Guided Resonances in Photonic Crystal Slabs for Cavity ...

Guided Resonances in Photonic Crystal Slabs for Cavity Optomechanics

Catvu Huu Bui

Submitted in partial fulfillment of the

requirements for the degree of

Master of Science

in the Fu Foundation School of Engineering and Applied Science

COLUMBIA UNIVERSITY

2011

© 2011

Catvu Huu Bui

All rights reserved

ABSTRACT

Guided Resonances in Photonic Crystal Slabs

for Cavity Optomechanics

Catvu Huu Bui

In this thesis, the phenomenon of Fano-type guided resonances found in photonic crystal slab

structures is investigated numerically and experimentally as a means to enhance the reflectivity

of an ultrathin silicon nitride micromechanical membrane while preserving its thickness, low

eigenmode mechanical mass and high mechanical Q factor. This enhancement finds its use in the

growing field of cavity optomechanics, particularly in the membrane-in-the-middle

configuration, in which a 50-nm-thick silicon nitride membrane with a high mechanical Q is

coupled to the optical field inside a high-finesse Fabry-Perot cavity excited by a 1064 nm laser.

It is predicted that ground state cooling and the observations of quantum behaviors in such

system could be made feasible should a more demanding set of requirements, among which is an

improved reflectivity of the membrane, be satisfied. Addressing this requirement for high

reflectivity, this thesis proposes and demonstrates the incorporation of photonic crystals, in the

form of small-area square-lattice hole arrays, into the freestanding micromechanical membranes.

At normal incidence plane wave illumination, Fano-type guided resonances in such photonic

crystal membrane are excited and provide a mechanism via which strong reflectivity peaks could

be tuned to target wavelengths. To accomplish this, transmission spectral line shapes, field

distributions inside the membrane and photonic band diagrams are studied using finite-difference

time-domain and plane wave expansion methods. Then, using electron beam lithography,

photonic crystal structures designed for a transmission dip near 1064 nm are patterned into

freestanding silicon nitride membranes. The fabricated membranes are tested optically for

normal-incidence transmission spectra using a broadband source. Guided resonances and

improved reflectivity near 1064 nm are verified against simulations. Preservation of the high Q

factors of mechanical modes of patterned membranes is confirmed experimentally via

mechanical ringdown measurements, verifying that the patterned membranes retain important

characteristics in the context of optomechanics, namely their small thicknesses, low eigenmode

mechanical masses and high Q factors. By realizing in the same device a good reflector and a

high-Q mechanical resonator, these results should lead to improved performances of

optomechanical systems of the membrane-in-the-middle type and present a path towards

motional ground state cooling of such membrane as well as observations of related quantum

effects.

i

Contents

List of Figures ................................................................................................................................ ii

Acknowledgments ......................................................................................................................... v

Chapter 1 Introduction................................................................................................................ 1

1.1 Background and Motivation .................................................................................................. 1

1.2 Thesis Organization............................................................................................................... 8

1.3 Credit and Acknowledgements ............................................................................................. 8

Chapter 2 Photonic Crystal Slab Design via Numerical Simulations .................................. 14

2.1 Preliminary Design Considerations ..................................................................................... 14

2.2 Structural Tuning ................................................................................................................. 15

2.3 Band Structure and Mode Shapes ....................................................................................... 20

Chapter 3 Fabrication ............................................................................................................... 24

Chapter 4 Measurements and Discussions .............................................................................. 29

4.1 Mechanical Ringdown Measurements and Discussions ..................................................... 29

4.2 Optical Transmission Measurements and Discussions ....................................................... 31

4.3 Optomechanical Performance ............................................................................................. 36

Chapter 5 Summary .................................................................................................................. 41

ii

List of Figures

Figure 1.1. Schematic of: (a) a photonic crystal slab and (b) a distributed Bragg reflector. The red

arrow denotes direction of illumination. Both devices, via different mechanisms, could act as

optical filters or mirrors. Note the compactness of the photonic crystal slab.

