todo el trabajo previo fue fundamental. También se haprobado la utilidad del correlador convergente para su uso en HDS.

• Se ha demostrado que un régimen puro HTM no es posible con los disposi-tivos PA-LCoS. Sin embargo, se ha propuesto una aproximación, la pseudo-HTM, que ofrece una buena homogeneidad en la intensidad de la página dedatos en el plano de la transformada de fourier.

• Se ha utilizado el fotopolímero PVA/AA como material de registro, y se hanobtenido unos buenos resultados en la fidelidad de la página de datos recu-perada.

En el futuro, me planteo algunos aspectos donde se puede extender la invest-igación. Será interesante ver si se puede inferir, de los modelos desarrollados, unentendimiento más profundo de la física de las moléculas del LC. Creo que sepodría modelar la inclinación de las moléculas, el grosor de la celda de LC, o in-cluso la birrefrigencia. Estos parámetros físicos pueden ser útiles para el desarrollode nuevos dispositivos y muy interesantes para la industria de las pantallas LCoS.

Como se ha visto hay otras fuentes de error como los efectos anamórficos o el“fringing field”, sería interesante entender estos fenómenos más profundamente oquizás poder modelarlos, con el propósito de minimizarlos o seleccionar el apro-piado diseño de las páginas de datos u otros DOEs.

Los resultados obtenidos de nuestro HDSS se han conseguido sin desenfocarla transformada de fourier. Será interesante combinar la codificación del pseudo-HTM con el desenfoque que se puede introducir con el correlador convergente.Se puede hacer una configuración del PA-LCoS más exhaustiva, ya que sabemosque solo se necesitan tres niveles de gris. Se podría encontrar una configuración

27


mejor si elegimos adecuadamente estos niveles de gris de forma que se reduzca el“flicker”. Hemos observado un empeoramiento del Bit Error Rate (BER) con lacodificación pseudo-HTM. Sin embargo, dado que presenta un orden cero menores fácil pensar que nos permitirá multiplexar más páginas de datos en la mismaregión del material, y quizás sea preferible este pseudo-HTM para nuestro HDSS.Se pueden hacer muchas mejoras para que la calidad óptica en el plano de recons-trucción del sistema mejore, lo que abre la posibilidad de reducir el BER de lacodificación pseudo-HTM. Cuando se pruebe la multiplexación angular tendremosmás datos para seleccionar el tipo de codificación. También quedaría por explorarla búsqueda asociativa de información en nuestro HDSS.

Otra técnica que hay que implementar, para poder explotar mejor el HDSS, esinsertar marcars fiduciales en las páginas de datos, lo que nos permitirá automatizarla detección y la recuperación de los datos almacenados.

Como el fotopolímero utilizado (PVA/AA) tiene muchas propiedades que sepueden modificar, será necesario una serie de pruebas para ver qué variantes puedenfuncionar mejor. Dado que el grupo también ha desarrollado otros fotopolímeroscomo el biophotopol, y ha trabajado también con HPDLC (“Holographic PolymerDispersed Liquid Crystal”), sería interesante utilizar estos polímeros en el sistemapara ver sus ventajas e inconvenientes.

28

CHAPTER

2Summary of the Thesis

2.1 Ph. D. Thesis ContextThis thesis presents an analysis on the use of modern parallel aligned Liquid Crystalon Silicon displays (PA-LCoS) in Diffractive Optics and Holographic Data Storage(HDS) applications. It is shown how this device can be modeled, characterized andthen optimally used in the applications.

In this thesis the PA-LCoS device acts as a Spatial Light Modulator (SLM)in three different roles, each of them of a high interest in many research fieldsin Optics and Photonics and other related Physics and Engineering research anddevelopment areas. Thus, it is shown its action as a state of polarization (SOP)conversor, as an amplitude-mostly modulator, and as a phase-only modulator. Spe-cifically, PA-LCoS devices have become widely used in Diffractive Optics and mayalso have a larger repercussion in HDS if the trend is to shift into multinary phase-only data page coding. Both of these important areas in Photonics are the subjectof exploration in this thesis, partly applying the knowledge gained from work donein modeling and characterizing PA-LCoS devices in the first period of the thesis.

PA-LCoS devices are a good example of the maturity achieved in Liquid Crys-tal (LC) technology, thus it is interesting to present the most relevant aspects of LCdevices physics shedding light into the actual context where this thesis has been

29

2. SUMMARY OF THE THESIS

developed. LC technology is mostly known by the general public because of the

many products existing in the displays consumer market. Developments in the

commercial arena have traditionally produced LC devices with ever increasing ca-

pabilities in the more specialized Optical and Photonics applications. To optimize

its application in this more sophisticated field proper modeling, characterization

and optimization of their performance is typically necessary. In the case of Dif-

fractive Optics, it is experiencing a continuous progress in the number of relevant

applications, even more since the field move also into the nano scale. In our case we

are interested in the application of the PA-LCoS to display programmable diffrac-

tive optical elements (DOE). These are optical devices which enable to perform

the same functions as conventional optical elements but applying the diffraction

phenomenon instead of the laws of refraction and reflection. Actually, they enable

further possibilities: for example they can perform simultaneously various actions

such as deflection and focusing, which are not possible by conventional optical

elements. Programmability comes from the possibility to change the optical func-

tion addressed in real time when the support for the DOE is a SLM, such as the

PA-LCoS. In this brief overview of the research fields encompassed in this work,

I want to bring into attention the exciting possibilities offered by Holography, in

its full integration of the amplitude and phase properties of electromagnetic fields.

In particular, Holographic Data Storage (HDS) is one of the most promising ap-

plications of Holography. Holographic Data Storage Systems (HDSS) have been

proposed since the invention of the laser in the 60’s. Competition with other data

storage technologies, typically cheaper and more robust, have until now resulted

in HDSS being only a reality in research labs. However it enables extremely high

density recording, with parallel readout of huge amounts of information. This para-

llelism is unique among information storage technologies which rely on sequential

writing and readout. In HDSSs two central elements are the SLM introducing the

data page and the holographic recording material where information is stored, both

of these aspects are of special interest in our group.

During this thesis two methods for characterizing the PA-LCoS device have

been developed, and they have been applied to solve several problems: the former

method is based on the classical linear polarimeter, and the second based on Stokes

30

2.1 Ph. D. Thesis Context

polarimetry. The first method, more widely available since no special instrument-

ation is required, offers a first approximation to the characterization. It can be

applied very straightforwardly to obtain custom configurations, such as selection

of stable signal addressing formats or optimizing the amplitud and/or phase dy-

namic ranges for an specific application. The second one is a full characterization

method, whose calibrated data enables to computer simulate the performance of

the PA-LCoS in a wide range of situations. In particular, I will test its predictive

capability in the realization of multinary phase-only diffractive optical elements

(DOEs). Eventually, I have applied the PA-LCoS in a full HDSS setup that I have

built in the last period of my thesis. This HDSS uses a convergent correlator geo-

metry in the object beam and an in-house made photopolymer material which we

intend to used various Optics and Photonics applications, one such is as a holo-

graphic memory. Using this scheme the Binary Intensity Modulation (BIM) an a

pseudo-Hybrid Ternary Modulation (HTM) will be compared.

My thesis work has resulted in five articles published in a series of international

journals in the Journal Citation Report of the ISI Web of Knowledge with impact

factor. They are the central material composing this Ph.D. Thesis that I present

in the form of compendium of publications. At the present moment there are also

an article dealing with holographic data storage in the peer review process, which

is also relevant for this thesis. Specifically, I will show the most relevant points

addressed in the HDSS manuscript, since it is the natural continuation of the pre-

vious five publications and connects with one of the main focus of interest of our

research group in the University of Alicante, working with photopolymer recording

materials and holographic memories.

In the next sections, I will describe the physics and application of liquid crystal

displays (LCD) and LCoS displays as SLMs together with the specific description

and characteristics of PA-LCoS devices. I will also develop the context for its

application in Diffractive Optics and HDSS. Then, I will present the goals of this

thesis, the description of the results achieved in the various publications. Eventua-

lly, I present a summary of the conclusions and future perspectives.

31


2.2 SLMs and Liquid Crystal on Silicon microdisplays

2.2.1 Liquid Crystal PhysicsLiquid Crystal (LC) materials correspond to a state of matter that shares typicalproperties from both, isotropic liquids and crystalline solids [1]. This means thatLCs can flow like a liquid but they also exhibit some orientational order in their mo-lecular distribution, which is analogous to the crystalline structure typically foundin solids. LCs can be classified by their molecular shape in two big groups: cala-mitic and discotic LCs. Calamitic ones are formed by rod-like molecules whereasdiscotic ones are formed by disk-like molecules. The majority of LCs interestingfor electrooptical applications are formed by molecules with a rod-like shape [2],which exhibit uniaxial anisotropic properties. For an ensemble of LC molecules,the average orientation of the long axes define a relevant electrooptical magnitudecalled the director axis, usually indicated by the vector ~n, which in these uniaxialmaterials corresponds to the orientation of the averaged extraordinary axis. Cala-mitic LCs belong to the thermotropic kind of LCs, which present different phasesdepending on the temperature: they are a crystalline solid at lower temperaturesand isotropic liquids at higher temperatures. In figure 2.1, the evolution with tem-perature is represented. In figure 2.2 it is shown how the director axis is defined ina calamitic and in a discotic LC material.

For calamitic materials in the liquid crystal phase various mesophases can beobserved [3], known as smectic phase, nematic phase and cholesteric phase, wheremesophase is defined as an intermediate state between crystalline and liquid state.We note that most of the LC materials applied in Optics and Photonics devices arenematic compounds. This is the case of the devices studied in this thesis.

In figure 2.3 the different mesophases are illustrated. In the nematic phase thereis a long range orientational order but no positional order, thus the director axis, ~n,remains equally oriented within a large volume in the material. In the smectic phasethe long axes of the molecules are parallel, but the molecules are also arranged inplanes, showing both a one dimensional translational order and an orientationalorder. In the cholesteric phase, the molecules are arranged in layers; each layeris rotated with respect the adjacent layers, then the director vector describes anhelix. The cholesteric phase corresponds to a nematic LC but formed by chiral

32


Figure 2.1: Transition from Crystalline state to Isotropic liquid.

Figure 2.2: Director exhibited by liquid crystal (a) Liquid Crystal formed by rod-likemolecules (b) LCs formed by disk-like molecules.

33


Figure 2.3: Phases of liquid crystals.

molecules [3]. The helicity of the helix formed by the director vector can be right

or left-handed depending on the molecular chirality.

Their relevant optical properties originate from the anisotropy showed by their

constitutive molecules. The dielectric constant in the material depends on the axis

considered, where the director axis indicates the orientation for the extraordinary

axis. This optical anisotropy or birefringence is what makes LCs so interesting for

optical applications. Birefringence will be defined and treated in more detail in

section 2.2.2.

Apart from the anisotropy at optical frequencies for the electromagnetic field,

LCs also have another interesting property. When an electric field or voltage is ap-

plied, it induces an electric dipole moment in nematic molecules, and they reorient

in order to align this dipole moment with the applied electric field. This is the so-

called field induced LC reorientation, which enables to tune the optical anisotropy

orientation via the application of an electric field [4]. This means that the director

axis changes orientation with the applied electric field, thus a light beam traversing

the LC layer sees a different effective extraordinary index which modifies the phase

difference between the extraordinary and ordinary components of its electric field.

This leads to a change in its state of polarization while traversing the LC material.

Using this, we can make devices that, for example, control the optical response

of a LC pixilated matrix with voltage. The first commercial device that used this

technology was an electronic calculator from Sharp presented on May 1973 [5].

34


Due to this special structure, that shares properties from liquid and crystals, LCsexhibit both elasticity and viscosity. If the LC orientation is distorted, restoringtorques will appear in opposition to the deformation. The forces in these processesare very weak compared with those in solids. In LCs an electric field is usuallyapplied to cause a reorientation of the molecules, and the final orientation of themolecules will depend on the interplay between the electrical torque and the resto-ring torque from the elastic constants. Any deformation can be divided into threebasic deformations. These are named splay, twist and bend, illustrated in figure2.4. These viscoelastic properties are very important to determine the behaviour ofLCs when a voltage is applied. The controlling voltage and the response times willbe defined by these constants [6].

Figure 2.4: Elastic deformations showed in LCs (a) Splay (b) Twist (c) Bend.

2.2.2 Birefringence and Optical PropertiesSince nematic LCs are uniaxially anisotropic, two refractive indices can be defined,one along the director axis, n‖, and the other perpendicular to that direction, n⊥,and optical birefringence is given by,

∆n = n‖ − n⊥ (2.1)

The axis parallel to the director axis is the extraordinary axis. LCs show diffe-rent refractive index depending both on the orientation measured between the LCdirector axis and the propagation direction of the incident beam, and also on itspolarization direction. As a result, one of the most evident characteristics that make

35


LCs useful in Optics and Photonics is their birefringence. This property allows us

to rotate the light oscillation plane, i.e. changes the State of Polarization (SOP) of

the light beam traversing the LC medium.

The electric field vector of a light beam interacting with the LC can be decom-

posed in two linearly polarized components: along the director axis and perpendi-

cular to it [5]. If there is a difference between the velocities for the two orientations

the two components will be affected in a different way, as represented in figure 2.5.

Due to the field induced LC reorientation the interaction of the light beam with the

anisotropy of the LC material can be controlled by voltage, and the SOP at the exit

will vary.

Figure 2.5: Light propagation in an anisotropic medium. The curves, i and ii, aresubject to different refractive index.

This electrically tunable birefringent property, that posses LC materials, makes

them very useful to be applied in displays, since the SOP variation can be converted

into an intensity modulation by inserting a linear polarizer at the exit. This tunable

birefringence property is also useful in Optics and Photonics to change locally the

properties of a wavefront, acting as Spatial Light Modulators (SLMs).

The large demand of LC devices in the display market has produced a large

development in the technology of LCs. Other big advances have been the access

to lithographic and semiconductor technology enabling the miniaturization of the

devices [7].

These advances are responsible for Liquid Crystal Displays (LCDs) to become

affordable devices. In the ’80s and early ’90s the interest of using these LC display

devices as SLMs largely increased. There are many important papers from those

36


years analyzing the use of commercial LC displays as SLM as noticed in references

[8–12].

In the next section a brief review of the technology and its evolution will be

presented. From the first developments to the more recent Liquid Crystal on Silicon

(LCoS) devices.

2.2.3 Liquid Crystal devices

There are different types of panels and structures for LC displays. Since the first

device was introduced in 1970 by Martin Schadt in Switzerland [13], the evolution

has advanced rapidly.

If we focus our attention in electronic control of the device, the first LCD was

a directly addressed one: device composed by several 8-segment displays, so that

you apply voltage independently to each segment. Right after the appearance of

the first device, in 1971, it was presented the concept of a matrix addressed display

for television (TV) using LCs [14]. Lechner et al. presented a 2x18 pixel display

[14], and Takata et al. just a year later, a 120x120 pixels LC matrix display [15].

In Takata’s display, rubbing at the internal faces of the windows of the display for

alignment of the LC molecules was used, and the authors claimed that this rubbing

technique reduced the crosstalk between pixels.

It is important to notice that in order to display an image it is mandatory to

have the possibility for addressing a complete matrix of LC cells. Obviously direct

addressing is not useful any more when the number of pixels exceeds a certain

value and multiplexed addressing is then applied: the image is addressed row by

row combined with vertical and horizontal synchronization signals. In this sense, it

has been developed, from the very beginning the Active Matrix (AM) control that

allows to send the image to the display by using an electronic control.

As early as in 1975 the first active matrix created using thin-film transistors

(TFT) and microelectronics techniques was introduced by T. Peter Brody [16]. This

display did not use LC but it was the first step towards the miniaturization of the

displays, and it was a breakthrough in technology.

Another center of attention will be the different LC cell geometries. It is inte-

resting to learn how the molecules can be aligned and ordered inside the LC cell. In

37


their natural state nematic LC molecules are organized with the longitudinal axes

almost parallel to each other. If we scrub some microgrooves on the alignment

layer in the inner face of the LC cell window, the LC molecules will align along the

direction of the grooves. The subsequent layers of LC in the bulk of the LC cell will

be aligned in that polished direction. But if we sandwich the LC material between

two glass plates on which we have practiced some microgrooves in orthogonal

directions, we force the LC to rotate its direction in a 90o helical twist. In figure

2.6 we see a schematic representation of this mode called Twisted Nematic (TN).

It is important to notice that the induced electric dipole moment does not have a

predetermined sense, so a preferred reorientation direction has to be defined. This

is done by generating a pre-tilt angle (Θ) in the alignment layer. This pre-tilt angle

is induced in the polishing sense along the microgrooves that have been generated

at the faces of the LC cell.

Figure 2.6: Representation of the 90o helical rotation of LC molecules in TwistedNematic cells.

This TN mode was the one used in the very first LCD used by Schadt [13]. An-

38


other operation mode used in display devices is the Super Twisted Nematic (STN)mode. In STN mode the molecules are twisted more than 90o, usually 180o, 240o or270o. This STN mode was introduced to improve the performance of LCD withoutthe use of TFT, and it improves the viewing angle too [3]. For achieving these twis-ted angles the nematic LC has to have an intrinsic rotating angle between layers, sonormally cholesteric LCs are used apart from the polishing directions on the glasswindows.

The main problem with TN or STN LCDs is that the phase and the amplitudeare coupled so that it is not possible to change the phase and the amplitude of thebeam independently. For Display applications it is not an important issue, becauseeyes are not sensitive to light phase, but for spatial light modulation it could betedious to manage [17].

Another LCD configuration used is parallel aligned one. In this type of devicethe LC molecules are aligned in the same direction in the two cell faces. With thisconfiguration we can make phase-only modulation when the incident light beamis linearly polarized along the director axis. They are generally known as ParallelAligned or Electrically Controlled Birefringence (ECB) devices.

Using this parallel alignment, different configurations can be found such as theParallel Aligned (PA) cells commented, but also, Vertical Aligned (VA) cells, andBend Aligned (BA) cells.

Figure 2.7: Schematic view of the three different PA cells

In figure 2.7 it can be seen the difference between the three configuration. InPA and BA the only difference is the pre-tilt angle alternation between the two cellfaces.

The first attempts to use LC devices as SLM, took advantage of the availableTN devices used in commercial TV displays [8–12, 17–19]. Nowadays new devices

39


have appeared that merge the two main advances introduced in the last section:

these concepts are the applicability of LCs for wavefront modulation, and the tech-

niques used in microelectronics industry to produce more sophisticated devices that

had proved interesting as early as in 1975 [16].

Thanks to the combination of these two important advances, Liquid Crystal

on Silicon (LCoS) devices were developed. They are reflective devices that use

the powerful and extended microelectronics technology to reduce pixel size and

integrate active matrix control in the same silicon substrate. Silicon can be polished

and can be metallized to obtain a good mirror. These devices add a LC layer that

can be used as SLM. The LCoS technology was initially developed for front and

rear-projection television systems (FPTV, RPTV) because of the high resolution

that can be achieved [20]. Despite of the downfall of the RPTV market, PA-LCoS

devices are very versatile components that can be used in many applications as a

phase-only modulator [20, 21].

Figure 2.8: Structure of phase-only LCoS device.

In figure 2.8 we can see a basic structure of an LCoS device. The SLM structure

formed by the LC layer is attached to a metallized aluminium layer that forms the

pixel structure. The aluminium layer voltage is controlled by the electronics em-

bedded in the silicon backplane. The backplane uses CMOS technology (CMOS is

an acronym for Complementary Metal-Oxide Semiconductor). The device showed

in figure 2.8 is a phase-only device because the LC layer is a parallel aligned one,

so we can use it as an ECB device.

The integration of high-performance driving circuitry allows a more accurate

control of the applied voltage for every pixel. The reflective mode can achieve a

40

2.3 Parallel Aligned LCoS

deeper phase modulation with the same LC thickness, and the fill factor can beoptimized [21].

Summing up, the advantages of reflective LCoS come from the integration ca-pability of CMOS technology, that is capable of producing high quality pixel mi-rrors too. They provide a small pixel size, and a high performance driving circuitrythat it is contained in the silicon chip.

In diffractive optics, a good characteristic for an SLM is to act only in the phaseregime, because modulating the amplitude always implies an energy loss, and thereare applications needing to modulate only the phase, such as in optical metrology[22], reconfigurable interconnects [23, 24], wavefront sensing of structured lightbeams [25], holographic optical traps [26], and a large etc. These PA-LCoS devicesare the last achievement available in the market, so it will be useful to know theirproblems and performance in different applications.

One of the goals of this thesis will be to use a commercial PA-LCoS device asan SLM so, these new devices need to be fully characterized. For that reason, someaspects of these devices will be presented in the next section.

2.3 Parallel Aligned LCoSIn this thesis a PA-LCoS device is used. It is important to present the requirementsthat are welcomed in a phase-only device, and the problems and limitations thatmay arise too. For example, to be used as a phase-only device it has to be capable ofmodulating the retardance at least from 0 to 2π. To achieve this interplay betweenthe cell thickness and the LC refractive indices is very important.

If the pre-tilt angle is ignored, and perpendicular incidence is supposed, it canbe found that the phase retardation difference between the two linearly polarizedcomponents of the light beam (along the director axis and along the ordinary axis),it is given by:

δ = 2π∆nd/λ (2.2)

where d is the LC cell thickness, ∆n is birefringence, as defined in equation 2.1,and λ is the incident light wavelength. If the LCoS devices works in reflectivemode, the total retardance will be doubled:

41


δ = 2π∆n2d/λ (2.3)

So, if the phase modulation retardance is required to be 2π we can obtain that

the thickness needed will be:

d = λ/2∆n (2.4)

We can conclude that in reflective mode a thinner LC cell can be used. This is

a better option if we need a rapid response in changing the phase since it is known

that the response time depends directly on “d2” [27, 28]. And even better, if we

only need a π modulation phase, the thickness can be reduced by half.

Because of the technology used in fabrication of LCoS the pixel can be smaller,

so you can deal with larger diffraction angles [21]. With this technique the fill

factor can be also optimized thanks to the resolution achieved by microelectronic

industry.

In the display industry the trend has been to produce digitally addressed devices

since they offer a more precise, repeatable and robust control than analogic signals.

A common problem presented in digitally addressed LC devices is the fluctuation,

or flicker, presented in the light modulation. This is due to the pulsed nature of

digital signals. In digitally addressed devices the voltage is defined by a Pulse

Width Modulation (PWM) signal. The viscosity of the LC limits the fluctuations,

but the molecules still appreciate a residual variation in the voltage signal applied.

This produces fluctuation of the LC orientation in the millisecond order, and affects

the optical response producing, for example, a decrease in efficiency [29, 30]. Other

authors have reduced this flicker by cooling the device since at a lower temperature

the LC viscosity increases [31].

Now we know that the digitally addressed displays present some level of flicker.

In this sense, an accurate measurement of the flicker is required. Therefore, during

this thesis we have developed some methods for evaluating this flicker. The charac-

terization of these fluctuations is important in order to evaluate the applicability of

the device for the different applications. We will notice that in more complex DOEs

the flicker effect is smaller. Then both digital and analog displays have other pro-

blems like anamorphic effects. This means, that the response is different depending

42

2.4 Application in Diffractive Optics and in Holographic Data Storage

on the orientation (rows and columns) of the variations in the voltage signal for the

addressed image [32]. This is due to the structure of the multiplexed signal along

the rows and columns addressed for displaying the image: the bandwidth is dif-

ferent for rows and columns. Another undesirable effect is known as fringe field

effect, which is due to the crosstalk produced between neighbouring pixels due to

the distribution of the electric field caused by voltage in the LC [33].

The digitally addressed LCoS display can use different addressing schemes

[34]. These schemes send the information (voltage) with a pulse width modula-

tion (PWM) or similar structure. So the flicker will be affected by the schemes

used. Some displays, as the one used in this thesis, can be configured for different

applications, and wavelength, just by changing the addressing scheme, and all of

these different schemes will present different flicker characteristics.

It is important, for different applications, to measure the flicker amplitude caused

by the active circuit used [35]. For example, in optical tweezers this flicker causes

instabilities in the position of the beam [26]. Because of this, this flicker measure-

ments is one of the main goals of this thesis.

There are many interesting modulation regimes depending on the application,

and many options that can be configured. Thus, it becomes necessary a profound

study of all of these characteristics, and how this configuration option affect the

flicker and performance of the display.

The SOP of the light wavefront is important for some specific applications [36],

for this reason it is relevant to look for a characterization based on the modulation

of SOPs and to measure how the flicker affects these applications. A method that

provides a prediction of the SOP, and the depolarization produced by flicker at the

exit of the LC devoce has been developed with the help of a polarimeter as I will

show.

2.4 Application in Diffractive Optics and in HolographicData StorageThe LCoS microdisplay has become widespread in diffractive optics. There are

many research groups working with these devices in any application we can ima-

43


gine.

One of these interesting applications is a holographic sensor. Meeser et al. [37]

developed a system that can adapt the reference wave for different object positions

without using a lens, the lens function is done by a PA-LCoS device. This allows to

change dynamically without changing any mechanical part of the system to adapt

to different object distances.

Another interesting use for SLMs is in holographic lithography, where it is used

to micro-manufacture 3D structures in a substrate [38]. In this application the SLM

is used to digitally generate the hologram that will correspond to a micro-pyramid.

The SLM allows to use multiple focused laser spots and at several distances. Jen-

ness et al. present an effective tool for arbitrary three dimensional laser-based

parallel micro-manufacturing [38].

Furthermore, the capability of LCoS devices to manage high light intensities

permits its use in high-power laser applications such as parallel micro-structuring

with laser ablation parallel at many points of an array [39].

Since the introduction by Hayasaki et al. in 1996 [20] of the possibility of

using SLM as an optical tweezer the use of LC-SLMs in this field have become

more intensive. In fact as early as 1998 Dufresne et al. present a method for

creating multiple optical tweezers from a single laser beam using diffactive optical

elements [40].

Many more applications can be found, such as in adaptive optics, holographic

projections, and many more, in a brief resume presented by Collings et al. [41].

But, eventually we want to highlight the use of SLMs and specially LC based SLMs

for holographic data storage. In this application the SLM is used as a data entry

point, used to modulate the optical signal in several regimes as, for example, Binary

Amplitude [42], Binary Phase, Ternary Modulation, an so on [43].

If we want to use these LC devices as SLM, we have to take into account the

modulation capabilities of the device, in figure 2.9 we present the typical complex

amplitude modulation domains that can be useful for some applications.

Figure 2.9 shows the different schemes that can be performed or are desirable to

accomplish. Some of these schemes will be available with different LCs devices,

but it is important to see the possible applications in order to choose the correct

device.

44

2.4 Application in Diffractive Optics and in Holographic Data Storage

Figure 2.9: Typical complex amplitude modulation domains. (a) Ideal full range ofmodulation. (b) Amplitude only. (c) Phase-only modulation with 2π depth (d) Phase-only modulation with π depth (e) Binary amplitude (f) Binary phase (g) Ternary (h)Coupled amplitude-phase

Figure 2.9(a) shows the ideal case where amplitude and phase is fully con-

figurable, this case it is impossible to accomplish because it only exists one free

parameter, the voltage. The other domains will be applicable or eligible depending

on the application.

As presented, the use of PA-LCoS as SLM for reproducing DOEs, that can be

used as a data page, plays an important role in the HDSS and all the development

of the LCoS industry opens the possibility of reducing the pixel size together with

the increase in the pixel number, and as a consequence it provides the way for in-

creasing the data storage density. HDSS have been a promising and very appealing

technology since many years ago [44, 45], and many deep efforts have been made

to control an improve HDSS as can be seen in the handbooks by Coufal et al. [43]

or more recently by Curtis et al. [46]. Due to the great advances in this area some

commercial prototypes have been developed along the years [47], even multibil-

lion companies [48, 49] are interested in long time archival storage nowadays [50].

From the birth of the holographic data storage idea, there were many problems that

have slowed down the expansion of this technology such as low performance light

source, data pager or spatial light modulator, camera, optical system, recording

material or multiplexing architecture [43, 46]. Some of these problems have been

45


partially or fully solved. The continuous progress being produced in SLM tech-

nology, where PA-LCoS microdisplays have replaced previous LCD technology in

most of optics and photonics applications, invites us to explore the implementation

of a HDSS incorporating this new device.

In this sense, the whole system tested has been designed using PA-LCoS as

a data pager input. With this device we have implemented different modulation

schemes, or types for coding the information. From the ones presented in figure 2.9,

in our experiments we have implemented the Binary Intensity Modulation (BIM)

(where only two levels of intensity are needed), figure 2.9(e), and a Hybrid Ternary

Modulation (HTM), figure 2.9(g). In HTM three gray level values are needed,

two of them with a high and equal intensity transmission and with a 180o relative

phase-shift, ON levels, and a third gray level with a low intensity transmission, OFF

level. Other schemes like binary phase, showed in figure 2.9(f), has the advantage

that there is more intensity available, but the detection process is more complicated

than just recording an intensity plane on a CCD camera.

Following with this whole HDSS design and characterization, it is important

to note that we have chosen the convergent correlator as an alternative to the 4-f

processor in holographic memory testing platforms. This optical system provides

an easy scaling of the Fourier transform dimensions. In addition I have compared

this scheme with the classical 4-f system [51, 52].

On the last place, but not less important, the recording material used in HDSS

system is a key element, as it can be seen if ref. [46]. There are some special

requirements for the holographic recording materials to be used as a base of holo-

graphic memories, such a low shrinkage (less than 0.5%) [53], high thickness,

high resolution, low noise, etc. In this sense photopolymers are appealing ma-

terials for this application, they exhibit good optical properties, low losses, low

scattering, high refractive index modulation and high density information storage

capability, and they are relatively inexpensive. In the field of photopolymers, our

research group has a wide experience in the preparation of thick photopolymers for

holographic data storage [54, 55]. In particular in this thesis a polyvinyl alcohol-

acrylamide (PVA) photopolymer has been used in the HDSS. About this material

our group has made some studies in order to multiplex many holograms even with

different modulation regimes [56, 57].

46

2.5 Main Goals of this Thesis

2.5 Main Goals of this ThesisBecause of the novelty of the LC device used the first goal I have to fulfill is the

characterization of the PA-LCoS device used. As it has seen this must be accom-

plished in order to control the configuration of the entry data point that is used in the

HDSS. This HDS system starts a new research line on holographic memories using

the novel reflective devices and the mentioned convergent correlator geometry.

We need a rapid and easy method to know the possibilities of every configura-

tion that the manufacturer provides. The device has many options but none of them

are optically characterized. In this thesis has been developed a characterization

method that can help us to configure the device for an specific application, or even

an specific modulation range.

