by Alex M. H. Wong - University of Toronto T-Space...Alex M. H. Wong Master of Applied Science The...

SUBWAVELENGTH FOCUSING VIA HOLOGRAPHIC METALLIC SCREENS

by

Alex M. H. Wong

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

The Edward S. Rogers Sr. Department of Electrical and Computer Engineering

University of Toronto

Copyright © 2009 by Alex M. H. Wong

ii

Abstract

Subwavelength Focusing via Holographic Metallic Screens

Alex M. H. Wong

Master of Applied Science

The Edward S. Rogers Sr. Department of Electrical and Computer Engineering

University of Toronto

2009

In this work we investigated a new class of subwavelength focusing device, termed the

holographic metallic screen. We first proposed a generalized procedure which takes a

holographic record of a subwavelength electromagnetic field distribution. Subsequently we

synthesized this record using two types of holographic metallic screen – the slot antenna

hologram (SAH) and the resonant slot antenna hologram (RSAH). We designed both

holograms and evaluated their performances through full-wave simulations, and

experimentally demonstrated subwavelength focusing for the RSAH. Simulations and

experiments illustrated various attractive properties of the subwavelength focusing RSAH,

which included (a) a tighter focal width than a single subwavelength aperture; (b) a focal

field amplitude surpassing the incident field amplitude; and (c) a simple design scalable to a

wide range of frequencies from microwave to optical. These properties should serve to

motivate further development on the holographic metallic screen towards potential

applications such as sensing, imaging and lithography.

iii

To His Highest

iv

Acknowledgements

Nearing completion of my master studies, I am most delighted to have time to properly write

this acknowledgement. No part of my studies can be truly considered an individual

accomplishment; I gratefully thank all who has made my studies a wonderful process of

discovery, learning and growth.

First and foremost I attribute thanksgiving to my God and my Lord, Who has made me grow

throughout the course of this research degree. I thank Him for His loving presence, His grace

and His providence in such small secular matters as these.

I must also express deep gratitude to my parents for their support in all aspects possible. I

will always be indebted to their love, their patience and the countless sacrifices they have

made for me.

I am thankful to my supervisor Prof. George V. Eleftheriades for more ways than I can

express. His invaluable guidance, motivation, and inspiration have made my research project

an exciting and rewarding experience. I look forward to learning much more from him as I

continue my Ph.D. studies under his supervision. I thank Prof. Costas D. Sarris for the help

and guidance he offered during the initial stages of my research. Moreover I thank my

professorial committee, which also included Prof. Li Qian and Prof. Sean V. Hum, for their

interest in my research, and their constructive comments regarding my work. I am also

grateful to have learnt from them in various graduate and senior undergraduate courses.

v

My colleagues in the electromagnetics group provided great company and solidarity

throughout the course of my master studies. I thank the senior students and researchers

within the group, in particular Rubyiat Islam, Marco Antoniades, Ashwin Iyer, Joshua Wong

and Dr. Michael Zedler for their help in various matters, from stimulating discussions to tips

on using simulation tools to help with course assignments. I especially thank Loїc Markley

and Yan Wang: I have enjoyed and benefited much from our collaboration on this project of

subwavelength focusing screens. I also thank Michael Studniberg and Jackie Leung, who

went through their master level studies the same time as I, and my close neighbours within

the student office, Jiang Zhu, Michael Selvanayagam, Muhammad Alam and Levent Kayili,

for their company throughout this time; time flies by when one has such good company. I

also wish to express appreciation to lab managers Micah Stickel and Tse A. Chan, for their

generous help with miscellaneous matters both in the office and in the laboratory.

Regrettably, for the sake of brevity I will not be able to acknowledge by name everyone who

has made a positive difference in my study. I sincerely thank all my colleagues and friends

within the university and abroad for their support, and wish them the best in my Lord Jesus

Christ.

vi

Contents

1. INTRODUCTION ................................................................................................................... 1

1.1 MOTIVATION TO SUBWAVELENGTH FOCUSING ................................................................ 1

1.2 CHAPTER ORGANIZATION ................................................................................................ 2

1.3 BACKGROUND: OVERCOMING THE DIFFRACTION LIMIT ................................................... 2

1.3.1 The Near-field Vicinity of a Source ...................................................................... 4

1.3.2 Negative Refractive Index (NRI) Metamaterial Superlens ................................... 5

1.3.3 Radiationless Interference Screen ........................................................................ 6

1.4 PROPOSAL ........................................................................................................................ 7

1.5 THESIS OUTLINE .............................................................................................................. 8

2. THE HOLOGRAM AND ITS EXTENSION ............................................................................... 9

2.1 INTRODUCTION TO CONVENTIONAL HOLOGRAMS ............................................................ 9

2.1.1 Recording a hologram ........................................................................................ 10

2.1.2 Reconstructing the Wavefront ............................................................................ 12

2.2 A SUBWAVELENGTH FOCUSING HOLOGRAM? ................................................................ 14

2.3 THE RESOLUTION LIMIT FOR CONVENTIONAL HOLOGRAMS .......................................... 15

2.3.1 Polarity of kz for Propagating Waves ................................................................ 17

2.3.2 Polarity of kz for Evanescent Waves .................................................................. 18

2.4 RECORDING A GENERALIZED HOLOGRAM VIA WAVEFRONT BACK-PROPAGATION........ 19

vii

3. THE GENERALIZED HOLOGRAM OF A SUBWAVELENGTH SOURCE ................................ 24

3.1 SPECTRUM BACK-PROPAGATION ................................................................................... 25

3.2 RESOLUTION CONSIDERATIONS FROM SPECTRAL TRUNCATION ..................................... 28

3.3 OBTAINING AN OBJECT WAVEFRONT AT THE SCREEN PLANE ........................................ 29

3.4 GENERAL PROPERTIES OF THE HOLOGRAPHIC PATTERN ................................................ 31

4. WAVEFRONT RECONSTRUCTION ..................................................................................... 33

4.1 CONVENTIONAL METHODS FOR MICROWAVE WAVEFRONT RECONSTRUCTION ............. 34

4.2 REVISITING THE TRANSMISSION FUNCTION .................................................................... 35

4.3 THE SLOT ANTENNA HOLOGRAM ................................................................................... 38

4.4 THE RESONANT SLOT ANTENNA HOLOGRAM ................................................................ 40

4.4.1 Transmission Behaviour of a Single Half-Wavelength Slot ............................... 41

4.4.2 Wavefront Synthesis by an Array of Slots .......................................................... 42

5. DESIGN AND SIMULATION ................................................................................................ 44

5.1 NON-RESONANT SLOT ANTENNA HOLOGRAM ............................................................... 44

5.1.1 Simulation Setup and Slot Dimensions............................................................... 45

5.1.2 Simulation Results .............................................................................................. 47

5.2 RESONANT SLOT ANTENNA HOLOGRAM ........................................................................ 51

5.2.1 Simulation Setup and Slot Dimensions............................................................... 51

5.2.2 Simulation Results .............................................................................................. 53

6. EXPERIMENTAL DEMONSTRATION .................................................................................. 60

6.1 SCREEN FABRICATION AND EXPERIMENTAL APPARATUS ............................................... 60

6.2 EXPERIMENTAL RESULTS ............................................................................................... 64

viii

7. CONCLUSION .................................................................................................................... 70

7.1 SUMMARY OF CONTRIBUTIONS ...................................................................................... 70

7.2 FUTURE DIRECTIONS ...................................................................................................... 73

APPENDIX A: ANALYTICAL FORWARD-PROPAGATION OF A HOLOGRAPHIC WAVEFRONT ............. 76

APPENDIX B: THE DIFFRACTION-LIMITED SINC DISTRIBUTION ................................................... 79

REFERENCES ........................................................................................................................... 80

ix

List of Tables and Figures

FIG. 2.1: A SCHEMATIC SHOWING THE PROCESS OF RECORDING A HOLOGRAM ......................... 11

FIG. 2.2: A PHOTOGRAPH OF AN OPTICAL HOLOGRAM [30] ...................................................... 11

FIG. 2.3: A SCHEMATIC OF THE WAVEFORM RECONSTRUCTION PROCESS. ................................ 12

FIG. 3.1: A SCHEMATIC OF THE BACK PROPAGATION PROCEDURE, WITH THE SCREEN AND IMAGE

PLANES DENOTED WITH SOLID OUTLINES, AND THE AUXILIARY PLANE DENOTED IN DOTTED

LINES. THE COORDINATES ARE GIVEN IN THE SCHEMATIC; THE ORIGIN IS DENOTED AS THE +

SIGN ON THE AUXILIARY PLANE. THE DOT ON THE IMAGE PLANE DENOTES THE POINT-

SOURCE, THE CONCENTRIC CIRCLES DEPICT THE CORRESPONDING WAVEFORM ON THE

AUXILIARY PLANE. THE BENT ARROW INDICATES FIELD PROPAGATION IN THE BACK-

PROPAGATION PROCEDURE: FROM THE IMAGE PLANE TO THE AUXILIARY PLANE TO THE

SCREEN PLANE. .................................................................................................................. 25

FIG. 3.2: THE BACK-PROPAGATED SPECTRUM OF THE OBJECT WAVEFRONT. WHEN THE OBJECT

WAVEFRONT IS BACK PROPAGATED FROM THE IMAGE PLANE (SOLID) TO A MID-POINT (DOT)

TO THE SCREEN PLANE (DASH), THE EVANESCENT COMPONENTS ( ) UNDERGO

EXPONENTIAL GROWTH, WHILE THE PROPAGATION COMPONENTS ( ) RETAIN THEIR

MAGNITUDE. THE PARAMETERS USED ARE: . ................ 27

FIG. 3.3: THE WAVEFRONT AT THE SCREEN PLANE FOR A POINT-SOURCE BACK-PROPAGATED

WITH THE PARAMETERS . .............................................. 27

FIG. 3.4: AN ANALYTICAL APPROXIMATION FOR THE BACK-PROPAGATED WAVEFRONT .......... 31

FIG. 3.5: A COMPARISON OF THE FOCUSING QUALITY AS OBTAINED BY NUMERICAL

CALCULATION (THICK) VS. ANALYTICAL DERIVATION (THIN), SHOWING THAT THE TWO

METHODS LEAD TO SIMILAR FOCUSING CAPABILITIES, ALTHOUGH THE LATTER PRODUCES

SLIGHTLY LARGER SIDE LOBES. .......................................................................................... 31

FIG. 4.1: A) THE MICROWAVE HOLOGRAM PROPOSED BY CHECCACCI IN [36]. B) OUR PROPOSED

SLOT ANTENNA HOLOGRAM AS AN EXTENSION TO CHECCACCI’S HOLOGRAM. ................... 39

x

FIG. 5.1: A SCHEMATIC OF THE SIMULATION SETUP FOR THE (NON-RESONANT) SLOT ANTENNA

HOLOGRAM, WITH A FRONT VIEW OF THE HOLOGRAM SHOWN ON THE RIGHT. THE

LOCATIONS OF THE HOLOGRAM, THE IMAGE AND AUXILIARY PLANES, THE COORDINATE

DEFINITION AND THE INCIDENT FIELD ARE AS DEFINED IN THE SCHEMATIC. THE WHITE AREA

IN THE SCHEMATIC DEFINES THE FREE-SPACE, TOTAL-FIELD REGION OF THE

COMPUTATIONAL DOMAIN, WHILE THE SHADED SURROUNDING REGION REPRESENTS

CARTESIAN PMLS. THE SLOT WIDTHS ARE AS GIVEN IN TABLE. 5.2 .................................. 46

TABLE 5.2: DIMENSIONS FOR SLOTS IN THE (NON-RESONANT) SLOT ANTENNA HOLOGRAM. .... 46

FIG. 5.3A: A PLOT OF , NORMALIZED AT EVERY Z-COORDINATE. HERE THE

SCREEN PLANE IS LOCATED AT , AND THE IMAGE PLANE IS SHOWN

AS THE DOTTED LINE. TO CLARIFY THE FOCUSING QUALITY THE ELECTRIC FIELD FWHM IS

OUTLINED IN BLACK. .......................................................................................................... 48

FIG. 5.3B: THE FIELD AMPLITUDE ALONG THE LINE . ............................................ 48

FIG. 5.4: A COMPARISON BETWEEN THE FOCUSING QUALITIES OF THE (NON-RESONANT) SLOT

ANTENNA HOLOGRAM (SOLID) AND A METALLIC SCREEN WHOSE ONLY APERTURE IS THE

CENTRAL SLOT IN THE HOLOGRAM (DOTTED), MADE AT THE IMAGE PLANE ( ).

SHOWN IN THE INSET IS THE TRANSMISSION AMPLITUDE OF THE SCREEN, WHICH IS

DOMINATED BY REAL COMPONENTS. .................................................................................. 49

FIG. 5.5: THE SIMULATION SETUP FOR THE RESONANT SLOT ANTENNA HOLOGRAM. THE SPATIAL

COORDINATES AND ORIENTATION OF THE INCIDENT WAVE ARE AS SHOWN IN THE DIAGRAM.