Figure 1.2. Schematic of: (a) the “dispersive” membrane-in-the-middle optomechanical setup

and (b) the traditional “reflective” optomechanical setup.

Figure 2.1. Relevant structural parameters of the photonic crystal slab designed in this section:

lattice constant a, hole diameter d and thickness t. These parameters are used to tune the slab to

produce guided resonances at the target locations and with the desired shapes.

Figure 2.2. Computational domain (cross-sectional view) for 3D finite-difference time-domain

calculation of the normal-incidence transmission spectrum of a photonic crystal slab. The

domain consists of a single unit cell of the photonic crystal slab. Perfectly matched layer

absorbing boundary condition is set for the top and bottom surfaces. Bloch periodic boundary

condition is set for the remaining four surfaces perpendicular to the slab. The excitation source is

a plane wave generated at normal incidence to the slab. Transmitted field amplitude is recorded

at a monitor point placed on the other side of the slab.

Figure 2.3. 3D finite-difference time-domain simulations for normal-incidence transmission

spectra of a photonic crystal slab with nSiN = 2.15 and d/a = 0.3 for: t/a = 0.3, 0.1, 0.052 (top to

bottom, respectively). When t is fixed at 50 nm, the bottom panel corresponds to a slab having a

= 967 nm, d = 290 nm, and producing a transmission dip (red dashed rectangular) at ~ 1064 nm.

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Figure 2.4. (a) Computed band structure for the designed photonic crystal slab. Dotted line

denotes the light line, above which modes are leaky. The red dashed region contains the leaky

mode at Γ point that is responsible for the dip on the transmission spectrum at the target

wavelength, shown earlier in Fig. 2.3(bottom). (b) Close-up view of band structure within the red

dashed region and calculated electric field intensity distributions of the leaky modes at Γ point

within this region: doubly degenerate mode A at 0.896 (2πc/a) and singly degenerate mode B at

0.886 (2πc/a). Mode A appears in Fig. 2.3(bottom) as the transmission dip.

Figure 3.1. Commercial 1 mm × 1 mm × 50 nm SiNx membrane suspended from a 200 µm thick

silicon frame (Norcada Inc.).

Figure 3.2. Fabrication process: (I) Membrane frame attached to substrate; (II) PMMA spin-

coating and e-beam lithography; (III) Pattern transfer using RIE etch; and (IV) PMMA and

substrate removal.

Figure 3.3. SEM images showing part of the fabricated hole array at the center of the membrane.

Inset shows close-up view of air holes.

Figure 4.1. (a) Measured Q factors and resonant frequencies at 10-6 Torr and room temperature

of various mechanical modes of the SiNx membrane after being patterned with a 250 × 250

photonic crystal hole array at the center. Modes are indexed as (i,j) where i and j are positive

integers relating the eigenfrequencies as 2 2, 1,1 ( ) / 2i jv v i j= + . (b) A typical mechanical ringdown

of the photonic crystal membrane, shown here for mode (3,2) at ν3,2 = 498 kHz with ~ 3.2 s 1/e

lifetime. Solid line is the exponential fit.

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Figure 4.2. Setup for transmission spectrum measurement: BL - halogen broadband source, MO -

objective lenses, PL - linear polarizer with adjustable polarizer angle θ, OSA - optical spectrum

analyzer. Membrane is illuminated at the photonic crystal site. Incident angle of illumination

could be varied by rotating the sample in the direction shown (red arrow). All measurements are

normalized to the same measurements with no sample in place.

Figure 4.3. Measured normal-incidence transmission spectrum of the fabricated photonic crystal

membrane, shown with finite-difference time-domain simulation.

Figure 4.4. (a) Measured transmission spectra at varied polarizer angles (θ ~ 0o to 90o). (b)

Magnitude and linewidth of the transmission dip associated with each polarizer angle.