The characterization of the SOP at the exit is very useful for diffractive appli-

cations. It would be interesting if an accurate enough model of the device could

be found with a predictive capability. In this thesis a complete polarization charac-

terization has been performed using a polarimeter. We have found a way to know

the flicker and retardance for every gray level. It is showed how this can be ac-

complish and how can be used to predict the behaviour of the devices in diffractive

applications.

The method proposed tries to be as general as possible, in fact it can be ap-

plied to any retarder device that presents flicker or variations on the retardance

introduced.

All of the methods proposed have to be validated and tested, and we have done

that in different ways.

After the PA-LCoS has been modeled, and characterized, it is applied in HDSS,

in which it plays the role of information (data pager) entry point, and the recording

material used is the well known and characterized by our research group PVA/AA.

The papers that compound this thesis can be found in Chapter 3. The main

goals covered by each paper are:

• In Section 3.1 and 3.2 we are presenting a characterization method based on

the classical linear polarimeter, traditionally applied to linear retarders. We

will see how this method fails with linear retarders that present fluctuations,

47


and how we can correct it and obtain information about the fluctuation am-

plitude. The method is presented and validated by means of real time mea-

surements of the retardance fluctuations.

• In the paper presented in Section 3.2 the extended method developed is

applied to generate custom configurations that can be useful for specific ap-

plications.

• In Section 3.3 we will present a characterization method based on Stokes

polarimetry. We will see how we can predict the State of Polarization (SOP)

at the exit and how we have fully characterize the device. We infer the a-

verage retardance and the flicker amplitude for all gray levels. This method

is validated using its prediction capability: it can predict the SOP at the exit

and the degree of polarization (DoP).

• Section 3.4 presents a paper where the applicability of the Mueller-Stokes

method has been applied to different device addressing schemes and different

laboratory geometries. Showing that the method is still valid and useful in a

variety of situations.

• In Section 3.5 we apply the method presented in Section 3.3 to a Diffractive

element as Blazed Grating. It shows a that it has a good prediction capability

even with complex elements.

• In Section 3.6 we use the PA-LCoS device as data pager in a complete

scheme for data storage. In this paper, that it has been submitted to Optic

Express journal and it is under revision at this moment, we mix three novelty

elements which are: the PA-LCoS used in HDSS, convergent correlator for

the object beam, and the PVA/AA photopolymer.

A brief description of the results obtained is presented in section 2.6. This

description has the intention to serve as a guide for the lecture and focus the main

topics attended in this thesis.

Finally in section 2.7 we will discuss the goals accomplished and some future

developments that can be afforded.

48

2.6 Results presented in the publications

2.6 Results presented in the publicationsTo achieve the goals of this thesis the first step was to characterized the PA-LCoS

display. As it acts as a tunable linear retarder the first logical approximation is to

apply the classical method used for fixed linear retarders. It was observed that this

method has its limitations when applied to retarders that present fluctuations. But

we demonstrate that these limitations with some conditions can serve us to measure

this fluctuation amplitude without using special equipment.

The method presented in reference [58] takes advantage from the extremal

points when we characterized the retardance introduced by the display as a function

of gray level. In these extremal points the retardance is known (multiple of 180o).

Knowing this we found a relation between the difference of the real retardance at

these points (180o or 0o) and the retardance measured. The relation found is a linear

relation within a big range of fluctuations amplitudes, so it becomes easy to correct

the retardance vs. gray level relation. And furthermore, we have an estimation

on the flicker amplitude for a specific configuration, allowing us to compare the

performance of different addressing configurations provided by the manufacturer.

In addition to the previous work, we learned how to profit the software provided

by the vendor to generate our own configurations. To do this, a profound study

and understanding of the device is needed. Understanding the parameters that the

manufacturer allows us to configure is crucial, for that reason we present the Vbrightand Vdark parameters. These are the high an low voltage respectively, that are

applied to the LC cell in a determined addressed sequence. Due to the pulse nature

of the digital signal it is easy to think that a lower difference between Vbright and

Vdark will result in a lower fluctuation amplitude. These voltages will define the

voltage range applicable in a configuration, and because there is a relation between

the voltage applied and the gray level, these voltages define the retardance range

disposable in a specific digital addressing scheme. The relation between voltage

applied and gray level is modulated by the gamma curve which is also configurable

in these devices. These gamma curves define a relation between the gray level

and the voltage applied to the cell by defining the digital sequence that has to be

addressed for every gray level. The shape of the gamma curve will define the shape

of the retardance introduced by the device as a function of gray level.

49


With the extended polarimeter method and with these concepts in mind, it has

been developed a method to linearize the retardance response of the device, or

even to create a novel configuration better adapted to the application [59]. The

method uses the correction presented in the previous paper [58] and generates the

appropriate gamma curve for the linearization process. This linearization can be

performed in the intensity domain or in the retardance domain. The appropriate

selection of Vbright and Vdark, and the way they affect the retardance response has

been presented too.

As presented in previous sections it is useful to know how the SOP is affected

by the device. In this sense we tried a Mueller-Stokes characterization process. The

polarimeter device used in this part of the thesis is a rotating wave-plate based one.

In the previous works we saw that the fluctuation frequencies are in the kHz range,

so with our polarimeter it is impossible to follow the time evolution of this fluctu-

ations produced in the SOP at the exit. But we have found that using an appropriate

averaged measures, we can estimate the flicker amplitude and the retardance intro-

duced by the device for every gray level. This allows a complete characterization

of the display. This method is presented in reference [60].

Due to the importance of this complete characterization method, a profound

study was needed. We wanted to know if the method is applicable to other electri-

cal configurations provided by the vendor, and if it was affected by the geometry

of the laboratory setup. In our case we use several setups: with perpendicular

incidence to the display (with the help a a non-polarizing beam splitter, NPBS),

with a quasiperpendicular incidence in order to separate the incidence beam from

the reflected one, and with a 45o incidence angle, just to know the limits and capa-

bilities of the setup and the PA-LCoS in the working geometries where it is used in

applications. We found that the average stokes method is robust. It presents a good

repeatability and reproducibility, so we can conclude that the method is useful in

many situations and different laboratory setups [61].

From the measurements taken for characterize the device we can predict any

SOP at the exit: if we know the SOP for the light beam impinging the device we

can calculate the SOP at the exit and the degree of polarization (DoP). These results

are presented in reference [60] too. Nevertheless, we went further in this prediction

capability and we test it with a complex DOE as a blazed grating.

50

2.6 Results presented in the publications

In reference [62] we show how the model predicts the diffraction efficiency

for multilevel phase blazed gratings, and not only predicts a loose of efficiency,

it also predicts the amplitude fluctuation in the first diffracted order. So we have

an static and a dynamic prediction for complex DOEs. Thanks to this work we

observed that: as the number of quantization levels increase the degradation due to

the flicker is reduced. In fact it is reduced until it shows an efficiency close to the

ideal one. This work also shows us that it has to take into account other degradation

sources such as crosstalk between neighbouring pixels, clearly exhibited when the

number of quantization levels is small.

All the knowledge about PA-LCoS accumulated during the realization of this

thesis has become useful in order to design a HDSS. With all the techniques learned,

we can configure the HDSS for using different codification for data pages. The last

paper that I have introduced in this thesis is submitted but it is still in revision pro-

cess in Optics Express. It has been included to a better illustration of the thesis

development.

In our HDSS we have tested the binary intensity modulation (BIM) and the

HTM modulation. HTM has been demonstrated and analysed with TN LCDs, how-

ever, there was no study with PA-LCoS devices so, we analyze the challenges that

have to be faced. We detected experimental and theoretical problems that are found

when you try HTM with a PA-LCoS. We saw that a pure HTM modulation was not

possible with this type of devices, but we developed a pseudo-HTM modulation

that can be performed with our device. We designed this new scheme that de-

creases the zero order intensity and maintain a good performance when the data

page is reconstructed.

In this last paper we also test the correlator convergent scheme, or VanderLugt

correlator, that provides more flexibility for scaling the Fourier transform of the

data page into the recording material [63], and allows to easily introduce a defocus

in the fourier transform that reduces the zero order intensity too [52]. It enables us

to increase the areal density of the pages stored. We used a versatile photopolymer

as data storage material. We chose it because it has a great potential for future

developments because of its highly tunable properties.

51


2.7 Conclusions and future developmentsSumming up all the work done and recent results we can extract some conclusions

of this thesis:

• It is important to know the characteristics and the operation of the PA-LCoS

device. For that reason, it has been developed a characterization method, easy

to apply, and without any special equipment needed that allows to compare

different addressing configurations. We obtain the retardance range provided

and we have an estimation on the flicker amplitude introduced, allowing us to

select the proper electrical configuration. Since we have measured the flicker,

the retardance range and how varies the retardance as a function of gray

level, we have concluded that the 5_5 sequences have less flicker, linearity

of the retardance vs gray level response depends on the gamma curve, and

the retardance range is related with the Vbright and Vdark values, the gamma

curve applied and wavelength.

• The method has been validated and proves that its estimated values are very

close to the real ones not only in average retardance respect but also with

flicker amplitude.

• It is mandatory to know how to adjust the device to a specific application.

In this sense, a method to design custom configurations has been presen-

ted. We provide a deeper understanding of several parameters that can be

configured by the vendor’s software. And as commented, this design can

be done without using any special equipment that could be not found in an

optical laboratory.

• The characterization method based on extending the classical polarimeter is

powerful but incomplete. So thinking in the importance of knowing the SOP

at the system exit, a complete polarimetric characterization has been done.

The average stokes method is a full characterization method. It predicts the

SOP at the exit and also the DoP, and it can be done for all gray levels.

52

2.7 Conclusions and future developments

• The prediction capability of the average stokes characterization method hasgone further predicting the result that can be obtained when applied to a com-plex DOE (Blazed Gratings). A good agreement has been observed and someother degradation sources have been presented (fringing field, anamorphic).Both a static and a dynamic evaluation of the diffraction efficiency is possibleusing our numerical approach.

• Two modulation schemes for a HDSS have been tested. To this goal all theprevious work was needed. A new scheme, as it is the convergent correlator,is used in our HDSS. The usefulness of the convergent correlator has beenproved for HDS.

• We have proved that pure HTM regime is not available with PA-LCoS devices.However, an approximation, the pseudo-HTM, has been proposed, which of-fers a good intensity homogeneity of the data page in the fourier transformplane.

• The PVA/AA has been used as a recording material showing good results inthe fidelity of the data page retrieved.

In a future, I think of some aspects where research can be extended or done.It will be interesting to see if a deeper understanding on the physics of the LCmolecules can be inferred from the models developed. I think that the inclinationsof the molecules, the thickness of the LC cell, or even the birefringence can beinferred. These physics parameters can be useful for developing new devices, andcan be interesting in display industry.

As it has been seen, there are other degradation sources such as anamorphic orfringing field. It will be interesting to have a deeper understanding or perhaps mo-delling these problems, in order to minimize them or select the appropriate designin data pages or other DOEs.

The results obtained with the HDSS used in this thesis have been extractedwithout defocus of the fourier transform. It will be interesting to combine thepseudo-HTM scheme presented and the defocussing that can be introduced withthe convergent correlator. And further work in configuring the PA-LCoS can bedone. Since, we know that only three gray levels are necessary, if we find a way

53


to smartly select these gray levels a better configuration in the PA-LCoS can beperformed to minimize the flicker. We have a decrease in Bit Error Rate (BER) withpseudo-HTM scheme. However, as it presents a smaller DC term it is easy to thinkthat it will permit us to multiplex more data pages in the same region, and perhapsthis makes the pseudo-HTM a preferred scheme in our HDSS. There is a lot ofwork to be done to improve the optical quality in the reconstruction plane and thisopens the possibility to reduce BER for pseudo-HTM. When angular multiplexingwill be tested there will be more data available to select the codification scheme.Another large field to be explored is the use of the associative search in our HDSSwhen angular multiplexing has been applied.

Another technique that has to be explored, to enhance the HDSS, is to insert fi-ducial marks in data pages, that will allow us to automate the detection and retrievalof the data saved.

As the PVA/AA photopolymer has many tunable properties, further investiga-tion will be needed. As the group has developed other photopolymer, the biopho-topol, and it has worked with HPDLCs (Holographic Polymer Dispersed LiquidCrystal) too, it will be interesting to apply these photopolymers in HDSS and seetheir performance.

54

CHAPTER

3Papers of this Thesis

3.1 Extended linear polarimeter to measure retard-ance and flicker: application to liquid crystal on sil-icon devices in two working geometries

55

Extended linear polarimeter to measure retardanceand flicker: application to liquid crystal on silicondevices in two working geometries

Francisco J. Martínez,a,b Andrés Márquez,a,b,* Sergi Gallego,a,b Jorge Francés,a,b and Inmaculada Pascualb,caUniversidad de Alicante, Dept. de Física, Ing. de Sistemas y T. Señal, Ap. 99, Alicante, E-03080, SpainbUniversidad de Alicante, I.U. Física Aplicada a las Ciencias y las Tecnologías, Ap. 99, Alicante, E-03080, SpaincUniversidad de Alicante, Dept. de Óptica, Farmacología y Anatomía, Ap. 99, Alicante, E-03080, Spain

Abstract. We focus on the evaluation of the applicability of the classical and well-established linear polarimeterto the measurement of linear retardance in the presence of phase flicker. This analysis shows that there are largeerrors in the results provided by the linear polarimeter when measuring the linear retardance of a device. Theseerrors depend on the specific retardance value under measurement. We show that there are some points wherethis limitation can be used to measure the fluctuation amplitude consistently. An elegant method is further pro-posed, enabling the measurement of the average retardance value, thus extending the applicability of theclassical linear polarimeter. Experimental characterization results are provided for various electrical sequencesaddressed onto a parallel aligned liquid crystal on silicon (LCoS) display. Good agreement is obtained withexperiment, thus validating the linear polarimeter methodology proposed. Furthermore, results are providedfor the LCoS in two reflection geometries, perpendicular incidence with and without nonpolarizing beam splitter,demonstrating robustness of the method. As a result, the evaluation of both phase modulation range and flickermagnitude for any electrical sequence addressed can be easily obtained, which is very important for optimal useof LCoS displays in applications. © 2014 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.53.1.014105]

Keywords: polarimeter; retardance measurement; liquid crystal on silicon displays; parallel aligned; phase-only modulation; spatiallight modulation; flicker; diffractive optics.

Paper 131700P received Nov. 5, 2013; revised manuscript received Dec. 20, 2013; accepted for publication Dec. 24, 2013; publishedonline Jan. 21, 2014.

1 IntroductionLiquid crystal (LC) microdisplays have become a centraldevice in a wide range of applications requiring spatialmodulation of a wavefront, like in diffractive optics,1 opticalstorage,2 or optical metrology.3 Liquid crystal on silicon(LCoS) displays have become the most attractive microdis-plays for these applications due to their very high spatial res-olution and very high light efficiency.4,5 However, severalauthors6–11 have detected that LCoS displays produce a cer-tain amount of phase flicker and/or depolarization. Amongthe different LCoS technologies typically available, parallelaligned LCoS (PA-LCoS) are especially interesting sincethey allow easy operation as phase-only devices withoutcoupled amplitude modulation. They can be thought ofas variable linear retarders; therefore, for incident linearlypolarized light along their neutral lines, no depolarizationoperation is available; however, they may still exhibitphase fluctuations. This is, for example, the case for digitallyaddressed devices12 due to the pulsed nature of the voltagesignal addressed.13,14 In these devices, if the digital sequenceformat is properly chosen, fluctuations can be significantlydecreased to acceptable levels.15 To this goal, a proper char-acterization not only of the phase modulation, but also ofthe amount of phase flicker in the optical signal needs tobe done. This reasoning is not restricted to PA-LCoS devices,but it can be applicable, in general, to electrooptic variable

linear retarders, such as the novel device proposed byRamirez et al.16

Different measurement methods have been proposed anddemonstrated to characterize the phase-shift versus voltagein LCDs and LCoS devices.12,17,18 Among these methods,the diffractive-based technique used by Lizana et al. inRefs. 12 and 17 allow for instantaneous phase-shift valuesmeasurement. In the case of parallel aligned devices, theyare totally characterized by their linear retardance versusvoltage values. Additional methods typically used in thecharacterization of wave plates become available to measurethe linear retardance. The most popular are ellipsometric,polarimetric, and interferometric ones. In interferometricmethods, we need a device introducing a carrier referencesignal (heterodyne interferometry), such as an He-NeZeeman laser19 or an electrooptic light modulator20 workingin the common-path geometry.21 We can also have someelement producing a phase-shift between the ordinary andextraordinary components of a beam (phase-shifting inter-ferometry), such as a birefringent wedge.22 Polarimetricmethods of a high precision generally rely on null measure-ments, such as in the Senarmont compensator.23,24 Othermethods are based on the variation of the intensity when wechange the angle of the elements in the polarimeter.25–29 Onecommon property of all these methods is that they assumethat the birefringence in the wave plate has a constant value,no fluctuations, during the measurement process.

*Address all correspondence to: Andrés Márquez, E-mail: [email protected] 0091-3286/2014/$25.00 © 2014 SPIE

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Recently some techniques to evaluate the average retard-ance in the presence of fluctuations in linear retarders havebeen demonstrated by our group15 and by Ramirez et al.,30

based, respectively, on the classical linear polarimeter and ona combination of linear and circular polarimeters. The linearpolarimeter technique in Ref. 15 can be implemented by anylab since it basically needs two linear polarizers and a radio-meter and only two sets of measurements are necessary toextract the retardance and flicker values. In Ref. 15, we con-centrated on the proposal of the characterization technique,but we did not provide the deeper insight, which is actuallyneeded to evaluate the range of validity of the linear pola-rimeter in the presence of phase flicker. Furthermore, thetechnique was presented for the combined system composedof the LCoS and a beam splitter in front.

In the present work, we perform a complete analysis ofthe applicability of the linear polarimeter to the measurementof linear retardance in the presence of phase flicker. Thisdeeper insight allows us to clearly define a valid and accuratemethodology enabling the use of the classical linear pola-rimeter to a new range of samples. First, in Sec. 2 we showthe basic theory for the linear polarimeter, its limitations, andits extension to samples exhibiting phase flicker. In Sec. 3,we proceed with the instantaneous and average measure-ments, using a PA-LCoS display and for various electricalsequences. We also consider two working geometries, withbeam splitter and in quasiperpendicular incidence, in orderto verify the validity of the measurements and technique intwo typical configurations. Eventually, the main conclusionsare given in Sec. 4.

2 Theory: Linear Polarimeter and Phase Flicker

2.1 Limitations in the Presence of Instabilities

The total retardance Γ of a wave plate is given by

Γ ¼ 2πΔnðλÞdλ

; (1)

where λ is the wavelength of the incident light, ΔnðλÞ is thedifference between the extraordinary and the ordinary indexof the birefringent material (e.g., the LC) at this wavelength,and d is the thickness of the wave plate. In general, most ofthe measurement methods provide the retardance value Γ0

modulo 2π, which is given by Γ ¼ Γ0 þm2π, where m isan integer, called the order of the wave plate. In the caseof the zero-order wave plate, m ¼ 0 and Γ0 ¼ Γ, which isthe case in many LCoS since thinner devices produce a fasterresponse.

The measurement method we propose is based on the li-near polarimeter,25–29 where the device under test, the linearretarder or wave plate, is situated between two linear pola-rizers (see Fig. 1), illuminated with a monochromatic lasersource. In the basic configuration, there is a radiometer at theoutput of the polarimeter.

By means of the procedure developed in Refs. 15 and 29,we know that a robust measurement of the retardance value Γis obtained from intensity measurements for polarizers withtheir transmission axis oriented at �45 deg with respect tothe neutral lines of the wave plate. The polarizers can still beparallel or perpendicular to each other, thus producing,respectively, the following intensity measurements:

IkOUT ¼ I02½1þ cos Γ�; (2)

I⊥OUT ¼ I02½1 − cos Γ�: (3)

The retardance value is contained in the cosine function,which can be properly isolated.

cos Γ ¼ IkOUT − I⊥OUTIkOUT þ I⊥OUT

: (4)

Finally, the retardance value Γ is given as

Γ ¼ cos−1

IkOUT − I⊥OUTIkOUT þ I⊥OUT

!: (5)

Measurements given in Eqs. (2) and (3) are probably themost typically used ones to obtain the retardance in linearretarder devices. Next, we want to analyze the range of appli-cability of the classical linear polarimeter in the presence ofinstabilities or fluctuations in the linear retardance. To incor-porate them, let us consider a triangular profile for the varia-tion of retardance with time ΓðtÞ (see Fig. 2). This isa reasonable assumption in the case of PA-LCoS as canbe seen from the instantaneous measurement plots that weshow in Sec. 3.1., which as a first approximation exhibita triangular shape. Furthermore, the triangular time-depen-dent profile represents actually a linear model; thus, thefirst option to try before more complex approaches maybe proven necessary. The triangular profile can be analyti-cally expressed by the following equation:

ΓðtÞ ¼(Γ̄ − aþ 2a

T∕2 t 0 ≤ t < T∕2Γ̄þ 3a − 2a

T∕2 t T∕2 ≤ t < T; (6)

Fig. 1 Linear polarimeter with the wave plate WP to be measured.P1 and P2 are the polarizers.

Fig. 2 Triangular profile considered for the temporal fluctuation ofthe linear retardance.


Martínez et al.: Extended linear polarimeter to measure retardance and flicker: application to liquid crystal. . .


where Γ̄ and a are, respectively, the values for the averageretardance and its fluctuation amplitude, and T is the periodfor the fluctuation instability.

To calculate the average value in Eqs. (2) and (3), weaverage the cosine and obtain

hcos ΓðtÞi ¼ sinðaÞa

cosðΓ̄Þ; (7)

where we appreciate the appearance of the sin c term modu-lating the cosine function. Equations (2) and (3) may berewritten in terms of the average value and the fluctuationamplitude as follows:

hIkOUTi ¼I02

�1þ sin a

acos Γ̄

�; (8)

hI⊥OUTi ¼I02

�1 −

sin aa

cos Γ̄�: (9)

Let us now combine Eqs. (8) and (9) to remove the nor-malization factor to obtain the following expression:

hIkOUTi − hI⊥OUTihIkOUTi þ hI⊥OUTi

¼ sin aa

cos Γ̄: (10)

Eventually, the average retardance Γ̄ is obtained byinverting Eq. (10).

Γ̄ ¼ cos−1�ðhIkOUTi − hI⊥OUTiÞ∕ðhIkOUTi þ hI⊥OUTiÞ

sin a∕a

�: (11)

In the case when no fluctuations exist (a ¼ 0o), Eqs. (8) to(11) transform into the classical results already given inEqs. (2) to (5).

To analyze the range of validity for the classical linearpolarimeter in the presence of phase flicker, we performthe following simulated experiment. We apply Eqs. (8)and (9) to calculate the experimental intensity values mea-sured in the presence of fluctuations in the retardance.The retardance value is recovered using Eq. (5), which isthe classical expression that does not take into accountthe existence of fluctuations. Analytically, the simulatedexperiment consists of applying Eq. (5) onto the experimen-tal result given by Eq. (10), that is,

Γ̄calc ¼ cos−1�sin aa

cosðΓ̄trueÞ�; (12)

where Γ̄calc is the calculated retardance, and Γ̄true and a arethe input values considered for the retardance and fluc-tuation, respectively. In Fig. 3, we show the calculated retar-dance Γ̄calc as a function of the true retardance Γ̄true, andfor various fluctuation amplitudes a, indicated on the plot.In Fig. 3, we represent the wrapped phase values in therange 0 to 180 deg, which is the definition dominium forthe cos−1 function in Eq. (12).

First thing we note is that maxima and minima, theextremals, in the calculated retardance occur at valuesmultiple of 180 deg of the true retardance independent ofthe flicker magnitude. We observe that there exist clear

deviations between the calculated and the true retardance,and the amount of this deviation varies largely as a functionof the true retardance value. The deviations magnify at trueretardance values multiple of 180 deg. Outside of thesepoints and if the fluctuation amplitude is not very large,we find that the calculated values are very close to thetrue retardance values. This allows us to say that the classicalmethod may still be valid in many cases. We note that, ingeneral, the fluctuation amplitude is not known, thus makingimpractical the application of Eq. (11), which, according tothe linear model developed, should provide the correct valuesfor the retardance. Then it is necessary to develop a methodto calculate the amplitude of the phase flicker. This is done inthe next section.

2.2 Extended Linear Polarimeter for Phase FlickerCharacterization

We have seen in Fig. 3 that maxima and minima, extremalpoints, in the calculated retardance occur at values multipleof 180 deg of the true retardance independent of the flickermagnitude. This invariance allows the a priori knowledge forthe average retardance at these points, enabling to calculatethe magnitude of the fluctuation amplitude a at these maximaand minima. This can be easily deduced from a close exami-nation of Eq. (10). From this expression we see that, in ge-neral, the intensity measurements are both dependent on theaverage retardance Γ̄ and the fluctuation amplitude a; thusthey cannot be uncoupled with only the parallel and crossedpolarizers intensity measurements. Yet at the extremals, theretardance value is known; it is a multiple of 180 deg, andthe intensity measurements become only dependent onthe fluctuation amplitude a; then its value can be obtained.Afterward the amplitude a can be introduced in Eq. (11) toproduce a more accurate estimation of the average retar-dance Γ̄.

To obtain the fluctuation amplitude a, the deviationbetween calculated and true retardance values can be used.If we apply Eq. (11) at the extremals, where the true retar-dance value is multiple of 180 deg, we find that the absoluteretardance difference Γdiff between the calculated and thetrue retardance value is given by

Fig. 3 Simulation of the retardance measurement experiment in thepresence of fluctuations considering the classical linear polarimeter.The calculated retardance is obtained applying the classical expres-sion, i.e., not taking into account the fluctuations introduced in thesimulation, indicated in the plots (30, 60, and 90 deg). The calculatedretardance coincides with the true retardance in the case of no fluc-tuations (continuous line).




Γdiff ¼ cos−1�sin aa

�: (13)

Let us represent in Fig. 4(a) the absolute retardance diffe-rence Γdiff [Eq. (13)] as a function of the fluctuation ampli-tude a. We see that for a fluctuation amplitude range till~110 deg, which is a reasonably high value, the retardancedifference Γdiff increases with a high linearity (dashedline; fitting equation written on the plot). If the coordinatesfor the values in Fig. 4(a) are reversed, then we obtainFig. 4(b). This plot enables to estimate the magnitude ofthe fluctuation a from the value of the retardance differenceΓdiff exhibited at the extremals. For the retardance differencerange considered, the relation is highly linear. On the figurewe show (dashed line) the linear fit to the curve. For the sakeof completeness, we show the squared correlation factor,which is very close to one. Using this linear relation, wecan quantitatively obtain the magnitude of the fluctuationfrom the value of Γdiff .

We note that other alternatives to estimate the magnitudeof the fluctuation amplitude are possible, such as in Ref. 15.However, some advantages are given by the procedure givenin this paper, which is based on the discussion of the simu-lated experiment presented in Fig. 3. On one side, we playdirectly with the retardance values, what offers a directinsight to estimate how accurate the retardance returned bythe classical linear polarimeter procedure may be. This canbe qualitatively inferred from the deviation between calcu-lated and true retardance in Fig. 3. On the other side, thequantitative applicability of the technique, when calculatingthe fluctuation amplitude, is greatly enhanced since we canwork with linear relations, as we have seen in Fig. 4(b). Oncethe amplitude of the fluctuation a has been obtained, thenEq. (11) can be used to obtain the average retardance Γ̄.

3 Results and Discussion

3.1 Experiment

We consider the characterization of a PA-LCoS in two of thetypical working geometries:31 perpendicular incidence with abeam splitter in front and quasiperpendicular incidence (nobeam splitter is necessary in front of the LCoS). In Fig. 5, theexperimental setup for the case with the beam splitter in frontof the LCoS is presented. It basically consists of a lightsource (He-Ne laser producing linearly polarized light at

λ ¼ 633 nm), the device under test (reflective LCoS), andthe necessary input and output linear polarizers, which areused in parallel or crossed configuration, and some intensitymeasuring device (radiometer). A quarter wavelength waveplate is added after the laser to assure that enough intensitypasses through the first polarizer. We introduce two nonpo-larizing cube beam splitters (NPBS, model 10BC16NP.4,from Newport): one of them to separate the incident andreflected beams from the LCoS and the other to enableamplitude division with a 50∕50 ratio of the reflectedbeam so that crossed and parallel intensity can be measuredsimultaneously (thus, two radiometers are introduced). Thisfurther enables synchronized measurement of the instantane-ous parallel and crossed polarizer intensity values simply byconnecting the two radiometers to the two channels of anoscilloscope (shown in the figure). The same setup is there-fore valid for the characterization of the average and theinstantaneous values. For the other working geometry con-sidered in this work, quasiperpendicular incidence, the onlychange introduced is the removal of the beam splitter in frontof the LCoS, and the beam impinges quasiperpendicularlyonto the LCoS so that the incident and reflected beamsbecome spatially separated.

In the experiments performed, the input polarizer trans-mission axis is at þ45 deg to the vertical of the lab,which is the x axis of our reference system as shown inFig. 1. The director for the liquid crystal (i.e., the extraordi-nary axis) in the LCoS is oriented along the horizontal, i.e., atþ90 deg with respect to the x axis. The director axis in ne-matic filled devices, as it is the case for our LCoS, typicallycorresponds to the slow axis. In the first working geometry,light impinges perpendicularly onto the LCoS and onto thetwo beam splitters. Perpendicular incidence onto the beamsplitters is important so that they do not introduce polariza-tion effects on their own. In the second working geometry,the beam splitter in front of the LCoS is removed and thebeam incides at quasiperpendicular incidence (∼3 deg).Typically the angular uncertainty on the orientation of theinput and output polarizers with respect to the neutrallines of the LCoS can be considered �1 deg.

We note that the method proposed in Sec. 2 can be appliedto any electrooptic variable linear retarder. In this paper, wefocus our attention on phase-only LCoS devices. We con-sider a modern electrically controlled birefringence LCoSdisplay. Some of its technical specifications are as follows:

Fig. 4 (a) Retardance difference with respect to the ideal a ¼ 0 deg as a function of the fluctuationamplitude, at retardance points multiple of 180 deg. (b) Reverse relation, valid to estimate the magnitudeof the fluctuation amplitude. We observe the goodness of the linear fit for the range of values consideredin the plot.