THE METALLIC (PEC) SCREEN IS (2Λ × 2Λ); THE SLOT DIMENSIONS ARE

GIVEN IN TABLE 5.6. THE OUTER BOX REPRESENTS SIMULATED FREE SPACE REGION; THE

PMLS WHICH SURROUND THIS FREE SPACE REGION HAVE BEEN REMOVED TO SIMPLIFY THE

DIAGRAM. .......................................................................................................................... 52

TABLE 5.6: DIMENSIONS FOR SLOTS IN THE (NON-RESONANT) SLOT ANTENNA HOLOGRAM. .... 52

FIG. 5.7: A) A SIMULATED PLOT OF FOR THE RSAH, NORMALIZED AT EVERY Z-

COORDINATE. HERE THE SCREEN PLANE IS LOCATED AT , AND THE IMAGE PLANE IS AT

IS SHOWN IN DOTTED LINE. TO CLARIFY THE FOCUSING QUALITY THE

ELECTRIC FIELD FWHM IS OUTLINED IN BLACK. B) THE FIELD AMPLITUDE OF THE RSAH

ALONG THE LINE . C) AND D) PLOT THE CORRESPONDING PROPERTIES OF A 1-

APERTURE SCREEN, CONTAINING ONLY THE CENTRAL SLOT OF THE RSAH. ....................... 53

FIG. 5.8: A COMPARISON OF THE SIMULATED FOCUSING QUALITY OF THE RSAH AND THE 1-

SLOT SCREEN, AT A LINE ACROSS THE IMAGE PLANE ( ). THE FWHM OF THE

RSAH (SOLID) MEASURES 3.9MM (0.13Λ), WHICH IS GREATLY IMPROVED FROM THE

FWHM OF THE 1-SLOT SCREEN, AT 6.29MM (0.21Λ). ........................................................ 55

xi

FIG. 5.9: A CROSS-SECTION PLOT SHOWING THE (A) AMPLITUDE AND (B) PHASE OF THE X-

DIRECTED ELECTRIC FIELD EMERGING FROM THE RSAH, COMPARED ALONGSIDE

CORRESPONDING QUANTITIES FOR THE 1-SLOT SCREEN (C AND D). .................................... 56

FIG. 5.10: A SEMI-LOG PLOT OF FIELD LEVELS ALONG THE LINE , FOR THE RSAH

(SOLID), THE RESONANT SLOT (FROM THE 1-SLOT SCREEN) (DOTTED), AND A NON-

RESONANT SLOT (DASH, 100× MAGNIFIED). ....................................................................... 57

FIG. 6.1: A PHOTOGRAPH SHOWING THE FABRICATED HOLOGRAPHIC SCREEN. THE INSET SHOWS

A CLOSE-UP OF THE FABRICATED 3-SLOT HOLOGRAPHIC PATTERN. THE SLOT SPACINGS AND

DIMENSIONS ARE GIVEN IN TABLE 5.6. ............................................................................... 61

FIG. 6.2: A SCHEMATIC OF THE EXPERIMENTAL APPARATUS. ................................................... 62

FIG. 6.3: THE PHOTOGRAPH ON THE LEFT SHOWS THE EXPERIMENTAL APPARATUS, SHOWING

THE ANTENNA HORN, THE DIELECTRIC LENS, THE METALLIC SCREEN AND THE SCANNING

PROBE. THE UPPER-RIGHT SHOWS A CLOSE-UP OF THE SCAN PROBE, PLACED DIRECTLY

ABOVE THE HOLOGRAPHIC PATTERN. ................................................................................. 62

FIG. 6.4: A CROSS-SECTION PLOT SHOWING THE (A) MEASURED AND (B) SIMULATED |EX|

EMERGING FROM THE RSAH, COMPARED ALONGSIDE CORRESPONDING QUANTITIES FOR

THE 1-SLOT SCREEN (C AND D). SPURIOUS COUPLING HAS LED TO THE MEASUREMENT OF AN

APPRECIABLE FIELD STRENGTH IN THE AREA ENCIRCLED BY THE ELLIPSE IN (A). THE FIELD

DISTRIBUTION ACROSS THE DOTTED LINE IS DISPLAYED IN FIG. 6.5. ................................... 65

FIG. 6.5: A EXPERIMENTALLY FOCUSING QUALITY OF THE RSAH (MEASURED AND SIMULATED),

THE 1-SLOT SCREEN, AND A DIFFRACTION-LIMITED DEVICE, AT A LINE ACROSS THE IMAGE

PLANE ( ). THE FWHMS OF THE CURVES ARE: ........................................ 66

FIG. 6.6: AN UNNORMALIZED PLOT OF |EX| ALONG , COMPARING

MEASUREMENT (SOLID) WITH SIMULATIONS CONDUCTED USING A PEC SCREEN (DOTTED)

AND A STAINLESS STEEL SCREEN (DASH). THE INCIDENT FIELD LEVEL IS ALSO DISPLAYED

WITH A THIN DASHED LINE. THE PEAK AMPLITUDES ARE: SIMULATION (PEC): 1.52V/M;

SIMULATION (STAINLESS STEEL): 1.43V/M; MEASUREMENT: 1.31V/M. .............................. 66

FIG. 6.7: S21 PLOTS OF THE MEASURED ELECTRIC FIELD AT 3 LOCATIONS: ,

, AND (ALL LENGTHS IN MM). ......................... 68

FIG. 6.8: A COMPARISON OF THE FOCUSING QUALITY FOR FIVE FREQUENCY POINTS WITHIN A

RANGE OF ±1% FROM 10GHZ, COMPARED ALONGSIDE THE 1-SLOT PATTERN. ................... 68

1

Chapter 1

Introduction

1.1 Motivation for Subwavelength Focusing

Subwavelength focusing (or subdiffraction focusing) refers to the tight spatial confinement of

electromagnetic radiation to dimensions smaller than those allowed by the limit of far-field

diffraction. While focusing within the diffraction limit has been sufficient for many

applications in conventional optics and electromagnetics, rapid technological developments

within the last few decades has led to increasing needs for subwavelength focused

electromagnetic wave distributions. Strongly focused electromagnetic waves, at frequencies

ranging from microwave to terahertz to optical regimes, have been extensively used in

various schemes of high resolution imaging and near-field sensing [1-3]. Developments in

subwavelength focusing techniques would certainly be helpful towards improving our

imaging and sensing capabilities in these applications. The booming biomedical community

has also proposed various interesting uses of strongly confined electromagnetic radiation;

here subwavelength focusing offers great help towards understanding biological processes,

diagnosing diseases, and even treating these diseases [3-5]. Finally subwavelength focused

optical radiation can potentially bring about dramatic improvements to lithographical

CHAPTER 1: INTRODUCTION 2

precision, which would greatly enhance our existing nano-fabrication capabilities [6, 7].

These and various potential applications of subwavelength focused electromagnetic radiation

have attracted strong recent interest towards this field of research.

1.2 Chapter Organization

In the remainder of the chapter, we will briefly review the origin of the diffraction limit, as

well as a few classes of existing devices which circumvent this limit. We will then introduce

the concept of the hologram and propose our subwavelength focusing device of investigation.

Finally we end this chapter by providing an overview of the remainder of this work.

1.3 Background: Overcoming the Diffraction Limit

In a nutshell, the diffraction limit arises from the inability to collect and refocus the

evanescent components of the object waveform. Consider the waveform from an arbitrary

monochromatic source as it diffracts over a long distance of homogeneous isotropic space.

One can decompose this arbitrary waveform into a basis formed by two types of plane

waves: (1) propagating waves which have transverse spatial periods longer than the

illumination wavelength in the medium of interest; and (2) evanescent waves which have

transverse spatial periods smaller than the illumination wavelength. In this decomposition the

propagating waves contain the low spatial frequency information of the source distribution


while the evanescent waves contain the high spatial frequency information. Both propagating

and evanescent waves are needed to form a sharp focus. However, if one solves Maxwell’s

equations for the situation at hand, one would find that propagating waves support an

imaginary propagation constant while evanescent waves support a real one. The implication

is that while propagating waves propagate (in a lossless medium) without attenuation,

evanescent waves decay exponentially as they travel away from their source. Practically, at

distances on the order of a wavelength from the source, most of the evanescent waves have

decayed to signal levels below the noise floor. The inability to detect these waves constitutes

the loss of the high resolution information of the source distribution. This loss of high

resolution information forms the diffraction limit.

Since the diffraction limit originates from the loss of high resolution information associated

with evanescent waves, to overcome the limit one must find some ways to maintain or

reintroduce evanescent waves. Unfortunately, Maxwell’s equations and the radiation

boundary condition dictate that evanescent waves must exponentially decay away from the

source (either a primary source or a part of the focusing device, which can be viewed as a

secondary source). Thus the formation of a subwavelength focus must necessarily be located

within the near-field of either the original source or the output interface of the focusing

device. Notwithstanding this limitation, near-field subwavelength focusing devices still have

great potential in diverse fields of applications. In the following we briefly survey a few

types of subwavelength focusing structures of ongoing research interest and highlight their

strengths and weaknesses.


1.3.1 The Near-field Vicinity of a Source

The most traditional subwavelength focusing device is a small source. Naturally, in the

vicinity of a highly-subwavelength electromagnetic source, one will find a subwavelength

focus of electromagnetic waves. Some examples of subwavelength electromagnetic sources

in the microwave regime include the electrically small dipole, the loop antenna and the small

aperture [8]; in the optical regime, electrically small sources can be achieved by making a

subwavelength hole in an opaque screen, or using a tapered optical fiber tip or a metallic

nanowire [9-12]. Following Synge’s proposal in 1928, these subwavelength sources are

scanned across an image area to provide subwavelength illumination to objects under

observation [13]. They have also been used in reciprocal procedures, whereby they collect

the scattered fields emitted from objects at highly selective spatial locations. These

techniques have been used in near-field scanning applications in microwave, and are key

concepts to scanning near-field optical microscopy (SNOM).

While subwavelength focusing is practically achievable with a near-field source, it has a

drawback of necessitating the physical presence of a source device at the location of the

focus. There often arise situations where it is impossible or undesirable to physically place a

source at a location where the focusing is needed. Also, from practical considerations, it is

often more convenient for the focusing device to remain a good working distance away from

the electromagnetic focus, so that it does not introduce any geometrical hindrance or

unwanted side effects. These reasons render the near-field source undesirable for many

applications where a subwavelength electromagnetic focus is needed.


1.3.2 Negative Refractive Index (NRI) Metamaterial Superlens

A subwavelength focusing device that attracted much theoretical and practical interest is the

negative-refractive-index (NRI) metamaterial superlens. In a nutshell, the NRI metamaterial

superlens is a planar slab of a medium with a negative refractive index which is equal in

magnitude, but opposite in sign, to the index of the surrounding material. While its intriguing

ability to focus propagating waves was proposed by Veselago four decades ago [14], its even

more surprising ability to also focus evanescent waves was only recently discovered by

Pendry, in 2000 [15]. The superlens focuses evanescent waves by compensating their decay

in free space with an exponential growth inside the lens. This growth of evanescent waves is

especially appealing, since it theoretically allows image formation with subwavelength

resolution at an arbitrary distance away from the superlens. Thus ever since its

conceptualization, much research effort has been expended towards superlens design,

simulation and fabrication. As of the time of this work, various working prototypes have

been demonstrated for the Veselago-Pendry superlens as well as a few superlens-inspired

subwavelength focusing devices [6, 16-21].

While the NRI metamaterials superlens is indeed a conceptual breakthrough in

subwavelength focusing, it does suffer from a few practical drawbacks. In particular,

causality dictates that the existence of a negative index is necessarily accompanied by

resonant losses, which in turn places rather stringent practical limits on the superlens’

imaging distance and resolution [22]. Typical experimental results from subwavelength

focusing devices have imaging distances within one wavelength from the output facet of the

superlens, and resolutions within a few times that of the diffraction limit [16-20]. Active


metamaterials may present a way to mitigate this problem of loss, but working prototypes of

improved focusing performances have yet to demonstrate the practicality of this method.

1.3.3 Radiationless Interference Screen

Very recently, another interesting alternative to ultra-tight focusing has been proposed by

Merlin [23] and demonstrated in a 2D microwave environment by co-researcher and

University of Toronto alumnus A. Grbic [24]. Their scheme uses a frequency selective

surface (FSS) to generate a prescribed distribution of current densities on a screen, which in

turn forms a sharp focus at the image plane. The radiationless interference screen differs

functionally from a metamaterial superlens in that while the superlens can form a one-to-one

image of an entire 2D plane, the radiationless interference screen is only designed to form a

sharp focal spot. In this perspective, this screen functions like a near-field scanning source,

except the near-field is effectively extended by interference effects. Despite its reduced

functionality as compared to the superlens, the radiationless interference screen compares

favourably to the superlens in one respect – that its focusing capability is unhindered by

material loss. This independence from loss allows the radiationless interference screen to

operate far beyond the far-field diffraction limit; a focus of λ/18 has been demonstrated in

Grbic’s experiment.

However, there are also some practical drawbacks to the radiationless interference screens.

Firstly, since the subwavelength focus is formed from an interference involving only


evanescent waves (hence the name “radiationless interference screen”), the focus must be

formed at a close proximity to the screen to avoid substantial decays of the highly evanescent

waves which form the sharp focus. In Grbic’s experiment the imaging distance was only

λ/15. Secondly, to construct the frequency selective surface, Grbic used a screen of mutually

coupling capacitor elements, with feature sizes on the order of λ/1000 (as deduced from a

figure in [24]). Thus while the focusing capability of the screen has been demonstrated at

microwave frequencies, it would be very difficult to fabricate such a screen at terahertz and

optical frequencies. These two serious drawbacks need to be mitigated before the

radiationless interference screen can become a practical device for subwavelength focusing.