Figure 4.5. Measured transmission spectra at varied incidence angles of illumination (0o to ~ 5o,

in arbitrary steps). Note the coupling to the singly degenerate mode as incidence angle moves

away from surface-normal direction.

Figure 4.6. Computed optomechanical detuning curves of the membrane-in-the-middle platform,

at normal and enhanced membrane reflectivity. ωcav(x) is given in units of 2π×FSR where FSR =

2.374 GHz.

v

Acknowledgments

First and foremost, I would like to thank my advisor, Professor Chee Wei Wong, for his support

and guidance. Working on this thesis is without a doubt one of my most inspiring and fruitful

experiences. The academic growth offered by and gained from this work is tremendously

meaningful to me. I am deeply indebted to Professor Wong for giving me this invaluable

opportunity to perform research at the Optical Nanostructures Laboratory.

It has been fortunate for me to join the wonderful team of students and researchers at the

Optical Nanostructures Laboratory. I would particularly like to thank Jiangjun Zheng. Over the

course of our collaboration, time and time again he has surmounted an impossible schedule to

deliver timely and valuable results that made this research effort complete. For his technical

expertise, his dedication to the task, and his willingness to help others, he has my deepest

admiration, respect and gratitude. I would also like to thank Lennon Lee, who had worked on

related simulations and fabrication before I came to the laboratory and has provided me with an

abundance of assistance during my early steps. I would like to thank Jie Gao and James

McMillan for their insightful discussions at various points during my time with the group. For all

other former and current members of the laboratory, it has been a privilege to be part of the team

and I wish you the best.

This work is done in collaboration with the research group of Professor Jack Harris at

Yale University. I would like to thank Professor Harris and his team for their valuable

contributions to the research. Within the team, I would like to thank Scott Hoch and Israel-Marc

Kositsky for their mechanical ringdown measurements, and Andrew Jayich for discussions

regarding the calculation of the detuning curve.

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I have received much help during my time working in the Columbia University

cleanroom and would like to thank the cleanroom support staff, in particular Geoff Goold and

Linus Fetter, for their training and fabrication advices. I would also like thank Hooman

Hosseinkhannazer from Norcada Inc. for useful information on their excellent silicon nitride

membranes.

Finally, I would like to thank my parents for their unconditional love and unwavering

support throughout my life. This thesis is thus dedicated to them.

vii

Dedicated to my parents, Bùi Hữu Lâm and Vũ Thị Ngọc Hương.

1

Chapter 1

Introduction

1.1 Background and Motivation

Photonic crystals, first introduced by Yablonovitch and John in 1987 [1], [2], are electromagnetic

media within which dielectric constants are periodically modulated [3]. A consequence of such

periodicity is the existence of photonic band gaps: frequency ranges in which electromagnetic

waves cannot propagate through the media. These gaps are the optical analogue to

semiconductor band gaps, which arise from atomic lattices acting on electron wavefunctions.

This ability of photonic crystals to prevent electromagnetic waves of specific frequencies to flow

in certain directions has opened up novel and potent ways to control and manipulate photons.

Such promising potential offered by photonic crystals has led to extensive research into their

theories, fabrication techniques and applications [3].

An important class of photonic crystals is the photonic crystal slab: a structure that has

two-dimensional periodicity in the plane of the slab but finite thickness in the third direction.

These hybrid structures are particularly practical since they retain many of the useful features of

true two-dimensional photonic crystals while their simpler planar geometries allow easier

fabrication using existing lithography techniques. Photonic crystal slabs are able to support

guided modes using band gaps for in-plane propagation and index guiding, a generalization of

total internal reflection, for confinement in the third dimension [3], [4]. Photonic crystal slabs

using in-plane waveguiding have been exploited to realize waveguides, cavities and other

building blocks of photonic components and circuits [3–8].