1920 × 1080 pixels, 0.7” diagonal, 8.0 μm of pixel pitch,and with a fill factor of 87%. It is a digitally addressedLCoS device. Different digital addressing sequences13,14

can be generated by the driver electronics. In particular,we will evaluate three addressing sequences, whose configu-ration files are provided with the software for the display,whose labels indicate their main properties: 18-6 default,18-6 2pi linear 633 nm, and 5-5 2pi linear 633 nm. Thefirst number indicates the quantity of equally weightedbit-planes, and the second number the quantity of binarybit-planes.13 This means that the sequence 18-6 is longerthan the one corresponding for the sequence 5-5. In princi-ple, for shorter sequences, the flicker gets smaller.13

However, a larger sequence provides a larger number of pos-sible phase levels: ð18þ 1Þ × 26 ¼ 1216 for the sequence18-6 and ð5þ 1Þ × 25 ¼ 192 for the sequence 5-5. In prin-ciple, the sequences 18-6 2pi linear 633 nm and 5-5 2pilinear 633 nm have been, respectively, optimized for thewavelength 633 nm to provide a phase depth of 2π radians,and quite a linear relation between phase value and graylevel. Sequence 18-6 default is a nonoptimized configura-tion, where the applied voltage range is not restricted to pro-vide a 2π radians phase depth and reduction of fluctuationshave not been taken care of. Therefore, it is a good configu-ration to test the theory and methodology presented in thepaper. Using these three different sequences, we can evaluatedifferent levels of flicker and ranges of retardance.

3.2 Instantaneous Retardance Measurements

As previously stated, the experimental setup enables thesynchronized measurement of the parallel and crossed pola-rizer intensities in real time, connected to a two-channeloscilloscope. Thus, we can obtain the instantaneous retar-dance value as a function of time for a fix gray-level value.This real-time capability is interesting in order to evaluate theprofile for the instabilities. Furthermore, the instantaneousvalues can be used to calculate the average retardanceand its amplitude fluctuation, and these values can be

compared with the ones obtained applying the extended li-near polarimeter approach presented in Sec. 2.

In particular, we have proceeded with the measurement ofthe instantaneous retardance for a series of gray levels (0,100, 200, and 255), which sample the applicable voltagerange. Results are shown in Figs. 6(a) to 6(c), respectively,for the sequences 18-6 default, 18-6 2pi linear 633 nm, and5-5 2pi linear 633 nm for the first working geometry (withNPBS), and in Figs. 6(d) to 6(f) for the second workinggeometry (quasiperpendicular). We obtain that the timeperiod (frequency) for the fluctuations in our PA-LCoSdevice is 8.66 ms (120 Hz). In the different plots we seethat there are no visible oscillations for gray level 0. Forthe other gray levels, the periodic temporal evolution actuallyresembles in a first approximation a triangular profile, asconsidered in the model according to Fig. 2, even though,in general, the profile is composed of various subpeakswithin each period, probably due to the pulsed nature ofthe digital sequence applied. This substructure varies forthe various gray levels within a graph. It is also visiblethat the amplitude of the oscillations is smaller for thesequence 5-5 2pi linear 633 nm and clearly larger for 18-6 default. Actually, in the latter sequence, the oscillationsclearly expand out of 0 to 180 deg dominium of definitionfor the cos−1 function: they are represented as folded againstthe 0-deg ordinate and 180-deg ordinate since we do nothave a consistent criterion to unfold the values. Comparisonbetween the results for the two working geometries showthat basically the values obtained are not influenced bythe insertion of the NPBS (first working geometry) or by thenonperpendicular incidence (second working geometry).

From the measurements in Fig. 6, we have elaboratedTable 1, where we show quantitative values for the averageretardance and its corresponding fluctuation amplitude, cal-culated as the half peak-to-peak amplitude of the temporaloscillation. No results are shown for the sequence 18-6default since we do not have a consistent criterion to unfoldthe retardance values to obtain the average retardance and itsamplitude. The same happens for some of the gray levels for

Fig. 5 Experimental setup for the working geometry corresponding to perpendicular incidence ontothe liquid crystal on silicon with a beam splitter in front. The setup allows to measure both averageand instantaneous values.




the two sequences presented in Table 1. We see that fluc-tuation amplitude increases with the gray level. Values ofretardance and fluctuation are, in general, quite similarbetween the two working geometries. Fluctuation is slightlylarger for the sequence 18-6 2pi linear 633 nm in comparisonwith the sequence 5-5 2pi linear 633 nm.

3.3 Average Retardance and Flicker Evaluation

Typically, intensity measurements provided by radiometerscorrespond to the integrated value in a time interval muchlarger than the few milliseconds period observed in the fluc-tuating signals in Fig. 6. In Figs. 7(a) and 7(c), we show thisaveraged intensity values measured for parallel (continuous)

and crossed (dashed) polarizers as a function of gray levelapplied to the LCoS and for the three electrical sequences.Using these curves and applying Eq. (5), we obtain thecalculated average retardance represented in Figs. 7(b) and7(d). First and second rows correspond, respectively, to theNPBS and to the quasiperpendicular working geometries.The analysis of the intensity curves (first column) was thefocus of the work we presented in Ref. 15. In the presentpaper, we get a more complete and straightforward analysisby focusing on the retardance results (second column). Wesee that the calculated retardance oscillates with maxima andminima clearly separated from values 0 or 180 deg, whichaccording to simulation in Fig. 3 indicate the existence ofphase fluctuations. Retardance difference Γdiff at maximaand minima is larger for the sequence 18-6 default whencompared with the other two sequences, indicating that fluc-tuations are also larger. We also see that there is an additionalextremal for the sequence 18-6 default, which means thatthe retardance range is 180 deg larger than for the othertwo sequences. We note the quantization steps in the caseof the sequence 5-5 2pi linear 633 nm, more visible inFig. 7(a), due to the limited number of levels, which is<256. In the case of the sequence 18-6 default, we observea series of peaks in the gray-level interval of 150 to 200,which is probably reflecting the switching off and on of dif-ferent bits in the pulsed signal as the gray level is increased.

From the retardance difference Γdiff at the extremals andtaking into account the linear equation in Fig. 4(b), we canobtain an estimation of the fluctuation amplitude. In Table 2,we show under the column “average values” both the retar-dance difference Γdiff and the estimated fluctuation ampli-tude at the various extremals, whose gray-level position isindicated in parentheses. The values are given for thethree sequences and for the two working geometries. Let uscomment the results for the NPBS geometry. We see that atthe first extremal, fluctuations are small. In the secondextremal, the sequence 18-6 2pi linear 633 nm contains a

Fig. 6 Temporal evolution of the retardance at gray levels 0, 100, 200, and 255, indicated on the figure.(a) and (d) 18-6 2pi linear 633 nm; (b) and (e) 18-6 2pi linear 633 nm; (c) and (f) 5-5 2pi linear 633 nm.First and second rows correspond, respectively, to the nonpolarizing cube beam splitter (NPBS) and tothe quasiperpendicular working geometries.

Table 1 Values for the average retardance and the magnitude ofthe amplitude of its fluctuation extracted from the instantaneousretardance measurements shown in Fig. 6.

SequenceGraylevel

With NPBSAvg:Ret:� Fluct.

(deg)

QuasiperpendicularAvg:Ret:� Fluct

(deg)

18_6 linear 0 141� 2 135� 3

100 51� 37 —

200 86� 54 79� 57

255 — —

5_5 linear 0 125� 1 126� 2

100 77� 24 75� 31

200 67� 37 85� 53

255 — —




fluctuation amplitude of ∼52 deg, whereas the sequence 5-52pi linear 633 nm stays at ∼38 deg. Fluctuation amplitude inthe case of the sequence 18-6 default at its second and thirdextremals are 84 and 92 deg, which are very large valuesindeed. Results are quite comparable with the ones presentedfor the quasiperpendicular geometry.

To validate the fluctuation amplitudes predicted fromthe average values, shown in Table 2, next we measurethe instantaneous retardance values, as done in Sec. 3.1,applying the gray levels where the extremals are located.In Figs. 8(a) and 8(b), respectively, for the first and secondworking geometries, we show the temporal evolution of theretardance for each of the sequences at the gray levels where

the second extremal is produced, indicated on the figure. Theoscillation amplitude is larger for the sequence 18-6 default.We note that the results shown in Fig. 8 are the folded valuesagainst the 0-deg ordinate. In these figures, we have a cri-terion to unfold the instantaneous retardance since weknow that the average value is in any case close to 0 deg;thus half of the peaks should be folded against the 0-ordinateaxis. Taking this into account, the fluctuation amplitudeactually corresponds to the peak value in each of the curves.These values are written in the column “inst. values” inTable 2. When comparing the results in column “average va-lues” and “inst. values,” we see that there is a good agree-ment between the fluctuation amplitude obtained with the

Fig. 7 Intensity measurements for parallel and crossed polarizers [(a) and (c)], and retardance calcu-lations [(b) and (d)], given by the classical expression in Eq. (5). (a) and (b) correspond to the setup withthe NPBS. (c) and (d) correspond to the quasiperpendicular incidence.

Table 2 Comparison between the fluctuation amplitude a obtained with the method proposed in the paper (average values column) andthe experimental value obtained from the instantaneous retardance values (inst. values column), and for the two working geometries.

Sequence

With NPBS Quasiperpendicular

Average values Inst. values Average values Inst. values

Γdiff (deg) Fluct. amplit. (deg) Fluct. amplit. (deg) Γdiff (deg) Fluct. amplit. (deg) Fluct. amplit. (deg)

18_6 default 11 20 (GL 16) 20 (GL 16) 15 27 (GL 17) 28 (GL17)

47 84 (GL 61) 92 (GL 61) 42 75 (GL 66) 88 (GL 66)

52 92 (GL 170) — 51 91 (GL 175) —

18_6 linear 6 11 (GL 16) 18 (GL 16) 10 18 (GL 22) 17 (GL 22)

30 52 (GL 131) 59 (GL 131) 27 47 (GL 130) 51 (GL 130)

5_5 linear 6 11 (GL 26) 17 (GL26) 11 19 (GL 32) 17 (GL32)

21 38 (GL 153) 44 (GL 153) 19 34 (GL 146) 37 (GL 146)




method we propose and the one obtained from the instanta-neous retardance values that we measure. This is true for thetwo working geometries.

Once the fluctuation amplitude values have been vali-dated, we can introduce them in Eq. (11) to calculate theaverage retardance. What we do is to extrapolate the fluc-tuation amplitude value calculated from the extremals toa wider gray-level range: the fluctuation amplitude valuefrom the first extremal is considered to be roughly validuntil half of the gray-level distance between the two extre-mals, and from there, the value from the second extremal isthe one considered valid, and so on.

In Figs. 9(a) to 9(c), we show, respectively, for the threesequences the average retardance value calculated with thisprocedure as a function of the gray level. They correspond tothe “corrected” curve (continuous line) in the legend. The“uncorrected” (dashed line) correspond to the calculatedretardance already plotted in Figs. 7(b) and 7(d). In thegraphs we plot the results for the two working geometriesconsidered, NPBS and quasiperpendicular. In the case ofthe sequence 18-6 default [Fig. 9(a)], deviations are verylarge between corrected and uncorrected curves for mostof the gray-level range. However, in the case of the sequen-ces 18-6 2pi linear 633 nm and 5-5 2pi linear 633 nm[Figs. 8(b) and 8(c)], deviations are only encountered inthe proximity of the extremals, which means that applicabi-lity of the classical expression given in Eq. (5) may still bepossible in many situations in spite of the existence of phaseflicker. We also appreciate that results obtained for bothworking geometries practically overlap. This indicates that

polarization effects produced by the NPBS in front of theLCoS and obliquity effects in the case of quasiperpendicularincidence are very small. Therefore, the extended linearpolarimeter technique we have proposed in the paper is arobust method to obtain the average linear retardance fordevices exhibiting fluctuations or instabilities in their linearretardance.

4 ConclusionsWe have shown the applicability of the classical and well-established linear polarimeter to the measurement of linearretardance in the presence of phase flicker. This analysisshows that there may exist large errors in the results providedby the linear polarimeter when measuring the linear retar-dance of a device. However, these errors depend on the spe-cific retardance value under measurement, being a maximumwhen at retardance points multiple of 180 deg. We show thatthese specific points, instead of being a limitation, offer theopportunity to measure the fluctuation amplitude consis-tently. An elegant method is further proposed enabling themeasurement of the average retardance value, thus extendingthe applicability of the classical linear polarimeter to a newrange of samples. The method is both simple and does notrequire especial or expensive equipment. Validation of thetechnique has been accomplished with a setup enablingboth average and instantaneous retardance measurements.We have analyzed the applicability of the technique fortwo different working geometries enabling to separate inci-dent and reflected beams. Results obtained indicate thatpolarization effects produced by the NPBS in front of the

Fig. 8 Temporal evolution of the retardance at gray levels, indicated on the figure, corresponding to thesecond extremals in the plots in Figs. 7(b) and 7(d) and for the three sequences. (a) and (b) correspond,respectively, to the setup with the NPBS and to quasiperpendicular incidence.

Fig. 9 Average retardance versus gray level when considering the classical (uncorrected curves) andthe extended (corrected curves) linear polarimeter technique proposed in the paper, for both the NPBSand the quasiperpendicular geometries. (a) 18-6 default; (b) 18-6 2pi linear 633 nm; (c) 5-5 2pi linear633 nm.




LCoS and obliquity effects in the case of quasiperpendicularincidence are very small. Therefore, the extended linearpolarimeter technique we have proposed in the paper is arobust method to obtain the average linear retardance fordevices exhibiting fluctuations or instabilities in their linearretardance. As a final remark, specifically for the LCoSanalysis performed, we note that the sequences consideredin this work enable to attain a dynamic retardance rangeof 360 deg or even larger. In the case of the sequence5-5 2pi linear 633 nm, the amplitude of the fluctuations isbasically <30 deg. In the case of blazed gratings, this ena-bles diffraction efficiencies >95% [see the work of Lizanaet al.,18 Fig. 3(b)]. Therefore, existence of fluctuations isnot in itself a problem. Proper characterization enables tofind the optimal sequence as shown in the paper.

AcknowledgmentsThis work was supported by the Ministerio de Trabajo yCompetitividad of Spain (projects FIS2011-29803-C02-01and FIS2011-29803-C02-02), by the Generalitat Valencianaof Spain (projects PROMETEO/2011/021 and ISIC/2012/013), and by Universidad de Alicante (project GRE12-14).

References

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11. J. García-Márquez et al., “Flicker minimization in an LCoS spatiallight modulator,” Opt. Express 20, 8431–8441 (2012).

12. A. Lizana et al., “Influence of the temporal fluctuations phenomena onthe ECB LCoS performance,” Proc. SPIE 7442, 74420G (2009).

13. A. Hermerschmidt et al., “Wave front generation using a phase-onlymodulating liquid-crystal based micro-display with HDTV resolu-tion,” Proc. SPIE 6584, 65840E (2007).

14. J. R. Moore et al., “The silicon backplane design for an LCOS polari-zation-insensitive phase hologram SLM,” IEEE Photon. Techol. Lett.20, 60–62 (2008).

15. A. Márquez et al., “Classical polarimetric method revisited to analysethe modulation capabilities of parallel aligned liquid crystal on silicondisplays,” Proc. SPIE 8498, 84980L (2012).

16. C. Ramirez et al., “Point diffraction interferometer with a liquid crystalmonopixel,” Opt. Express 21, 8116–8124 (2013).

17. A. Lizana et al., “Time fluctuations of the phase modulation in a liquidcrystal on silicon display: characterization and effects in diffractiveoptics,” Opt. Express 16, 16711–16722 (2008).

18. A. Lizana et al., “The minimum Euclidean distance principle applied toimprove the modulation diffraction efficiency in digitally controlledspatial light modulators,” Opt. Express 18, 10581–10593 (2010).

19. L. Yao, Z. Zhiyao, and W. Runwen, “Optical heterodyne measurementof the phase retardation of a quarter-wave plate,” Opt. Lett. 13, 553–555 (1988).

20. L.-H. Shyu, C.-L. Chen, and D.-C. Su, “Method for measuring theretardation of a wave plate,” Appl. Opt. 32, 4228–4230 (1993).

21. A. Márquez et al., “Phase measurements of a twisted nematic liquidcrystal spatial light modulator with a common-path interferometer,”Opt. Commun. 190, 129–133 (2001).

22. S. Nakadate, “High precision retardation measurement using phasedetection of Young’s fringes,” Appl. Opt. 29, 242–246 (1990).

23. H. G. Jerrard, “Transmission of light through birefringent and opticallyactive media: the Poincaré sphere,” J. Opt. Soc. Am. 44, 634–640(1954).

24. P. Kurzynowski, “Senarmont compensator for elliptically birefringentmedia,” Opt. Commun. 197, 235–238 (2001).

25. G. Goldstein, Polarized Light, Marcel Dekker, New York (2003).26. N. G. Theofanous, “Error analysis of circular polarizer-analyzer

systems for phase retardation measurements,” J. Opt. Soc. Am. A 4,2191–2200 (1987).

27. P. A. Williams, A. H. Rose, and C. M. Wang, “Rotating-polarizerpolarimeter for accurate retardance measurement,” Appl. Opt. 36,6466–6472 (1997).

28. M. Born and E. Wolf, Principles of Optics, 7th ed., pp. 823–826,Cambridge University, Cambridge (1999).

29. A. Márquez et al., “Characterization of the retardance of a wave plateto increase the robustness of amplitude-only and phase-only modula-tions of a liquid crystal display,” J. Mod. Opt. 52, 633–650 (2005).

30. C. Ramirez et al., “Polarimetric method for liquid crystal displayscharacterization in presence of phase fluctuations,” Opt. Express 21,3182–3192 (2013).

31. A. Lizana et al., “Influence of the incident angle in the performance ofliquid crystal on silicon displays,”Opt. Express 17, 8491–8505 (2009).

Francisco J. Martínez received his BS degree in electronic engineer-ing fromUniversity of Valencia in 1996 and his BS degree in physics in1999. Currently, he is a research assistant with the Applied Physics toScience and Technology Institute from University of Alicante, andassistant professor of electronics in Miguel Hernandez University.His research interests include diffractive optics and holography.

Andrés Márquez received his MSc and PhD degrees in physics fromUniversidad Autónoma de Barcelona in 1997 and 2001, respectively.In 2000, he joined the Universidad de Alicante, where he is an asso-ciate professor of applied physics. His research interests include holo-graphic recording materials, liquid crystal spatial light modulators,optical image processing, and diffractive optics.

Sergi Gallego obtained his degree in physics at University ofValencia in 2001 and his PhD at the University of Alicante, Spain,in 2005, where he works as lecturer. His research interests includeholographic recording materials, diffraction, liquid crystal displaysapplied to holography, diffractive elements, and photopolymers. Hehas authored or coauthored one patent and more than 60 publicationsin renowned international journals (journals in the JCR of ISI).

Jorge Francés received his PhD degree at the University of Alicantein 2011. He received his MSEE in 2009 and his BSEE in 2006, bothfrom the Technical University of Valencia, Valencia, Spain. He hasbeen working as an assistant lecturer with the University of Alicantesince 2008. His main research interests include physical optics,sound and vibration, and numerical simulation.

Inmaculada Pascual received her MSc degree in physics fromthe University of Granada in 1985 and her PhD degree from theUniversity of Valencia in 1990. She is a full professor of optics atthe University of Alicante. She has carried out research in holography,mainly on holographic recording material, holographic optical ele-ments, and optical data storage. She has published more than 125papers and presentedmore than 170 papers in scientific conferences.




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3.2 Electrical dependencies of optical modulation capabilities in digitallyaddressed parallel aligned liquid crystal on silicon devices

3.2 Electrical dependencies of optical modulation cap-abilities in digitally addressed parallel aligned liquidcrystal on silicon devices

67

Electrical dependencies of optical modulationcapabilities in digitally addressed parallelaligned liquid crystal on silicon devices

Francisco Javier Martínez,a,b Andrés Márquez,a,b,* Sergi Gallego,a,b Manuel Ortuño,a,b Jorge Francés,a,b

Augusto Beléndez,a,b and Inmaculada Pascualb,caUniversidad de Alicante, Dept. de Física, Ing. de Sistemas y T. Señal, Ap. 99, Alicante E-03080, SpainbUniversidad de Alicante, I.U. Física Aplicada a las Ciencias y las Tecnologías, Ap. 99, Alicante E-03080, SpaincUniversidad de Alicante, Dept. de Óptica, Farmacología y Anatomía, Ap. 99, Alicante E-03080, Spain

Abstract. Parallel aligned liquid crystal on silicon (PA-LCoS) displays have found wide acceptance in applica-tions requiring phase-only modulation. Among LCoS devices, and PA-LCoS as a specific case, digital address-ing has become a very common technology. In principle, modern digital technology provides some benefits withrespect to analog addressing such as reduced interpixel cross-talk, lower power consumption and supply volt-age, gray level scale repeatability, high programmability, and noise robustness. However, there are also somedegradating issues, such as flicker, which may be enhanced. We analyze the characteristics of the digital pulsewidth modulated voltage signals in relation to their effect on the optical modulation capabilities of LCoS displays.We apply calibration techniques developed in our laboratory, basically the classical linear polarimeter extendedto take into account the existence of flicker. Various digital sequence formats are discussed, focusing the analy-sis on the variations in the magnitude of the applied voltages across the LC layer. From this analysis, we obtainhow to amplify the retardance dynamic range and how to enhance linearity in the device without enhancingflicker and without diminishing the number of available quantization levels. Electrical configurations intendedfor phase-only and intensity modulation regimes, useful in diffractive optics, are given. © 2014 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.53.6.067104]

Keywords: liquid crystal on silicon displays; parallel aligned; retardance measurement; phase-only modulation; spatial light modu-lation; flicker; diffractive optics; pulse width modulation.

Paper 140570P received Apr. 4, 2014; revised manuscript received May 20, 2014; accepted for publication May 22, 2014; publishedonline Jun. 24, 2014.

1 IntroductionLiquid crystal (LC) microdisplays have found widespreaduse as spatial light modulators (SLM) in applications suchas in diffractive optics,1 optical storage,2 optical metrology,3

reconfigurable interconnects,4,5 or quantum optical compu-ting.6 Among microdisplays, liquid crystal on silicon (LCoS)devices have become the most attractive system for theseapplications due to their very high spatial resolution andvery high light efficiency.7–9 LCoS displays are technologi-cally complex devices where the electrical signal appliedcontrols the optical transmission in each pixel. A propercharacterization and configuration of the main electricalparameters are crucial to generate the required optical modu-lation regimes in SLM applications, as exemplified by Luand Saleh10 with the previous analog liquid crystal displays(LCDs) taking into account the applied voltage dynamicrange and bias voltage. Interplay between electrical and opti-cal performance parameters has also been useful for identi-fying the origin for degradation effects in LCDs, such as theanamorphic and frequency dependent modulation,11 alsoreported in both digital12 and analog13 LCoS devices. Anadditional degradation effect detected in LCoS displays byseveral authors14–17 is that they produce phase flicker and/or depolarization, especially the ones with a digital back-plane18 due to the pulsed nature of the voltage signal

addressed.8,9,19,20 Among the different LC layer geometries,parallel aligned LCoS (PA-LCoS) devices are especiallyinteresting since they allow easy operation as phase-onlydevices without coupled amplitude modulation.8,9 No depo-larization operation is also available; however, they may stillexhibit phase fluctuations.18 Recently, we proposed a methodto characterize not only the linear retardance but also theamount of phase flicker in the optical signal21 valid for pa-rallel aligned LC devices. This method, the extended linearpolarimeter, has demonstrated its applicability to PA-LCoSdisplays21 exhibiting phase flicker; therefore, it can beused in the analysis of the influence of the electrical para-meters on the optical modulation performance propertiesof LCoS displays. This method is an elaboration of theclassical linear polarimeter which may be found to measurethe birefringence in LCs22 and in waveplates.23

As described by Lueder,24 pixel circuitry in the LCoS back-plane8,24–26 can be based on dynamic random-access memory(DRAM) or on static random-access memory (SRAM), whichis accordingly related to the differentiation between analogand digital addressing.27,28 If a memory is provided at thepixel in the form of an analog storage capacitor (DRAM)or digital memory circuit (SRAM), the array can be frame-addressed, i.e., all pixels are addressed simultaneously. Thishas the advantage of reducing the low frequency flickerof the line-addressed arrays, which was typical in LCDs.11

*Address all correspondence to: Andrés Márquez, E-mail: [email protected] 0091-3286/2014/$25.00 © 2014 SPIE

Optical Engineering 067104-1 June 2014 • Vol. 53(6)

Optical Engineering 53(6), 067104 (June 2014)


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Various reasons have motivated the introduction of digitalbackplanes in LCoS technology. The digital scheme is usu-ally more stable than analog and shows a repeatable perfor-mance. Field inversion is possible at the kilohertz range (e.g.,for each modulation pulse) without image retention. Thescheme does not suffer from electrical cross-talk (i.e., nolow pass filtering of the signal).29 A lower power consump-tion and a lower supply voltage are possible with the digitalbackplane.8,20 Digital driving can control the precise graylevel scale. Another advantage of pulse modulation is thatthe LC is driven between zero and saturation voltages in atime multiplexed fashion,26–28 so that the slow response ofswitching between adjacent voltage states is avoided. Thenematic LC responds to the root-mean-square (RMS) of theapplied waveform integrated over the switching time of theLC. However, the time varying AC modulation of the drivingvoltage is partially transferred to the LC and produces a tem-poral fluctuation or flicker of the reflected light as describedin Refs. 18–20, i.e., the electro-optical response of the SLMis not constant over the frame period. If the digital sequenceformat is properly chosen, fluctuations can be significantlydecreased to acceptable levels.21 To this goal, a proper cha-racterization not only of the phase modulation but also of theamount of phase flicker in the optical signal needs to bedone. This reasoning is not restricted to PA-LCoS devices,but it can be applicable in general to electro-optic variablelinear retarders such as the novel device proposed byRamirez et al.30 Apart from the sequence format, otheroptions (applied binary voltage value, selection of pulsecodes, and number of gray scale levels) offered by digitaladdressing need to be analyzed to produce proper phasedepth, enhanced linearity, number of quantization levelsavailable, etc.

In this work, we take advantage of the ease of imple-mentation and the effectiveness of the calibration methodproposed in Ref. 21 to characterize the basic electricalparameters controlling the digital signal addressed to a PA-LCoS display and their effect on the retardance dynamicrange and the magnitude of the flicker. Preliminary resultswere presented in Ref. 31. This enables us to produce elec-trical configurations specifically optimized for phase-only orfor amplitude-mostly modulation regimes.

2 Characterization Using the Extended LinearPolarimeter

In the following, we introduce the extended linear polari-meter proposed in Ref. 21, where the linear retarder undertest, in our case the PA-LCoS, is situated between two linearpolarizers illuminated with a monochromatic laser source. Inthe analysis shown in Ref. 21, it was demonstrated that theclassical technique based on measuring the intensity with aradiometer at the exit of the system for the parallel andcrossed configurations of the polarizers, both oriented at45 or −45 deg with respect to the neutral lines of thewaveplates, is not only a simple method but also the mostrobust to misalignments and other types of noise sourcesin the system. With respect to the classical technique, theextended linear polarimeter21 enables us to characterizethe retardance magnitude in the presence of retardance insta-bilities or flicker. Other techniques may also be applied;32,33

however, the present method is very straightforward to

implement and accurate enough for the study to be donein this work.

In the case of instabilities or fluctuations in the linearretardance, we demonstrated21 that as a first approximationfor PA-LCoS displays we may consider a model with a linearvariation of retardance with time ΓðtÞ. In this case, the aver-aged parallel and crossed polarizers intensities become

hIjjOUTi ¼I02

�sin aa

cos Γ̄�; (1)

hI⊥OUTi ¼I02

�sin aa

cos Γ̄�; (2)

where Γ̄ and a are the values for the average retardance andits flicker fluctuation amplitude, respectively. When combi-ning the two expressions, we obtain

hIjjOUTi − hI⊥OUTihIjjOUTi þ hI⊥OUTi

¼ sin aa

cos Γ̄; (3)

where we see that the cosine term becomes modulated by thepresence of a sinc function dependent on the magnitude ofthe fluctuation a. This sinc function represents the deviationfrom the classical expression typically used to obtain the li-near retardance value.21 Eventually, the average retardance Γ̄is obtained by inverting Eq. (3)

Γ̄ ¼ cos−1�hIjjOUTi − hI⊥OUTi∕hIjjOUTi þ hI⊥OUTi

sin a∕a

�: (4)

In the case when no fluctuations exist (a ¼ 0 deg), thenEqs. (1)–(4) simplify into the classical results. We note thatEq. (4) returns the wrapped phase values in the range 0 to180 deg, which is the definition domain for the cos−1 func-tion. In general, the average retardance Γ̄ and the fluctuationamplitude a are coupled in Eqs. (1)–(4), yet at the maximaand minima (the extremals) the retardance value is known; itis a multiple of 180 deg (0 or 180 deg in the wrappeddomain). At these points, we may calculate the value for thefluctuation amplitude a and then it can be introduced inEq. (4) to produce a more accurate estimation of the averageretardance Γ̄ out of the extremals (what in Fig. 4 will becalled the “corrected” values). To obtain a, it was demon-strated that at the extremals, the absolute retardance diffe-rence Γdiff between the calculated and the true retardance val-ues is

Γdiff ¼ cos−1�sin aa

�: (5)

By inverting the set of numerical values (Γdiff ; a) providedby Eq. (5), we produce a fitting function enabling us to cal-culate a as a function of the value measured for Γdiff. Thisfitting function was calculated in Ref. 21 for a fluctuationamplitude a interval covering the range (0 deg, 110 deg)and is given by the linear expression


Martínez et al.: Electrical dependencies of optical modulation capabilities. . .


a ¼ 1.7795Γdiff : (6)

In Fig. 1, we present the experimental setup that we use inthis work to obtain the parallel and crossed polarizers inten-sities. The unexpanded beam from a He-Ne laser (633-nmwavelength) is incident onto a linear polarimeter where thereflective LCoS is the linear retarder to be measured. A quar-ter wavelength waveplate is added after the laser so thatenough light passes through the first polarizer irrespectiveof the orientation of its transmission axis. We consider qua-siperpendicular incidence at a 3-deg incidence angle onto theLCoS. We introduce one nonpolarizing cube beam splitter(NPBS) (model 10BC16NP.4, from Newport) to enableamplitude division with a 50/50 ratio of the reflectedbeam so that crossed and parallel intensities can be measuredsimultaneously (thus, two radiometers are introduced). Thisfurther enables synchronized measurement of the instantane-ous parallel and crossed polarizers intensity values simply byconnecting the two radiometers to the two channels of anoscilloscope (shown in this figure). The same setup is, there-fore, valid for the characterization of the average and theinstantaneous values. We note that in Ref. 21, we showedthat insertion of NPBS did not introduce significant polari-zation effects in the calibration when perpendicular inci-dence onto the beam splitter is considered as will be the casein this paper.