1.4 Proposal

It would be ideal to have a near-field focusing device which (1) performs ultra-tight near-

field focusing comparable to or surpassing subwavelength sources of similar dimensions; but

(2) supports much extended working distances of λ/10 and beyond; and (3) can be extended

across a wide range of frequencies, from microwave to optical. This work proposes to

achieve all these objectives through the holographic transmission screen. A hologram is a

complete record of an electromagnetic wavefront, from which the original wavefront can be

retrieved upon the incidence of a reference beam. As conventional holography involves far-

field wavefronts within the diffraction limit, we will begin by extending holographic

techniques to near-field wavefronts beyond the diffraction limit. Building on this


investigation, we will then design and fabricate a metallic transmission screen synthesizing

the desired hologram, and examine its focusing capabilities.

1.5 Thesis Outline

The remainder of this work is organized as follows. Chapter 2 introduces the concept of

holography in more detail, and discusses a generalized procedure in recording a hologram.

Chapter 3 follows this procedure to record a hologram of a point-source. Subsequently,

Chapter 4 discusses different alternatives to holographic wavefront reconstruction, with a

specific focus on reconstructing a near-field subwavelength source. Chapter 5 draws on the

generalized holographic techniques from Chapters 3 and 4 to design two proof-of-principle

holographic transmission screens capable of subwavelength focusing, and presents

corresponding simulation results. The screen with superior performance – namely the

resonant slot antenna hologram – is then fabricated and tested; Chapter 6 presents

experimental results demonstrating subwavelength focusing, and discusses the salient

features of this holographic transmission screen as a subwavelength focusing device. Finally,

Chapter 7 concludes this work, and provides some thoughts on improvements and future

directions.

9

Chapter 2

The Hologram and its Extension

In this chapter, we derive the formulation of a near-field hologram. We begin with a concise

introduction to holography, emphasizing key concepts and general procedures. We then

focus on the first step in holographic image reproduction – the recording of a hologram – and

examine the resolution limit encountered by traditional methods for recording a hologram.

Subsequently, we introduce an alternative analytical method to accurately record a near-field,

subwavelength hologram.

2.1 Introduction to Conventional Holograms

A hologram is a complete record of an electromagnetic wavefront, comprising information

for the wavefront magnitude as well as its phase. It was invented in 1948 by Dennis Gabor

[25], who proposed that one can record the wavefront of an object by interfering it with a

reference wavefront. Gabor further demonstrated that illuminating this hologram in a

prescribed manner allows one to completely regenerate the object wave. Even though

Gabor’s initial interest was towards electron microscopy, his concept was generally

applicable to electromagnetic waves of all frequencies. Following Gabor’s work, gradual but

continued improvements to holographic techniques, as well as advancements in related

CHAPTER 2: THE HOLOGRAM AND ITS EXTENSION 10

scientific fields, have dramatically improved the quality and versatility of holographic

wavefront reconstruction [26, 27].

A general holographic process involves two major stages. First an interference pattern is

formed and the hologram is recorded, then the original wavefront is reconstructed. In the

following we provide a brief overview to these two major stages using a scalar mathematical

formulation. While a vectorial formulation will be needed for a general 3D holographic

process, the much simpler scalar formulation provides clear illustration to the salient features

of the holographic process. In addition, this scalar formulation can be directly and rigorously

applied to a 2D environment, which we shall consider in Chapter 3.

2.1.1 Recording a hologram

Fig. 2.1 depicts the process of recording a hologram, using means that are relevant to the

microwave regime. An incident wave is provided by the microwave source of illumination,

which in this case is a horn antenna. A part of the incident wave travels directly to the screen

plane. This is the reference wavefront, and will be denoted by Aref. Another part of the

incident wave gets scattered by an object and ultimately also reaches the screen plane. This is

the object wavefront, and will be denoted by Aobj. Hence along the screen plane, we have an

interference of the object and reference waves. The intensity profile of this interference is

given by

. (2.1)


This intensity profile will be recorded as the hologram. For microwave frequencies, the

recording is done either by noting the field intensity as a function of position with a scanning

probe [28], or by timed exposure to a photographic film, as suggested by [29].

Fig. 2.1: A schematic showing the process of recording a hologram

Fig. 2.2: A photograph of an optical hologram [30]


Fig. 2.3: A schematic of the waveform reconstruction process.

2.1.2 Reconstructing the Wavefront

While it is of scientific interest to obtain the interference pattern, we are ultimately interested

in reconstructing the object wave – the wave formed when an incident wave scatters off an

object. Reproducing this wave will give us much information about the object itself; a look at

the optical hologram in Fig. 2.2 will readily convince us that a well reconstructed wave

closely resembles the waveform scattered by the original object. Fig. 2.3 is a diagram of the

hologram reconstruction process. In this process the hologram is present in the functional

form of a transmission screen, containing the interference pattern obtained in the previous

subsection. A typical screen takes on transmission values between 0 and 1, and can be related

to the recorded interference pattern by


, (2.2)

where β serves as an appropriate scaling factor. We will delay to chapter 4 a detailed

discussion on how this transmission screen can be practically synthesized. If, for the moment,

we assume the existence of such a screen, then illuminating the screen with the original

reference wavefront Aref will allow us to obtain a reconstructed wavefront, Arecon, which is

mathematically described as follows:

.

(2.3)

When the radiation source is sufficiently far away from the screen plane, Aref can be

considered as a wavefront of a plane wave. (In a practical setup, a lens can be used if

necessary to satisfy this far-field condition). Making this simplification, we have

reconstructed a scaled factor of the object wavefront in the second term in equation (2.3).

This wavefront propagates as if it emerged from the object at a location to the left of the

screen. On the other hand, if we illuminate the screen with Aref*, we will reconstruct the

following wavefront:

.

(2.4)


Here, we have obtained a scaled factor of the wavefront Aobj* in the third term of (2.4). Aobj*

is the conjugate object wavefront. Unlike the divergent object wavefront, the conjugate

wavefront converges back into the object, forming a real image of the object to the right of

the screen. While at the moment the reconstructed object wave (original or conjugate

depending on the reconstructing wavefront) is buried within the superposition with other

wavefronts, various techniques exist which minimize or isolate the spurious contributions,

allowing one to retrieve the desired object wave pattern.

2.2 A Subwavelength Focusing Hologram?

As mentioned in the previous section, observing a successfully reconstructed object wave is

in many ways the same as observing the actual object under illumination. One might view the

hologram in Fig. 2.2 as an optical illusion, since the wavefront suggests the presence of an

object which is actually not there. In this regard, one might imagine doing more with a

hologram than just reproducing existing objects. If the interference record of a certain

wavefront can somehow be produced, then by the same reconstruction process described

above, one can construct both the magnitude and the phase of the desired wavefront upon the

incidence of the pre-decided reference wave. From this viewpoint, the concept of holography

becomes a powerful tool for synthesizing arbitrary electromagnetic waveforms.

In this context, we begin with an attempt to answer the following question: Is it possible to

create a hologram of a near-field point-source, and retrieve from it a subwavelength focused

waveform, which closely mimics the presence of a point-source? First we consider the


problem of obtaining a holographic record of such a near-field, subwavelength waveform.

We then devise ways of synthesizing the required transmission function which allows us to

retrieve the reconstructed waveform in the near-field.

2.3 The Resolution Limit for Conventional Holograms

In the previous section, we have proposed recording, then reconstructing, the waveform for a

near-field point-source. In this section we begin a detailed investigation in recording such a

waveform. In such consideration, we first revisit the general process for recording holograms

(which we have described in section 2.1.1), and point out the relevant inherent resolution

limitations.

We have seen from equations (2.1) – (2.3) that the wavefront Aobj (or Aobj*) can be

reproduced given that (a) a perfect interference record is captured, and (b) the ideal

transmission function can be synthesized. However, up until this point we have implicitly

equated the successful reconstruction of the object wavefront to the ultimate reconstruction

of the entire object waveform.1 Must we necessarily retrieve the entire object waveform

when we reproduce the wavefront Aobj (or Aobj*) along the screen plane?

1 In this discussion and hereafter, we use the word “waveform” to describe the 2D/3D distribution of

electromagnetic field in the environment of consideration, and the word “wavefront” to describe the

1D/2D field distribution along a transverse line/plane, at a specific longitudinal (z) location.


We facilitate this investigation via the method of plane wave decomposition. While we

establish the mathematical framework for the object wave Aobj, the same formulation applies

for the conjugate wave Aobj*. We decompose the wavefront Aobj into 2D plane waves by

performing a 2D Fourier transform at the screen plane, after the convention

;

. (2.5)

The spectrum describes the set of 2D plane waves which form the distribution Aobj in

the screen plane.

The half space in the +z direction of the screen plane is a homogeneous, isotropic space,

which can be decomposed as a linear composition of 3D plane waves, with corresponding 3D

wavevectors , where is the spatial frequency of the

illumination source. We can readily express this 3D waveform in relation to . Using the

dispersion relationship for a homogeneous, isotropic space,

, (2.6)

we can determine the z-directed spatial frequency of each 2D plane wave component in ,

and thus end up with a collection of 3D plane waves which make up the waveform on the

halfspace. However, the dispersion relation (2.6) only uniquely determines ; it doesn’t

specify the polarity of kz. In the following, we will examine how the polarity of kz can be


determined for propagating and evanescent waves, and its implication on the resolution of the

reconstructed waveform.

2.3.1 Polarity of kz for Propagating Waves

For transverse spatial frequencies smaller than or equal to the overall spatial frequency

, the dispersion relation (2.6) requires , and thus kz takes on a real

value. Hence the resultant plane wave will be a propagating wave in the z-direction.

However, depending on the polarity of kz, the wave may propagate either in the +z direction

or the -z direction. The polarity of kz plays a crucial role in determining the wave that is

reconstructed, since it controls the phase progression and hence the interference

characteristics amongst propagating plane waves. For example, in Fig. 2.3, we have depicted

a wave traveling in the +z direction, and coming to focus at the original object location to the

right of the screen. Conversely, if the reconstruction wave were to travel in the –z direction,

it would form a diverging object wave, and one observing the object wave from the left side

of the screen would see the wavefront as if there is an object at its original location behind

the screen.

Although in the above discussion, we have concentrated on the wavefront Aobj, obviously the

same is also true for the wavefront Aref. That is to say, in the reconstruction process, one can

illuminate the transmission screen with a wavefront Aref in two ways: either with a wave

propagating it the +z direction, or with a wave propagating in the –z direction. Furthermore,

the polarity of kz in this reconstruction wave directly influences the corresponding quantity

for the reconstructed object wave.


In the reconstruction process as described in section 2.1.2, the polarity of kz of the object

wave will follow that of the reconstruction wave, since the hologram is synthesized as a

transmission screen. In the geometry as depicted in Fig. 2.1 and Fig. 2.3, the object wave

travels to in –z direction during the recording phase, and +z direction during the

reconstruction phase, hence the original waveform is recovered, as far as propagating waves

are concerned. In general, the polarity of kz of the reconstructed object wave can be

controlled by geometrical parameters, and by illuminating the hologram with an

appropriately chosen reference wave. Hence, given the ability to reproduce the wavefront

Aobj at the screen plane, perfect electromagnetic wave reconstruction can be achieved for

waves propagating in the direction normal to the screen.

2.3.2 Polarity of kz for Evanescent Waves

We shall now direct our attention to evanescent waves. For waves with transverse spatial

frequencies larger than the overall spatial frequency , the dispersion relation

(2.6) requires . Hence kz becomes imaginary, resulting in a wave that is evanescent in

the z-direction. Once again we are met with the question as to the polarity of kz – whether kz

assumes a positive imaginary value or a negative one. For evanescent waves, the polarity of

kz determines the direction of decay for the evanescent plane wave, and thus controls the

relative amplitudes amongst evanescent plane waves, and hence their interference

characteristics.


Unfortunately, unlike the case for propagating waves, the direction of decay for evanescent

plane waves cannot be easily controlled. The radiation boundary condition requires that

electromagnetic fields decay away from their sources. Thus in the hologram recording stage,

the evanescent component of the object wave decays from the object to the screen plane,

while in the wavefront reconstruction stage, the evanescent component of the reconstructed

object wavefront Aobj decays again from the screen plane back to the original location of the

object. This “double-path” decay seriously limits the resolution of the reconstructed

waveform, since the high spatial frequency components of the object wave are essentially

lost to evanescent decay for object-screen distances greater than a fraction of the wavelength.

Thus we see that one cannot obtain a sub-diffraction focus through conventional holography.

2.4 Recording a Generalized Hologram via Wavefront Back-

Propagation

In light of the resolution limit to the conventional holographic process as described in the

previous section, we would like to develop a general process for recording a hologram, which

would allow one to reconstruct the original 3D object waveform with subwavelength

precision. Specifically, this hologram would mitigate the problem associated with the

evanescent decay of high spatial frequency (and hence high resolution) plane waves.

In section 2.1.1, we described the process of recording a conventional hologram by

physically interfering an object wavefront with a reference wavefront. While that remains a

popular and practical approach to recording a hologram, one can also record a hologram


following an alternative approach – the computer generated hologram (CGH). The CGH has

found various niche applications since its introduction in the 1960’s [31, 32]; it is especially

useful for reconstructing waveforms which for various reasons cannot be easily recorded

with the conventional method. With today’s high-performance computers, one can

numerically calculate the interference pattern between an object wave and a reference wave

at a desired screen location, thus digitally forming a hologram. Once the hologram pattern is

calculated, the corresponding spatial filter can be physically synthesized in a manner

appropriate to the wavelength of illumination and the geometry involved.