2

In contrast to the ability to support guided modes, which are completely confined in the

plane of the slab, photonic crystal slabs also support discrete leaky modes, or guided resonances,

that can couple to the continuum of free-space radiation modes [9], [10]. Such guided resonances

could be excited by and interfere with out-of-plane external incident light, giving rise to complex

asymmetrical line shapes on the normal-incidence reflection and transmission spectra. Such

resonant spectral features are often sharp over narrow frequency ranges and therefore useful for

filtering functions. When used in this fashion, i.e. illuminated with free-space out-of-plane beams

[Fig. 1.1(a)], photonic crystal slabs offer new and promising ways to realize a wide array of

compact devices in sensing and communication applications and, as a result, have been a subject

of significant interest recently [9]. Guided resonances in photonic crystal slabs have been studied

over a wide selection of wavelength ranges including visible [10–12], near-infrared [10], [13],

infrared [14], [15], submillimeter [16], [17], and millimeter [18] wavelengths, with important

applications including optical filters [15], [19], [20], broadband mirrors [21], polarization

splitters [22], displacement sensors [23], [24], acoustic sensors [25] and more efficient designs of

light-emitting diodes [26] and lasers [27]. A major advantage of photonic crystal slabs, compared

to the traditional implementation of thin film dielectric filters and mirrors, i.e., distributed Bragg

reflectors, is their compactness (Fig. 1.1). In contrast to a multilayer dielectric stack, which

imposes strict requirements on each layer’s thickness and material stress as well as restrictions

on mass and size, a photonic crystal slab requires only a single layer and therefore offers relative

ease of fabrication and integration into microelectromechanical systems.

3

Figure 1.1. Schematic of: (a) a photonic crystal slab and (b) a distributed Bragg reflector. The red

arrow denotes direction of illumination. Both devices, via different mechanisms, could act as

optical filters or mirrors. Note the compactness of the photonic crystal slab.

(a) (b)

4

In this thesis, I present a new application of guided resonances in the field of cavity

optomechanics, namely the use of photonic crystal hole-array structures as a means to enhance

normal-incidence reflectivity of an ultrathin high-mechanical-Q micromechanical membrane, a

central element in certain optomechanical setups.

The field of cavity optomechanics, one where the coupling of optical and mechanical

degrees of freedom via radiation pressure is studied, has seen significant progress recently [28],

[29]. Thanks to the availability of high-finesse mirrors and advances in micro- and

nanofabrication, a series of innovative experimental systems where observations of

optomechanical coupling are possible has been realized. While current designs are diverse with

mechanical elements employing a wide range of masses (typically from 10-12 to 1 g) and

frequencies (typically from 1kHz to 100MHz) [29], a relatively common approach involves a

setup that consists of a conventional high-reflectivity Bragg mirror on one end and a movable

mirror on the other end, forming an optical cavity [Fig 1.2(b)]. The movable mirror serves as the

mechanical element within the optomechanical system, providing mechanical degrees of

freedom. Such mirror has been experimentally realized as a suspended gram-scale mirror [30],

micrometer-sized mirror on an atomic force microscopy cantilever [31], and doubly-clamped

coated micromirror [32], [33], to name a few. Another important geometry employs a

microtoroid optical cavity, where the optical whispering gallery modes couple to mechanical

breathing modes [34], [35].

5

Figure 1.2. Schematic of: (a) the “dispersive” membrane-in-the-middle optomechanical setup

and (b) the traditional “reflective” optomechanical setup.

(a)

(b)

Movable membrane

Movable end mirror

6

Central to this thesis is yet another optomechanical experimental platform: the so-called

‘membrane-in-the-middle’ configuration [Fig 1.2(a)], where an ultrathin SiNx membrane is

placed inside a high-finesse Fabry-Perot cavity with two fixed end-mirrors excited by a

stabilized 1064 nm laser [36–38]. Such geometry offers two important features compared to

other setups. First, optical and mechanical functionalities are separated into different structures.