3 Digital Electrical Signal Addressing ParametersLet us provide the basic driving procedure to understand thegeneral parameters configuring the pulsed signal addressedin digital backplane LCoS displays. There are two steps:23,25

data addressing to the pixel SRAMmemory in the backplaneand LC driving. First, the input signal is processed by thecontroller and turned into bit-plane format. In this step, alook-up table (LUT) is used to determine the correct bitsequence for a target gray level. The first bit plane image iswritten to the pixel SRAM of the digital backplane, wherethe pixel memory has only “on” or “off” information. Theremay be a master and a slave memory enabling data to betransferred to a pixel at the same time that the previously

loaded data are being displayed. After the data-addressingstep is finished, the next step is to drive the LC wherethree voltage signal values, V0 and V1, low and high voltagesignals on the pixel backplane, and VITO, on the ITO trans-parent electrode on the glass substrate, are used. These volt-ages are applied to all pixels of the display area at the sametime so that every pixel is driven by the same condition. TheV0 and V1 signals are selected by the SRAM information ofthe pixel. Then, the driving voltage across the LC layer VLC

is given by VITO − V1 or VITO–V0, in the following Vbright

and Vdark, respectively. A schematic diagram for the absolutevalue of the digital signal VLC is shown in Fig. 2 (upper line),where the amplitudes for Vbright and Vdark are represented. Inthis driving scheme, the LC layer is driven by high frequencythat approximates a square wave running at several kilohertz.The nematic LC responds to the RMS value of the drivingvoltage over the switching time of the LC and the residualAC modulation is responsible for the flicker described in di-gital LCoS devices, represented in Fig. 2 (lower line). This iswhy it is generally understood that flicker noise can be mini-mized by high modulation rates.25,34

Fig. 1 Experimental setup used to measure the linear retardance as a function of the applied voltage(gray level) for a parallel aligned liquid crystal on silicon (LCoS). The setup allows both to measure aver-age and instantaneous values.

Fig. 2 Schematic diagram for the different voltage magnitudesinvolved in the digital addressing of the LCoS. Upper line driving volt-age V LC applied across the LC layer. Lower line voltage as seen bythe LC molecules.




The analysis in this paper deals with a digitally addressedphase-only LCoS, specifically the electrically controlledbirefringence LCoS display distributed by the companyHOLOEYE (Berlin, Germany). It is an active matrix reflec-tive mode device with 1920 × 1080 pixels and a 0.7 in.diagonal named the PLUTO SLM, with a pixel pitch of8.0 μm and a fill factor of 87%. The signal is addressedvia a standard digital visual interface signal. By means ofthe RS-232 interface and its corresponding provided soft-ware, we have access to the basic electrical parameters ofthe device19,20 previously described, such as the digitaladdressing sequence format (bit planes), the gamma curve,and the voltage dynamic range of the pulse width modulated(PWM) signal (through two digital potentiometers), i.e., volt-ages Vbright and Vdark.

The software for the SLM is provided with a series ofconfiguration files corresponding to two different digitaladdressing sequences: the so-called 18-6 and 5-5 digitalsequences where the first number indicates the quantity of“equally weighted” bit planes and the second numberindicates the quantity of “binary” bit planes.19 This translatesinto a number of distinct voltage levels as follows: ð18þ 1Þ×26 ¼ 1216 for the sequence “18-6” and ð5þ 1Þ × 25 ¼ 192for the sequence “5-5.” The larger number of bit planesmakes the sequence 18-6 longer than the one correspondingfor the sequence 5-5. In principle, the shorter the sequence,the smaller the flicker,19 but on the other side, a largersequence provides a larger number of quantization levels.For each of the sequences, we find a series of configurationfiles which have been, in principle, optimized by the vendorfor different wavelengths to enable SLM users a fast accessto a proper operating configuration for the device adapted tothe specific application and illuminating wavelength. In ge-neral, these configuration files enable phase-only operationwith a 2π linear phase dynamic range for the specified wave-length. To this goal, the digital potentiometer values and thegamma curve, which corresponds to the LUT, need to beproperly selected.

In Sec. 2, we have presented an efficient and effectivetechnique able to discriminate, in terms of the average retar-dance dynamic range and the magnitude of the flicker,between the digital addressing sequences given with the soft-ware and driver electronics. Finer adjustment of the retar-dance dynamic range and its linearity is then feasible bymodifying the gamma curve and changing the digital poten-tiometers values. The simplicity of the linear polarimetersetup on which the method is based allows SLM users tohave access to adapt the calibration to their specific working

geometry and illuminating wavelength, adding novel pos-sibilities to the configuration files provided by the vendor.

In the rest of this paper, we focus on sequence 5-5 since itproduces less flicker than sequence 18-6, as previously com-mented on and as already demonstrated in a number ofpapers.19,35 By applying different gamma curves and differ-ent Vbright and Vdark voltages to a specific sequence we mayobtain very different retardance versus gray level curves. InFigs. 3(a)–3(c), we show three examples of gamma curvesfor the 5-5 sequence, corresponding, respectively, to the con-figuration files named “5-5 default ramp,” “5-5 linear 2pi633 nm,” and “5-5 linear 2pi 405 nm.” As we will showin Sec. 4.1, the first configuration provides the largest vol-tage modulation range available, thus enabling us to estimatethe maximum retardance dynamic range provided by theLCoS unit. The second and third configurations span aretardance range of 2π and π radians, respectively, which arethe necessary ranges to produce, respectively, a phase-onlyregime and an amplitude-mostly regime when using a va-riable linear retarder.

Figure 3(a) shows a linear mapping between the input andoutput values where all the 192 possible values associatedwith the 5-5 sequence are available. In Fig. 3(b), the mappingis highly nonlinear, especially for lower gray levels, and only95 out of the 192 possible values are available. In Fig. 3(c),the mapping is rather linear and a large quantity of gray le-vels, 175, are available.

With respect to the voltages Vbright and Vdark, we note thatboth the configuration file “5-5 default ramp” and “5-5 linear2pi 633 nm” have the same values, which are Vbright ¼3.82 V and Vdark ¼ 0.03 V. In the case of the configuration“5-5 linear 2pi 405 nm,” these values are Vbright ¼ 2.02 Vand Vdark ¼ 1.36 V. A change in these values affects thevoltage amplitude of the PWM pulses, thus affecting theaccumulated voltage experienced by the LC molecules in awhole frame period. We note that the largest voltage rangepossible is 3.82 to 0.03 V.

4 Results and Discussion

4.1 Initial Configurations

Some additional details are necessary to complement theexperiment description given in relation to Fig. 1. The in-tensities for both the parallel and crossed polarizer measure-ments have been obtained with the input polarizer transmis-sion axis at þ45 deg to the vertical of the laboratory, whichis the X-axis of our reference system. We note that the direc-tor for the LC (i.e., the extraordinary axis) in the LCoS isoriented along the horizontal, i.e., at þ90 deg with respect

Fig. 3 Gamma curves for the PLUTO LCoS for the configuration files: (a) “5-5 default ramp,” (b) “5-5linear 2pi 633 nm,” and (c) “5-5 linear 2pi 405 nm.”




to the X-axis. The director axis in nematic filled devices, as itis the case for our LCoS, typically corresponds to the slowaxis. Typically, the angular uncertainty on the orientation ofthe input and output polarizers with respect to the neutrallines of the LCoS can be considered to be �1 deg.

In Fig. 4, we plot the retardance (wrapped values) calcu-lated using the classical approach (uncorrected curves),which considers no flicker, and the extended calibrationmethod (corrected curves). The plots in Figs. 4(a)–4(c) cor-respond, respectively, to the configurations “5-5 defaultramp,” “5-5 linear 2pi 633 nm,” and “5-5 linear 2pi 405 nm.”From the retardance difference at the extremals in the uncor-rected curves, and applying the polynomial in Eq. (6), thefluctuation amplitude a can be calculated and the correctedcurves are obtained. These values are presented in Table 1.We see that the fluctuation amplitude increases with theapplied gray level: for example, for the configuration “5-5default ramp,” the successive extremals exhibit fluctuationamplitudes of 14, 34, and 43 deg.

If the retardance is unwrapped in Fig. 4(a), we obtain adynamic range of about 560 deg, exceeding the necessary360 deg for phase-only modulation. We observe fromFig. 4(a) that the uncorrected and corrected retardance curvesonly deviate at values which are multiples of 180 deg. Wealso see that the retardance presents a nonlinear relation withthe gray level. In Fig. 4(b) for the configuration “5-5 linear2pi 633 nm,” the retardance dynamic range is slightly higherthan 360 deg, and the relation with the gray level is very

linear. Therefore, the gamma curve in Fig. 3 and the Vbright

and Vdark values have been properly tuned and this configu-ration may be used for phase-only operation: this requiresincident light linearly polarized along the molecular directorof the LCoS (along the horizontal). In Fig. 4(c), the curveshows a retardance range o about 180 deg. This range isappropriate to obtain amplitude-mostly operation as weshowed in Fig. 6(b) in Ref. 31.

4.2 Control of the Modulation Range and Flicker

In Sec. 3, we described the various parameters governing themagnitude and structure of the digital signal addressed. Inthis section, we present the effect observed when the valuesin the parameters are changed and how this can be used tocontrol the modulation range to our benefit. We take advan-tage of the fact that the software interface for the digitalLCoS model we use enables us to change the values inthese parameters, in particular Vbright and Vdark. We will con-sider the gamma curve and sequence corresponding to con-figuration “5-5 default ramp” which provides the largerdynamic modulation range, as shown in Fig. 4(a).

We want to study the effect of the magnitude of Vbright andVdark on the retardance value. To start with we want toremove the pulsed nature of the digital signal addressed.To this goal, Vbright and Vdark must be equal. In fact, the mini-mum difference available with our LCoS device turns out tobe 0.06 V, i.e., slightly different from zero.

In Fig. 5, we show the instantaneous retardance values,measured with the setup in Fig. 1, for a series of configura-tions with Vbright ≅ Vdark and at gray level 100. The first and

Fig. 4 Retardance (wrapped values) calculated using the classical approach (uncorrected curves) andthe extended calibration method (corrected curves) for the configurations: (a) “5-5 default ramp,” (b) “5-5linear 2pi 633 nm,” and (c) “5-5 linear 2pi 405 nm.”

Table 1 Fluctuation amplitude a (third column) calculated for threeelectrical configuration files (first column), using the values for theretardance difference (second column) at the extremals for the paral-lel (or crossed) polarizers intensity curves as given by the calibrationmethod in Ref. 21.

Configuration Γdiff (deg) a (deg)

5-5 default ramp 7.9 (GL 40) 14

19.2 (GL 80) 34

24.5 (GL 140) 43

5-5 linear 2pi 633 nm 6.6 (GL 26) 12

20.6 (GL 155) 36

5-5 linear 2pi 405 nm 6.6 (GL 210) 12 Fig. 5 Retardance temporal evolution for gray level ¼ 100 forV dark ≅ V bright.




second values in each pair in the legend correspond, respec-tively, to Vdark and Vbright, ranging from (0.03 V, 0.09 V) to(3.40 V, 3.46 V), thus covering a wide voltage range. We seethat the curves show no flicker, in agreement with the factthat the pulse character in the digital signal is lost: boththe on and off bits addressed have equal voltage amplitudes.The fact that there are no temporal instabilities is also indica-tive that other origins for the flicker, such as thermal noise,can be disregarded. When applying other gray levels, theretardance values obtained remain the same and stable withtime. We note that there is a threshold value below which themolecules do not react. This can be seen because of the over-lap between the retardance curves for 0.03 and 0.39 V. Forhigher voltage values, retardance curves start to evolve anddo not overlap.

Next, let us consider different values for the parametersVdark and Vbright, so that retardance evolves as a function ofgray level. The goal is to analyze how the retardance dynamicrange as a function of gray level is affected. It is important tokeep in mind that, according to the explanation in Sec. 3, thetime structure for the pulsed sequence associated with eachgray level is not affected by the changes in Vdark andVbright and only the accumulated applied voltage must vary.

In Figs. 6(a) and 6(b), we plot the average retardance as afunction of gray level when Vdark and Vbright are, respectively,changed. We note that in Figs. 6(a) and 6(b), voltages Vbright

and Vdark are fixed, respectively, at 3.82 and 0.03 V, whichare the values for the results shown in Fig. 4(a). In Fig. 6(a),as we increase Vdark the resulting effect is the increase of theaccumulated voltage applied associated with each gray level:maxima in the curves shift to lower gray levels as we increaseVdark. At high applied voltages (high gray levels), we observea saturation process and all the curves tend to a commonaverage retardance value of about 20 deg. At gray levelzero, the increase of Vdark changes the starting retardancevalue which, combined with the saturation process, resultsin a reduction of the total retardance range. We note thatthe retardance at gray level zero is equal to the value obtainedin the previous Fig. 5 for equal Vdark voltages, thus indicatingthat the pulsed sequence associated with gray level zero hasall the bits in the off-state.

In Fig. 6(b), as Vbright increases the maxima in the curvesshift to lower gray levels, i.e., there is an increase in the accu-mulated voltage applied associated with each gray level. Acombination of the conclusions from Figs. 6(a) and 6(b) pro-vides a guide to tune the parameters Vdark and Vbright to pro-duce specific retardance dynamic range spans as a functionof gray level. For example, maximizing the retardancedynamic range means that the accumulated voltage valuesapplied at gray level zero and at high gray levels are, respec-tively, below LC threshold and close to saturation voltage. Tothis goal, Vdark must be higher than zero but lower than thethreshold and Vbright must be high enough to saturate the LCreorientation: we see in Fig. 6(a) that this is what is obtainedwith the values (Vdark ¼ 0.51 V, Vbright ¼ 3.82 V) and themaximum retardance span with our LCoS is about560 deg (unwrapped) at the 633-nm wavelength.

Let us now analyze the retardance difference at theextremals in Figs. 6(a) and 6(b) to learn about the effectof parameters Vdark and Vbright on the flicker magnitude.Comparison between the two maxima exhibited by eachof the curves shows that are smaller at the first maximum,occurring at lower gray level values, i.e., lower accumulatedvoltages. On the other side, if we look at the second maxi-mum, we see that may partly depend on the voltage diffe-rence between Vbright and Vdark: a smaller voltage differenceseems to produce smaller values, i.e., decrease in the flicker.This can be more clearly analyzed in Figs. 7(a) and 7(b)where the average retardance as a function of gray levelis plotted for various combinations of Vbright and Vdark

providing different relations of the voltage difference. InFig. 7(a), we see that the curve (0.81 V, 3.16 V) providesa smaller value at the second maximum. In Fig. 7(b), diffe-rence Vbright− Vdark is kept constant at the maximum valueallowed by the software, which is 3.80 V: Γdiff remains con-stant independently of the voltage parameters Vbright andVdark. What is clear from Figs. 6 and 7 is that the retardancedifference cannot be diminished from about 20 deg by sim-ply varying Vbright and Vdark, so other strategies should beconsidered, such as the one reported by García-Márquezet al.,17 consisting of reducing the temperature of theLCoS to increase the viscosity of the LC thus reducing itstime response.

Fig. 6 Average retardancemeasured for a series of configurations withV dark ≠ V bright: (a) changing V dark(V bright ¼ 3.82 V) and (b) changing V bright (V dark ¼ 0.03 V). Gamma curve and sequence correspondingto “5-5 default ramp;” λ ¼ 633 nm.




4.3 Conventional Modulation Regimes andSpecialized for Applications

We can use all the previous analyses to generate the particu-lar modulation configurations needed for each specific appli-cation. We will consider two typical cases to exemplify ourapproach: phase-only modulation and maximum intensity oramplitude-mostly modulation. We want to produce thesemodulation regimes with a high number of quantization le-vels, high linearity as a function of gray level, and with mi-nimal flicker influence.

If we want to use the PA-LCoS as a phase-only modulator,linearly polarized light along the director axis must illumi-nate the device and a 360-deg retardance range as a functionof gray level should be produced. This is enabled by configu-ration “5-5 linear 2pi 633 nm” as shown in Fig. 4(b); how-ever, it provides only 95 out of the 192 possible values whichare available for the 5-5 sequence format. To this goal, ourstarting point is the configuration “5-5 default ramp” whichcovers the maximum retardance range of 560 deg. To limitthe total range to 360 deg, the first step is to accordinglyreduce the Vbright voltage. In Fig. 8(a), we show the averageretardance for the original and for the reduced voltage range.

Fig. 7 Average retardance measured for a series of configurations where the voltage difference betweenV bright and V dark is the magnitude of interest: (a) Different voltage differences and (b) keeping the maxi-mum voltage difference. Gamma curve and sequence corresponding to “5-5 default ramp;” λ ¼ 633 nm.

Fig. 8 (a) Retardance range comparison when V bright is changed in the “5-5 default ramp” configuration.(b) Linearization look-up table (LUT).

Fig. 9 Configuration for phase-only modulation linearized with themethod described, shown against the configuration provided byvendor.




Changing Vbright reduces our retardance range to the ne-cessary value, but the curve is highly nonlinear. To linearizethe curve first we calculate the corrected retardance by apply-ing the extended calibration method21 as was shown inFig. 4. Then the average retardance and gray level axesare inverted and the corresponding linearization LUT isobtained, shown in Fig. 8(b), where the whole range of192 quantization levels is maintained.

In Fig. 9, we show the average retardance as a function ofgray level for the configuration “5-5 linear 2pi 633 nm” pro-vided by the vendor and the result obtained by applying theconfiguration we have generated with the method described,which we call “5_5 user linearized.” We plot both the uncor-rected and the corrected measurements for both configura-tions. Our novel configuration maintains all the 192 levelsavailable and is highly linear and due to the reduction inthe difference between Vbright and Vdark, it shows less flickerthan the configuration provided by the manufacturer.

Next, we take advantage of the understanding alreadygained to produce a totally new configuration, able to pro-duce intensity modulation with a high contrast. Now, insteadof being linear in average retardance, we need a linear inten-sity curve as a function of gray level. The starting point isonce again the configuration “5-5 default-ramp” whichhas all the 192 levels available. First of all, we have to reduce

the retardance range to 180 deg, enough to produce a fullswing between zero and maximum intensity modulationfor the PA-LCoS inserted between crossed polarizers withtheir transmission axes at 45 deg with respect to the directoraxis of the LCoS. This is the actual configuration presentedin Sec. 2 for the extended linear polarimeter. We find thatapplying Vdark ¼ 1.23 V and Vbright ¼ 1.90 V reduces theaverage retardance to a value close to the demanded180 deg range as we show in Fig. 10(a). Linearization isnow obtained by inverting the intensity transmission versusgray level, and the resulting linearization LUT to be appliedis shown in Fig. 10(b). The number of quantization levels is172, very close to the available 192. A retardance rangeslightly larger than 180 deg was previously tolerated andthis results in a small cut in the number of quantization le-vels. Finer adjustment may be possible if necessary.

In Fig. 11, we plot the normalized intensity transmissionas a function of gray level obtained with the LCoS betweencrossed (and parallel) polarizers with their transmission axesat 45 deg with respect to the director axis. We see both thecurve before and after the linearization LUT is applied. Avery linear curve switching from zero to maximum intensityis obtained showing the feasibility of our proposal.

5 ConclusionsIn this work, we have applied the extended linear polarimetermethod to analyze the influence on the optical signal of thevarious electrical parameters governing the signal addressedonto digital LCoS devices. The analysis has been performedin terms of the average retardance dynamic range and themagnitude of its flicker. Proper choice of the digital addres-sing sequence, together with the fine adjustment available viathe software options given by the gamma curve and the vol-tage parameters Vdark and Vbright, enable the control of theretardance dynamic range, the linearity in the response ofthe device, the number of quantization levels, and to someextent the magnitude of the flicker amplitude. Applyingthe results from this analysis, optimum electrical configura-tions intended for phase-only and intensity or amplitude-mostly modulation regimes, useful in diffractive optics, havebeen obtained.

Fig. 10 (a) Average retardance for the novel intensity modulation configuration compared with thedefault configuration. (b) Linearization LUT.

Fig. 11 Intensity versus gray level before and after linearization.




AcknowledgmentsThis work was supported by the Ministerio de Trabajo yCompetitividad of Spain (projects FIS2011-29803-C02-01and FIS2011-29803-C02-02), by the Generalitat Valencianaof Spain (projects PROMETEO/2011/021 and ISIC/2012/013), and by the Universidad de Alicante (project GRE12-14).

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Francisco Javier Martínez received his BS degree in electronic engi-neering from University of Valencia in 1996 and his BS degree inphysics in 1999. Currently, he is a research assistant with the AppliedPhysics to Science and Technology Institute from University of Ali-cante, and assistant professor of electronics at Miguel HernandezUniversity. His research interests include diffractive optics andholography.

Andrés Márquez received his MSc and PhD degrees in physics fromUniversidad Autónoma de Barcelona in 1997 and 2001, respectively.In 2000, he joined the Universidad de Alicante, where he is an asso-ciate professor of applied physics. His research interests include holo-graphic recording materials, liquid crystal spatial light modulators,optical image processing, and diffractive optics.

Sergi Gallego obtained his Msc degree in physics at University ofValencia in 2001 and his PhD degree at the University of Alicante,Spain, in 2005, where he works as a lecturer. His research interestsinclude holographic recording materials, diffraction, liquid crystal dis-plays applied to holography, diffractive elements, and photopolymers.He has authored or coauthored 1 patent and more than 60 publica-tions in renowned international journals.

Manuel Ortuño received his MSc degree in organic chemistry fromthe University of Murcia, Spain, in 1993. He received his MSc degreein chemical engineering in 1999 and a PhD degree in physics from theUniversity of Alicante, Spain, in 2005. In 2004, he joined theUniversidad de Alicante, where he is an associate professor. Hisresearch interests include light sensitive materials, photopolymers,holographic recording materials, liquid crystals, dyes and greenmaterials.

Jorge Francés received a PhD degree at the University of Alicante in2011. He received his MSEE in 2009 and his BSEE in 2006, both fromthe Technical University of Valencia, Valencia, Spain. He has beenworking as an assistant lecturer with the University of Alicantesince 2008. His main research interests include physical optics,sound and vibration, and numerical simulation.

Augusto Beléndez received his MSc degree and PhD degree inphysics from the University of Valencia, Spain, in 1986 and 1990,respectively. In 1996 he became full professor of applied physicsat the University of Alicante. He is mainly interested in holography,holographic recording materials, holographic optical elements, opticalprocessing, nonlinear oscillations and teaching of physics and engi-neering. He has published more than 250 technical papers and pre-sented more than 210 papers at scientific conferences.

Inmaculada Pascual received her MSc degree in physics from Uni-versity of Granada in 1985 and her PhD degree from University ofValencia in 1990. She is a full professor of optics at the Universityof Alicante. She has carried out research in holography, mainly onholographic recording material, holographic optical elements, andoptical data storage. She has published more than 125 papers andpresented more than 170 papers at scientific conferences.




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3.3 Retardance and flicker modeling and characterization of electro-opticlinear retarders by averaged Stokes polarimetry

3.3 Retardance and flicker modeling and character-ization of electro-optic linear retarders by averagedStokes polarimetry

79

Retardance and flicker modeling and characterizationof electro-optic linear

retarders by averaged Stokes polarimetryFrancisco J. Martínez,1,2 Andrés Márquez,1,2,* Sergi Gallego,1,2 Jorge Francés,1,2

Inmaculada Pascual,2,3 and Augusto Beléndez1,2

1Dept. de Física, Ing. de Sistemas y Teoría de la Señal, Universidad de Alicante, Ap. 99, E-03080 Alicante, Spain2I.U. Física Aplicada a las Ciencias y las Tecnologías Universidad de Alicante, Ap. 99, E-03080 Alicante, Spain

3Dept. de Óptica, Farmacología y Anatomía, Universidad de Alicante, Ap. 99, E-03080 Alicante, Spain*Corresponding author: [email protected]

Received October 16, 2013; revised December 22, 2013; accepted January 12, 2014;posted January 14, 2014 (Doc. ID 199630); published February 12, 2014

A polarimetric method for the measurement of linear retardance in the presence of phase fluctuations is presented.This can be applied to electro-optic devices behaving as variable linear retarders. The method is based on an ex-tended Mueller matrix model for the linear retarder containing the time-averaged effects of the instabilities. As aresult, an averaged Stokes polarimetry technique is proposed to characterize both the retardance and its flickermagnitude. Predictive capability of the approach is experimentally demonstrated, validating the model and thecalibration technique. The approach is applied to liquid crystal on silicon displays (LCoS) using a commercialStokes polarimeter. Both the magnitude of the average retardance and the amplitude of its fluctuation are obtainedfor each gray level value addressed, thus enabling a complete phase characterization of the LCoS. © 2014 OpticalSociety of AmericaOCIS codes: (120.2040) Displays; (120.5410) Polarimetry; (230.3720) Liquid-crystal devices; (230.6120) Spatial light

modulators; (230.2090) Electro-optical devices.http://dx.doi.org/10.1364/OL.39.001011

The linear variable retarder is a common electro-opticaldevice that may be used to generate or to detect specificstates of polarization (SOP) in systems such as Stokes orMueller matrix polarimeters [1–3]. More complex devi-ces, such as parallel aligned liquid crystal on silicon(PA-LCoS) displays [4,5] or devices such as the liquidcrystal-based point diffraction interferometer proposedby Ramirez et al. [6], can be assimilated to linear variableretarders. Since these electro-optic devices are totallycharacterized by their linear retardance values, thenmethods typically used in the characterization of wave-plates become available. The most popular are ellipso-metric, polarimetric, and interferometric ones [1,7].One property common to all linear retardance charac-

terization methods is that they assume that the birefrin-gence in the waveplate has a constant value, nofluctuations, during the measurement process. In thepresence of fluctuations or instabilities, measured for ex-ample in modern LCoS devices [8–13] these methods mayprovide erroneous results [14]. Furthermore, the ampli-tude of the retardance fluctuation becomes a magnitudeof interest for a more accurate characterization and mod-eling of the device under test. Recently some techniqueshave been demonstrated by our group [14] and byRamirez et al. [15], based respectively on the classicallinear polarimeter and in a combination of linear andcircular polarimeters.In the present Letter, we provide a more general frame-

work to analyze the existence of fluctuations in the re-tardance of variable linear retarders, which may bedirectly applied to the widely used PA-LCoS devices.We use the Mueller–Stokes formalism and through thisanalysis we demonstrate a straightforward calibrationtechnique, based on a conventional Stokes polarimeter,to measure the average retardance and the magnitude of

its fluctuation amplitude as a function of the applied volt-age. We further demonstrate the predictive capability ofthis theoretical framework and its associated calibrationmethodology. First, we will introduce the theoreticalframework. Then, average retardance and flicker willbe experimentally calibrated for a commercially avail-able PA-LCoS, using a commercial rotating waveplateStokes polarimeter. Eventually, predictive capability ofour proposal will be demonstrated by comparing exper-imental and simulated results.

The methodology that we propose is based on theMueller–Stokes formalism [1], which enables one to dealboth with polarized and with unpolarized light. A numberof electro-optic devices, such as PA-LCoS displays, canbe modeled as a variable retardance waveplate, whoselinear retardance Γ depends on the voltage applied. Spe-cifically, the Mueller matrix MR�Γ� of a linear retarderwith a retardance value Γ, with its fast axis along thex axis, is given by

MR�Γ� �

0BB@1 0 0 00 1 0 00 0 cos Γ sin Γ0 0 − sin Γ cos Γ

1CCA. (1)

This well-known expression considers a constantvalue for the retardance. In order to incorporate theexistence of fluctuations or instabilities, let us considera triangular profile for the variation of retardance withtime Γ�t�. This is a reasonable assumption in the caseof PA-LCoS as can be seen from the instantaneous meas-urement plots shown in a number of papers [9,–14].Furthermore, the triangular time-dependent profile rep-resents actually a linear model, thus, the first option to

February 15, 2014 / Vol. 39, No. 4 / OPTICS LETTERS 1011

0146-9592/14/041011-04$15.00/0 © 2014 Optical Society of America

http://dx.doi.org/10.1364/OL.39.001011

try before more complex approaches may be provennecessary. The triangular profile can be analytically ex-pressed by the following equation:

Γ�t� �� Γ̄ − a� 2a

T∕2 t 0 ≤ t < T∕2Γ̄� 3a −

2aT∕2 t T∕2 ≤ t < T

; (2)

where Γ̄ and a are respectively the values for the averageretardance and its fluctuation amplitude, and T is theperiod for the fluctuation instability.Using the linear time-dependent fluctuation model we

can calculate a more general expression for the Muellermatrix of the linear retarder. Typically fluctuations, ifexistent, are considerably faster than the integration timeof detector systems; thus the expression of interest inmost applications is the averaged Mueller matrix forthe linear retarder. Thus, we need to calculate the aver-age values for the cosine and the sine functions in Eq. (1):

hcos Γ�t�i � �sin a∕a� cos�Γ̄�; (3)

hsin Γ�t�i � �sin a∕a� sin�Γ̄�: (4)

Taking into account Eqs. (3) and (4) we can describeour averaged matrix for the linear retarder as

hMR�Γ̄; a�i �

0BB@1 0 0 00 1 0 00 0 �sin a∕a�cos Γ̄ �sin a∕a� sin Γ̄0 0 −�sin a∕a� sin Γ̄ �sin a∕a�cos Γ̄

1CCA:

(5)

This expression provides a more realistic and precisemodel of the linear retarder, where the average retard-ance and its fluctuation need to be characterized.Now we are interested in analyzing the output SOP,

produced by a linear retarder fulfilling Eq. (5). Two dif-ferent working geometries may be considered forelectro-optic devices: in transmission and in reflection.In the latter case, an inversion of the horizontal axis isproduced between the corresponding forward and thebackward (right-handed) reference systems, which inthe Mueller–Stokes formalism is expressed by the inver-sion matrix as follows:

Inv �

0BB@1 0 0 00 1 0 00 0 −1 00 0 0 −1

1CCA: (6)

Then, in the case of reflective devices, the averagedoutput SOP can be calculated as follows:

hSouti � Inv · hMR�Γ̄; a�i · Sin: (7)

For transmissive devices, no inversion matrix needs to beconsidered. The resultant average Stokes vector is

hSouti �

0BB@

S0

S1

��sin a∕a��S2 cos Γ̄� S3 sin Γ̄��sin a∕a��−S2 sin Γ̄� S3 cos Γ̄�

1CCA; (8)

where the Stokes components for the input SOP are S0,S1, S2, and S3. The plus and minus signs before the thirdand fourth Stokes components apply for transmissiveand reflective devices respectively. Let us calculate thedegree of polarization (DoP) for this average output SOP:

DoP ��S1�2 � �sin a∕a�2��S2�2 � �S3�2�

q∕S0: (9)

We note that the DoP depends on the input SOP and onthe amplitude a of the fluctuations, but it does not de-pend on the average retardance Γ̄. According to themodel proposed, Eqs. (5), (8), and (9) provide all the in-formation necessary to calculate the performance of theelectro-optic device under test and the resulting outputlight wavefront.