In the following, we derive an analytical method for recording a hologram, which can be

viewed as an extension to the CGH. In the spirit of the CGH, we calculate the interference

pattern between the object and reference waves. However, in departure from the CGH, we

analytically grow the evanescent components of the object wave in a manner which perfectly

compensates for their decay in the reconstruction process. In this way, upon hologram

synthesis and wavefront reconstruction, we will recover both propagating and evanescent

components of the object wave, and achieve sub-diffraction resolution at the location of the

object.

To begin our derivation, we first relate Aobj to a corresponding waveform Fobj, where at the

screen plane we have . Furthermore, we define a quantity to be the

2D Fourier transform of Fobj at a specific z-plane:

, (2.7)


such that at the screen plane we have . Applying Helmholtz’s

equation in the spatial domain, we have

(2.8)

Performing plane-wave decomposition via a 2D inverse Fourier transformation, we have

(2.9)

(2.9) is the 2D inverse Fourier transform of the term . Since a zero function

Fourier transforms to itself, we conclude that


. (2.10)

Thus for any arbitrary pair of , the plane wave component described by Fobj obeys

the 1D Helmholtz equation for a homogeneous isotropic space in the z-direction. The

solutions to this well known wave equation can be written as

, (2.11)

where C- and C+ are functions of kx and ky, but constant with respect to z and kz. Since we

deal with a traveling wave at the reconstruction stage, we simplify the notation by expressing

as only one term,

, (2.12)

while reserving full generality with the sign flexibility on kz.

We shall now choose the appropriate sign on kz. For real values of kz, choosing a positive

sign leads to a forward propagating plane wave, whose phase increases in the –z direction

(i.e. when the hologram is being recorded) and decreases in the +z direction (i.e. when the

object waveform is being reconstructed). For imaginary values of kz, choosing kz as a

negative imaginary number will lead to an amplitude growth in the –z direction and an

amplitude decay in the +z direction. In summary, for the purpose of recording the hologram,

we define kz as follows,

. (2.13)


With the mathematical framework in place, we are now ready to calculate the interference

pattern of a generalized hologram. We begin with the original wavefront of an object, located

on an xy-plane at location . Direct substitution into (2.12) gives

. (2.14)

Thus, at the screen plane where the hologram is recorded, the analytically calculated

wavefront is given by,

. (2.15)

In the above we have analytically back-propagated an object wave and obtained a

corresponding wavefront which we need to reconstruct at the screen plane. The interference

between this wavefront and the reference wavefront can now be calculated using equation

(2.1). Through this process, we have analytically obtained a generalized hologram of an

object wave – a hologram which contains propagating and evanescent components of the

object wave, in artificially designed proportions and phase offsets which compensate effects

inherent to a wavefront reconstruction process. Such a generalized hologram will allow the

near-perfect reconstruction of an object waveform, including that of a sub-diffraction

electromagnetic source.

24

Chapter 3

The Generalized Hologram of a Subwavelength Source

In the previous chapter, we have already seen how to record a generalized hologram, the

reconstruction of which allows one to synthesize a desired waveform with sub-diffraction

resolution. In this chapter, we would like to apply this back-propagation procedure to record

a generalized hologram of a subwavelength source. Our purpose for this derivation is two-

fold. Firstly, we aim to circumvent problems with singularities and spectral divergences, and

in doing so, obtain a practical wavefront at the screen plane, the reconstruction of which

forms our desired subwavelength focus at the image plane. Secondly, through a practical

example, we aim to gain insight on salient properties of the generalized holographic pattern,

which, besides being of theoretical interest, would also help us in devising methods of

hologram synthesis.

Fig. 3.1 shows a diagram of the back-propagation procedure and defines the coordinate

system. For the sake of consistent nomenclature, in the following derivation the script will

be used to denote a 2D waveform, while E will represent a 1D waveform along the x-

direction; as before, a tilde represents the corresponding spectrum, obtained by a 1D Fourier

transform in the x-direction. As previously noted, we will work with a 2D back-propagation

formulation, in which the focusing is 1D; nonetheless the same holographic principle can be

extended to obtain 2D focusing in a 3D environment.

CHAPTER 3: THE GENERALIZED HOLOGRAM OF A SUBWAVELENGTH SOURCE 25

Fig. 3.1: A schematic of the back propagation procedure, with the screen and image planes

denoted with solid outlines, and the auxiliary plane denoted in dotted lines. The coordinates

are given in the schematic; the origin is denoted as the + sign on the auxiliary plane. The dot

on the image plane denotes the point-source, the concentric circles depict the corresponding

waveform on the auxiliary plane. The bent arrow indicates field propagation in the back-

propagation procedure: From the image plane to the auxiliary plane to the screen plane.

3.1 Spectrum Back-Propagation

Our choice for the subwavelength source is the point-source function

. (3.1)

On the image plane, this source has the distribution

, (3.2)


which is strongly focused around the singularity . In practice, any function with a sharp

focus can be chosen as the subwavelength source. In the case where the chosen function is

singular along the image plane, the (source) singularity can be circumvented by

reconstructing the waveform at an auxiliary plane, at a small distance s away from the image

plane. For our subwavelength source the field distribution and the spatial spectrum at the

auxiliary plane are, respectively, [33]

, and (3.3)

, (3.4)

where is the zeroth-order modified Bessel function of the second kind, and kz is defined

as , which is a 2D specialization to (2.13). From here, applying (2.14)

readily gives the back-propagated spectrum of the object (3.5), which evaluates to (3.6) at the

screen plane:

, (3.5)

. (3.6)

Fig. 3.2 shows the spectral change as the object wave is back-propagated from the auxiliary

plane towards the screen plane. Clearly, the evanescent field components have been

substantially grown, allowing the retrieval of full subwavelength information upon

reconstruction. However it is also clear, both from Fig. 3.2 and from equation (3.6), that


Fig. 3.2: The back-propagated spectrum of the object wavefront. When the object wavefront

is back propagated from the image plane (solid) to a mid-point (dot) to the screen plane

(dash), the evanescent components ( ) undergo exponential growth, while the

propagation components ( ) retain their magnitude. The parameters used are:

.

Fig. 3.3: The wavefront at the screen plane for a point-source back-propagated with the

parameters .


diverges as . Since a diverging spectrum cannot be Fourier transformed, we

truncated to a maximum spatial frequency km, such that

. (3.7)

The truncated spectrum can now be inverse Fourier transformed to determine the

wavefront which we seek to reconstruct at the screen plane. Fig. 3.3 shows the

resultant wavefront at the screen plane upon choosing the set of parameters

.

3.2 Resolution Considerations from Spectral Truncation

Before we continue with the derivation we shall briefly examine how the resolution is

affected by a truncation of the field spectrum. Clearly, when we truncate the spectrum to km

in the previous section, we degrade the achievable spatial resolution since we are effectually

discarding high spatial frequency field components (with ). Nonetheless, despite

this loss of resolution we may still perform sub-diffraction focusing, since our truncated

spectrum includes significant evanescent components. Specifically, the width of the truncated

spatial spectrum is widened by a factor of

(3.8)


compared to the diffraction-limited spectrum. Therefore it would still achieve an R-fold

resolution improvement from the diffraction limit. In principle, one can choose km as large as

possible, as long as the resultant distribution can be practically synthesized. Hence we see

that when km is appropriately chosen, a spectral truncation does not prevent one from

obtaining focusing properties far beyond the diffraction limit. Instead the choice of km can

serve as a tool through which one specifies the resolution one desires to achieve, irrespective

of the image distance. Thus in chapters 5, we will use this resolution improvement factor to

assess the performance of holographic screens, by comparing simulated focal widths with the

theoretically achievable focal widths for the chosen spectral truncation. In addition, since we

are ultimately performing subwavelength focusing in the near-field, we will also compare our

focusing results (both in terms of focal amplitude and focal width) with near-field

electromagnetic radiations from subwavelength apertures of similar dimensions.

3.3 Obtaining an Object Wavefront at the Screen Plane

We shall now continue towards obtaining the object wavefront at the screen plane. As

indicated towards the end of section 3.1, we can do so numerically by performing an inverse

Fourier transform to in (3.7). However, in the following we also provide and

validate an analytical derivation for , which would help us anticipate the distribution

pattern of a general hologram with sub-diffraction resolution, and suggest sampling strategies

to hologram synthesis.


We begin with an assumption that . Since a significant portion of the spectrum lies in

a region with large , where , we employ the following approximation for

(3.6):

, (3.9)

where in order to write the second equality sign a large argument approximation was used for

the modified Bessel function [34]. Clearly, for large the exponential term

within (3.9) dominates the spectral variation with respect to the wavelength. Thus we make a

further approximation,

. (3.10)

With now defined by (3.10), (3.7) can be analytically inverse Fourier transformed

into a distribution on the screen plane: [33]

. (3.11)

Here the second term within the square is dropped, since the denominator of both terms

concentrates the distribution to small values of x, where a first order Taylor expansion shows

that the second term is second order small compared to the first. Equation (3.11) is plotted in

Fig. 3.4, for the parameters .


Fig. 3.4: An analytical approximation for the

back-propagated wavefront

Fig. 3.5: A comparison of the focusing

quality as obtained by numerical calculation

(thick) vs. analytical derivation (thin),

showing that the two methods lead to similar

focusing capabilities, although the latter

produces slightly larger side lobes.

While we have made a few assumptions in deriving the object wavefront on the screen plane,

we analytically show in Appendix A that these assumptions do not compromise the focusing

capability of the hologram. Moreover, Fig. 3.5 compares the focal spots resulting from

reconstructing the screen plane wavefront exactly (i.e. the inverse Fourier transform of (3.7))

versus analytically (from (3.11)), for a hologram with a screen-to-image distance of 0.1λ.

3.4 General Properties of the Holographic Pattern

Equations (3.10) and (3.11) provide valuable insight towards the nature of the holograms

formed from the above procedure. We observe from (3.11) that the screen plane wavefront


contains a rapid sinusoidal oscillation underneath a comparatively slow varying envelope.

Thus its interference with a normally incident plane wave forms a holographic fringe pattern

bearing apparent resemblance to conventional holograms. These holographic fringes exist

due to the dominance of high spatial frequency components, especially of those components

close to km, in the spectrum at the screen plane (3.10); this dominance of high spatial

frequency components, in turn, is caused by the exponential growth of evanescent

components inherent to our back propagation procedure. Hence, while in the above we’ve

only dealt the case of a point-source, we deduce that similar fringe patterns appear for all

target wavefronts which contain appreciable subwavelength (i.e. high spatial frequency)

variations.

The similarity between generalized and conventional holograms hints at possibilities of

adapting well established synthesis methods for conventional holograms towards

synthesizing generalized holograms. However, the difference between the two also warrants

attention. In a generalized hologram, evanescent wave components form the fringes,

whereas in a conventional hologram, propagating wave components form the fringes. Hence

the fringe oscillations in a generalized hologram are highly subwavelength in nature, and

much more rapid (in the spatial sense) compared to the fringe oscillations in conventional

holograms. This presents an additional challenge to synthesizing a generalized hologram. In

the following chapter, we will examine practical methods through which one can synthesize

these generalized holograms.

33

Chapter 4

Wavefront Reconstruction

In the previous chapters, we have developed a method to analytically record a generalized

hologram, and have specifically applied it to record the wavefront of a subwavelength

source. In this chapter, we turn our attention towards the subsequent task of synthesizing a

transmission function representing a generalized hologram, keeping in mind that eventually

we would like to synthesize a transmission function to reconstruct a near-field subwavelength

source. We begin this chapter by reviewing common methods for microwave wavefront

reconstruction, and explaining their inadequacies in meeting the challenges involved in

reconstructing a near-field, subwavelength wavefront. We then take a closer look at the

waveform generated at a general reconstruction procedure, and identify possible alternative

transmission functions which adequately reconstruct our waveform of interest. Finally, we

propose two slot antenna holograms – the non-resonant hologram and the resonant hologram

– capable of producing the desired transmission functions.

CHAPTER 4: WAVEFRONT RECONSTRUCTION 34

4.1 Conventional Methods for Microwave Wavefront Reconstruction

Although the recording of microwave holograms have been reported in various imaging

applications [35], there has been considerably less work on physically reconstructing a

microwave wavefront. Instead, holograms which are recorded in microwave (often as part of

an imaging procedure) are often reconstructed digitally on a computer, or optically after a

scale-down procedure [35]. Notwithstanding, early works on microwave reconstruction (or,

in some cases, synthesis) emerged in the late 1960’s, under the title “holographic-antennas”

[28, 36-38]. In Checcacci’s holographic antennas, metallic pieces in forms of strips and rods

were used to “zero out” the electromagnetic field at appropriate locations in order to

represent the ideal transmission function. Iizuka proposed similar ideas for holographic

antennas, and further suggested that the transmission amplitude could be controlled by the

width of metallic slots. Subsequent improvements in holography, in particular in the field of

binary holograms, have consistently improved our ability to generate microwave holograms.

Notwithstanding these establishments in reconstructing microwave holograms, the

subwavelength and near-field features of our target waveform present great challenges

towards holographic reconstruction. The subwavelength variations in the hologram we wish

to synthesize invalidates the simplistic view of the metallic screen as a binary hologram. At

length scales of smaller than or approximately equal to the wavelength, no longer can the

transmittance be considered unity across the entire area of the apertures on the screen.