Other examples listed above, along with most optomechanical systems described in the literature,

all rely on a single structure to provide both optical and mechanical performances, i.e. a high

optical finesse (typically from 103 to 105) and a high mechanical quality factor (typically from

103 to 105) [29]. This requirement of integrating mechanical elements, which tend to be pliable

and have small masses and sizes, into optical cavities, which tend to be bulky, represents a

technical difficulty. The membrane-in-the-middle geometry circumvents this issue by utilizing a

50-nm-thick SiNx membrane with high mechanical-Q (up to 106) as the mechanical element and

a fixed Fabry-Perot cavity with high-finesse (up to 104) for optical functionality [36]. Second, the

membrane-in-the-middle approach allows a quadratic dependence of cavity resonances ωcav on

membrane displacement x, permitting the possibility of quantum non-demolition readout of

membrane’s phonon number [36–39]. This quadratic coupling, where ωcav(x) ∝ x2, occurs when

the membrane is placed at a node or antinode of the intracavity standing wave. The strength of

such optomechanical coupling is determined by the curvature of the detuning ωcav(x) [36]. This

curvature in turns depends on the reflectivity |rm|2 of the membrane, which must approach ~0.99

for quantum effects to be observable [36]. The nearly-transparent 50-nm-thick SiNx used in

reference [36] thus represents an important technical obstacle to having sufficient coupling

strength. The difficulty comes from the fact that traditional solutions for increasing reflectivity,

such as metal coatings or multilayer distributed Bragg reflectors, would not be appropriate in this

7

case for they would compromise the remarkably high mechanical Q [40], [41], low motional

mass and low optical absorption of the SiNx membrane, all of which are essential to the

performance of the membrane-in-the-middle system. For cavity optomechanics in general, high-

Q and low-mass mirrors of various types have been realized in different experimental setups

[42–44]. However, such elements so far have not achieved a comparable combination of

motional mass and Q factor seen in thin SiNx membranes used in references [36], [40], [41].

Such challenge of increasing reflectivity while retaining the high Q factor of the SiNx

membranes has also motivated alternative approaches which seek to circumvent such

requirement by using avoided crossings between cavity modes to increase the curvature of the

detuning ωcav(x) [38], [39].

In this thesis, guided resonances in photonic crystal slab structures are demonstrated as a

means to optimize ultrathin high-Q SiNx membranes for increased reflectivity, thus allowing

enhanced optomechanical coupling in systems of membrane-in-the-middle type. The photonic

crystals are designed using numerical simulations including 3D finite-difference time-domain

method for transmission spectra and plane wave expansion method for band diagrams. The

photonic crystal slabs are structurally tuned to produce normal-incidence transmission dips near

the target wavelength of 1064 nm, which is the laser wavelength used in reference [36]. Electron

beam lithography is used to pattern small-area hole arrays in 50-nm-thick commercial SiNx

membranes as a reflectivity-enhancing modification. Transmission measurements verify guided

resonances and transmission dips near 1064 nm. The membranes’ high Q factors are confirmed

by mechanical ringdown measurements to be preserved in the presence of photonic crystal

structures. By realizing a good optical reflector and a high Q, low mass, ultrathin mechanical

resonator in the same device, these results would allow an improved performance of the

8

membrane-in-the-middle system via increased optomechanical coupling, thus presenting a path

towards ground state cooling and observations of related quantum effects in such system.

1.2 Thesis Organization

Chapter 2 of the thesis discusses results from numerical simulations of relevant aspects of

photonic crystal slabs, which lead to the final structural design of the photonic crystal slab.

Chapter 3 describes the fabrication process where photonic crystal structures are patterned into

SiNx membranes. Chapter 4 discusses results from mechanical ringdown and optical

transmission measurements of the fabricated photonic crystal membranes.