In the rest of the Letter, to simplify the discussion, wewill restrict our attention to reflective devices, whichmeans that in Eq. (8) we have to consider the minus signaccompanying the third and fourth Stokes components.Extension to transmissive devices is then straightfor-ward. Now let us take a closer look into Eqs. (8) and (9):We may find specific input SOPs that may prove useful tomeasure the two parameters in the model, Γ̄ and a. In thissense, if the beam impinging the retarder corresponds tolinearly polarized light at�45° with respect to the X axis,i.e., (S0 � 1, S1 � 0, S2 � 1, S3 � 0), the average SOPand DoP at the output will be expressed as follows:

hSouti �

0BB@

10

−�sin a∕a� cos Γ̄�sin a∕a� sin Γ̄

1CCA; (10)

DoP � �sin a∕a�: (11)

We note that the output S1 component is zero inde-pendently of the retardance and its fluctuation ampli-tude. The expression for DoP is also straightforward andis directly related with the fluctuation amplitude. Equa-tions (10) and (11) can be used to measure both the aver-age retardance value Γ̄ and its fluctuation amplitude a.This can be easily accomplished using Eq. (11) to obtainthe fluctuation amplitude a, and the ratio between thethird and fourth Stokes vector components, i.e.,−hS3i∕hS2i � tg�Γ̄�, to obtain Γ̄.

The model and the accompanying calibration tech-nique that we propose in this Letter may be applied toany device that can be modeled as a variable waveplateretarder. Specifically in this Letter, we analyze a commer-cial PA-LCoS display, model PLUTO distributed by thecompany HOLOEYE. It is a nematic liquid crystal filled,with 1920 × 1080 pixels and 0.7 inch diagonal. It is elec-tronically configured to obtain a 360° retardance range atthe 633 nm wavelength.

1012 OPTICS LETTERS / Vol. 39, No. 4 / February 15, 2014

The polarimetric measurements have been obtainedwith a Stokes polarimeter, model PAX5710VIS-T distrib-uted by the company THORLABS. This is a rotating wave-plate-based polarimeter, which belongs to time-divisionmode polarimeters [16]. They are not able to provide re-liable instantaneous values if the SOP is changing morerapidly than its measurement time interval. In particular,the polarimeter is able to provide the values for theStokes vector after half a rotation of the waveplate,which corresponds to a measurement time interval of3 ms (maximum rotation frequency is 333 Hz). On theother hand, the software allows the measurement timeinterval to be enlarged to obtain an averaged signal.When enough rotations are considered, and if the periodsof fluctuation and half-rotation are not multiple, then foreach angular position of the rotating waveplate theamount of samples collected is representative ofthe time-varying SOP generated by the fluctuations inthe device. Taking into account that the time period(frequency) for the fluctuations in our PA-LCoS deviceis 8.66 ms (120 Hz), we have tried different averagingtime-interval options with the polarimeter software.The polarimeter averaging time considered in the paper,600 ms, is much larger than actually needed to obtainfully stable and repeatable SOP measurements.In Fig. 1 we show the average Stokes vector compo-

nents and the DoP measured with the Stokes polarimeteras a function of the gray level value addressed onto theLCoS device. The LCoS setup includes an input polarizerwith its transmission axis at�45° with respect to the lab-oratory vertical (x axis for the right-handed system usedin this work). The unexpanded beam of a fully polarizedHe–Ne laser impinges quasi-perpendicularly to the LCoS,at 3° incidence angle to separate the input and the re-flected beam. The director axis of the LCoS is alongthe horizontal. Typically, the director axis in nematic de-vices corresponds to the slow axis. From the results inFig. 1 we note that parameter S1 is close to zero in clearconfirmation of the predicted result in Eq. (10). Valuesmeasured for DoP vary between 1.003 and 0.939 as graylevels increase. Some depolarized light may be producedby scattering or multiple reflections [8]. This was mea-sured with the LCoS switched off and DoP values higherthan 1 were obtained, so this origin for unpolarized canbe disregarded in the present case.In Fig. 2, we plot the calculated average retardance and

its fluctuation amplitude. We observe that the retardancerange is about 360° with a very good linearity. The

fluctuation amplitude is close to 0° for lower gray levelsand increases to about 35° for higher gray levels. Thevarious jumps encountered in the fluctuation amplitudereveal the pulsed nature of the digital signal addressedonto the LCoS. For DoP > 1, nonphysical values, we haveconsidered that fluctuation amplitude is 0°.

Once the retardance and its fluctuationmagnitude havebeen calibrated, we can apply these values in Eqs. (5), (8),and (9) to predict the output Stokes vector for any inputSOP. To this goal we havemeasured the reflected SOP foran incident light beam linearly polarized at 15° with re-spect to the lab vertical. In Figs. 3 and 4 we plot respec-tively the normalized Stokes parameters and the DoP andboth the experimental (dotted lines) and the simulated(continuous lines) values are compared.

We see that there is a very good agreement betweenthe model and the experimental results. In Fig. 3 we notethe constancy of the S1 component and the oscillatorybehavior of S2 and S3 components, in accordance withthe expression in Eq. (8) where the retardance affectsthe third and fourth components through the cosineand sine functions. In Fig. 4 we also see how the modelis able to predict the DoP with a good accuracy acrossthe whole gray level range. This is a very important resultsince this proves the enhanced validity of the model pro-posed in Eq. (2) for the time fluctuations. Therefore, thecalibration performed enables a full prediction of theaverage output SOP.

For a more complete analysis of the predictive capabil-ity of our model, we have also obtained measurementsfor a circularly polarized incident light beam. In particu-lar, we have measured the reflected SOP for right-handedcircularly polarized light, i.e., (S0 � 1, S1 � 0, S2 � 0,

S1

S2

S3

DoP

-1-0,8-0,6-0,4-0,2

00,20,40,60,8

1

0 50 100 150 200 250Gray level

Fig. 1. Experimental values for the Stokes parameters andDoP, for input SOP linear at �45° and λ � 633 nm.

0

5

10

15

20

25

30

35

0

60

120

180

240

300

360

420

0 50 100 150 200 250

Flu

ctuatio

n am

plitu

de (º)

Ave

rag

e re

tard

ance

(º)

Gray level

Fig. 2. Calculated values for the average retardance and thefluctuation amplitude for λ � 633 nm.

0 50 100 150 200 250Gray level

0

0.5

1

sekotS

marap.

S3expS3simS2expS2simS1expS1sim

–

–

Fig. 3. Experimental and simulated values for the Stokesparameters, for input SOP linear at �15° and λ � 633 nm.

February 15, 2014 / Vol. 39, No. 4 / OPTICS LETTERS 1013

S3 � 1). In Figs. 5 and 6 we show equivalent plots to theones previously presented in Figs. 3 and 4. Once again,we find very good agreement between experiment andsimulation. In the present case, we note that the expres-sion for the output SOP greatly simplifies

hSouti �

0BB@

10

−�sin a∕a� sin Γ̄−�sin a∕a� cos Γ̄

1CCA: (12)

The DoP dependence turns out to be the same as forthe input SOP used for calibration, i.e., Eq. (11). Circu-larly polarized light could also be a good option to beused for calibration purposes. However, it is more robustto generate linearly polarized light so that no additionalpolarization elements, such as a quarter wavelengthwaveplate, are necessary. We note that both configura-tions, linear at �45° (−45°) and circular right-handed(left-handed), are the most sensitive input SOPs, sincethey provide the largest oscillation amplitude in themeasurement of the third and fourth Stokes components,as can be seen in Figs. 1 and 5, together with the maxi-mum variation of the DoP, shown in Fig. 6.In conclusion, we have demonstrated a polarimetric

method able to predict the performance of modernelectro-optical devices showing retardance instabilities.We have extended the applicability of Stokes polarimetryto the characterization of these devices by a proper aver-aging process. This enables a fast and simple approachfor full phase and flicker evaluation across the whole ap-plied voltage range. From a more general perspective,

having a stochastic model for the polarization behaviormay help, on one side, to refine the understanding ofthe dynamics of liquid crystal devices and, on the otherside, to widen their applicability in polarization control,as with experiments dealing with unconventional polari-zation states [17].

Work supported by Min. de Trabajo y Competitividadof Spain (projects FIS2011-29803-C02-01 and FIS2011-29803-C02-02), by Gen. Valenciana of Spain (projectsPROMETEO/2011/021 and ISIC/2012/013), and by Univer-sidad de Alicante (project GRE12-14).

References

1. G. Goldstein, Polarized Light (Marcel Dekker, 2003).2. A. De Martino, Y. K. Kim, E. Garcia-Caurel, B. Laude, and B.

Drévillon, Opt. Lett. 28, 616 (2003).3. A. Peinado, A. Lizana, J. Vidal, C. Iemmi, and J. Campos,

Opt. Express 18, 9815 (2010).4. S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays

(Wiley, 2005).5. N. Collings, T. Davey, J. Christmas, D. Chu, and B.

Crossland, J. Disp. Technol. 7, 112 (2011).6. C. Ramirez, E. Otón, C. Iemmi, I. Moreno, N. Bennis, J. M.

Otón, and J. Campos, Opt. Express 21, 8116 (2013).7. A. Márquez, C. Cazorla, M. J. Yzuel, and J. Campos, J. Mod.

Opt. 52, 633 (2005).8. J. E. Wolfe and R. A. Chipman, Appl. Opt. 45, 1688

(2006).9. A. Hermerschmidt, S. Osten, S. Krüger, and T. Blümel, Proc.

SPIE 6584, 65840E (2007).10. J. R. Moore, N. Collings, W. A. Crossland, A. B. Davey, M.

Evans, A. M. Jeziorska, M. Komarčević, R. J. Parker, T. D.Wilkinson, and H. Xu, IEEE Photon. Technol. Lett. 20, 60(2008).

11. A. Lizana, I. Moreno, A. Márquez, C. Iemmi, E. Fernández, J.Campos, and M. J. Yzuel, Opt. Express 16, 16711 (2008).

12. A. Lizana, I. Moreno, A. Márquez, E. Also, C. Iemmi, J.Campos, and M. J. Yzuel, Proc. SPIE 7442, 74420G-1 (2009).

13. J. García-Márquez, V. López, A. González-Vega, and E. Noé,Opt. Express 20, 8431 (2012).

14. A. Márquez, F. J. Martínez, S. Gallego, M. Ortuño, J.Francés, A. Beléndez, and I. Pascual, Proc. SPIE 8498,84980L (2012).

15. C. Ramirez, B. Karakus, A. Lizana, and J. Campos, Opt.Express 21, 3182 (2013).

16. C. Flueraru, S. Latoui, J. Besse, and P. Legendre, IEEETrans. Instrum. Meas. 57, 731 (2008).

17. T. G. Brown and Q. Zhan, Opt. Express 18, 10775(2010).

0 50 100 150 200 250Gray level

0.98

0.985

0.99

0.995

1

PoD

DoPexp

DoPsim

Fig. 4. Experimental and simulated DoP, for input SOP linearat �15° and λ � 633 nm.

0 50 100 150 200 250Gray level

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

PoD

DoPexp

DoPsim

Fig. 6. Experimental and simulated DoP at output, for inputSOP right-handed circular and λ � 633 nm.

0 50 100 150 200 250Gray level

0

0.5

1

sekotS

marap.

S3expS3simS2expS2simS1expS1sim

–

–

Fig. 5. Experimental and simulated values for the Stokesparameters, for input SOP right-handed circular andλ � 633 nm.

1014 OPTICS LETTERS / Vol. 39, No. 4 / February 15, 2014

3.4 Averaged Stokes polarimetry applied to evaluate retardance and flicker inPA-LCoS devices

3.4 Averaged Stokes polarimetry applied to evaluateretardance and flicker in PA-LCoS devices

85

Averaged Stokes polarimetry applied to evaluate retardance and flicker in PA-LCoS devices

Francisco J. Martínez,1,2 Andrés Márquez,1,2,* Sergi Gallego,1,2 Manuel Ortuño,1,2 Jorge Francés,1,2 Augusto Beléndez,1,2 and Inmaculada Pascual2,3

1Dept. de Física, Ing. de Sistemas y Teoría de la Señal, Univ. de Alicante, P.O. Box 99, E-03080, Alicante, Spain 2I.U. Física Aplicada a las Ciencias y las Tecnologías Univ. de Alicante, P.O. Box 99, E-03080, Alicante, Spain

3Dept. de Óptica, Farmacología y Anatomía, Univ. de Alicante, P.O. Box 99, E-03080, Alicante, Spain *[email protected]

Abstract: Recently we proposed a novel polarimetric method, based on Stokes polarimetry, enabling the characterization of the linear retardance and its flicker amplitude in electro-optic devices behaving as variable linear retarders. In this work we apply extensively the technique to parallel-aligned liquid crystal on silicon devices (PA-LCoS) under the most typical working conditions. As a previous step we provide some experimental analysis to delimitate the robustness of the technique dealing with its repeatability and its reproducibility. Then we analyze the dependencies of retardance and flicker for different digital sequence formats and for a wide variety of working geometries.

© 2014 Optical Society of America

OCIS codes: (120.2040) Displays; (120.5410) Polarimetry; (230.3720) Liquid-crystal devices; (230.6120) Spatial light modulators; (230.2090) Electro-optical devices.

References and links

1. J. Turunen and F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, 1997).

2. H. J. Coufal, D. Psaltis, and B. T. Sincerbox, eds., Holographic Data Storage, (Springer-Verlag, 2000). 3. W. Osten, C. Kohler, and J. Liesener, “Evaluation and application of spatial light modulators for optical

metrology,” Opt. Pura Apl. 38, 71–81 (2005). 4. M. A. F. Roelens, S. Frisken, J. A. Bolger, D. Abakoumov, G. Baxter, S. Poole, and B. J. Eggleton, “Dispersion

trimming in a reconfigurable wavelength selective switch,” J. Lightwave Technol. 26(1), 73–78 (2008). 5. M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M.

Bigot-Astruc, L. Provost, and G. Charlet, “Mode-division multiplexing of 2 100 Gb/s channels using an LCOS-based spatial modulator,” J. Lightwave Technol. 30(4), 618–623 (2012).

6. M. A. Solís-Prosser, A. Arias, J. J. M. Varga, L. Rebón, S. Ledesma, C. Iemmi, and L. Neves, “Preparing arbitrary pure states of spatial qudits with a single phase-only spatial light modulator,” Opt. Lett. 38(22), 4762–4765 (2013).

7. S. T. Wu and D. K. Yang, Reflective Liquid Crystal Displays (John Wiley, 2005). 8. N. Collings, T. Davey, J. Christmas, D. Chu, and B. Crossland, “The applications and technology of phase-only

liquid crystal on silicon devices,” J. Display Technol. 7(3), 112–119 (2011). 9. J. E. Wolfe and R. A. Chipman, “Polarimetric characterization of liquid-crystal-on-silicon panels,” Appl. Opt.

45(8), 1688–1703 (2006). 10. A. Hermerschmidt, S. Osten, S. Krüger, and T. Blümel, “Wave front generation using a phase-only modulating

liquid-crystal-based micro-display with HDTV resolution,” Proc. SPIE 6584, 65840E (2007). 11. J. R. Moore, N. Collings, W. A. Crossland, A. B. Davey, M. Evans, A. M. Jeziorska, M. Komarčević, R. J.

Parker, T. D. Wilkinson, and H. Xu, “The silicon backplane design for an LCOS polarization-insensitive phase hologram SLM,” IEEE Photon. Technol. Lett. 20(1), 60–62 (2008).

12. I. Moreno, A. Lizana, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive optics,” Opt. Express 16(21), 16711–16722 (2008).

13. A. Lizana, I. Moreno, A. Márquez, E. Also, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the temporal fluctuations phenomena on the ECB LCoS performance,” Proc. SPIE 7442, 74420G (2009).

14. J. García-Márquez, V. López, A. González-Vega, and E. Noé, “Flicker minimization in an LCoS spatial light modulator,” Opt. Express 20(8), 8431–8441 (2012).

#211179 - $15.00 USD Received 30 Apr 2014; revised 29 May 2014; accepted 1 Jun 2014; published 11 Jun 2014(C) 2014 OSA 16 June 2014 | Vol. 22, No. 12 | DOI:10.1364/OE.22.015064 | OPTICS EXPRESS 15064

15. G. Lazarev, A. Hermerschmidt, S. Krüger, and S. Osten, “LCOS spatial light modulators: trends and applications,” in Optical Imaging and Metrology: Advanced Technologies, W. Osten and N. Reingand, eds. (John Wiley, 2012).

16. F. J. Martínez, A. Márquez, S. Gallego, M. Ortuño, J. Francés, A. Beléndez, and I. Pascual, “Electrical dependencies of optical modulation capabilities in digitally addressed parallel aligned LCoS devices,” Opt. Eng. (accepted for publication).

17. A. Márquez, F. J. Martínez, S. Gallego, M. Ortuño, J. Francés, A. Beléndez, and I. Pascual, “Classical polarimetric method revisited to analyse the modulation capabilities of parallel aligned liquid crystal on silicon displays,” Proc. SPIE 8498, 84980L (2012).

18. F. J. Martínez, A. Márquez, S. Gallego, J. Francés, and I. Pascual, “Extended linear polarimeter to measure retardance and flicker: application to liquid crystal on silicon devices in two working geometries,” Opt. Eng. 53, 014105 (2014).

19. C. Ramirez, B. Karakus, A. Lizana, and J. Campos, “Polarimetric method for liquid crystal displays characterization in presence of phase fluctuations,” Opt. Express 21(3), 3182–3192 (2013).

20. F. J. Martínez, A. Márquez, S. Gallego, J. Francés, I. Pascual, and A. Beléndez, “Retardance and flicker modeling and characterization of electro-optic linear retarders by averaged Stokes polarimetry,” Opt. Lett. 39(4), 1011–1014 (2014).

21. A. Lizana, N. Martín, M. Estapé, E. Fernández, I. Moreno, A. Márquez, C. Iemmi, J. Campos, and M. J. Yzuel, “Influence of the incident angle in the performance of liquid crystal on silicon displays,” Opt. Express 17(10), 8491–8505 (2009).

22. G. Goldstein, Polarized Light (Marcel Dekker, 2003). 23. C. Flueraru, S. Latoui, J. Besse, and P. Legendre, “Error analysis of a rotating quarter-wave plate Stokes’

polarimeter,” IEEE Trans. Instrum. Meas. 57(4), 731–735 (2008). 24. T. G. Brown and Q. Zhan, “Introduction: unconventional polarization states of light,” Opt. Express 18, 10775–

10776 (2010).

1. Introduction

In recent years liquid crystal on silicon (LCoS) displays have become the most attractive microdisplays for all sort of spatial light modulation applications, like in diffractive optics [1], optical storage [2], optical metrology [3], reconfigurable interconnects [4,5], or quantum optical computing [6], due to their very high spatial resolution and very high light efficiency [7,8]. Among the different LCoS technologies, parallel aligned LCoS (PA-LCoS) are especially interesting since they allow easy operation as phase-only devices without coupled amplitude modulation. From a modeling point of view PA-LCoS displays can be assimilated to linear variable retarders [7,8], then the magnitude of interest to characterize these devices is their linear retardance.

It is known that LCoS and more specifically PA-LCoS exhibit some flicker or fluctuations [9–14]. This is generally true in digital backplane devices due to the pulsed digital signal addressed [10,11,15,16]. Typical methods used to characterize linear variable retarders may provide erroneous results [17,18] since they typically assume that the birefringence in the waveplate has a constant value, no fluctuations, during the measurement process. Furthermore, the amplitude of the retardance fluctuation becomes a magnitude of interest for a more accurate characterization and modeling of the device under test. Recently appropriate techniques to obtain both retardance and flicker values have been demonstrated by our group [18] and by Ramirez et al. [19], based respectively on the classical linear polarimeter and in a combination of linear and circular polarimeters. These are easier techniques to implement, specially the extended linear polarimeter in [18]. However, a more detailed characterization can be obtained by the average Stokes polarimetry method we recently proposed in [20]. This technique in combination with a Mueller matrix based model allows us to predict the response of the device for every gray level and any kind of state of polarization (SOP) at the system entry.

A first parameter to be evaluated in the performance of a digital backplane LCoS is the sequence format addressed. This may affect the number of available quantization levels and also the amount of flicker exhibited by the device [16]. Another important aspect is the working geometry where the LCoS is used: angle of incidence, use of a beam-splitter, may change the retardance dynamic range, linearity in the response [21].


In this work we apply the average Stokes polarimetry method for the evaluation of the linear retardance and flicker amplitude in PA-LCoS devices. We present a study about the robustness of the method, focused on reproducibility and repeatability capabilities of the technique. Then, a complete analysis of the performance of the PA-LCoS for a series of different sequence formats addressed and for various working geometries is undertaken.

2. Theory and characterization method

The average Stokes polarimetric technique [20] is based on the Mueller-Stokes formalism [22], which enables to deal both with polarized and with unpolarized light. The approach is valid for the modeling and characterization of linear variable retarders whose linear retardance exhibits instabilities, such as PA-LCoS displays. In principle the Mueller matrix

( )RM Γ of a linear retarder with a retardance value Γ, with its fast axis along the X-axis is

given by,

( )

1 0 0 0

0 1 0 0

0 0 cos sin

0 0 sin cos

RM

Γ = Γ Γ − Γ Γ

(1)

We showed in [20] that a reasonable assumption in the case of PA-LCoS is that temporal evolution of fluctuations ( )tΓ can be approximated by triangular time-dependent profile,

characterized by its average retardance Γ and its fluctuation amplitude a , defined as half the maximum-to-minimum value for the fluctuation. Taking into account this time-dependent linear model we can describe the averaged matrix for the linear retarder as:

( ) ( )( ) ( )

1 0 0 0

0 1 0 0( , )

0 0 sin cos sin sin

0 0 sin sin sin cos

RM aa a a a

a a a a

Γ = Γ Γ − Γ Γ

(2)

This expression provides a more realistic and precise model of the linear retarder, where the average retardance and its fluctuation need to be characterized.

Since PA-LCoS displays are reflective devices, in order to analyze the output SOP, an inversion of the horizontal axis must be considered between the corresponding forward and the backward (right-handed) reference systems, which in the Mueller-Stokes formalism is expressed by the inversion matrix as follows,

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Inv

= − −

(3)

Then, the averaged reflected SOP outS can be calculated as,

( , )out R inS Inv M a S= ⋅ Γ ⋅ , (4)

where inS corresponds to the input SOP. We may find specific input SOPs which may proof

useful to measure the two parameters in the model, Γ and a [20]. In this sense, if the beam impinging the retarder corresponds to linearly polarized light at + 45° with respect to the X


axis, i.e. (S0 = 1, S1 = 0, S2 = 1, S3 = 0), the average SOP and the degree of polarization, DoP, at the output of the device will be expressed as follows:

( )( )

1

0

sin cos

sin sin

outSa a

a a

= − Γ Γ

(5)

( )sinDoP a a= (6)

We note that the output S1 component is zero independently of the retardance and its fluctuation amplitude. The expression for DoP is also straightforward and is directly related to the fluctuation amplitude. Equations (5) and (6) can be used to measure both the average

retardance value Γ and its fluctuation amplitude a . This can be easily accomplished using Eq. (6) to obtain the fluctuation amplitude a , and the ratio between the 3rd and 4rth Stokes

vector components, i.e. 3 2 ( )S S tg− = Γ , to obtain Γ .

3. Calibration and robustness results

3.1 Calibration and comparison with instantaneous values from the linear polarimeter

The model and the accompanying calibration technique may be applied to any device which can be modeled as a variable waveplate retarder. We apply the technique to a commercial PA-LCoS display, model PLUTO distributed by the company HOLOEYE. It is a nematic liquid crystal filled, with 1920x1080 pixels and 0.7” diagonal, and digitally addressed. By means of a RS-232 interface and its corresponding provided software, we can configure the modulator for different applications and wavelengths. Besides, different pulse width modulation (PWM) addressing schemes (digital addressing sequences) can be generated by the driver electronics [10,16]. We have selected two electrical sequences exhibiting a clearly different scale of fluctuations, whose configuration files are provided with the software. They correspond to the configurations labeled as “18-6 633 2pi linear” and “5-5 633 2pi linear”.

The averaged polarimetric measurements have been obtained with a Stokes polarimeter, model PAX5710VIS-T distributed by the company THORLABS. This is a rotating waveplate-based polarimeter, which belongs to time-division mode polarimeters [23]. Its software allows different time interval options so as to obtain an averaged signal. When enough rotations are considered, and if the periods of fluctuation and half-rotation are not multiple, then for each angular position of the rotating waveplate the amount of samples collected is representative of the time-varying SOP generated by the fluctuations in the device. The polarimeter averaging time considered in the paper, 600 ms, is much larger than actually needed to obtain fully stable and repeatable SOP measurements. Note that the time period (frequency) for the fluctuations in our PA-LCoS device is 8.66 ms (120 Hz).

In Figs. 1(a) and 1(b) we show respectively the diagrams for the characterization setup associated with the two generic working geometries used with the PA-LCoS display, that is, with and without a beam-splitter in front. The unexpanded beam from a laser (He-Ne laser at 633 nm in this work) incides onto a polarizer with its transmission axis at + 45° with respect to the laboratory vertical (X-axis for the right-handed system used in this work). When using polarized laser light an additional waveplate must be inserted before the polarizer to secure that enough light traverses the polarizer. In Fig. 1(a) light impinges perpendicularly to the LCoS and a non-polarizing cube beam-splitter (model 10BC16NP.4, from Newport, in this work) separates the input and the reflected beam, which eventually is detected by the polarimeter head. Thus, strictly speaking the characterization corresponds to the combination of LCoS and cube. In Fig. 1(b) the polarimeter head measures directly the reflected beam from the LCoS. We note that the director axis (extraordinary axis) in nematic based LCoS


generally corresponds to the slow axis. In the present LCoS the director axis is along the horizontal. We may appreciate the simplicity of these setups.

Laser633 nm

LCoS BeamSplitter

LinearPolarizer

QuarterWaveplate

Polarimeter

Laser633 nm

LCoS

LinearPolarizer

QuarterWaveplate

Polarimeter

(a)

(b)

Fig. 1. Experimental setup used to measure linear retardance and flicker with the averaging Stokes polarimetric technique and for the two generic working geometries: (a) with a beam-splitter, and (b) without.

In Figs. 2(a) and 2(b) we show respectively for the two sequences the average Stokes vector components and the DoP measured with the Stokes polarimeter and for different gray level values addressed onto the LCoS device. These measurements have been obtained for the working geometry with a beam-splitter in front of the LCoS (see Fig. 1(a)). From the results in Fig. 2(a) we note that parameter S1 is close to zero in clear confirmation of the result in Eq. (5). We also see that DoP, Fig. 2(b), is larger for the 5-5 sequence.

In Fig. 3, we calculate the fluctuation amplitude and the average retardance. We observe that the retardance range is about 360° for both sequences with a very good linearity. The fluctuation amplitude is clearly smaller for the 5-5 sequence with maximum values about 30°. The various jumps encountered in the fluctuation amplitude in both sequences reveal the pulsed nature of the digital signal addressed onto the LCoS: more jumps are seen in the 18-6 sequence. For DoP>1, non-physical values, we consider that fluctuation amplitude is 0°.


Fig. 2. Experimental values for the: (a) Stokes parameters; (b) the DoP. For input SOP linear at + 45°, λ = 633nm, and for sequences “18-6 633 2pi linear” (dashed) and “5-5 633 2pi linear” (continuous). For working geometry with a cube beam-splitter in front of the PA-LCoS.

Fig. 3. Calculated values for the average retardance and the fluctuation amplitude for λ = 633nm, and for sequences “18-6 633 2pi linear” (dashed) and “5-5 633 2pi linear” (continuous). For working geometry with a cube beam-splitter in front of the PA-LCoS.

Laser633 nm

LinearPolarizer

Radiometer

LinearPolarizer

BeamSplitter

LCoS BeamSplitter

LinearPolarizer Quarter

Waveplate

Radiometer

Oscilloscope

Fig. 4. Experimental setup for the extended linear polarimeter [18], which further enables to measure instantaneous values. For working geometry with a cube beam-splitter.

Recently we have also proposed another technique to obtain both retardance and flicker values [18] based on an adapted version of the classical linear polarimeter, which we have called the extended linear polarimeter. A more detailed characterization can be obtained by


the average Stokes polarimetry, however the extended linear polarimeter can be more widely applied by any lab since only two linear polarizers are needed. At this moment we want to take advantage that this setup also enables the measurement of the instantaneous values for the retardance as we showed in [18]. We want to compare the average Stokes polarimetry results against the values calculated from the instantaneous values of retardance.

In Fig. 4 we show the experimental setup corresponding to the linear polarimeter [18], where the necessary input and output linear polarizers to be used in parallel or crossed configuration can be seen. The working geometry considered in the figure is the perpendicular one where a non-polarizing cube beam splitter is used to separate the incident and reflected beams. There is a second one to enable amplitude division of the reflected beam so that crossed and parallel intensity can be measured simultaneously. Instantaneous measurements can be obtained by connecting the two radiometers to the two channels of an oscilloscope.

Table 1. Average retardance and fluctuation amplitude obtained with the average Stokes polarimetric method (columns 3 and 4) and with the instantaneous measurements

(colums 5 and 6).

Avrg. Stokes Pol. Instant. Values Seq. Gray

level Avrg.

ret. (°) Fluct.

Amp. (°) Avrg.

ret. (°) Fluct.

Amp. (°) 18_6 0 409 0 399 2

100 243 40 231 37 200 101 49 94 54 255 36 35 36 36

5_5 0 420 0 414 1 100 263 18 257 24 200 113 28 113 37 255 39 29 44 44

In Table 1 we show the values for the average retardance and its fluctuation amplitude obtained by means of average Stokes polarimetry (columns 3 and 4), plotted in Fig. 3, against the corresponding values calculated from the instantaneous values of retardance (columns 5 and 6), and measured with the setup in Fig. 4. The results have been obtained for the two sequences and for a series of applied gray levels sampling the whole voltage range. There exists a good agreement between the average Stokes polarimetry method results and the ones related with instantaneous values measurement. This provides an alternative validation of the average Stokes polarimetry characterization technique proposed, which complements the validation already presented in [20], which was based on its capability to predict the Stokes vector for the SOP reflected by the LCoS for arbitrary input SOPs.