Instead, at these apertures, one must account for dramatic field oscillations due to diffraction

[39], which greatly complicate screen design, and limit the screen’s capability to synthesize

arbitrary transmission functions. Moreover, the subwavelength and near-field nature of our


target waveform disallows one from isolating the object waveform from other diffraction

orders using conventional spatial separation techniques. Whereas in the far-field, diffraction-

limited regime, one can spatially separate these orders by adjusting the geometry of the

reconstruction wave, a near-field waveform with broad spatial bandwidth cannot be isolated

by traditional approaches. Thus together, the subwavelength and near-field features of our

target waveform hinder us from directly applying conventional methods in its reconstruction.

4.2 Revisiting the Transmission Function

In the previous section, we have established that for the near-field, subwavelength source of

our consideration, the reconstructed object wave cannot be isolated from the other diffractive

orders. Thus in this section, we will examine the effects of all terms in the transmission

function (2.2), particularly when illuminated by a plane wave such as Aref or Aref*.

In chapter 2, we have introduced a transmission function as a scaled factor of the interference

pattern between the object and reference wavefronts, which can be mathematically described

as:

, (4.1)

Furthermore, upon illumination with a plane wave Apln, we obtain


.

(4.2)

We have already discussed that Aobj is the object wavefront while Aobj* is the conjugate

wavefront, and that they would be exactly reproduced when the reconstructing plane wave

Apln is equal to Aref and Aref* respectively. However, in the more general case of wavefront

reconstruction with an arbitrary plane wave, as described in (4.2), we see that both object and

conjugate wavefronts are actually reconstructed, though each is angularly shifted by the

corresponding plane wave products and . In addition to these two

wavefronts, the first term of the transmission function generates a third wavefront

. This term is known as the DC term since it has a constant phase across

the transmission screen; it diffracts a transmitted wave along a principal axis parallel to that

of kpln.

In conventional holograms, the object and conjugate waves, known as the twin image [27],

are often isolated by exploiting this difference in angular shift. Using an obliquely incident

plane wave as the reference and reconstruction beams, one can reconstruct the object and

reference wavefronts with an angular separation of . In the same scheme, the principle

axis of the DC term is also shifted by ; thus the overlapping effect with this constant

background wave is also somewhat reduced. However, as discussed in the previous section,

this angular separation method provides little help in separating near-field waveforms with

broad transverse spatial components. Thus in our case of point-source reconstruction, while


Aobj* would produce the desired sub-diffraction focus at a prescribed image distance in front

of the transmission screen, it would be overlapped with Aobj, which represents the diverging

waveform from the conjugate location behind the screen, as well as a background

contribution from the DC term.

Having concluded that the overlap between reconstructed waveforms in unavoidable, we

simplify the analysis and design process by using a normally incident plane wave as the

reference wavefront and reconstruction illumination. Hence equation (4.2) turns into

.

(4.3)

It would be ideal to synthesize a transmission function containing just the Aobj* term, which

would yield an uncontaminated, sharply focused electromagnetic distribution. However,

while such a transmission can be conceptualized, it is complex-valued in general and is thus

difficult to synthesize. An alternative would be to synthesize a transmission function of the

real part of Aobj. As seen in (4.3), such a transmission would be free from overlap with the

DC term. While it is true that we are still left with the overlap between the object and the

conjugate waveform, the effect of the overlap should be minimal, because at the vicinity of

the imaged source, the convergent conjugate waveform must attain field strengths much

higher than the divergent object waveform. Nonetheless, there remains considerable

difficulty in synthesizing such a transmission screen, since in general it possesses both

positive and negative transmission values. In the following two sections, we propose two


practical methods to synthesize transmission screens which allow one to reconstruct close

approximations of the transmission function Re{Aobj}. Within the limited scope of this work,

we only propose 1D sub-diffraction focusing transmission screens. Notwithstanding, the

ideas we propose below can be generalized for 2D sub-diffraction focusing in a 3D

environment.

4.3 The Slot Antenna Hologram

Although traditional microwave reconstruction techniques are inadequate in general for

reconstructing a near-field, subwavelength waveform, they do provide general concepts upon

which improvements can be made. In this regard, our proposal of the slot antenna hologram

represents the extension of a useful concept from an earlier work in microwave holography.

In his work on the holographic antenna [36] (see Fig. 4.1a), Checcacci synthesized, with

some success, a microwave transmission screen by placing metallic strips located at the full-

width-half-maximums of interference fringes, thus padding the high intensity locations with

metal. We propose to modify Checcacci’s idea as follows (see Fig. 4.1b): We would

synthesize the transmission function Re{Aobj} by placing metallic strips, parallel to the

direction of the electric field, at locations where the electric field is negative-valued.

Structurally, such a collection of metallic strips can also be viewed as a collection of slots, or

slot antennas of infinite length, cut on a ground plane. We have adopted the latter viewpoint,

and hence named the structure the “slot antenna hologram”. [40]


Fig. 4.1: a) The microwave hologram proposed by Checcacci in [36]. b) Our proposed slot

antenna hologram as an extension to Checcacci’s hologram.

The functionality of the slot antenna hologram can be reasoned as follows. Firstly, it is clear

that for areas covered by metal the electric field will not be transmitted. However, for areas

not covered by metal, some electric field can be transmitted, with a spatial profile dependent

upon the width of the slot apertures. For the waveforms of interest, the slots attain widths of

subwavelength dimensions since the wavefront Aobj contains subwavelength oscillations. At

these subwavelength dimensions, numerical simulations have shown that these narrow slots

transmit electric field at a spatial profile similar to a half-cycle of a sinusoidal function –

which, as it happens, nicely approximates the oscillatory profile of Re{Aobj}. We must admit,

however, that with this hologram one does not have complete freedom to choose both the slot

widths and the relative transmission amplitude through the slots, since the latter is directly

related to the former. Nevertheless, after some appropriate fine-tuning, one can synthesize a

reasonable approximate to the transmission function,


. (4.4)

Obviously, this transmission function has its own drawbacks. Particularly, its inability to

generate negative field transmission coefficients implies the necessary existence of a “DC”

component, which would again contribute to background noise. Nonetheless, amidst the

noise, this realizable transmission function should allow one to reconstruct a near-field,

subwavelength waveform. In the following chapter we shall investigate its capability to

reconstruct a near-field subwavelength source.

4.4 The Resonant Slot Antenna Hologram

As we have seen from the previous section, the inability of the holographic screen to generate

negative transmission coefficients led to the transmission of a constant background. Thus it is

of interest to investigate whether it is possible to generate negative transmission coefficients

for certain locations on a transmission screen. With binary transmission screens, it is

naturally impossible to generate negative transmission coefficients. Indeed there are methods

to transmit a complex valued wavefront through designing the reference wave as well as the

screen [31][27], but such methods involve approximations which only apply for

reconstructing far-field waveforms with slow spatial oscillations (compared to the

wavelength), and are thus not applicable for reconstructing a near-field subwavelength

source. However, as discussed earlier, due to the subwavelength dimensions of the features

on the screen, a metallic screen can no longer be considered a binary transmission screen.


Instead, as suggested by our nomination – the slot antenna hologram – the metallic screen is

more aptly seen as an array of antenna elements arranged in a subwavelength fashion. In this

regard, it is potentially possible to control both the amplitude and the phase of the

transmission coefficient. Furthermore, it is also potentially possible to achieve a transmission

coefficient greater than unity even with a passive screen. This last fact greatly enhances the

efficiency of the reconstructed waveform.

With these perspectives in mind, we propose another type of hologram – the resonant slot

antenna hologram – in the following subsections.

4.4.1 Transmission Behaviour of a Single Half-Wavelength Slot

While the electromagnetic behaviour of slot antennas with arbitrary dimensions and

excitation methods has mostly been studied by numerical methods, the behaviour of a very

narrow, center-fed slots has been theoretically studied in many texts as a Babinet

complement to the thin wire dipole [8]. Thus it is well known that just like the thin wire

dipole, the narrow slot exhibits resonant behaviour at approximate multiples of half

wavelength. For our purposes we focus on the fundamental resonance achieved when the

slot’s length is around half of the wavelength of illumination. Whereas a dipole has a

negative reactance when its length is below half wavelength, and a positive reactance when

above, a slot, as a Babinet complement, exhibit the opposite behaviour: It is inductive (i.e.

has a positive reactance) when its length is below half wavelength, and capacitive (i.e. has a


negative reactance) when its length is above half wavelength. This flip in impedance

characteristic lends directly to a phase flip in the slot transmission behaviour. Additionally, it

is also widely known that both dipole and slot antennas radiate most effectively at resonance.

Hence by detuning of the slot length from resonance, one can also change the slot’s

transmission amplitude. In summary, a basic understanding on the slot antenna’s

electromagnetic behaviour suggests an intuitive handle on its transmission phase and

amplitude; namely, they can both be tuned by adjusting the slot length around the half

wavelength resonance.

4.4.2 Wavefront Synthesis by an Array of Slots

We have already seen how the amplitude and polarity (capacitive vs. inductive) of the

transmitted field through a half wavelength slot antenna can be tuned by slightly adjusting

the slot length. Then, it is only logical to envision a closely spaced array of such slots, with

lengths individually adjusted to synthesize “sample points” of a desired wavefront. Since the

widths of the slots are very narrow, they can be close-packed in a subwavelength manner,

generating a dense array of sample points capable of representing subwavelength variations

of a high-resolution wavefront. When such an array of half wavelength slot antennas are

illuminated by a normally incident plane wave, the excitation approximates that of in-phase

excitation to the center of each slot within the array. Thus the “sample points” of the desired

wavefront will be simultaneously reconstructed. Furthermore, even though each sample point

would initially resemble a discrete spike in space, this wavefront discreteness withers quickly


as the corresponding ultra high spatial frequency components are lost to evanescent decay.

Therefore, within a very short distance from the screen what remains would be a smoothened

wavefront resembling an interpolation of the sample points. At this point we would have

reconstructed a close approximate to our desired waveform.

We must admit that we have yet to provide a rigorous grounding to the above proposed idea

of a resonant slot antenna hologram. In the above discussion we have assumed the electric

field transmission behaviour at the near-field to be related to the radiation impedance – an

inherently far-field quantity; we have yet to quantitatively investigate any coupled effects of

phase and amplitude adjustment; we have also excluded an analysis on the mutual coupling

amongst slots in the closely spaced array. These factors, amongst others, must be investigated

before one can obtain a deep understanding of the capabilities and limits with the resonant

slot antenna hologram. Unfortunately, a thorough investigation of all these effects lies

beyond the scope of the present work. Notwithstanding the absence of a rigorous analysis, in

this section we have argued that the resonant slot antenna hologram is capable of generating

amplitude as well as phase modulations to an incident illumination, and can thus be used as a

transmission screen to reconstruct a near-field subwavelength source, and thereby perform

sub-diffraction focusing. This will be demonstrated by a design example in the next two

chapters.

44

Chapter 5

Design and Simulation

In this chapter, we design a slot antenna hologram (hereafter referred to as the “non-

resonant” slot antenna hologram for clarity) and a resonant slot antenna hologram to achieve

subwavelength focusing at near-field imaging distances. For comparison purposes, we have

chosen to design both holograms with a back-propagation distance of , and a spectral

truncation at . We give the final dimensions of each design, as optimized using

full-wave simulators. We also display the various field plots, through which we evaluate the

focusing capabilities of the holograms and discuss their salient features.

5.1 Non-Resonant Slot Antenna Hologram

We begin with results on a non-resonant slot antenna hologram designed to focus a 3GHz

incident plane wave into a subwavelength spot at an image distance2 of 10mm ( ). We

will first describe our simulation setup and give our final dimensions.

2 In Chapters 5 and 6, we define the “image distance” as the back-propagation distance used in the

hologram recording stage, as explained in chapter 3. Accordingly, we define the “image plane” as a

plane parallel to, but one image distance away from the screen. From a design perspective, the image

distance is the designed longitudinal location of the focal spot. However, from a practical perspective,

the image distance can be defined as a range of longitudinal distances where (1) all side lobes have

diminished to a reasonable level, and (2) the main lobe remains tightly focused. Thus in chapter 6 we

provide experimental results for an “extended” image plane at 0.15λ away from the screen.

CHAPTER 5: DESIGN AND SIMULATION 45

5.1.1 Simulation Setup and Slot Dimensions

The non-resonant slot antenna hologram was simulated using Comsol Multiphysics version

3.4 – a commercial full-wave simulation package which solves 2D and 3D electromagnetic

problems using the finite element method (FEM). A schematic of the simulation setup is

shown in Fig. 5.1. We ran a 2D simulation with in-plane TE waves (i.e. with E pointing

perpendicular to the simulation domain), and represented the metallic screen as a line of

perfect electric conductor (PEC) at z = 0, separated by gaps of subwavelength widths. Since

the structure extends to infinity in the y- (out of plane) direction, each gap represents a slot of

infinite length, in conformation to the description of the slot antenna hologram in section 4.3.

In our simulation, a 3GHz plane wave generated by the total-field-scattered-field (TFSF)

formulation emerges from the left and forms normal incidence to the hologram. Scattering is

absorbed by perfect matching layers in the scattered-field regions on the perimeters of the

computational domain. To avoid unphysical effects caused by a singular intersection between

the PEC and the TFSF boundary, a two step simulation procedure is used. First, a plane

wave is sent normally incident onto an infinitely extending PEC screen (with no aperture),

thus generating a standing wave pattern on the incident side of the screen. This standing

wave pattern is then used as the incident wave to the hologram in the TFSF simulation.