3.2 Robustness analysis

Next we will consider the electrical configuration “5-5 2pi linear 633nm”, which is optimized for 633nm laser wavelength, enabling 360° of retardance dynamic range. The working geometry corresponds to quasi-perpendicular incidence at 3°, i.e. no beam-splitter (Fig. 1(b)). To evaluate the repeatability of the average Stokes polarimetric method we have taken a total of 10 successive measurements under the same conditions: this means that first we aligned all devices and without changing conditions we have taken the data several times in a row. The ten series of measurements were taken in a short time interval and environmental conditions stayed stable. We evaluate the associated standard deviation as the figure of merit to estimate the repeatability.

In Figs. 5(a) and 5(b) we plot respectively the average retardance (10 curves) and its associated standard deviation. Then in Figs. 5(c) and 5(d) we show the equivalent plots for the fluctuation amplitude (10 curves) and its associated standard deviation. We see that the results for the various series for the average retardance and for the fluctuations amplitude, Figs. 5(a) and 5(c), are very similar, practically overlapping, with standard deviations, Figs. 5(b) and 5(d), around 1-2°.


Fig. 5. Evaluation of the repeatability. (a) Average retardance (10 curves), and (b) its associated standard deviation. (c) Fluctuation amplitude (10 curves), and (d) its associated standard deviation. For λ = 633nm, sequence “5-5 633 2pi linear”, and incidence at 3°.

Next we have analyzed the reproducibility of the method by comparing 3 different series of measurements acquired in different years. The experimental setup has been rebuilt from scratch in each case, so as to evaluate the deviations that can be introduced by the alignment tolerances produced by the human operator.

Fig. 6. Evaluation of the reproducibility. (a) Average retardance and (b) fluctuation amplitude results. The setup has been rebuilt from scratch in each case. For λ = 633nm, sequence “5-5 633 2pi linear”, and incidence at 3°.

In Figs. 6(a) and 6(b) we see the results obtained respectively for the average retardance and the fluctuations amplitude for the three measurements. We remark that we have no temperature room control, which may affect the liquid crystal properties, so the small difference that we observe may be partly due to different temperature, together with the inherent uncertainty that we introduce in the orientation of the polarizer or in the alignment of the other various elements in the setup. In any case curves practically overlap in the case of


average retardance, Fig. 6(a), and are also quite similar in the case of the fluctuations amplitude, Fig. 6(b).

The results in Figs. 5 and 6 provide an overview of the robustness of the method, which is very high for the average retardance measurements, and still quite remarkable in the case of the fluctuations amplitude measurements.

4. Evaluation of sequence formats and working geometries

Once the technique has been presented and its robustness has been analyzed, next we consider the characterization of the PA-LCoS for the two electrical sequences and in three typical working geometries [21]: perpendicular incidence with a beam-splitter in front (see Fig. 1(a)), quasi-perpendicular incidence at about 3°, and right-angle (45°) incidence. In the two latter geometries no beam-splitter is necessary in front of the LCoS, see Fig. 1(b). We have added a fourth characterization geometry corresponding to the setup used in previous Fig. 4 for the instantaneous measurements, where two beam-splitters are considered. In this case the polarimeter is located in transmission after the second beam-splitter. We want to estimate the influence of the insertion of the second beam-splitter. This is used not only for the instantaneous measurements but also to ease the measurements acquisition in the extended linear polarimeter, proposed in [18].

To evaluate the applicability of the average Stokes polarimetric method for each of the four geometries, first we evaluate if the Stokes parameter S1 is close to zero, as given by the model, Eq. (5) in Section 2. In Figs. 7(a) and 7(b) we plot the results for S1 vs. gray level respectively for the electrical configurations “18-6 633 2pi linear” and “5-5 2pi linear 633nm”. The curves for each of the four geometries are conveniently identified in the legend in the graphs. We see that the parameter S1 lies in all the geometries within ± 0.1, which is a small value and the average Stokes polarimetric method may be applied. We note that the S1 values are a bit larger in the geometries at right-angle (45° incidence) and with 2 cubes, probably due to the effect produced in the first case by the Fresnel transmission coefficients, and in the second case to the cumulative polarization effects generated by the two beam-splitter cubes.

Fig. 7. Stokes parameter S1 for the two sequences and the 4 geometries: (a) Sequence “18-6 2pi linear 633nm”, (b) Sequence “5-5 2pi linear 633nm”.

In Figs. 8(a) and 8(b) we show for the electrical configuration “18-6 633 2pi linear” respectively the results obtained for the average retardance and the fluctuation amplitude and for the four geometries considered. On the X-axis we have also added on the left of the gray level values the OFF-state corresponding to the LCoS switched off. In Fig. 8(a) we see that all the average retardance curves practically overlap with each other except the one corresponding to 45° incidence whose dynamic range is clearly smaller: 257° versus 370° for the other 3 geometries. This dynamic range can be increased and the linearity enhanced applying the methodology we present in [16]. We also note that for these three geometries the


retardance is about 20° higher for the OFF-state than for the 0-gray level, i.e. the LC molecules are already experiencing an applied voltage at 0-gray level.

In Fig. 8(b) we see that once again the results for geometry at 45° incidence are different from the ones for the other three geometries: at 45° incidence fluctuation amplitude is lower than about 40° whereas the other geometries reach values over 50°. We also note that in the OFF-state both at 45° incidence and with 2-cubes the value for the fluctuation amplitude is not zero. These non-zero values are not actually related with electrical fluctuations but they show that the DoP is smaller than one. The origin is not clear but it could be due to a larger influence of multiple reflections at these geometries. Small vibrations easily change path lengths and produce fluctuations in the detected SOP.

Fig. 8. Results for the 4 geometries and for the sequence “18-6 2pi linear 633nm”: (a) Average retardance, (b) Fluctuation amplitude.

In Figs. 9(a) and 9(b) we show the graphs equivalent to the ones presented in previous Fig. 8 but now corresponding to electrical configuration “5-5 633 2pi linear”. As in previous figure we see that the 45° incidence geometry shows less retardance dynamic range and smaller fluctuation retardance with respect to the three geometries which show a very similar behaviour. Now in Fig. 9(a), for these three geometries, there is a smaller difference between the OFF-state and the 0-gray level retardance values, about 10°. In Fig. 9(b), as in previous Fig. 8(b), we see the non-zero value in the OFF-state for the fluctuation amplitude both at 45° incidence and with 2-cubes.

Fig. 9. Results for the 4 geometries and for the sequence “5-5 2pi linear 633nm”: (a) Average retardance, (b) Fluctuation amplitude.

5. Conclusions

We can conclude that average Stokes polarimetry is a valid and robust method to characterize the average retardance and its flicker amplitude. We have showed that the results show a high degree of repeatability and reproducibility. The time interval between measurements and uncertainties in the alignment of all elements in the setup do not produce significant deviations.


We have also shown results for different system architectures, angular incidences, and for different digital sequence formats. This detailed analysis permits to select the most convenient electrical configuration and system architecture for the LCoS in a specific application. From this analysis we have seen that the configuration “5-5 633 2pi linear” shows smaller fluctuation amplitudes. We have also seen that the 45° incidence geometry provides a smaller retardance dynamic range, whereas the other 3 geometries practically overlap their average retardance curves and further they show a close behavior in their fluctuation amplitudes. With respect to the addition of a second cube beam-splitter, useful in the instantaneous measurements setup and in the extended linear polarimeter [18], we have seen that it does not noticeably influence the results.

From a more general perspective, the characterization provided by the average Stokes polarimetric technique may be useful, on one side, to refine the understanding of the dynamics of liquid crystal devices and, on the other side, to widen their applicability in polarization control, as with experiments dealing with unconventional polarization states [24], where the predictive capability of our Mueller-Stokes model already demonstrated in [20] may proof very useful to calculate the reflected SOPs by the LCoS.

Acknowledgments

Work supported by Ministerio de Trabajo y Competitividad of Spain (projects FIS2011-29803-C02-01 and FIS2011-29803-C02-02), by Generalitat Valenciana of Spain (projects PROMETEO/2011/021 and ISIC/2012/013), and by Univ. de Alicante (project GRE12-14).


3.5 Predictive capability of average Stokes polarimetry for simulation ofphase multilevel elements onto LCoS devices

3.5 Predictive capability of average Stokes polarimetryfor simulation of phase multilevel elements onto LCoSdevices

99

Predictive capability of average Stokes polarimetry forsimulation of phase multilevel elements

onto LCoS devices

Francisco J. Martínez,1,2 Andrés Márquez,1,2,* Sergi Gallego,1,2 Manuel Ortuño,1,2Jorge Francés,1,2 Inmaculada Pascual,2,3 and Augusto Beléndez1,2

1Dept. de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, P.O. Box 99, E-03080 Alicante, Spain2Instituto Universitario Física Aplicada a las Ciencias y las Tecnologías, Universidad de Alicante,

P.O. Box 99, E-03080 Alicante, Spain3Dept. de Óptica, Farmacología, y Anatomía, Universidad de Alicante, P.O. Box 99, E-03080 Alicante, Spain

*Corresponding author: [email protected]

Received 1 October 2014; revised 28 December 2014; accepted 14 January 2015;posted 14 January 2015 (Doc. ID 224110); published 16 February 2015

Parallel-aligned (PA) liquid-crystal on silicon (LCoS) microdisplays are especially appealing in a widerange of spatial light modulation applications since they enable phase-only operation. Recently we pro-posed a novel polarimetric method, based on Stokes polarimetry, enabling the characterization of theirlinear retardance and the magnitude of their associated phase fluctuations or flicker, exhibited by manyLCoS devices. In this work we apply the calibrated values obtained with this technique to show theircapability to predict the performance of spatially varying phase multilevel elements displayed onto thePA–LCoS device. Specifically we address a series of multilevel phase blazed gratings. We analyze boththeir average diffraction efficiency (“static” analysis) and its associated time fluctuation (“dynamic”analysis). Two different electrical configuration files with different degrees of flicker are applied in orderto evaluate the actual influence of flicker on the expected performance of the diffractive optical elementsaddressed. We obtain a good agreement between simulation and experiment, thus demonstrating thepredictive capability of the calibration provided by the average Stokes polarimetric technique. Addition-ally, it is obtained that for electrical configurations with less than 30° amplitude for the flicker retard-ance, they may not influence the performance of the blazed gratings. In general, we demonstrate that theinfluence of flicker greatly diminishes when the number of quantization levels in the optical elementincreases. © 2015 Optical Society of AmericaOCIS codes: (050.1970) Diffractive optics; (120.2040) Displays; (120.5410) Polarimetry; (230.3720)

Liquid-crystal devices; (230.6120) Spatial light modulators; (230.2090) Electro-optical devices.http://dx.doi.org/10.1364/AO.54.001379

1. Introduction

Phase-only modulation is necessary for many spatiallight modulation applications, such as diffractiveoptics [1], optical storage [2,3], optical metrology [4],reconfigurable interconnects [5,6], wavefront sensingof structured light beams [7], holographic opticaltraps [8], or quantum optical computing [9]. Among

the available spatial light modulator (SLM) technol-ogies [10], parallel-aligned (PA) liquid-crystal on sil-icon (LCoS) microdisplays are especially appealingsince they enable phase-only operation withoutcoupled amplitude modulation. They are electroopticdevices which can be assimilated to linear variableretarders. Therefore, their performance is character-ized by their linear retardance as a function of theapplied voltage. However, it has been found by nu-merous researchers [11–18] that in general, LCoS(both parallel-aligned and twisted nematic) produces

1559-128X/15/061379-08$15.00/0© 2015 Optical Society of America

20 February 2015 / Vol. 54, No. 6 / APPLIED OPTICS 1379

phase flicker and/or depolarization. This is truefor the ones with a digital backplane [13] due tothe pulsed nature of the voltage signal addressed[12,13], but it may also be true in analogicallyaddressed due to charge leakage in the pixel betweenconsecutive refreshing frames.

Recently some methods have been proposed tocharacterize not only the linear retardance but alsothe magnitude of phase flicker in the optical signal[19–21] specifically valid for parallel alignedliquid-crystal devices (LCDs). In particular, we havedemonstrated the predictive capability of a time-averaged Stokes polarimetric technique [21], andwe have analyzed its robustness and its applicabilityto characterize PA–LCoS devices in a wide variety ofworking conditions [22].

In Refs. [21,22] the PA–LCoS was addressed by auniform voltage image; however, in spatial lightmodulation applications spatially varying signalsare actually applied. Then it would be interestingboth to learn about the influence of the phase flickerand also to check the predictive capability of the cali-brated values we have obtained when used in morecomplex situations. SLMs are widely used in diffrac-tive optics, and one of the most extensively useddiffractive optical elements (DOEs) is the blazedgrating. Due to its multilevel structure and to itswell-defined spatial frequency it is a very proper in-crease in complexity with respect to uniform voltageimages. The effects of flicker in blazed gratings wereanalyzed in [23]. However, estimated flicker ampli-tude values used were not that precise as the onesenabled by average Stokes polarimetry. Additionally,only a very specific long-period blazed grating wasconsidered [23]. Focus was also given to the applica-tion of the minimum Euclidean approach to phase-only SLMs exhibiting flicker and with a phasedynamic range smaller than 2π. In this paper thephase dynamic range is 2π and we consider a muchfavorable and precisely measured flicker amplitudescenario. This enables us to quantitatively analyzethe validity of the calibration previously performedwith the Stokes average method and to use thesemore exact values to show the actual influence offlicker in the behavior of the PA–LCoS device in aspatially varying spatial light modulation applica-tion. To this goal we will address 2π phase depthblazed gratings, with different spatial periodicitiesand with different numbers of quantization levels.Specifically we will apply one pixel per level, so thatan N-level blazed grating has an N-pixel period.Both the static performance and the dynamic varia-tion of the diffraction efficiencies will be analyzed,and a comparison between experiment and simula-tion will provide us with some profitable conclusions.

2. Retardance and Flicker Calibration

The phase flicker exhibited by LCoS devices withinthe frame period has been found to be properly ap-proximated by a triangular time-dependent profileas shown in Refs. [16,21]. If we consider the averaged

value for a measurement integration time equal orlarger than the addressing frame period what weget is that the state of polarization (SOP) for the lightbeam reflected by the LCoS may not be fully polar-ized. This leads us to use the Mueller–Stokes formal-ism, which enables us to deal both with polarized andunpolarized light [24] to describe the action of PA–LCoS displays on the light beam. Applying thisformalism and taking into account the triangulartime-dependent profile for the linear retardance,we obtained Matrix (1) for a linear variable retarderwith flicker

hMR!Γ̄;a"i#

0BBB@

1 0 0 0

0 1 0 0

0 0 !sin a∕a"cos Γ̄ !sin a∕a"sin Γ̄0 0 −!sin a∕a"sin Γ̄ !sin a∕a"cos Γ̄

1CCCA;

(1)

where Γ̄ is the average retardance within the frameperiod, and a its fluctuation amplitude (defined ashalf the maximum-to-minimum value for the fluc-tuation). To calibrate the values for these two param-eters in the model we measured the averaged Stokesvector parameters for an input light beam linearlypolarized light at $45° with respect to the x-axis(i.e., S0 # 1, S1 # 0, S2 # 1, and S3 # 0). The ana-lytical expressions for the reflected averaged Stokesvector hSouti and for the degree of polarization (DoP)are very simple in this case

hSouti #

0BB@

10

−!sin a∕a" cos Γ̄!sin a∕a" sin Γ̄

1CCA; (2)

DoP # !sin a∕a": (3)

Eventually, the calibration can be easily accom-plished using Eq. (3) to obtain the fluctuation ampli-tude a and the ratio between the third and fourthStokes vector components, i.e., −hS3i∕hS2i # tg!Γ̄",to obtain Γ̄ [21].

In the present work we consider a commercialPA–LCoS display, model PLUTO distributed by thecompany HOLOEYE. It is filled with a nematicliquid crystal, with 1920 pixels× 1080 pixels and0.7 in. diagonal, 8 μm pixel pitch, 83% fill factor,and digitally addressed. By means of a RS-232 inter-face and its correspondingly provided software,we can configure the modulator for different applica-tions and wavelengths. Besides, different pulsewidth modulation (PWM) addressing schemes (digi-tal addressing sequences) can be generated by thedriver electronics [12,25]. We consider two electricalsequences exhibiting a clearly different scale of fluc-tuations, whose configuration files are provided withthe software. They correspond to the configurationslabeled as “18-6 633 2pi linear” and “5-5 633 2pi

1380 APPLIED OPTICS / Vol. 54, No. 6 / 20 February 2015

linear.” The actual number of quantization valuesavailable with these configuration files is respec-tively 256 and 92. If necessary, in the case of the5-5 sequence it is possible to increase the numberof available quantization levels up to 192 as shownin [25].

The working geometry considered in this papercorresponds to quasi-perpendicular incidence [22](actual angle of incidence is 3°). We note that the di-rector axis (extraordinary axis) in nematic-basedLCoS generally corresponds to the slow axis. Inthe present LCoS the director axis is along the hori-zontal. In Fig. 1 we show the calibrated values for theaverage retardance and its flicker amplitude versusthe gray level addressed (voltage), and for the twodigital sequences considered. We observe that theretardance range is about 360° for both sequenceswith a very good linearity, especially for the sequence“5-5 633 2pi linear.” The fluctuation amplitude isclearly smaller for the 5-5 sequence with maximumvalues slightly higher than 30°. The various jumpsencountered in the fluctuation amplitude in bothsequences reveal the pulsed nature of the digitalsignal addressed onto the LCoS: more jumps are seenin the 18-6 sequence. For DoP > 1, nonphysicalvalues, we consider that fluctuation amplitude is 0°,as can be seen for low gray-level values [21].

3. Numerical and Experimental Realization:Blazed Gratings

The calibrated values will be used to simulate thediffraction efficiency for multilevel phase DOEs, inparticular blazed gratings. We want to check the in-fluence of the fluctuations. We consider a series ofgratings with different number of quantization lev-els, which due to the pixelated structure of LCoS de-vices translates into different spatial periodicities:specifically in this work we consider 1 pixel/level.In particular the range considered varies from twoto 20 quantization levels, which approaches the limitof a continuous profile. In all cases the gratings havea 360° phase depth, so that maximum diffraction

efficiency in the first order (η1) can be reached, whichcan be analytically calculated with Eq. (4) [1]

η1 #!sin!πq"

πq

"2

; (4)

where q is the number of quantization levels. For ex-ample, η1 # 0.987 for q # 16 levels, i.e., basicallyperforms as a continuous profile. On the lower dif-fraction efficiency limit, η1 # 0.405 for q # 2 levels,which actually corresponds to a binary phase gratingwith a 180° phase depth. We also note that for a 360°phase depth blazed grating the zero-order diffractionefficiency is η0 # 0. As an additional remark, in thesimulation, even though the calibration curves forthe retardance in Fig. 1 show a good linearity, wehave applied the appropriate linearization look-uptable for each of the two sequences, so that theresidual nonlinearity is compensated via software.

We want to note that, as shown in Fig. 1, when dis-playing quantized multilevel elements the availablephase domain may be smaller than 360°, i.e., thisconstraint may be relaxed if necessary. In Fig. 2we show the phase depth required as a function ofthe number of quantization levels of the multileveldiffractive element. The lower limit corresponds tothe case of the binary phase element, where a 180°phase depth domain is required. For 6 and 12 quan-tization levels, the phase depth necessary is, respec-tively, 300° and 330°. We note that first diffractionefficiency for blazed gratings with 6 and 12 quantiza-tion levels reaches such high values as η1 # 91% andη1 # 98%, respectively.

To perform the experiments in the lab, a lightbeam from a He–Ne laser at 633 nm is expanded us-ing a spatial filter and then is collimated with a lens.As given in Section 2, the angle of incidence onto theLCoS is 3°. The collimated beam of light passesthrough a polarizer whose transmission axis is ori-ented so that the incident SOP corresponds to lin-early polarized light along the lab-horizontal, i.e.,parallel to the director axis of the liquid crystal inthe LCoS: this is the phase-only configuration. Wehave checked that there is no need for an analyzer

0

10

20

30

40

50

60

0

60

120

180

240

300

360

420

0 50 100 150 200 250

Flu

ctuatio

n am

plitu

de (º)

Ave

rag

e re

tard

ance

(º)

Gray level

Fig. 1. Calculated values for average retardance and fluctuationamplitude for λ # 633 nm, and for sequences “18-6 633 2pilinear” (dashed) and “5-5 633 2pi linear” (continuous) at quasi-perpendicular incidence at 3°.

Fig. 2. Phase depth domain necessary as a function of thenumber of quantization levels of the multilevel phase element.


at the output since the reflected SOP is fully linearlypolarized along the lab-horizontal. A lens focuses thereflected beam to its focal plane where a detectorconnected to a radiometer measures the diffractedintensity.

We havemeasured both zero- and first-order inten-sities in the following (I0 and I1, respectively) for aseries of blazed gratings with a spatial frequencyof 2, 3, 4, 8, 12, and 16 pixels/period, thus coveringboth a wide spatial frequency and number of quan-tization levels range. A diaphragm in front of the de-tector is needed to filter out any unwanted orders inthe measurements. Vertical blazed gratings areaddressed, i.e., the spatial frequency corresponds torows/period, since along this direction potential devi-ations due to the anamorphic frequency-dependenteffect [26–28] are less important. In this paper wewant to isolate as much as possible the flicker fromother degradation effects which may be exhibited bySLMs. In this sense the anamorphic frequency-dependent effect has been found both in transmissiveLCDs [26] and in modern LCoS devices, both analog[27] and digitally [28] driven.

The values obtained, I0 and I1, need to be normal-ized by the total energy reflected by the LCoS. To thisgoal we apply a uniform zero gray-level value screen,then all the light reflected by the LCoS is sent to thezero order. We call this intensity value IT1. We notethat due to the pixelated structure of the device,multiple replicas of the diffraction orders of theblazed grating are generated, each replica centeredat the position of one of the diffraction orders pro-duced by the periodic pixelated structure. This is awell-known consequence of the sampling theory[29]. We restrict our attention to the zero-order rep-lica, i.e., the one centered at the position of the zeroorder produced by the periodic pixelated structure.In principle the relative energy deflected to each ofthe replicas does not depend on the image being ad-dressed but only on geometrical characteristics of thedisplay, basically on the fill factor. However, duringthe realization of the experiment in the lab we ob-served a very clear change of brightness between dif-ferent replicas when a uniform screen was sent withrespect to sending the blazed gratings. Therefore wemeasured the total energy in the zero-order replica inthe following IT2 when addressing a blazed grating:we open the aperture of the diaphragm in front of thedetector so that only the set of diffraction orders inthe zero-order replica are gathered. Specifically, wehave measured IT2 for the lowest spatial frequencyblazed grating, 16 pixels/period, since aliasing be-tween neighboring replicas is then reduced; thusthe IT2 value is more robust. In principle, accordingto the sampling theory the values IT1 and IT2 shouldbe the same; however we found their ratio to beR # IT2∕IT1 # 0.92, i.e., almost a 10% of the energywas deflected into higher-order replicas. Therefore,additional care has to be taken when normalizingthemeasurements to compensate for this unexpectedexchange of energy between replicas. Then, the

normalized intensity (or diffraction efficiency) can beobtained as

ηi #1R

IiIT1

; (5)

where i is the number of the diffraction order consid-ered for the blazed grating. Probably the differentexchange of energy between replicas when changingthe image being addressed onto the LCoS is dueto the pixel crosstalk caused by the fringing fields,i.e., the gradual voltage changes across the borderof neighboring pixels and by elastic forces in theliquid-crystal material preventing abrupt spatialvariations in the phase modulation [30–33]. Both ef-fects cause the realized phase modulation of a pixelto depend also on the voltage applied over adjacentpixels. This crosstalk phenomenon increases as pix-els get smaller and as voltage changes betweenneighboring pixels are larger.

In Figs. 3(a) and 3(b) we show respectively for thesequences “18-6 633 2pi linear” and “5-5 633 2pi lin-ear” the results for the measurements of the instan-taneous first-order diffraction efficiencies for thesix blazed gratings. In the legend we show the quan-tization number of levels, q, in each case. All the gra-tings aremeasured and optimized for the wavelength633 nm. The instantaneous values have been ob-tained by connecting a radiometer model 1830-Cfrom Newport, that has an analog bandwidth greater

Fig. 3. Instantaneous values for the first-order diffracted inten-sity for blazed gratings with different quantization levels (q shownin the legend); (a) sequence “18-6 633 2pi linear;” (b) sequence “5-5633 2pi linear.” Data for λ # 633 nm and for quasi-perpendicularincidence at 3°.


than 30 kHz, to a digital oscilloscope model TDS-1012b from Tektronix, with a 100 MHz analog band-width. The first thing we note is that the diffractedintensity fluctuates, with a period of 8.33 ms, i.e.,frequency 120 Hz. The amplitude of fluctuations inFigs. 3(a) and 3(b) is smaller as the number ofquantization levels increases. Additionally, the aver-age diffraction also increases with q which is thetheoretically expected result. If we compare the plotsfor the two sequences, we find that the amplitude inthe oscillations is smaller for the 5-5 configura-tion file.

Curves in Figs. 3(a) and 3(b) have been acquiredwith arbitrary triggers and they do not have thesame starting instant. This is not relevant for theanalysis in the present paper where the variousquantized gratings are not to be addressed simulta-neously. However it is interesting to note that thereare applications where proper synchronization of ac-quisition times in the analysis may be an issue asgiven by [34], which shows empirically that for poly-chromatic illumination applications, fluctuationsgiven by a diffraction grating for the various wave-lengths may be partly synchronized by addressingthe diffraction grating with the appropriate graylevels for each separate wavelength.

For the simulations, firstwe calculate the gray levelg to be applied to produce the desired phase valueφ ateach step in the multilevel blazed grating. To thisgoal we consider the calibrated values Γ̄!g" given inFig. 1, and for each desired φ we look for the closestavailable average retardance value Γ̄. Second, theframe period T is divided in time intervals smallenough so that the fluctuating phase profile of theblazed grating is accurately sampled. At each timesample, the staircase phase profile is partly distortedfrom the ideal quantized linear ramp since each step,associated with a specific gray level g, is fluctuatingwith a different amplitude a!g", given in Fig. 1. Thevarious phase steps in the staircase profile fluctuatein a correlated form with each other, in accordancewith the periodic behavior exhibited by the instanta-neous values in Fig. 3. As a starting value, at timet # 0 we have chosen for each phase step φ the phasevalue Γ̄!g" − a!g" which then increases linearly withtime until Γ̄!g" $ a!g". Another starting set of valuesmay be chosen, resulting in different instantaneousprofiles, but since we are interested in averagedeffects this has no relevance on the results. As a thirdand final part of the simulation, at each time samplethe Fourier transform of the blazed grating profile iscalculated: this provides the instantaneous diffrac-tion efficiency. From these instantaneous values wecalculate the averaged one and the amplitude andstandard deviation for its fluctuation. Quantitativecomparisons between simulations and experimentare presented in the next section.

4. Results and Discussion

We want to perform a complete analysis of the suit-ability of the calibration performed using the average

Stokes polarimetric technique and also to obtain adeeper insight on the actual degree of influence offluctuations on the performance of blazed gratings,as a widespread example of a multilevel phaseelement. To fulfill these two goals we look both intoaverage diffraction efficiencies and also the magni-tude of its variations, i.e., we plan both static and dy-namic analyses. Depending on the application it isnot only important to maximize first-order diffrac-tion efficiency and/or to reduce the importance ofthe zero order, but it may be also important to gen-erate stable spots. One such application is holo-graphic optical tweezers where both intensity andpositional stability is an issue [8]. In the case of anoptical communication link, where the LCoS is play-ing the role of a reconfigurable optical add–drop mul-tiplexer (ROADM) [5], these fluctuations lead to adegradation of the signal-to-noise ratio and a nonde-sirable increase in the bit-error rate (BER) in thecommunication link.

Let us start with the results dealing with the staticanalysis, i.e., average diffraction efficiencies. InFigs. 4(a) and 4(b) we show the diffraction efficiencyfor the zero order and for the first order, respectivelyas a function of the quantization levels of the blazedgrating and for the two configuration files addressed.Lines and markers correspond, respectively, to simu-lated and experimental results. The simulatedresults take into account both the average retard-ance and the flicker amplitude values shown in Fig. 1.The experimental results correspond to the averagevalue delivered on the display of the radiometer. In

Fig. 4. Realistic simulations and experimental results for the dif-fraction efficiency for the two configuration files; (a) zero order;(b) first order.


relation with Fig. 4(a) let us remember that for a 2πphase depth blazed grating the theoretical diffrac-tion efficiency for the zero order is η0 # 0. The theo-retical simulation taking into account the existenceof flicker provides values that are nonzero, beingslightly more intense for the 18-6 sequence. We notethat the simulated values decrease with the numberof quantization values. The same tendency can be ap-preciated in the experimental results, which showvalues equal or smaller than 2% for eight quantiza-tion levels and above, which is a small residual zeroorder. For less than five quantization levels thedeviation between simulation and experiment in-creases. This may be partly due to the aliasing effectbetween diffraction orders from neighboring pixela-tion replicas.

In Fig. 4(b) we show the equivalent plot corre-sponding to the first-order diffraction efficiency. Wesee that the experimental values show a very goodagreement with simulations, especially for quantiza-tion levels equal or larger than 4. We also note in theexperimental values that the efficiency is slightlylarger for the 5-5 sequence, which agrees with thesimulation results. The theoretical curve given byEq. (4) is also included, “ideal” label in the legend,to show the actual effect of fluctuations on thediffraction efficiency. We see that for the sequence5-5, fluctuations have basically no effect in diminish-ing the diffraction efficiency. To make this importantresult more evident, in Fig. 5 we represent thedeviation between the ideal diffraction efficiencyand the realistic numerical simulation, which is nu-merically calculated for each of the two sequences in-cluding the calibrated fluctuations, normalized bythe ideal diffraction efficiency. We see that the nor-malized difference for the sequence 18-6 decreasesfrom 4.5% to 3% as the number of quantization levelsincrease. In the case of the sequence 5-5, the normal-ized difference shows a constant behavior and thevalues are smaller than 1%. Thus, it can be concludedthat for the latter sequence fluctuations are not ac-tually degrading the performance of the element.If the amplitude of fluctuations is larger, as it isthe case for the sequence 18-6, it is important to note

that their effect decreases as the number of quanti-zation levels increase, approaching reasonable val-ues for five quantization levels or larger.

A possible criticism to the previous analysis re-garding deviations between ideal values [given byEq. (4)] and the realistic simulated values is that thelatter are not only affected by fluctuations, but alsoby the mismatch between the required and the avail-able phase value for each level. This mismatch isinduced by the limited and nonequally spaced quan-tization levels in the available phase domain. Let usnote that the number of phase levels is limited to 256for configuration 18-6 and 92 for configuration 5-5 ascommented in Section 2, and they do not need to beequally spaced in the phase domain. The small pla-teaus for the average retardance in Fig. 1 show thislimited and nonequal spacing in the available phasedomain. In Fig. 6 we show for the two sequences, thenumerically calculated difference between the idealdiffraction efficiency and the one actually given bythe available phase domain, normalized by the idealdiffraction efficiency. In these simulations we con-sider that there are no fluctuations and only the cali-brated average retardance values are used, thus theonly deviations result from the mismatch betweenrequired and available phase values for each of thephase levels. We see that the normalized deviationis smaller than 0.1%, being slightly larger for the se-quence 5-5, which is the one with a smaller numberof phase levels. This means that the phase levels mis-match effect can be disregarded when discussing theinfluence of fluctuations on the diffraction efficiency.

In the first paragraph in this section we com-mented about the necessity for a “static” and a “dy-namic” analysis of the influence of flicker on theperformance of the elements addressed. Previousplots dealing with average diffraction efficienciesfocus on the “static” analysis. Next we show inFigs. 7(a) and 7(b), for the first diffraction order ofthe displayed blazed gratings, respectively, theamplitude of the fluctuations and the standarddeviation values, normalized by the average diffrac-tion efficiency. Experimentally, the average and thefluctuating estimation parameters are obtained from

Fig. 5. For each of the two sequences, deviation between the idealfirst-order diffraction efficiency and the realistic numerical simu-lation (i.e., including the calibrated fluctuations), normalized bythe ideal diffraction efficiency.

Fig. 6. For each of the two sequences, deviation between the idealfirst-order diffraction efficiency and the numerical simulation(including only the calibrated average retardance), normalizedby the ideal diffraction efficiency.


the measurements acquired with the digital oscillo-scope, shown in Fig. 3. Numerically, as commented atthe end of Section 3, the instantaneous values aresimulated applying the triangular time evolutionof the fluctuating retardance using (for each graylevel) the average retardance Γ̄ and its fluctuationamplitude a (characterized in Fig. 1). From these dif-fraction efficiency instantaneous values we calculatethe average value, the amplitude, and the standarddeviation for its fluctuation. The amplitude andstandard deviation are then normalized by theaverage value.

In Fig. 7(a) simulation and experiment show agood agreement when using more than eight quanti-zation levels. Both numerically and experimentallywe obtain that sequence 5-5 shows smaller fluctua-tions. In Fig. 7(b), where the figure-of-merit is thestandard deviation, the agreement between experi-mental and numerical results is very good, especiallyfor more than eight quantization levels. The ten-dency is clear: to reduce fluctuations generated bythe optical element addressed the number of quanti-zation levels has to be larger than 5. In this case thetime stability in the spot reaches a low value for thesequence 18-6 and becomes very good for the se-quence 5-5. We can conclude that standard deviationis a much more robust figure-of-merit since it takes amore averaged estimation of the time fluctuations. Itis important to remark that the results in Fig. 7 arenot dependent on the external normalization factor Rshown in Eq. (5), which is due to the crosstalk effect

between neighboring pixelation replicas. This giveseven more relevance to the excellent agreement be-tween simulation and experiment in Fig. 7(b) sincethey are self-referenced measurements without anadditional parameter which may introduce furtheruncertainty in the results.

5. Conclusions

We have shown that the average retardance and itsfluctuation value calibrated by means of the averageStokes polarimetry technique can be used to predictthe diffraction efficiency for multilevel phase blazedgratings, not only its average (“static” analysis) butalso its time fluctuation (“dynamic” analysis). This isvery important since it enables calculating in ad-vance if the specific LCoS device and/or the configu-ration of the electrical parameters governing itsperformance may be the appropriate ones for a spe-cific application. An interesting result obtained isthat the degradation produced by a sequence pos-sessing a large amount of flicker can be largelydecreased, even to acceptable values, if the numberof quantization levels in the element is increased, aswe have seen for the sequence 18-6: for five or morequantization levels the decrease both in the first-or-der diffraction efficiency and in the magnitude oftime fluctuations was kept small. Then, for the se-quence 5-5, whose retardance flicker amplitude islower than 30°, we have obtained that the diffractionefficiency results are very close to the ones that maybe expected in the ideal case with no fluctuations. Asa byproduct of the present work we have seen thatcare has to be taken in the experiments due to thecrosstalk between neighboring pixelation replicas.Proper normalization enabled us to uncouple thiscrosstalk from the analysis of flicker influence inspatially varying phase elements.

This work was supported by the Ministerio deTrabajo y Competitividad of Spain (ProjectsFIS2011-29803-C02-01 and FIS2011-29803-C02-02),by the Generalitat Valenciana of Spain (ProjectsPROMETEO/2011/021 and ISIC/2012/013), and bythe University de Alicante (Project GRE12-14).

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Fig. 7. For the two configuration files, realistic simulations andexperimental results for the magnitude of the time fluctuation inthe first-order diffraction efficiency normalized by its averagevalue; (a) amplitude; (b) standard deviation.


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3.6 [Unpublished] Exploring binary and ternary modulations on a PA-LCoSdevice for holographic data storage in a PVA/AA photopolymer

3.6 [Unpublished] Exploring binary and ternary mod-ulations on a PA-LCoS device for holographic datastorage in a PVA/AA photopolymer

109

Exploring binary and ternary modulations on a PA-LCoS device for holographic data storage in

a PVA/AA photopolymer Francisco J. Martínez,1,2 Roberto Fernández,1,2 Andrés Márquez,1,2, * Sergi Gallego,1,2

Mariela L. Álvarez,1,2 Inmaculada Pascual2,3 and Augusto Beléndez1,2 1Dept. de Física, Ing. de Sistemas y Teoría de la Señal, Univ. de Alicante, P.O. Box 99, E-03080, Alicante, Spain 2I.U. Física Aplicada a las Ciencias y las Tecnologías Univ. de Alicante, P.O. Box 99, E-03080, Alicante, Spain

3Dept. de Óptica, Farmacología y Anatomía, Univ. de Alicante, P.O. Box 99, E-03080, Alicante, Spain * [email protected]

Abstract: We focus on the novelty of three elements in holographic data storage systems (HDSS): the data pager, where we introduce a parallel-aligned liquid crystal on silicon (PA-LCoS) microdisplay; the recording material, where we consider the highly versatile PVA/AA photopolymer; and also in the architecture of the object arm, where a convergent correlator system is introduced. We show that PA-LCoS devices cannot implement pure hybrid-ternary modulated (HTM) data pages but a rather close approximation. Validation of the HDSS expressions for the convergent correlator and comparison with the widespread 4-f system is performed. Experimental results with PVA/AA material showing bit-error rates (BER) in the range of 10-3, further show its potential application for HDSS, and also demonstrate the validity of the testing platform and PA-LCoS calibration and optimization. © 2015 Optical Society of America OCIS codes: (090.2900) Optical storage materials, (210.2860) Holographic and volume memories, (230.3720) Liquid-crystal devices, (230.6120) Spatial light modulators.

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17. F.J. Martínez, A. Márquez, S. Gallego, J. Francés, I. Pascual and A. Beléndez, “Retardance and flicker modeling and characterization of electro-optic linear retarders by averaged Stokes polarimetry,” Opt. Lett. 34, 1011-1014 (2014).

18. F. J. Martínez, A. Márquez, S. Gallego, M. Ortuño, J. Francés, A. Beléndez, and I. Pascual, “Averaged Stokes polarimetry applied to evaluate retardance and flicker in PA-LCoS devices,” Opt. Express 22, 15064-15074 (2014).

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the geometry of a holographic memory setup,” Proc. SPIE 8429, 84291Y-1–10 (2012) 22. M. Schnoes, B. Ihas, A. Hill, L. Dhar, D. Michaels, S. Setthachayanon, G. Schomberger, and W. L. Wilson,

"Holographic data storage media for practical systems," Proc. SPIE 5005, 29-37 (2003). 23. A. Márquez, C. Neipp, A. Beléndez, S. Gallego, M. Ortuño, I. Pascual, “Edge-enhanced imaging with

polyvinyl alcohol/acrylamide photopolymer gratings,” Opt. Lett. 28, 1510–1512 (2003). 24. A. Márquez, S. Gallego, M. Ortuño, E. Fernández, M.L. Álvarez, A. Beléndez; I. Pascual, “Generation of

diffractive optical elements onto a photopolymer using a liquid crystal display,” Proc. SPIE 7717, 77170D-1-12 (2010).

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30. F. J. Martínez, A. Márquez, S. Gallego, M. Ortuño, J. Francés, A. Beléndez, and I. Pascual, “Electrical dependencies of optical modulation capabilities in digitally addressed parallel aligned LCoS devices,” Opt. Eng. 53, 067104 (2014).

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http://scholar.google.es/citations?user=VA6XaDsAAAAJ&hl=en&oi=sra

1. Introduction

Holographic data storage systems (HDSS) have been a promising and very appealing technology since the first laser developments in the sixties [1], enabling true 3-D storage of information and also associative memory retrieval [2]. Many scientific and engineering aspects need to be addressed for HDSS to be viable in the commercial arena as reported in the handbooks by Coufal et al. [3] or more recently by Anderson et al. [4]. Progress has taken advantage from technological advancements in the various components (light source, data pager or spatial light modulator (SLM), camera, and recording material) and from the proposal of novel multiplexing and signal conditioning strategies [3,4]. A number of testing platforms and commercial prototypes have been developed along the years [5], with long time archival storage as the main application in focus nowadays [6].

Impact of ongoing advances in the various components needs to be explored in its specific application to HDSS. In this sense continuous progress is being produced in SLM technology where parallel-addressed liquid crystal on silicon (PA-LCoS) microdisplays [7] have replaced previous liquid-crystal display (LCD) technology in most of optics and photonics applications. PA-LCoS microdisplays are high-resolution reflective devices which enable phase-only operation without coupled amplitude modulation, what makes them ideal for binary or multinary phase-only data pages [8,9] which leads to DC term cancellation when recording the Fourier transform of the data page. More conventional binary intensity modulated (BIM) data pages can also be addressed with PA-LCoS devices as we will show in the paper. They produce large DC terms, saturating the dynamic range of the material [10,11], if no pseudo-random phase masks [12] or defocus of the recording plane [10,13] is implemented. The main benefit of BIM is that it needs a simple on-off intensity detection system. Hybrid-ternary modulation (HTM) is another scheme which combines the ease of detection of BIM pages together with DC term cancellation of phase-only modulation. HTM has been demonstrated and analysed with twisted-nematic LCDs [14-16], however, no study has been addressed with PA-LCoS devices. In the paper we show the challenges, both theoretical and experimental to be faced. Average Stokes polarimetry [17,18] is applied to characterize the PA-LCoS microdisplay, then the amplitude transmission and phase-shift versus gray level is calculated, thus predicting the complex amplitude modulation provided by the device.

In order to calculate the performance of HDSS we have built a holographic memory simulator based on the Fourier optics formalism [19,20]. In the holographic recording process, the interference between the Fourier transform of the object beam, carrying the data page introduced by a SLM, and the reference beam is produced. Most of the testing platforms and commercial prototypes [5,6] proposed for holographic memory systems are based on the so-called 4-f system in the object beam. This is an optical system typically found in optical processing applications [19,20], which allows for exact cancellation of quadratic phase terms arising in the free space beam propagation. Recently we presented the convergent or VanderLugt correlator as an alternative to the 4-f system in the object beam [21]. In optical signal processing it enables easy scaling of the dimensions of the Fourier transform of the scene to the scale of the filter in the Fourier plane [20]. In material testing platforms in HDSS, the flexibility provided to change the size of the Fourier transform of the data page into the recording material, simply shifting axially the plane of the data page, enables to increase the areal density of the pages stored in the memory and relaxes the requirements of distances and focal lengths when compared with the 4-f system. In this paper we derive the theoretical expressions for the most general situation with the convergent correlator architecture when the recording material is located out of focus of the Fourier plane. The simulator includes as well the effect of the aperture size of the beams in the recording process (Nyquist aperture) and the realistic complex amplitude values implemented by the SLM. Simulated results show the magnitude of the expected DC term in the Fourier plane together with estimation for the

quality of the signal-to-noise ratio and bit-error rate (BER) in the retrieval process under various defocus, Nyquist apertures, and SLM modulations, thus enabling for deeper understanding of the performance of the experimental setup and the modulations implemented onto the PA-LCoS device.

One of the most demanding components in HDSS is the recording material [22]. Photopolymers are the typical material of choice for write-once read-many (WORM) applications such as long time archival storage. Among photopolymers, polyvinil alcohol/acrylamide (PVA/AA) materials have been long studied for a wide range of applications, such as holographic and diffractive applications [23,24] and also in HDSS [25,26], since they combine good optical properties, ease of fabrication, self-development capability and they offer the capability to establish thick recording layers [25]. There are some studies about the capability of this material to multiplex many holograms, diffraction gratings, and to calculate the dynamic range of the material with promising results [25,26]. Nevertheless few efforts have been done in order to optimize this material for holographic memories [27]. We will introduce a PVA/AA photopolymer compound in the holographic memory testing platform to provide experimental results for storage and retrieval of information both with the BIM and the HTM data pages, showing its potential and interest for future research focused on this material with highly tunable properties.

In Section 2 we will develop the necessary expression for the recording and reconstruction steps with the convergent correlator architecture in the object beam. These expressions incorporate the Nyquist aperture and defocus of the material with respect to the Fourier transform of the data page. These expressions are then used in the holographic memory simulator we build to simulate the system performance. In Section 3, we focus on the calculation of the binary intensity and hybrid-ternary modulation regimes with a PA-LCoS device, to be used in HDSS. Numerical and experimental results are presented in Section 4, where a PVA/AA photopolymer is used as the recording material. Eventually, main conclusions are summarized in Section 5. An Appendix is added at the end to accompany the theoretical derivations in Section 2.

2. HDSS simulator with defocus based on the convergent correlator architecture

Recently we proposed [21] the use of the convergent correlator architecture for the object beam instead of the traditional 4-f system typically used in HDSS. It is an interesting alternative providing a larger range of flexibility in the design of HDSS testing platforms. We derived the theoretical expressions for the specific situation when the recording material is located on the Fourier plane for the data page. In the theoretical expressions we found that when compared with the 4-f system, a quadratic phase term emerges which can be disregarded if the data page is not codified as phase-only. Further, the scales in the various planes were obtained. Then, in the analysis in [21] we obtained a detailed validation of its utility. In this paper we calculate the most general case when the material is out of focus of the Fourier plane, which includes as a particular case the derivation in [21]. Next we present the main theoretical results in the body of the paper, leaving for the Appendix the identities and properties used in this derivation.

In Fig. 1 we show the generic scheme for the holographic memory system that we want to model and the names for the various magnitudes and variables. In the recording step the reference beam and the object beam interfere onto the holographic recording material. The object beam setup corresponds to a convergent correlator architecture where the data page on the SLM is illuminated with a converging beam produced by lens L1. The recording material is located a distance zΔ out of focus of the plane where this beam converges. We will see that at this plane we obtain the defocused Fourier transform of the data page. In the reconstruction step, the object beam is blocked and only the reference beam is used to illuminate the material. The diffracted beam from the material is then Fourier transformed by

lens L2 onto the CCD plane. In the figure the basic components in the system are shown: beam-splitter, spatial filters, mirror, lenses, spatial light modulator, storage material, and CMOS/CCD cameras. Even though it is not indicated in the figure, we also take into account the placement of a diaphragm to limit the beam aperture incident onto the material, the so called Nyquist aperture [13], thus limiting the total area where actual recording of a data page takes place thus increasing the areal density of stored information.

Fig. 1. Scheme for the experimental setup we model in this work where the various elements and also the names for the magnitudes and variables are introduced.

Let us first deduce the theoretical expression describing the information stored in the

material in the recording step. For the reference beam, we consider a plane wavefront, given by ( )rkj ref

rr⋅exp , where rr

ref

is the position vector of a point in the wavefront in a certain

reference system, and kr

is the wave vector indicating the direction and propagation sense for the plane wavefront. We name the object beam incident onto the material as . Therefore, onto the material we obtain the interference between both wavefronts. In the paper we consider the ideal situation where a linear recording of the interference pattern is produced onto the material. This enables to analyse the validity of the introduction of the convergent correlator instead of the 4-f system.

( )333 y,xf

To calculate the expression for ( )333 y,xf we take into account free-space propagation in the Fresnel approximation of the wavefront (see Appendix) along distances

1,

2 and

, and transmission through lens L1 of focal length , which introduces a quadratic phase onto the wavefront (see Appendix). Apart from some constant factor, the propagation from plane O to O’ can be expressed as the following sequence of convolutions,

z z

zz Δ+3 1f

( ) [ ][ ] ( )[ ] )Zy,(xy,xf)Zy,(x)Fy,(x)Zy,(xy,xf 33322222111

*111333 ';;;; ψψψψ ⊗⊗= (1)

where corresponds to the data page displayed onto the SLM and ⊗ indicates convolution. To ease development of the expressions we apply notation described in the Appendix for quadratic phase terms as

( 22 y,xf )

( )( )λπψ /exp; 221 y+xZj )zZy(x, ≡= − , where λ is the wavelength used in the illumination. This enables to apply the properties presented by A. VanderLugt [20], summarized in the Appendix, which provide the following result,

( ) ( ) ( )∫∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛+

Δ+−

−+−+

2222323

322

121

22

32223333332exp';';

P

dydxyyxxzz

jy,x)fFZZ

ZZZy,(x)Zy,(x=)y,(xf λ

πψψ (2)

This is the defocused Fourier transform of the data page ( )22 y,xf . The derivation process is very similar to the one shown for the non-defocus case in [21], where the sequence followed to apply the properties 3, 4, 6 and 9 in the Appendix can be viewed. If the plane for the material O’ is defocused with respect to the conjugate for plane O, then and if

, then the quadratic term inside the integral in Eq. (2) can be simplified as 0≠Δz

3zz <<Δ

( )( ))/(exp; 23

22

22

2322

* zy+xzj)Zzy,(x λπψ Δ−=Δ

20W

Ryxyx NN /),(),( 2222 =

20W

. We will show next how to rewrite this term as a function of the defocus term , which is the notation typically used in analysis and modeling of optical systems. This is very interesting since provides an invariant parameter not dependent on the specific distances of a certain system. We note that this is not the notation found in HDSS literature and it is a novel approach that we develop in the paper. To this goal first we need to normalize the aperture coordinates by the total aperture of the system, i.e. . Then, if the quadratic term is now compared with the generic expression for the defocus term ,

20W

R

( ) (( 22N +x

20

)) (3) 22202exp NyWj+ π−=2

2Ny ⎟⎟⎠

⎞22Nx2

3

2

expzzRj

λπ

⎜⎜⎝

⎛ Δ−

, then we obtain the relation between the invariant parameter W and the magnitudes for the specific parameters in the experiment,

( ) λ2220 NAzΔ=W (4)

where corresponds to the numerical aperture for the system. Then, Eq. (2) can be expressed as.

NA= 3/ zR

[ ( )( ) ( )]( ) ( )z

vzz

xRuNNN Rxfy+xj,(xf =Δ+

=3

3 ,22

22

233λλ

N Ry,2W202π)FTZy,(x=)y 3333 ';ψ −exp (5) z

yRΔ+3

3

Additionally, we consider the insertion of a stop ( )333 y,xp

)Z 3'

, also called Nyquist aperture, in front of the material, to limit the extent of the area used. Then what is actually registered is the product . Then, in Eq. (5) we observe two differences with respect to the 4-f system: the quadratic phase term

( ) ( 333333 y,xpy,xf )(x3ψ y, 3 ; , which shifts the exit plane if the

optical system continues after plane P3; the scale of the Fourier transform can be adjusted, thus the areal density, simply changing distance . We note that the present derivation is valid independent of the codification (binary intensity, binary phase, hybrid ternary modulation,…) considered for the data page.

3z

Let us now analyse the reconstruction step, where we consider that the recording material is illuminated by the reference beam and is able to diffract the recorded wavefront described by . We calculate the result obtained in the final plane P5 (Fig. 1), where the camera is located. The sequence followed by the wavefront from plane P3 to plane P5 can be expressed, dropping constant factors, as follows,

( )333 y,xf ( )333 y,xp

( ) ( ) ( )[ ][ ][ ])Zy,(xFy,(x)Zy, 55544*

443 ;;; ψ )2 ψ⊗(x4ψ⊗y,xp 33y,xfy,xf 333555 = (6)

In the case when the SLM and CCD planes, i.e. planes P2 and P5, are conjugate with each other, i.e.

( )( ) 02542

4 =−+ FZZZ4 −+ Z3Z

( )

(7)

, and applying various properties (3, 4, 6 and 9) in the Appendix then we obtain,

( )vu,⎭⎬⎫

vu

PZ

v,)fZzZ

v,Z

u()y,xf,

33

23

33

*

255 '

;''⎩

⎨⎧

⊗⎟⎟⎠

⎞−Δ

λλλψZ

u

3'⎜⎜⎝

⎛ − λFZ

Z

5

25

−ZZy,(x

45555 ;

+−=ψ

)v

(8)

where is the Fourier Transform of and the scale is given by, (u,P3 )y,(xp 333

( ) ( )254

554

254

554 , FZZyZZ

vFZZxZZ

u −+=−+= λλ (9)

The quadratic phase factor dependent on in Eq. (8) does not appear in a 4-f system. It may be dropped when only the amplitude of the wavefront is of interest as it is normally the case. If the data page is codified as a phase-only function, e.g. binary or multinary phase [

)y,(x 55

8,9], then, depending on the detection method used in plane P5, we may have to leave the quadratic phase term. In the ideal case of infinite aperture, , then )

1333 =)y,(xp

( ) ( vu,vu,P δ=3, which is the delta Dirac function, and the convolution in Eq. (8) reduces

to ( )

vuZ

v,Z

ufy,xf,33

555 '' ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

λλ (10)

, where we have dropped both the quadratic phase factor dependent on and the one dependent of the defocus distance

)y,(x 55

zΔ . As just said this is possible when only the amplitude of the wavefront is of interest. The result in Eq. (10) corresponds to the magnified and inverted image of the data page given by lens L2 in plane P2. If the aperture is finite, what we have in Eq. (8) is that widens and therefore the convolution smoothes the data page which is also multiplied by the defocus quadratic phase term. The edges of the bits in the data page are then blurred increasing the probability of wrong detection of 1s and 0s. We then see that the convergent correlator architecture allows obtaining the reconstructed signal as in the case of the 4-f system. Differences may arise if the data page is introduced as a non-amplitude-only function, since in this case there is a resultant quadratic phase factor dependent on which does not appear in the 4-f system.

)y,(x 33p3

( vu,P3 )

)y,(x 55

Function acts as a low-pass filter in the frequency domain whose aperture defines the cutoff frequency. In this sense, since the data page contains binary information, we have that its fundamental frequency, or Nyquist frequency, is produced at a distance

)y,(xp 333

dzRN 2/3λ= , or Nyquist radius, in the plane P3, where corresponds to the size of the bits in the data page. Then the radius of function must be at least equal to the Nyquist radius not to produce too much cross-talk between neighboring bits [

d)y,(xp 333

12]. In Section 4 we will show some numerical results where this will become clearer.

3. Modulation curves for HDSS with a PA-LCoS microdisplay

Our aim is to construct a holographic data storage testing platform based on the convergent correlator architecture for the object beam, and where the data page is introduced by a PA-LCoS microdisplay. These devices are well adapted to display phase-only elements since they enable phase-only modulation without coupled amplitude. This can be achieved when illuminated by linearly polarized light parallel to the director axis. In the paper, we investigate that PA-LCoS devices can also be used to display the widely applied BIM data pages. We also investigate HTM data pages, which are very much demanding on the phase and amplitude modulation properties of an SLM and which have not yet been studied with PA-LCoS devices. HTM data pages combine the ease of detection of BIM data pages, together with the large reduction of the DC term of the Fourier Transform of the data page possible with phase-only data pages. This is necessary to avoid saturation of the dynamic range of the recording material [10,15,16].

To analyse the modulation capabilities of PA-LCoS devices, we use the model and characterization technique presented in Ref. [17,18]. These devices can be considered as

pixilated waveplates whose linear retardance can be tuned by the voltage applied. The technique we demonstrated, the time average Stokes polarimetry, enables to obtain not only the retardance versus applied voltage but also its flicker amplitude, which is exhibited by a number of LCoS microdisplays [28,29], especially the ones digitally addressed [30]. Then, first we calibrate the linear retardance, and then we use these values to calculate the complex amplitude modulation of the PA-LCoS device inserted between external polarization elements (polarizers and/or waveplates). In all these modeling and calculations in the paper, we apply the Jones formalism for polarization, since it enables calculation of both the amplitude and phase-shift. In previous papers, where the interest was to calculate the state of polarization (SOP) and the degree of polarization (DoP), we employed the Mueller-Stokes formalism.

Next, we provide the basic expressions necessary for the complex amplitude calculations in the Jones matrix formalism. The basic elements are the matrices for linear polarizer and linear retarder, the former given by,

(11) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

0001

XP

which expresses the linear polarizer with its transmission axis along the X-axis, which we consider along the vertical of the lab. The matrix for a linear retarder of linear retardance φ with its slow axis along the X-axis in the reference system is given by,

( ) ( )( )⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=

2exp002expφ

φφ

jj

W (12)

When polarization elements are rotated an angle θ with respect to the X-axis, then the two dimensional rotation matrix is necessary,

(13) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−

=θθθθ

θcossinsincos

R

The basic configuration that we need to produce both BIM or HTM is with the PA-LCoS device inserted between two rotated linear polarizers at angles 1θ and 2θ , i.e. the complex

amplitude for the electric field OUTEr

at the output of the system is given by the following sequence,

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅Γ⋅⋅=

1

12 sin

cosθθ

θ PAXOUT WRPEr (14)

, where OUTEr

corresponds to linearly polarized light at an angle 2θ with respect to the X-axis. In the expression ( )ΓPAW is the matrix for the PA-LCoS device, which is given by Eq. (12), and whose retardance Γ varies with the applied voltage ( )VΓ . The retardance value that we consider is the average value measured with the average Stokes polarimetric technique. Its flicker amplitude will not be considered in the calculations: in the calibration we have taken care to select electrical configurations minimizing the existence of this flicker in the retardance [18,30]. The intensity transmission is given by the hermitic product of OUTI OUTE

r

and the phase-shift OUTϕ corresponds to the argument. Then, once a certain working configuration is selected for the PA-LCoS device, i.e. ( )VΓ is fixed, we can optimize in the computer the angles 1θ and 2θ to produce the best BIM and HTM regimes.

The specific PA-LCoS device considered in this work is a commercially available PA-LCoS microdisplay, model PLUTO distributed by the company HOLOEYE. It is filled with a nematic liquid crystal, with 1920x1080 pixels and 0.7” diagonal. The pixel pitch is of 8.0 μm

and the display has a fill factor of 87%. The signal is addressed via a standard DVI (Digital Visual Interface) signal. By means of the RS-232 interface and its corresponding provided software, we have access to the basic electrical parameters of the device [28,29], such as the digital addressing sequence format, the gamma curve, and the voltage dynamic range of the pulse width modulated (PWM) signal (through two digital potentiometers), i.e. the so-called voltages Vbright and Vdark, expressing the maximum and minimum amplitudes of the pulsed voltage applied across the LC layer. In particular the software for the SLM is provided with a series of configuration files corresponding to two different digital addressing sequences: the so-called 18-6 and 5-5 digital sequences. The first number indicates the quantity of “equally weighted” bit-planes, and the second number the quantity of “binary” bit-planes [28]. This means that the sequence 18-6 is longer than the one corresponding for the sequence 5-5. In principle the shorter the sequence the smaller the flicker [28] thus in the following we restrict our attention to sequence 5-5 since it produces less flicker.

Retardance measurements have been taken at an angle of incidence of 11.5º, i.e. we consider no beam-splitter in front of the SLM, and for the wavelength 532 nm, to which the PVA/AA photopolymer is sensitized. These are the working conditions of the PA-LCoS device in the HDSS setup that we will present in Section 4. The PA-LCoS device is addressed with two different electrical configurations adapted in each case for the retardance dynamic range necessary for BIM and for HTM and minimizing the possibility of flicker. In Ref. [30] we introduced how to adapt the electrical configuration as a function of the application. In this sense we consider different pair of values for the configuration voltages Vbright and Vdark: for BIM and for HTM they are respectively (Vbright = 2.02V, Vdark = 1.11V) and (Vbright = 3.82V, Vdark = 0.03V). These values enable to produce a π radians retardance range for BIM, which is enough to generate an amplitude transmission configuration with minimum and maximum intensity transmission values for the specified wavelength. In the case of HTM, we consider voltages producing a larger retardance range, close to 2π radians, since we not only need a low and high transmission intensity values, but we also need two equally high transmission intensity points with a relative phase-shift difference as close to π radians as possible. In the case of twisted-nematic LCDs [15,31] we showed that these conditions were better met if the modulation dynamic range available is increased. In this Section we will analyse if this is still the case with PA-LCoS devices.

In Fig. 2 we show the retardance (left axis) and flicker amplitude (right axis) measurements as a function of gray level obtained applying the time-average Stokes polarimetric technique [17,18] for both the BIM and HTM adapted electrical configurations. In the case of the BIM (continuous curves) the retardance range is slightly larger than 180º and the associated flicker amplitude is about 10º, which is a low value, in most of the range. For the HTM (dashed curves) the retardance range is larger than 360º and the flicker amplitude reaches values close to 40º, even though only at high gray levels.

Fig. 2. Calculated values for the average retardance and the fluctuation amplitude for λ=532nm, at an angle of incidence of 11.5º, for sequence 5-5 with configuration voltages (Vbright = 2.02V, Vdark = 1.11V) for BIM (continuous) and (Vbright = 3.82V, Vdark = 0.03V) for HTM (dashed).

05101520253035404550

300350400450500550600650700750800

0 50 100 150 200 250

Fluctuation amplitude (º)Av

erag

e re

tard

ance

(º)

Gray level

In the case of BIM the goal is to generate the maximum intensity contrast between the ON and OFF values, i.e. OFFONcontrast III = . In Ref. [12] they calculate that an acceptable raw BER can be obtained with a contrast of 1:20, even though the system performance can be further improved for higher contrast values. Maximum contrast can be obtained with only polarizers, if they are parallel or crossed with each other and at 45º with respect to the director axis of the PA-LCoS device. Using the average retardance values ( )VΓ characterized in Fig. 2, and for crossed polarizers and at -45º (+45º) with respect to the director axis (lab vertical) in the PA-LCoS (which is along the lab horizontal) we obtain the intensity transmission and phase-shift curves as a function of the gray level displayed in Fig. 3(a), which are also plotted in Fig. 3(b) as a phasor representation in the complex plane.