As explained in section 4.3, a consequence of having subwavelength slot widths was that the

transmission amplitude was sensitive to these widths. Thus these parameters were fine-tuned

through multiple simulations to best reconstruct the target wavefront, which would be the

substitution of the screen plane wavefront (3.11) into the slot antenna hologram transmission

function (4.4):


Fig. 5.1: A schematic of the simulation setup for the (non-resonant) slot antenna hologram,

with a front view of the hologram shown on the right. The locations of the hologram, the

image and auxiliary planes, the coordinate definition and the incident field are as defined in

the schematic. The white area in the schematic defines the free-space, total-field region of the

computational domain, while the shaded surrounding region represents Cartesian PMLs. The

slot widths are as given in Table. 5.2

Center Width

Central Slot 0mm (0λ) 14.4mm (0.144λ)

Side Slot (1st pair) ±24.1mm (0.241λ) 6.2mm (0.062λ)

Side Slot (2nd pair) ±45.0mm (0.450λ) 6.2mm (0.062λ)

Side Slot (3rd pair) ±65.8mm (0.658λ) 7.5mm (0.075λ)

Table 5.2: Dimensions for slots in the (non-resonant) slot antenna hologram.


, (5.1)

where for this design we obtained from performing an inverse Fourier transform on

. The final locations and widths of the designed 7-slot screen are given in Table 5.2.

[40]

5.1.2 Simulation Results

Fig. 5.3a shows a plot of the y-directed electric field amplitude on the output side of the

screen. To obtain a clear display of the focusing quality, we have normalized the electric

field at each z coordinate, and used a black solid outline to indicate its full-width-half-

maximum (FWHM). The electric field amplitude along the principle axis ( ) is plotted in

Fig. 5.3b. The decay of the field amplitude is expected; it is characteristic of the existence of

evanescent wave components which lead to subwavelength focusing.

The electric field profiles at the screen plane ( ) and the image plane (

, denoted in Fig. 5.3a with a white dotted line) are shown in Fig. 5.4. The FWHM of the

electric field at the image plane measures 18.6mm, or 0.186 wavelengths, which is more than

thrice improved in comparison with the electric field FWHM of 0.603λ, for a diffraction-

limited sinc function (see Appendix B for the derivation of this value). However, while we

achieved sub-diffraction focusing, we did not realize the full focusing potential for our spatial

bandwidth , since our simulated focus compares unfavourably with the

improvement factor as calculated from (3.8).


Fig. 5.3a: A Plot of , normalized at every z-coordinate. Here the screen plane

is located at , and the image plane is shown as the dotted line. To

clarify the focusing quality the electric field FWHM is outlined in black.

Fig. 5.3b: The field amplitude along the line .


Fig. 5.4: A comparison between the focusing qualities of the (non-resonant) slot antenna

hologram (solid) and a metallic screen whose only aperture is the central slot in the hologram

(dotted), made at the image plane ( ). Shown in the inset is the transmission

amplitude of the screen, which is dominated by real components.

Fig. 5.4 also compares the electric field profile at the image plane with the corresponding

profile from another simulation, in which the hologram is replaced by a metallic screen

which contains only one slot – corresponding to the central slot in the hologram. It has been

well known that a single subwavelength aperture readily produces evanescent waves, and this

fact has been utilized in various near-field sub-diffraction imaging devices [2]. While a

purpose of this work would be to achieve a superior alternative to sub-diffraction focusing, in

terms of either focusing quality or imaging distance, this purpose has not been achieved with

the non-resonant slot antenna hologram. As evident in Fig. 5.4, the image obtained from the


hologram is very similar to that obtained by the single aperture screen. While in the

hologram, the presence of laterally shifted slots slightly increased the peak focal amplitude,

they also slightly widened the focal width. Thus the inclusion of laterally shifted slots in this

hologram only facilitates a small tradeoff between the focal amplitude and the focal width.

This result is not surprising, though, since the electric field transmitted by the hologram’s

slots (shown in the inset of Fig. 5.4) are more or less invariant in phase, their superposition in

the extreme near-field resembles a scalar addition of field amplitudes. From this perspective

it makes intuitive sense that in-phase additions with laterally shifted field distributions

couldn’t possibly improve the focal width. Of course, this intuition does not apply for a

hologram with (a) a larger imaging distance, or (b) more lateral slots, where the distance

between outer slots and the focal point approaches half-wavelength. However for these cases

the transmitted evanescent field becomes extremely weak, and thus do not make major

contributions to the focal point, unless they are somehow amplified in the transmission

process.

In summary, while our designed non-resonant slot antenna hologram does achieve

subwavelength focusing of 0.186λ at an imaging distance of 0.1λ, its focusing characteristics

is very similar to that of a screen with a single subwavelength aperture. In particular we see

that to obtain a performance better than that of a single subwavelength aperture, interference

effects need to be present and exploited. This latter point should become clear in our

following discussion on the simulation results for the resonant slot antenna hologram.


5.2 Resonant Slot Antenna Hologram

We now show simulation results of a resonant slot antenna hologram designed to focus a

10GHz incident plane wave into a subwavelength spot at an image distance of 3mm (0.1λ)

[33]. As in the previous section, we first describe our simulation setup and give our slot

dimensions.

5.2.1 Simulation Setup and Slot Dimensions

The simulations of the resonant slot antenna hologram were conducted using Ansoft HFSS

version 10.1, which is an FEM solver for 3D electromagnetic problems. A schematic of the

simulation setup is given in Fig. 5.5. In this simulation, an x-polarized plane wave forms

normal incidence to the hologram, which is located within a total field region of a TFSF

simulation terminated with Cartesian PMLs on all sides. In this simulation we have opted for

a finite ground plane of size 60mm by 60mm (2λ × 2λ) to show that an infinite (or extremely

large) ground plane is not needed.

For the ease of design we have conducted a proof-of-principle experiment involving only

three slots, spaced λ/10 apart according to (3.11). While this may seem to be a very small

number of slots, we have confirmed through numerical calculations that for our imaging

distance, a properly designed three-slot hologram already achieves subwavelength focusing

close to λ/10. Several trial simulations are performed to optimize the slot lengths and widths

for focusing at the prescribed imaging distance. The final slot dimensions are shown in Table

5.6, and the full-wave simulation results are given in the following subsection.


Fig. 5.5: The simulation setup for the resonant slot antenna hologram. The spatial coordinates

and orientation of the incident wave are as shown in the diagram. The metallic (PEC) screen

is (2λ × 2λ); the slot dimensions are given in table 5.6. The outer box

represents simulated free space region; the PMLs which surround this free space region have

been removed to simplify the diagram.

Center Length Width

Central Slot 0mm (0λ) 13.2mm (0.44λ) 1.2mm (0.04λ)

Side Slots ±3mm (0.1λ) 17.6mm (0.59λ) 0.6mm (0.02λ)

Table 5.6: Dimensions for slots in the (non-resonant) slot antenna hologram.


5.2.2 Simulation Results

As in the previous section we begin with a set of figures showing field distribution in the xz-

plane. Fig. 5.7a shows the amplitude of the x-directed electric field on the xz-plane, as the

electromagnetic wave emerges from the hologram. Again, to elucidate the focusing quality,

we normalized the field amplitude at each z coordinate, and outlined the FWHM contour in

black. The field amplitude along the line z-axis is plotted in Fig. 5.7b. For comparison, to the

immediate right, we have shown the corresponding plots for a metallic screen in which only

the central slot exists (see Fig. 5.7c and Fig. 5.7d).

Fig. 5.7: a) A simulated plot of for the RSAH, normalized at every z-

coordinate. Here the screen plane is located at , and the image plane is at

is shown in dotted line. To clarify the focusing quality the electric field FWHM is

outlined in black. b) The field amplitude of the RSAH along the line . c) and d)

plot the corresponding properties of a 1-aperture screen, containing only the central slot of

the RSAH.


Comparing Fig. 5.7a with Fig. 5.7c, we see that the FWHM of the resonant slot antenna

hologram is clearly improved from the corresponding 1-slot screen. Specifically, we observe

that within a range of about 5mm from the screen plane, the width (FWHM) of the electric

field emerging from the hologram expands at a much slower rate compared to that for the

electric field emerging from the 1-slot screen. This can be readily understood as the

consequence of near-field interference between the central slot and the satellite slots, whose

field transmission are about π-shifted in phase. Beyond that 5mm mark, however, the fields

emerging from the satellite slots have undergone significant decay and spread, and thus have

a diminished effect in their interference with the field emerging from the central slot. As a

result we see a region where the width of electric field increases comparatively rapidly,

“catching up” with the field emerging from the 1-slot screen. We note that the reduction in

field-widening at is due to the interference from fringing fields, which are formed

by the incident field rapping around the edges of the metallic plate. At the designed image

plane ( ), as denoted by the white dotted lines in Figs. 5.7a and 5.7c, the FWHM

measures 3.9mm (0.13λ), which closely approaches to the theoretical optimal value of 0.12λ,

obtained by dividing the FWHM of the diffraction-limited sinc function by the resolution

improvement factor . Fig. 5.8 compares the field focus of the hologram to that of the

one-slot screen, and shows that a clear resolution improvement is achieved by the presence of

the satellite slots. Moreover, from Fig. 5.7c we also observe some flexibility in the device’s

operation focal length: for , the side lobes of transmitted field have are

largely diminished, while the width of the main beam expands slowly, and remains focused

to about half the width of that of the single-slot metallic screen. Thus one can achieve

superior subwavelength focusing by operating within this range of image distance.


Fig. 5.8: A comparison of the simulated focusing quality of the RSAH and the 1-slot screen,

at a line across the image plane ( ). The FWHM of the RSAH (solid)

measures 3.9mm (0.13λ), which is greatly improved from the FWHM of the 1-slot screen, at

6.29mm (0.21λ).

We shall also examine the field evolution amongst transverse cross-sections at the image

plane ( ). Fig. 5.9 shows the x-directed electric fields (amplitude and phase)

emerging from the hologram, in comparison to the same field quantities emerging from the

1-slot screen. These plots capture the field inversion amongst adjacent slots, and show how

the resulting field pattern is squeezed along the x-axis at the focal plane. We note that for the

current design, subwavelength focusing only occurs in the x-direction; our concluding

remarks will discuss ideas towards focusing in both x- and y-directions, and work

progressing in that direction.


Fig. 5.9: A cross-section plot showing the (a) amplitude and (b) phase of the x-directed

electric field emerging from the RSAH, compared alongside corresponding quantities for the

1-slot screen (c and d).

Finally, we shall take a closer look at the field decay characteristics in the longitudinal

direction to obtain more insights regarding the functionality of the resonant slot antenna

hologram. Fig. 5.10 reproduces the field amplitudes along the z-axis as plotted in Figs. 5.7b

and 5.7d, but overlays them in a logarithmic scale for comparison. Here we have also plotted

a 100× magnification of the electric field transmission from a slot with infinite length but a


Fig. 5.10: A semi-log plot of field levels along the line , for the RSAH (solid),

the resonant slot (from the 1-slot screen) (dotted), and a non-resonant slot (dash, 100×

magnified).

width corresponding to that of the central slot of the hologram.3 The field transmission

through the infinite slot simulates typical instances of field transmission through non-

resonant subwavelength apertures. In this case the field amplitude transmitted at the screen

plane is about 10% of the incident field strength; this field level further drops to about 1% of

the incident field strength at 3mm (λ/10) away from the screen. A similar rate of decay in the

transmitted field is also observed for the resonant 1-slot screen, while an even more dramatic

decay is observed from the 3-slot RSAH. In fact this rapid field decay is a necessary artefact

3 We obtained this last result with the simulation setup described in the previous section. In this case we

used an incident plane wave with a y-polarized electric field of unit strength.


in subwavelength focusing, as it is caused by the dominant existence of evanescent wave

components, which ultimately achieve subwavelength focusing. Nonetheless, despite this

rapid field decay, the suitable exploitations of slot antenna resonances tremendously boost

the transmitted field amplitude, allowing field amplitude transmission to be “extraordinary”

(higher than the incident field) for image distances up to 5.7mm ( ~0.19λ) away from the

screen. Furthermore, comparing field amplitudes for the two resonant cases yields a

surprising result. Since the 3-slot resonant hologram achieves its tight focusing from

destructive interference between π-phase-shifted electric fields, one may, at first glance,

expect its transmission amplitude to be lower than that of the single resonant slot metallic

screen. While the field transmission amplitude for the resonant slot antenna hologram indeed

undergoes a more rapid decay than its 1-slot counterpart, it still transmits a stronger field

amplitude for image distances within about 5mm from the screen, thanks to stronger mode

coupling from the incident plane wave. Hence we see that resonant slot antenna holograms

can simultaneously achieve subwavelength focusing and increase the focal field strength, as

compared to both a non-resonant subwavelength slot, and a metallic screen with one resonant

slot of subwavelength width.

We note once again that we have only conducted a proof-of-principle screen, with the main

purpose of demonstrating subwavelength focusing through near-field interference. A

reduction in inter-slot spacing and the addition of more slots with appropriately weighted

field transmission will lead to tighter focusing, or similar focal widths at extended image

distances; also a design with slot lengths even closer to resonance will improve the

transmitted field strength. We have nonetheless demonstrated that both subwavelength


focusing much beyond the diffraction limit, and a resonant enhancement of the focal field

strength, are achievable over a range of distances with a resonant slot-antenna hologram.