04080120160200240280320360400

00,10,20,30,40,50,60,70,80,9

1

0 50 100 150 200 250

Phase-shift (º)

Nor

mal

ized

inte

nsity

Gray level -1

-0,5

0

0,5

1

-1 -0,5 0 0,5 1

Im

Re

(a) (b) Fig. 3. Simulation for BIM. (a) Intensity transmission and phase-shift; (b) Phasor evolution in the complex plane. Only polarizers at +45º with respect to the X-axis (lab vertical).

-100-50050100150200250300350400

00,10,20,30,40,50,60,70,80,9

1

0 50 100 150 200 250

Phase-shift (º)

Nor

mal

ized

inte

nsity

Gray level -1

-0,5

0

0,5

1Im

1Re

-1 -0,5 0 0,5

(a) (b) Fig. 4. Simulation for pHTM. (a) Intensity transmission and phase-shift; (b) Phasor evolution in the complex plane. Input and output polarizers at +55º and -45º with respect to the X-axis.

From Fig. 3(a) the low and high intensity transmission points occur respectively at gray

levels 12 and 239. From the values in Fig. 3(a), we obtain that the theoretical contrast tends to infinity, and the phase –shift values are respectively 270º and 360º. We note that at these two gray levels, from Fig. 2 the modulus-2π retardance values are respectively very close to 0º and 180º: thus, at these two gray levels the PA-LCoS behaves respectively as a zero-wave plate and as a half-wave plate, thus the SOP is orthogonal and parallel respectively to the transmission axis of the output polarizer. In the experimental measurements we obtain that the low and high transmission points occur slightly displaced at gray levels 14 and 248, and the contrast we measure is about 1:50. The theoretical contrast value is idealistic since the various degradation effects in the PA-LCoS have a direct impact in the minimum intensity transmission value: e.g. inhomogeneities across the aperture of the PA-LCoS and the flicker will produce a spatial and time fluctuation of the retardance from its average value and as a result leakage of light between the crossed polarizers. The phasor representation (Fig. 3(b)) is very useful to see the topology of the complex amplitude evolution with gray level, which

contrastI

follows a circular trajectory. We see that the 180º phase-jump at gray level 12 (Fig. 3(a)) is produced by the phasor excursion in the vicinity of the origin. We have verified that independently of the orientation of the transmission axis of the polarizers we always obtain a circular trajectory, however both the passage through the origin (which is responsible for the intensity minimum) and the radius of the circle (which will decrease) will be affected lowering the contrast parameter . The same effect is produced when adding waveplates to the polarizers at the entrance or/and at the exit of the PA-LCoS device: we always obtain a circular trajectory.

contrastI

In the case of HTM we need to find three gray level values, two of them with a high and equal intensity transmission and with a 180º relative phase-shift (ON levels), and a third gray level with a low intensity transmission (OFF level). After performing a series of optimizations and simulations we have found that the PA-LCoS device cannot fully meet these requirements. In the case of twisted-nematic LCDs it was possible, even though lowering the intensity transmission for the two ON levels, as we reported in [15,16,31]. Twisted-nematic LCDs provided a coupled amplitude and phase-shift modulation which enabled to produce spiral and other arbitrary complex amplitude trajectories (complex plane representation) as shown in [31]. In the case of PA-LCoS devices, we have found that the trajectory is always a circle, as discussed in Fig. 3(b). This means that if the OFF level is close to the origin, two equidistant points in the circle with a 180º phase-shift difference and with some intensity transmission are not possible. A compromise can be found if the circular trajectory is slightly shifted. Then at the cost of producing some leakage of light in the OFF value, we can produce two ON values with an appreciable intensity transmission and a relative phase-shift not far from 180º. Therefore, we can state that PA-LCoS devices do not produce a pure HTM modulation, but they enable an approximate trade-off, which we will call in the rest of the paper pseudo-HTM modulation (pHTM). We want to study if the pHTM can still be useful in its application to HDSS.

In Fig. 4 we show the complex amplitude modulation for one of the possible pHTM configurations obtained in our simulations. It corresponds to input and output polarizers at 55º and -45º with respect to the X-axis (lab vertical). In Fig. 4(a) we show the intensity and the phase-shift versus gray level, and in Fig. 4(b) we plot the phasor evolution in the complex plane where we see that the circular trajectory is not traversing the origin. The two ON gray levels considered are 105 and 168, with amplitude transmission values of 0.28 and phase-shift values of 75º and 281º respectively, i.e. a phase-shift difference of 206º. The OFF gray level is 140 with an amplitude transmission of 0.03 and phase-shift 170º. The intensity contrast

is then 1:10. This is a low value, however increasing the contrast means almost crossing the two polarizers to produce a smaller intensity transmission for the OFF value. This has the drawback that the phase-shift becomes rather close to a 180º phase-jump at the OFF value, and the phase-shift difference between two ON levels about the OFF level rapidly becomes much larger than 180º if we want a reasonable intensity transmission at the ON levels. In the next Section we will evaluate numerically the pHTM in comparison with the idealistic HTM and BIM. We will also obtain experimental results both with BIM and pHTM.

contrastI

4. Numerical and experimental results

In Section 2 we verified theoretically the validity of the convergent correlator as an alternative to the 4-f processor in holographic memory testing platforms. We obtained the expressions considering the recording of the defocused Fourier transform of the data page and the possibility of a limiting aperture, the Nyquist aperture, on the recording plane. Furthermore we can simulate the realistic complex amplitude values for the levels implemented on the SLM to address the data page. These realistic values for the BIM and pHTM data pages can be obtained from the calculations exposed in Section 3. Next we will show numerical results

using the HDSS simulator, and they will be compared with experimental results using the testing platform in our lab that we now present.

LCoS

BS

SF

CCDcamera

WP

RecordingMaterial

RotatingMirror

LP1

Mirror

Mirror

LP2

LP3

Stop

ShutterNyquist

filter

defocus

HWP

L1L2L3

L4

L5

L6 L7

L8

Fig. 5. Diagram for the experimental HDSS testing platform. System described in the text.

In Fig. 5 we show the scheme for the experimental holographic memory setup used in our

lab. We consider the 532 nm beam from a Nd:YVO4 laser, to which the PVA/AA photopolymer is sensitized. An additional waveplate, WP, is inserted before the shutter and spatial filter, SF, to secure that enough light traverses the polarizers in the object and reference beams. After the SF, the beam is collimated, and a non-polarizing beam-splitter produces the object and reference beams. The intensity ratio between both beams is controlled with one attenuator in each arm (not shown in the figure). The two linear polarizers LP1 and LP3 are used to produce the appropriate SOP for the object beam, and for the reference beam (along the vertical of the lab, to produce TEM mode interference on the material). In the reference beam, a stop limits the aperture of the beam to about a diameter of 1 cm. We have built an afocal system with lenses L6 and L7 so that the rotating mirror and the recording material are at conjugate planes, enabling angular multiplexing simply by rotation of the mirror. In the object beam, we have combined a divergent (L2) and convergent (L3) lens so that we can control the curvature of the converging beam onto the PA-LCoS. At the convergence plane the Fourier transform of the data page is produced and a stop, the Nyquist aperture, is located, as indicated in the figure. In our setup Nyquist aperture is at 15 cm from the PA-LCoS device. Then we have built a relay system to image the Nyquist filter plane onto the recording plane, where lens L4 collimates the beam before inciding onto polarizer LP2, which plays the role of exit polarizer for BIM and pHTM. The half-wave plate (HWP) is oriented so that the object beam is linearly polarized along the lab vertical. In this way we produce interference between TEM modes both in the object and reference beams, which generates maximum visibility of the interference fringes. The position of the image for the Fourier transform of the data page is produced on the focal plane of lens L5, with a focal length of 15 cm. The angle between the object and reference beams is 37º. In this work we will produce a symmetrical recording, i.e. the angle of incidence of either beam with respect to the normal to the plane of the material is 18.5º, which in the case of two plane waves interference would be equivalent to the recording of a transmission diffraction grating with 1193 lines/mm, which is well within the range characterized with PVA/AA photopolymers [25]. Taking into account that the beam size is about 1 cm diameter, and that the second lens in the object beam has a focal length of 15 cm, we obtain that the numerical aperture, NA, to be used in Eq. (4) is about 0.03, and for a defocus parameter the lab defocus is 120 =W mmz 1=Δ .

In the reconstruction step, the recording material is illuminated with the reference beam and the lens L8 produces the Fourier transform of the object beam retrieved onto the plane of the CCD camera (pco.1600 model from pco.imaging). This is a high dynamic 14 bits cooled CCD camera system with a resolution of 1600x1200 pixels, and a pixel size of 7.4x7.4 μm2.

The magnification of the PA-LCoS plane onto the camera plane is about a factor x2. In this paper, we consider data pages on the PA-LCoS with 64x64 bits and with 8 pixels per bit. Many LCDs and LCoS devices exhibit cross-talk effects, such as the anamorphic and frequency-dependent effect [32,33], when there are fast transitions in the image. To limit these effects we have preferred to work with this bit size.

To numerically simulate the system we use the fast Fourier transform algorithm using the MATLAB software. In the simulations we consider data pages with 256x256 bits. To accurately simulate the optical system we have to consider both oversampling and zero padding of the data page bits [21]. According to all these factors, the size in pixel number of the Nyquist radius is 512 pixels.

NRIn the simulations the figures of merit that we consider to evaluate the fidelity in the

retrieval are bit-error rate (BER), which is given by the number of incorrectly detected ones and zeros with respect to the total. Then we also use the so-called Q-factor, which is an estimate for the quality of the signal-to-noise ratio, typically used in digital systems as for example in optical fiber communications [35], which is given by,

01

01

σσμμ

+−

=Q (15)

, where 1μ and 0μ are the mean value in the histograms produced for the gray level

distribution of ON and OFF bits respectively, and where 1σ and 0σ are the corresponding

standard deviations. BER below 10-2-10-3 are necessary so that further reduction to practical levels is obtained through additional error correction codes. To evaluate the homogeneity of the intensity distribution in the recording plane we consider the ratio between the DC-term (zero frequency) intensity value with respect to the intensity averaged across the recording plane,

DCI avrgI

( )avrgDCDCterm IIR 10log10= (16)

, given in decibels (dB). This figure of merit is a variation of the ones defined in [12,13] Next we show some simulated results. We apply gaussian additive noise with a variance

of 0.005 and mean zero, which is a good estimation of the thermal noise of the CCD camera used in the experiments. In the BIM we consider random bit data pages with an equal number of 0s and 1s, whereas in the pHTM the 1s are equally distributed between the two ON levels.

(a) Fig. 6. Fourier plane where the limiting stop is double the length of the Nyquist radius, which is 512 pixels. (a) Image plane; (b) Radial cut across a diameter crossing the center of the Fourier plane. The representation is logarithmic. We have considered a 128x128 bits BIM data page.

(b)

In Fig. 6 we show the resulting pattern in the Fourier plane for an ideal BIM data page,

when considering a limiting stop with a length double the Nyquist radius, , that is the aperture radius is 1024 pixels, therefore it transmits both the fundamental frequency and the first harmonic for the data page. We can see, especially in radial cut in Fig. 6(b), the

pixelsR N 512=

much larger intensity of the zero frequency (or DC term) in comparison with the rest of the frequency components in the Fourier plane. This is what happens when using BIM data pages and results in saturation of the dynamic range of the recording material in the location of the DC term, therefore deteriorating the data page information when retrieved and reducing the dynamic range of the recording material.

(c) (b) (a) Fig. 7. Simulation for BIM and for 128x128 bits data page. (a) Homogeneity in the recording plane; (b) Q-factor and (c) BER, to evaluate the quality of the reconstruction. In the legend, size of the Nyquist aperture in Nyquist radius units.

(c) (b) (a) Fig. 8. Simulation for ideal HTM and for 128x128 bits data page. (a) Homogeneity in the recording plane; (b) Q-factor and (c) BER, to evaluate the quality of the reconstruction. In the legend, size of the Nyquist aperture in Nyquist radius units.

(b) (a) (c) Fig. 9. Simulation for pHTM and for 128x128 bits data page. (a) Homogeneity in the recording plane; (b) Q-factor and (c) BER, to evaluate the quality of the reconstruction. In the legend, size of the Nyquist aperture in Nyquist radius units.

In Fig. 7 we show the simulated results for various figures of merit as a function of

defocus, and for various Nyquist apertures (in units of the fundamental frequency, in the legend), for the case of ideal BIM. In Fig. 7(a) we observe the large dynamic range necessary to both register the DC-term and the higher frequencies, whose value will tend to the average

value. We see that one way to reduce this dynamic range is to defocus the Fourier transform of the data page onto the recording material. In Fig. 7(b) and 7(c) we show respectively the Q-factor and the BER. We observe that the reconstruction deteriorates for smaller Nyquist apertures. BER results better than 10-2 can be obtained for apertures larger than 1.5 the Nyquist radius, which is related with Q factor values better than 1. We note that

avrgI

we have also done the simulation using the experimental values for the BIM, which provide a limited contrast of about 1:50 (discussed with Fig. 3). The simulations obtained are basically the same as the plots in Fig. 7.

In Fig. 8 we show the plots equivalent to the ones in Fig. 7, but now for the case of ideal HTM. We see that is more than 40 dB smaller than in the BIM case. Actually the value varies with each realization of the simulation since we are considering a random data page, then we can even obtain negative values for . Defocus (Fig. 8(a)) does not have an influence in the intensity homogeneity in the recording plane. It is interesting to observe that now to obtain Q factor better than 1 (Fig. 8(b)), the Nyquist aperture needs to be close to 2. This is in correspondence with necessary Nyquist aperture to obtain a BER smaller than 10-

2 (Fig. 8(c)). The BER value obtained for a Nyquist aperture value of 2 is about 50 times better in the case of BIM with respect to HTM. We believe that the low-pass filtering due to the Nyquist aperture is producing a larger cross-talk between neighbouring pixels in the case of HTM, since larger spatial frequencies increase their relative importance with respect to the DC-term.

DCtermR DCI

DCtermR

In Fig. 9 we show the plots equivalent to the ones in Fig. 7 and 8, but now for the case of pHTM. In Fig. 9(a) we observe that for the non-defocused situation, the homogeneity has worsened with respect to ideal HTM, but it is still 20 dBs (2 orders of magnitude) better than for BIM. The results for the Q-factor and BER (Fig. 9(b) and (c)) are very close to the ones in Fig. 8, thus there is no degradation in the quality of the reconstructed page when compared with HTM.

In the previous numerical simulations we considered an ideal linear material, thus not reflecting the impact that saturating DC peaks produce in the fidelity of the reconstructed data page. Next we show the experimental results obtained with a PVA/AA photopolymer as recording material. In the paper we restrict our attention to the non-defocus situation, i.e.

. The effective Nyquist aperture in the experiment is in principle better than 2, i.e. both the fundamental frequency and the first harmonic are not filtered. The PVA/AA photopolymer composition we use is similar to the one used in previous works [

0=Δz

34]. This photo-chemical composition is characterized by the presence of yellowish eosin (YE) as dye and N,N’-methylene-bis-acrylamide (BMA) as crosslinking monomer. It also contains a co-sensitizer which is triethanolamine (TEA), the acrylamide (AA) monomer and the PVA as a binder. The thickness of the samples is around 90±2µm and the refractive index modulation achieved for a single holographic grating with a spatial frequency of 1150 lines/mm is 0.005.

First we plot the results for the image retrieved when addressing the BIM data page to the PA-LCoS and with no recording material. This allows evaluating the performance of the setup. In Fig. 10(a) and (b) we show respectively the data page captured with the CCD camera and the distribution of 0s and 1s in the histogram. We note that no errors are obtained, i.e. BER=0. In Fig. 10(a) a good contrast between ON and OFF bits is visible. In Fig. 10(b) we see the large separation between the histograms for the 0s and 1s, with no overlap.

In Fig. 11 we show the retrieved results when the PVA/AA photopolymer is used to record the data page. It corresponds to a beam intensity ratio about 1:400, where the intensity incident onto the recording material is 3.16 mW/cm2 and 8µW/cm2 respectively for the reference and object beam, and for an exposure time of 6 seconds. The number of errors detected is 52, i.e. BER=1.3·10-2. This is larger than the numerical BER value in Fig. 7(c) for the Nyquist radius of 2. Other beam ratios and exposure times have been tried, and results in Fig. 11 are in the range of the best results obtained in the various batches. We see the good contrast between 0s and 1s in Fig.11(a). In Fig. 11(b) we see a slight overlap between the 0s and 1s histograms.

Next we concentrate on the results obtained when the pseudo-HTM data page is addressed. In Fig. 12 we show the retrieved results when no photopolymer is in the system, i.e. only the effect of the holographic memory setup and implementation on the PA-LCoS is

considered. We detect 8 errors, which corresponds to a BER of 2.0·10-3. This is worse than in the BIM case in Fig. 10, which is probably due to the lower contrast of the data pages in the PA-LCoS, as commented when discussing the contrast parameter in Fig. 4. In Fig. 12(a) it can be seen a lower contrast than in Fig. 10(a). The histograms (Fig. 12(b)) have a slight overlap, especially when compared with Fig. 10(b).

contrastI

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

2

4

6

8

10

12

14

0s

1s

(b) (a) Fig. 10. Experimental results BIM data page and no material. (a) Data page; (b) Histograms.

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

2

4

6

8

10

12

0s

1s

(b) (a) Fig. 11. Experimental results BIM data page and PVA/AA. (a) Data page; (b) Histograms.

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

2

4

6

8

10

12

0s

1s

(b) (a) Fig. 12. Experimental results pHTM data page and no material. (a) Data page; (b) Histogram.

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

1

2

3

4

5

6

7

0s 1s

(b) (a) Fig. 13. Experimental results pHTM data page and PVA/AA. (a) Data page; (b) Histogram.

Now the photopolymer is used to record the pseudo-HTM data page and we plot the

results in Fig. 13. Beam intensity ratio is about 1:800, where the intensity incident onto the

recording material is 3.16 mW/cm2 and 4µW/cm2 respectively for the reference and object beam, and for an exposure time of 10 seconds. There are 229 errors detected, which corresponds to a BER of 5.6·10-2. Visually, the image in Fig. 13(a) is not showing a much worse contrast or definition, even though the BER is worse than in previous cases, which is clearly appreciated in the histograms overlap in Fig. 13(b). When compared with results for Fig. 11(b) the BER is about 5 times larger. In the case of the numerical simulations, corresponding to a linear material, for the Nyquist aperture value of 2, an analogous comparison between the two modulations produced a BER ratio of about 50. Therefore, the experimental result for the pHTM is 10 times better than the numerical one. We believe that the better experimental ratio is due to the impact of the larger DC peak in the case of the BIM, which causes a nonlinear recording of the data page due to saturation of the dynamic range of the material, which is an effect not included in the numerical simulation. Further experimental work has to be done with PVA/AA photopolymer to get a deeper understanding of these saturation effects in this material. In general, we have obtained good results with the PVA/AA material, improving the fidelity of the reconstructed data page with respect to previous works in the literature [25,26], thus showing its potential in data storage.

5. Conclusions

We have built a holographic data storage testing platform, where the novelty in three of its elements has been analysed. On one hand, we have introduced a last generation PA-LCoS device to display the data page. We have verified its good performance to display BIM data pages. We have also shown that pure HTM data pages cannot be obtained with PA-LCoS devices, however, a rather close performance is obtained when implementing the pseudo-HTM data pages that we have proposed in this paper. Another novelty is that instead of the typical 4-f system, we have studied both theoretically, numerical and experimental, the use of the convergent correlator architecture for the object beam. The expressions obtained, based on the Fourier Optics formalism, have been used to construct a holographic storage simulator which includes the possibility of a Nyquist aperture and defocus of the Fourier transform of the data page with respect to the recording material. Differences with respect to the 4-f system may arise if the data page is introduced as a phase-only function, since in this case there is a resultant quadratic factor which does not appear in the 4-f system. Then, we have also analysed experimentally the potential of PVA/AA photopolymer to be used in HDSS applications. We have obtained promising results with BER values in the range of 10-3 thus showing the interest of further studies with this phopolymer in data storage.

Appendix

Next we show the basic Fourier Optics notation needed for the theoretical derivations in Section 2. In the Fourier Optics formalism [19,20] optical systems are composed of two basic units: propagation in air (or free-space) and propagation through lenses. In the case of free-space propagation the impulse response, in the Fresnel approximation, is given by,

( ⎟⎠⎞

⎜⎝⎛ 22exp y+x

2zkj

zje = y)h(x,

jkz

λ) (A1)

where k is the wave number, k=2π/λ, with λ the light wavelength. In the case of a lens, it produces the change in the curvature of the incoming beam thus generating a spherical wavefront, and for a lens with a focal length the effect on the phase of the incoming wavefront is represented by,

f

( ⎟⎟⎠

⎞⎜⎜⎝

⎛− 22exp y+x

fj = y)(x,t L λπ ) (A2)

where, as in Eq. (A1), we have a parabolic expression. This is a very good approximation as long as we move in the paraxial regime, i.e. not very large numerical apertures. For complex optical systems it becomes very useful to use the notation introduced in pages 117-118 in the book of A. VanderLugt [20]. Using the definition,

( ⎟⎠⎞

⎜⎝⎛≡= 22exp1; y+xZj )

zZy(x,

λπ )ψ (A3)

we rewrite Eq. (A1) and (A3) as follows,

Z)y(x,Zj

e = y)h(x,jkz

;ψλ

(A4)

F)y(x, = y)(x,tL −;ψ (A5)

These expressions verify a series of very useful properties which are used intensively in the theoretical calculations in Section 2. For self containment of the paper we list them as follows, 1º) Z)y(x, = Z)y(x, −;; *ψψ

2º) Z)y(x, = Z)yx,( ;; ψψ −−

3º) )ZZy(x, = )Zy(x, )Zy(x, 2121 ;;; +ψψψ

4º) )ZZy(x, =)ZZy(x, = )Zy(x, )Zy(x, 12*

212*

1 ;;;; −− ψψψψ

5º) Z)cy(x, = Z)ycx,(c 2;; ψψ

6º) ( ) yvxuz

jZ)y(x,Z)v(u, = Z)vyu,(x ⎟⎟⎠

⎞⎜⎜⎝

⎛+−−−

λπψψψ 2exp;;;

7º) = Z)y(x,Z 1;lim *0ψ→

8º) ( ) yx = Z)y(x,ZZ ,;lim δψ∞→

9º) ( ) )Z

Zv(u,

Zj

=dydx yvxuz

j)Zy(x,1

22*

121 ;2exp; ψ

λλπψ ⎟⎟

⎠

⎞⎜⎜⎝

⎛+−∫ ∫

+∞

∞−

+∞

∞−

,

( ) )Z

Zv(u,

Zj

=dydx yvxuz

j)Zy(x,1

22

121

* ;2exp; ψλ

λπψ

−⎟⎟⎠

⎞⎜⎜⎝

⎛+−∫ ∫

+∞

∞−

+∞

∞−

Acknowledgements

Work supported by Ministerio de Trabajo y Competitividad of Spain (projects FIS2011-29803-C02-01 and FIS2011-29803-C02-02), by Generalitat Valenciana of Spain (projects PROMETEO/2011/021 and ISIC/2012/013), and by Univ. de Alicante (project GRE12-14)

APPENDIX

4Complete list of publicationsrelated with the Ph.D. Thesis

Apart from the five papers that form this thesis by published compilation arti-

cles, and the one unpublished but introduced in the previous section. I want to list

all the works that has been presented in international or national conferences that

are related with this Ph.D. Thesis.

4.1 Papers in Journals• Francisco J. Martínez, Andrés Márquez, Sergi Gallego, Manuel Ortuño, Jorge

Francés, Inmaculada Pascual and Augusto Beléndez,“Predictive capability

of average stokes polarimetry for simulation of phase multilevel elements”,

Applied Optics, 54, no6, 1379-1386. 2015

• Francisco J. Martínez, Andrés Márquez, Sergi Gallego, Manuel Ortuño, Jorge

Francés and Inmaculada Pascual, "Averaged Stokes polarimetry applied to

evaluate retardance an flicker in PA-LCoS devices", Optics Express, 22,

no12, 15064-15074. 2014

129

4. COMPLETE LIST OF PUBLICATIONS RELATED WITH THE PH.D.THESIS

• Francisco J. Martínez, Andrés Márquez, Sergi Gallego, Jorge Francés, In-maculada Pascual and Augusto Beléndez, "Retardance and flicker modeling

and characterization of electro-optic linear retarders by averaged Stokes po-

larimetry", Optics Letters, 39, no4, 1011-1014. 2014

• Francisco J. Martínez, Andrés Márquez, Sergi Gallego, Manuel Ortuño, JorgeFrancés and Inmaculada Pascual, "Electrical dependencies of optical modu-

lation capabilities in digitally addressed parallel aligned liquid crystal on

silicon devices", Optical Engineering, 53, no6, 067104-1:9. 2014

• Francisco J. Martínez, Andrés Márquez, Sergi Gallego, Jorge Francés andInmaculada Pascual, “Extended linear polarimeter to measure retardance

and flicker: application to liquid crystal on silicon devices in two working

geometries”, Optical Engineering, 53, no1, 014105-1:9. 2014

• Francisco J. Martínez, Roberto Fernández, Andrés Márquez, Sergi Gallego,Mariela L. Álvarez, Inmaculada Pascual and Augusto Beléndez “Exploring

binary and ternary modulations on a PA-LCoS device for holographic data

storage in a PVA/AA photopolymer”, Optics Express, Under Revision

• Francisco J. Martínez, Andrés Márquez, Sergi Gallego, Jorge Francés, In-maculada Pascual and Augusto Beléndez “Effective angular and wavelength

modelling of parallel aligned liquid crystal devices”, Optics and Laser inEngineering, Accepted for publication 2015

4.2 Papers in Conferences• Sergi Gallego, Roberto Fernández, Andrés Márquez, Francisco J. Martínez,

Cristian Neipp, Manuel Ortuño, Jorge Francés, Augusto Beléndez and In-maculada Pascual, "Influence of the photopolymer properties in the fabrica-

tion of diffractive optical elements", SPIE, SPIE Optics and Photonics. ISSN:0277-786X , Vol. 9216, 921619-1–8, San Diego (EE.UU), 2014

• Andrés Márquez, Francisco J. Martínez, Sergi Gallego, Manuel Ortuño, JorgeFrancés, Augusto Beléndez and Inmaculada Pascual, "Averaged Stokes po-

larimetry applied to characterize parallel-aligned liquid crystal on silicon

130

4.2 Papers in Conferences

displays", SPIE, SPIE Optics and Photonics. ISSN: 0277-786X , Vol. 9216,

92160H-1–9, San Diego (EE.UU), 2014

• Francisco J. Martínez, Andrés Márquez, Sergi Gallego, Manuel Ortuño, Jorge

Francés, Augusto Beléndez and Inmaculada Pascual. "Robustness of average

Stokes polarimetry characterization of digitally addressed parallel-aligned

LCoS displays". International Comitee of Optics. ICO’23. ISBN: 978-84-

697-1027-2,3. Santiago de Compostela, 2014

• Andrés Márquez, Francisco J. Martínez, Sergi Gallego, Manuel Ortuño, Jorge

Francés, Augusto Beléndez and Inmaculada Pascual, "Study of the modula-

tion capabilities of parallel aligned liquid crystal on silicon displays", SPIE,

SPIE Optics and Photonics. ISSN: 0277-786X , Vol. 8855, 885504-1–9, San

Diego (EE.UU), 2013

• Sergi Gallego, Andrés Márquez, Roberto Fernández, Álvaro Piera, Francisco

J. Martínez, Manuel Ortuño, Jorge Francés, Augusto Beléndez and Inmacu-

lada Pascual, "Analysis of the fabrication of diffractive optical elements in

photopolymers", SPIE, SPIE Optics and Photonics. ISSN: 0277-786X , Vol.

8855, 88550V-1–8, San Diego (EE.UU), 2013

• Francisco J. Martínez, Andrés Márquez, Sergi Gallego, Manuel Ortuño, Au-

gusto Beléndez, Inmaculada Pascual, "Caracterización del retardo en pres-

encia de fluctuaciones en pantallas de cristal líquido sobre silicio con alin-

eación paralela", RSEF, XXXIV Reunión Bienal de la RSEF. Valencia (Es-

paña),2013

• Andrés Márquez, Francisco J. Martínez, Sergi Gallego, Elena Fernández,

María L. Álvarez and Inmaculada Pascual, "Holographic memory simulator

based on the convergent correlator architecture". Sociedade Portuguesa para

la Óptica e Fotónica. VIII Iberoamerican Optics Meeting. Libro de abstracts,

RIA100-199. Porto (Portugal), 2013

• Francisco J. Martínez, Andrés Márquez, Sergi Gallego, Manuel Ortuño, Au-

gusto Beléndez and Inmaculada Pascual, "Caracterización y configuración

131

4. COMPLETE LIST OF PUBLICATIONS RELATED WITH THE PH.D.THESIS

de pantallas PA-LCoS", SEDOPTICA Comité de Optoelectrónica, VIII Re-unión española de Optoelectrónica. ISBN: 978-84-88754-21-9, 223-227. Al-calá de Henares (Madrid), 2013

• A. Márquez, E. Fernández, F.J. Martínez, S. Gallego, M. Ortuño, A. Belén-dez, I. Pascual, "Analysis of the geometry of a holographic memory setup",SPIE, SPIE Optics Photonics Europe. ISSN: 0277-786X , Vol. 8429, 8429Y-1–10, Bruselas (Bélgica), 2012

• Andrés Márquez, Francisco J.Martínez, Sergi Gallego, Manuel Ortuño, JorgeFrancés, Augusto Beléndez and Inmaculada Pascual, "Classical polarimetric

method revisited to analyse the modulation capabilities of parallel aligned

liquid crystal on silicon displays", SPIE, SPIE Optics and Photonics. ISSN:0277-786X , Vol. 8498, 84980L-1– 11, San Diego (EE.UU), 2012

• Francisco J. Martínez, Andrés Márquez, Elena Fernández, Sergi Gallego,Jorge Francés, Augusto Beléndez and Inmaculada Pascual, "Efecto del desen-

foque en el registro y en la reconstrucción de información en memorias holo-

gráficas basadas en el correlador convergente", SEDOPTICA, X ReuniónNacional de Óptica. Libro de Abstracts, 393-396, Zaragoza, 2012

132

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Declaration

I herewith declare that I have produced this work without the assistanceof third parties and without making use of aids other than those spe-cified; notions taken over directly or indirectly from other sources havebeen identified as such. This work has not previously been presentedin identical or similar form to any examination board.

The dissertation work was conducted from 2010 to 2015 under thesupervision of Andrés Márquez Ruiz and Sergi Gallego Rico at Uni-versity of Alicante.

Alicante,

This dissertation was finished writing in Alicante on Tuesday 9th June, 2015

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