60

Chapter 6

Experimental Demonstration

In this final chapter of the thesis body we present experimental results for 1D subwavelength

focusing using the resonant slot antenna hologram presented in chapter 5. We first describe

the screen fabrication and the experimental apparatus, then present and discuss our

experimental results, which verify simulation results and demonstrate subwavelength

focusing. [41]

6.1 Screen Fabrication and Experimental Apparatus

Our designed holographic metallic screen, with slot dimensions as described in Table 5.6,

was fabricated by laser-cutting the designed slots from a 22” by 22” ( )

plate of stainless steel with a thickness of 6mil (0.152mm). The screen was then stretched

onto a frame to add structural strength and ensure its flatness. Multiple hologram patterns

were printed onto the screen, with a center-to-center separation distance of 140mm (4.67λ) –

which exceeds both the side length of the simulated ground plane and the Gaussian beam

waist of the illumination. Hence the plate can be considered as effectively infinite in extent,

and all transmission effects can be attributed to the single holographic pattern under

CHAPTER 6: EXPERIMENTAL DEMONSTRATION 61

illumination. Fig. 6.1 shows a photograph of the fabricated screen; the inset of Fig. 6.1 shows

a close up of the holographic pattern with which the experiment was conducted.

Figs. 6.2 and 6.3 show a schematic and a photograph of our experimental apparatus. We first

used an X-band horn to generate a spherical wave with the E-plane in the xz-direction. Then,

a Rexolite dielectric lens collimated this spherical wave to form a normally incident Gaussian

beam with a 90mm beam waist located at the screen plane. The biconvex Rexolite lens has a

refractive index ; to optimize beam collimation, its input facet was placed 144mm

from the horn aperture, and its output facet was placed 306mm from the sample. As shown in

Fig. 6.3, in the experiment the horn and lens were positioned onto a wooden frame to

generate the Gaussian beam incidence from beneath the screen.

Fig. 6.1: A photograph showing the fabricated holographic screen. The inset shows a close-

up of the fabricated 3-slot holographic pattern. The slot spacings and dimensions are given in

table 5.6.


Fig. 6.2: A schematic of the experimental apparatus.

Fig. 6.3: The photograph on the left shows the experimental apparatus, showing the antenna

horn, the dielectric lens, the metallic screen and the scanning probe. The upper-right shows a

close-up of the scan probe, placed directly above the holographic pattern.


On the output side of the screen, the x-directed E-field was measured by a probe made with a

semi-rigid co-axial cable with inner and outer conductors having diameters of 0.31mm and

1.19mm respectively. To minimize spurious coupling (coupling through the outer conductor

of the cable), the co-axial cable approached the screen from the +y direction; at its tip the

inner conductor extended outwards for a length of 3mm, and bent in the +x direction to form

a short electric dipole. This dipole probe was mounted onto a Newark xyz-translation station,

which allowed us to position the probe with sub-millimeter precision.

Using an Agilent E8364B Programmable Network Analyzer, we performed automated scans

over the xy-plane at a fixed distance above the screen, and measured the x-directed electric

field at 401 frequency sample points, ranging from 8GHz to 12GHz. In general we performed

three types of scans: (1) a scan over the image plane, with all 3 slots unobstructed; (2) a scan

over the image plane with the two satellite slots covered with copper tape, leaving only the

central slot; and (3) a scan over the screen plane (without the presence of screen) for

calibration purposes. As discussed in section 5.2, while the simulated screen was designed

for an image distance of , it achieved desirable focusing over the range of

. For this chapter we have chosen to present focusing results at an image plane of

(0.15λ), as we found that at this larger imaging distance there is a reduced level

of spurious probe coupling, thus allowing more accurate field measurements and more

meaningful comparisons with simulation.


6.2 Experimental Results

Fig. 6.4 compares normalized amplitude profiles of the x-directed electric field at the image

plane of 4.5mm (0.15λ), for both the resonant slot antenna hologram and the 1-slot metallic

screen. One clearly observes the holographic focusing effect for the x-direction, while the

focus in the y-direction remains at about λ/2 as expected. Here we also see that the measured

amplitude profiles correspond well with the simulated profiles, with the exception of some

unpredicted asymmetric lobes located on the lower right, as circled in white. We deduced, by

varying the probe geometry, that these asymmetries are caused by electromagnetic fields

coupling spuriously into the body of the co-axial cable, and are thus artefacts unrelated to the

fields at the probed location. Fig. 6.5 plots the field profile (measured and simulated) at the

line (labelled in Fig. 6.4 in white), and compares them with the

diffraction-limited sinc function and measured field from the 1-slot metallic screen. Despite

the aforementioned asymmetric smoothening caused by probe imperfection, the FWHM of

the electric field measures 5.2mm (0.17λ) – which agrees well with the simulation value of

0.16λ, and shows a clear improvement from the FWHM of 9.8mm (0.33λ) obtained with the

side slots covered. Furthermore, this measured FWHM is about 3.5 times improved over the

far-field diffraction limit of 0.603λ.

Fig. 6.6 compares the measured field amplitude with the simulated amplitudes for a

hologram cut on a PEC, as well as one cut on stainless steel. The peak field strengths for each

case are labelled, and the horizontal line indicates the incident field level. From the two

simulated amplitudes, we observe that conductor loss slightly reduces the focal field strength,

but has minimal effects on the focusing quality. As for the measured field amplitude, Fig. 6.5


experimentally demonstrates that conductor loss does not affect the focusing quality of the

resonant slot antenna hologram. Anyhow, we do see a further amplitude decrease in the

measured field profile in comparison to simulated profiles (with conductor loss), which is

possibly due to a slight near-field perturbation caused by the measuring probe. Nevertheless,

despite the slight amplitude drop, the field strength at this image plane still exceeds the

incident field strength by about 30%.

Fig. 6.4: A cross-section plot showing the (a) measured and (b) simulated |Ex| emerging from

the RSAH, compared alongside corresponding quantities for the 1-slot screen (c and d).

Spurious coupling has led to the measurement of an appreciable field strength in the area

encircled by the ellipse in (a). The field distribution across the dotted line is displayed in Fig.

6.5.


Fig. 6.5: A experimentally focusing quality of the RSAH (measured and simulated), the 1-

slot screen, and a diffraction-limited device, at a line across the image plane (

). The FWHMs of the curves are:

RSAH (measured): 5.2mm (0.17λ); RSAH (simulated): 4.9mm (0.16λ);

1-slot (measured): 9.8mm (0.33λ); diffraction limit: 18.0mm (0.60λ).

Fig. 6.6: An unnormalized plot of |Ex| along , comparing measurement

(solid) with simulations conducted using a PEC screen (dotted) and a stainless steel screen

(dash). The incident field level is also displayed with a thin dashed line. The peak amplitudes

are: simulation (PEC): 1.52V/m; simulation (stainless steel): 1.43V/m; measurement:

1.31V/m.


While we have designed the resonant slot antenna hologram to perform subwavelength

focusing at one specific frequency, in the following we examine its frequency characteristic,

which yields further insight to its frequency of operation and gives practical intuition on its

frequency sensitivity. Fig. 6.7 shows S21 plots for three probe locations on the image plane:

, , and (all lengths in mm).

Conventional antenna theory tells us that the shorter central slot resonates at a frequency

slightly above 10GHz, while the longer satellite slots resonate at slightly below 10GHz. The

S21 plots show the superposition characteristics of fields emanating from all slots; in

particular the transmission dips are caused by total destructive interference of fields

emanating from the central and side antennas. Since location A is close to the central slot, the

transmission dip (or the frequency for total destructive interference) is slightly red-shifted

from 10GHz to about 9.8GHz, at which frequency the radiation from the central slot weakens

while those from the satellite slots strengthen. However, if we consider |S21| at 10GHz as

move from location A through location B to location C, we can reason that the field

contribution from the central slot wanes because we move away from it, while the field

contribution from one of the satellite slots strengthens because we move closer to it. As a

result, destructive interference takes on a much more dramatic effect for frequencies around

10GHz, resulting in a rapid shift in transmission dip towards 10GHz, which in turn results in

sharp focusing for a narrow band of frequencies surrounding 10GHz.

Fig. 6.8 plots the field profile at , for a few different frequencies around 10GHz.

It can be seen that within a small range of frequencies around 10GHz, the change in

frequency amounts to a tradeoff between the focal width and side lobe levels. However,


Fig. 6.7: S21 plots of the measured electric field at 3 locations: ,

, and (all lengths in mm).

Fig. 6.8: A comparison of the focusing quality for five frequency points within a range of

±1% from 10GHz, compared alongside the 1-slot pattern.


outside this range, the focusing quality degrades rather rapidly. If we define the operating

bandwidth as a frequency window of desirable focusing, within which the focal width

remains within 15% of the width at 10GHz, and the side lobe levels remain below 30% of the

focal peak, then our measurements show that the resonant slot antenna hologram has an

operational bandwidth of 0.9%, stretching from 9.96GHz to 10.05GHz. While one can

possibly widen the operating bandwidth by operating all three slots farther from their

respective resonances, this may not be most desirable since this will lead to corresponding

reductions in the transmitted field strength. We therefore conclude that the resonant slot

antenna hologram is an intrinsically narrowband subwavelength focusing device.

70

Chapter 7

Conclusion

7.1 Summary of Contributions

This work has presented a few contributions to the design of subwavelength focusing

devices. We shall briefly review them in this section.

Formulation of the Generalized Hologram

We began our work by asking whether one can store and retrieve a subwavelength waveform

through the principle of holography. After a close examination on holographic process, we

concluded that this is indeed possible in the near-field, as long as one sufficiently

compensates for the exponential decay of evanescent waves, which contain the high-

resolution information of a waveform. We then proceeded to formulate a generalized method

for recording a hologram. In this generalized method, a wavefront at an image plane is

decomposed into propagating and evanescent plane waves, which are back-propagated to a

screen plane. We paid special attention to the sign of the phase during the back-propagation

procedure, and showed that when the correct choice is made one can arrive at a suitable

wavefront at the screen plane, the reproduction of which retrieves a desired image waveform.

CHAPTER 7: CONCLUSION 71

Finding a Generalized Hologram for a Point-source

In this work we were ultimately interested in the subwavelength focusing of electromagnetic

waves. As such, we used the aforementioned theory to record the generalized hologram of a

point-source on the image plane. We observed that the resultant wavefront on the screen

plane had rapidly varying fringes under a slow varying envelope – features which curiously

resembled a wavefront recorded for a conventional hologram, even though they arose from

vastly different physical mechanisms. Moreover we reasoned that these features would apply

to generalized holograms of a large class of waveforms. Recognizing the generality of these

features, we went on to devise transmission screens which synthesize wavefronts with these

features.

The Non-Resonant Slot Antenna Hologram

We proposed the non-resonant slot antenna hologram as a first try to synthesize near-field,

holographic patterns of the type mentioned above. Made of closely packed, infinite slots of

subwavelength thickness, this hologram was found, by full-wave simulation, to have a rather

weak electric field transmission. Nonetheless, we demonstrated this hologram’s ability to

facilitate small tradeoffs between the amplitude and width of the focus, as compared to a

similar screen with only the central aperture.

The Resonant Slot Antenna Hologram

We then proposed an alternate version of the slot antenna hologram, where slots close to half

wavelength replaced those of infinite length. In this design, operating near resonance


dramatically improves field transmission. Furthermore, fine-tuning the lengths of the closely

packed slots above and below half wavelength leads to phase flips in the transmitted field,

allowing the synthesis of positive and negative transmission values. We conducted

simulation on a 3-slot proof-of-principle resonant slot antenna hologram and found that at the

designed image distance of 0.1λ, the azimuthal x-directed electric field forms a focus of

0.13λ – roughly fivefold improved from the far-field diffraction limit, and about twofold

improved from a 1-slot aperture of similar dimensions. The field strength at this focal

location was also enhanced to threefold the incident radiation. We then fabricated this screen

and performed field focusing experiments at an image distance of 0.15λ. Despite small

systematic errors attributable to our probe configuration, our results still agreed very well

with simulation. At this extended distance the measured focal width was 0.17λ, while peak

field strength reached 1.31λ. We again note that further optimization of various parameters

should lead to holograms with better focusing performances. Nonetheless, through our proof-

of-principle design we confirmed the capability of the resonant slot antenna hologram to

perform near-field subwavelength focusing, and demonstrated its attractiveness in the

following aspects:

An ability to achieve ultra-tight subwavelength focusing, far surpassing the far-field

diffraction limit and superior to focusing using a single near-field subwavelength

aperture;

A focusing quality unaffected by metallic loss;

A very high field transmission; and


A simple design directly scalable from microwaves to terahertz to near optical

wavelengths. [42]

These advantages render the resonant slot antenna hologram a desirable alternative for

potential applications in near-field subwavelength focusing.

7.2 Future Directions

Besides showing early signs of promise towards subwavelength focusing with holographic

metallic screens, this work has also manifested several directions of subsequent development

towards a practical and versatile tool for subwavelength focusing. Below we outline a few of

such directions which are most important in our opinion.

A procedure for resonant slot design

One important direction of further research would be to develop a systematic method for

synthesizing resonant slot antenna holograms to achieve an arbitrary target waveform. While

the corresponding analysis problem –calculating field transmission given the slot dimensions

and spacing parameters – are relatively straightforward, we have yet to come up with a

systematic procedure for screen synthesis. In this work we have arrived at suitable slot

dimensions through intuitions regarding inductances and capacitances of individual slots, and

iterative optimizations using screen analysis tools. However for screen designs involving

more slots, one needs to pay closer attention to mutual impedances amongst slots. To that


end, a recent work by Markley et al. [43] attempted to design slots dimensions by

considering mutual coupling amongst slots via the induced EMF method. While this

development certainly represented progress in the forward direction, the latter stages of

screen design still heavily relied on optimizations using full-wave simulations. Thus a

method which has sufficient rigor to account for the excited antenna modes, but yet

incorporates appropriate assumptions so as to allow a systematic procedure for slot design,

would prove greatly useful.

2D Subwavelength focusing

This work has dealt with the focusing of the electric field in the x-direction. While 1D

focusing by itself is useful for the raster scanning type of imaging schemes, further

developments towards achieving 2D subwavelength focusing would be greatly desired.

Again, recent work by Markley et. al. [43] has demonstrated 2D focusing using a resonant

slot antenna hologram where the slots are shifted in both x- and y-directions. However, as

half-wavelength slot antennas must be of a resonant length in the transverse directions, a

resonant slot antenna hologram is ultimately limited from achieving drastic subwavelength

2D focusing. Some ideas for possible improvements from the existing resonant slot antenna

hologram include using dielectric or circuit element loading, which can perhaps shrink the

resonant slot size as compared to the free space wavelength. However, perhaps a solution

with greater potential would be to construct holographic screens with other elements which

are resonant at a transverse size much smaller than half wavelength. For example the

longitudinal dimension can be used to achieve resonance; a combination of positive and


negative index materials can also be considered for structures resonant on the subwavelength

length scale.

Subwavelength Focusing at Optical Frequencies

Finally, it would be very desirable to extend the present work from microwave to optical

frequencies, where it will find applications in optical communications, biomedical imaging

and therapy, and optical lithography. As it was pointed out in the previous section, the simple

design of this screen allows for direct scalability to essentially all frequency ranges where the

perfect conductor assumption remains valid. While indeed we have shown that unlike the

metamaterial superlens, the resonant slot antenna hologram performs focusing independent

of small losses in the structure, the plasmonic nature of metals at optical frequencies greatly

alter the physics of the problem. Nevertheless it has been shown that after suitable rescaling

of antenna dimensions one can obtain similar responses from the so called nanoantennas, as

one does from their microwave counterparts. To that end, recent work by the author has

shown working simulations for a plasmonic subwavelength focusing slot antenna hologram

operating at optical frequencies [42]. We envision further improvements upon this structure,

and eventually developments into 2D optical subwavelength focusing, to bring much

excitement in improving current subwavelength focusing capabilities for various

applications.

76

Appendix A

Analytical Forward-Propagation of a Holographic

Wavefront

In chapter 3, we have obtained an analytical approximation of a back-propagated wavefront,

, (3.11), (A.1)

where km refers to the spectral truncation value as defined in (3.7), and d, s, and x are as

introduced in Fig. 3.1. In this appendix we will analytically show that if this wavefront can

be retrieved, it forward-propagates into a tightly focused distribution.

We first take a Fourier transform of , which becomes [44]

. (A.2)

We now forward-propagate this wavefront from the screen plane to the image plane,

following a rearrangement of (2.15):

.

(A.3)

APPENDIX A: ANALYTICAL FORWARD PROPAGATION OF A HOLOGRAPHIC WAVEFRONT 77

Applying (A.3) towards (A.2) gives

. (A.4)

We now take the inverse Fourier transform of (A.4) to obtain the electric field profile on the

image plane.

APPENDIX A: ANALYTICAL FORWARD PROPAGATION OF A HOLOGRAPHIC WAVEFRONT 78

(A.5)

We see from (A.5) that all terms decay with increasing x, thus forming a focus along .

Moreover, when we concern ourselves only with small values of x, we can approximate

(A.5) with its 1st order Taylor expansion,

. (A.6)

Clearly, for small x and large km, the profile is dominated by last term, which represents the

sinc function in (A.5). Furthermore this sinc function has a sinusoidal term that varies at R

times the frequency of a diffraction-limited sinc function, where R is as defined in equation

(3.8) in section 3.2. Thus we have retrieved a tightly focused sinc-like profile, with a focal

width reduction by the factor R in accordance to our discussion in section 3.2.

79

Appendix B

The Diffraction-Limited sinc Distribution

For far-field imaging in a homogeneous isotropic environment, with an illumination with

spatial frequency , we have at our disposal the propagating electromagnetic waves, for

which the transverse spatial frequency lies in the range (where ). An optimal

way of using this spectral range to generate a sharp focus is to have an in-phase interference

of equal amplitudes from all components across this spectral range, i.e.:

. (B.1)

An inverse Fourier transform takes into its corresponding spatial distribution, which is

[45],

. (B.2)

(B.2) is known as the diffraction-limited sinc function. It can be found that

. (B.3)

Thus the FWHM of the diffraction-limited sinc is

. (B.4)

80

References

[1] P. C. Lauterbur, "Image Formation by Induced Local Interactions: Examples Employing

Nuclear Magnetic Resonance," Nature, vol. 242, pp. 190-191, 1973.

[2] E. A. Ash and G. Nicholls, "Super-resolution aperture scanning microscope," Nature, vol.

237, pp. 510-512, Jun. 1972.

[3] D. W. Pohl, W. Denk and M. Lanz, "Optical stethoscopy: Image recording with

resolution λ /20," Applied Physics Letters, vol. 44, pp. 651-653, Jan. 1984.

[4] S. Nie and S. R. Emory, "Probing Single Molecules and Single Nanoparticles by Surface-

Enhanced Raman Scattering," Science, vol. 275, pp. 1102-1106, Feb. 1997.

[5] C. Loo, A. Lin, L. Hirsch, M. H. Lee, J. Barton, N. Halas, J. West and R. Drezek,

"Nanoshell-Enabled Photonics-Based Imaging and Therapy of Cancer," Technology in

Cancer Research & Treatment, vol. 3, pp. 33-40, Feb. 2004.

[6] N. Fang, H. Lee, C. Sun and X. Zhang, "Sub-diffraction-limited optical imaging with a

silver superlens," Science, vol. 308, pp. 534-537, April. 2005.

[7] Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun and X. Zhang, "Far-field

optical superlens." Nano Letters, vol. 7, no. 2, pp. 403-408, Feb. 2007.

[8] C. A. Balanis, Antenna Theory: Analysis and Design. ,3rd ed.Hoboken, New Jersey: Jon

Wiley & Sons, Inc., 2005, pp. 151-281, 653-738.

[9] J. J. Greffet and R. Carminati, "Image formation on near-field optics," Progress in

Surface Science, vol. 56, pp. 133-237, Nov. 1997.

[10] K. Iizuka, Elements of Photonics. , vol. 1, New York: Wiley-Interscience, 2002, pp.

150-162.

[11] P. Muhlschlegel, H. J. Eisler, Martin, O. J. F., B. Hecht and D. W. Pohl, "Resonant

optical antennas," Science, vol. 308, pp. 1607-1609, Jun. 2005.

REFERENCES 81

[12] J. J. Greffet, "Nanoantennas for light emission," Science, vol. 308, pp. 1561-1563, Jun.

2005.

[13] E. H. Synge, "A Suggested Method for Extending Microscopic Resolution into the

Ultra-Microscopic Region," Philosophical Magazine, vol. 6, pp. 356-362, Aug. 1928.

[14] V. G. Veselago, "The electrodynamics of substances with simultaneously negative

values of eps and mu," Soviet Physics Uspheki, vol. 10, pp. 509-514, Jan. 1968.

[15] J. B. Pendry, Negative refraction makes a perfect lens. Physical Review Letters, vol. 85,

no. 18, pp. 3966-3969, Apr. 2000.

[16] G. V. Eleftheriades, A. K. Iyer and P. C. Kremer, "Planar Negative Refractive Index

Media Using Periodically L-C Loaded Transmission Lines," IEEE Transactions on

Microwave Theory and Techniques, vol. 50, pp. 2702-2712, Dec. 2002.

[17] A. Grbic and G. V. Eleftheriades, "Overcoming the diffraction limit with a planar left-

handed transmission-line lens," Physical Review Letters, vol. 92, pp. 117403, Mar.

2004.

[18] P. Alitalo, S. Maslovski and S. Tretyakov, "Experimental verification of the key

properties of a three-dimensional isotropic transmission-line superlens," Journal of

Applied Physics, vol. 99, pp. 124910, Jun. 2006.

[19] K. Aydin, I. Bulu and E. Ozbay, "Subwavelength Resolution with a Negative-Index

Metamaterial Superlens," Applied Physics Letters, vol. 90, pp. 254102, Jun. 2007.

[20] A. K. Iyer and G. V. Eleftheriades, "Mechanisms of subdiffraction free-space imaging

using a transmission-line metamaterial superlens: An experimental verification,"

Applied Physics Letters, vol. 92, pp. 131105, Mar. 2008.

[21] M. J. Freire and R. Marques, "Planar magnetoinductive lens for three-dimensional

subwavelength imaging," Applied Physics Letters, vol. 86, pp. 182505, May. 2005.

[22] A. Grbic and G. V. Eleftheriades, "Practical Limitations of Subwavelength Resolution

Using Negative-Refractive-Index Transmission-Line Lenses," IEEE Transactions on

Antennas and Propagation, vol. 53, pp. 3201-3209, Oct. 2005.

REFERENCES 82

[23] R. Merlin, "Radiationless electromagnetic interference: evanescent-field lenses and

perfect focusing," Science, vol. 317, pp. 927-929, Aug. 2007.

[24] A. Grbic, L. Jiang and R. Merlin, "Near-field plates: Subdiffraction focusing with

patterned surfaces," Science, vol. 320, pp. 511-513, Apr. 2008.

[25] D. Gabor, "A new microscopic principle," Nature, vol. 161, pp. 777-778, May. 1948.

[26] E. N. Leith and J. Upatniek, "Reconstructed Wavefronts and Communication Theory,"

Journal of the Optical Society of America, vol. 52, pp. 1123-1130, 52. 1962.

[27] J. Goodman, Introduction to Fourier Optics. ,3rd ed.Englewood, Colorado: Roberts and

Company, 2004, pp. 297-395.

[28] P. F. Checcacci, V. Russo and A. M. Scheggi, "Holographic Antennas," Proceedings of

the IEEE, vol. 56, pp. 2165-2167, Dec. 1968.

[29] K. Iizuka, "Microwave hologram by photoengraving," Proceedings of the IEEE, vol. 57,

pp. 813-814, May. 1969.

[30] M. Chaerani, "Hologram at MIT Museum," Aug. 2006.

[31] A. W. Lohmann and D. P. Paris, "Binary Fraunhofer Holograms, Generated by

Computer," Applied Optics, vol. 6, pp. 1739-1748, Oct. 1967.

[32] B. R. Brown and A. W. Lohmann, "Computer-generated Binary Holograms," IBM

Journal of Research and Development, vol. 13, pp. 160-168, Mar. 1969.

[33] G. V. Eleftheriades and Wong, A. M. H., "Holography-inspired screens for sub-

wavelength focusing in the near field," Microwave and Wireless Component Letters,

vol. 18, pp. 236-238, Apr. 2008.

[34] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With

Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972, pp. 378.

[35] K. Iizuka, Engineering Optics. ,2nd ed.Germany: Springer-Verlag, 1987, pp. 313-340.

[36] P. F. Checcacci, V. Russo and A. M. Scheggi, "Holographic antennas," IEEE

Transactions on Antennas and Propagation, vol. 18, no. 6, pp. 811-813, Nov. 1970.

REFERENCES 83

[37] K. Iizuka, "Microwave holograms and microwave reconstruction," IEE Electronics

Letters, vol. 5, pp. 26-28, Jan. 1969.

[38] K. Iizuka, M. Mizusawa, S. Urasaki and H. Ushigome, "Volume-type holographic

antenna," IEEE Transactions on Antennas and Propagation, vol. 23, no. 6, pp. 807-810,

Nov 1975.

[39] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation,

Interference and Diffraction of Light. ,7th ed.Cambridge: Cambridge University Press,

1999, pp. 633-673.

[40] A. M. H. Wong, C. D. Sarris and G. V. Eleftheriades, "Metallic transmission screen for

sub-wavelength focusing," IEE Electronics Letters, vol. 43, pp. 1402-1404, Dec. 2007.

[41] A. M. H. Wong and G. V. Eleftheriades, "Experimental verification of sub-wavelength

focusing via a holographic metallic screen," in IEEE Antennas and Propagation Society

International Symposium, 2008,

[42] L. Markley, A. M. H. Wong, Y. Wang and G. V. Eleftheriades, "Spatially Shifted Beam

Approach to Subwavelength Focusing," Physical Review Letters, vol. 101, pp. 113901,

Sep. 2008.

[43] L. Markley and G. V. Eleftheriades, "Two-dimensional subwavelength focusing using a

slotted meta-screen," Microwave and Wireless Component Letters, 2009. (in press)

[44] F. Oberhettinger, Tables of Fourier Transforms and Fourier Transforms of

Distributions. Germany: Springer-Verlag, 1990, pp. 21.

[45] A. V. Oppenheim, A. S. Willsky and S. Hamid, Signals and Systems. ,2nd ed.Upper

Saddle River, New Jersey: Prentice Hall, 1996, pp. 329.

by Alex M. H. Wong - University of Toronto T-Space...Alex M. H. Wong Master of Applied Science The...

Documents

Transcript of by Alex M. H. Wong - University of Toronto T-Space...Alex M. H. Wong Master of Applied Science The...