1.3 Credit and Acknowledgements

This thesis is partially adapted from the manuscript “High-reflectivity, high-Q micromechanical

membranes via guided resonances for enhanced optomechanical coupling” by Catvu H. Bui,

Jiangjun Zheng, S. W. Hoch, Lennon Y. T. Lee, J. G. E. Harris and Chee Wei Wong, at this time

under review by Applied Physics Letters. Catvu H. Bui and Jiangjun Zheng contribute equally.

The work in this thesis was done in collaboration between the above authors. Jiangjun

Zheng and I are co-first authors. I performed the simulations (Chapter 2), fabrication (Chapter 3)

and prepared the manuscript. Jiangjun Zheng set up and performed the optical measurements

(Section 4.2, Chapter 4). S. W. Hoch and J. G. E. Harris performed the mechanical ringdown

measurements (Section 4.1, Chapter 4). Lennon Y. T. Lee provided assistance during simulation

and fabrication steps. Chee Wei Wong supervised the research effort and edited the manuscript.

Fabrication was performed at the Columbia University cleanroom.

9

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[38] J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nature Physics, vol. 6, no. 9, pp. 707–712, Jun. 2010.

[39] J. C. Sankey, A. M. Jayich, B. M. Zwickl, C. Yang, and J. G. E. Harris, “Improved ‘Position Squared’ Readout Using Degenerate Cavity Modes,” in Proceedings of the XXI International Conference on Atomic Physics, Storrs, CT, USA, 2008, pp. 131–149.

[40] B. M. Zwickl, W. E. Shanks, A. M. Jayich, C. Yang, A. C. Bleszynski Jayich, J. D. Thompson, and J. G. E. Harris, “High quality mechanical and optical properties of commercial silicon nitride membranes,” Applied Physics Letters, vol. 92, no. 10, p. 103125, 2008.

[41] D. Wilson, C. Regal, S. Papp, and H. Kimble, “Cavity Optomechanics with Stoichiometric SiN Films,” Physical Review Letters, vol. 103, no. 20, Nov. 2009.

[42] D. Kleckner, W. Marshall, M. de Dood, K. Dinyari, B.-J. Pors, W. Irvine, and D. Bouwmeester, “High Finesse Opto-Mechanical Cavity with a Movable Thirty-Micron-Size Mirror,” Physical Review Letters, vol. 96, no. 17, May 2006.

[43] G. D. Cole, S. Gröblacher, K. Gugler, S. Gigan, and M. Aspelmeyer, “Monocrystalline Al[sub x]Ga[sub 1−x]As heterostructures for high-reflectivity high-Q micromechanical resonators in the megahertz regime,” Applied Physics Letters, vol. 92, no. 26, p. 261108, 2008.

[44] U. Kemiktarak, M. Metcalfe, M. Durand, and J. Lawall, “Mechanically Compliant High Contrast Grating Mirrors for Radiation Pressure Cooling,” presented at the Conference on Lasers and Electro-Optics 2011, Baltimore, MD, USA, 2011.

14

Chapter 2

Photonic Crystal Slab Design

via Numerical Simulations

Guided resonances in photonic crystal slabs are modes that have their electro-magnetic power

weakly confined within the slab and could couple to external radiation [1], [2]. This interference

coupling between discrete, leaky modes supported by the slab and the continuum of free-space

modes produces out-of-plane spectra with complex asymmetrical line shapes with sharp resonant

features. Such line shapes are characteristic of Fano resonances, often observed in atomic or

condensed-matter systems [3].

In this section, we design, via ab initio simulations, a photonic crystal slab suitable for

use in a membrane-in-the-middle optomechanical platform. The slab design would support a

guided resonance that would in turn produce a normal-incidence transmission dip near 1064 nm,

while maintaining a mechanical performance equal to or better than previously demonstrated in

references [4–7].

2.1 Preliminary Design Considerations

Without loss of generality, a square lattice of air holes in a dielectric slab would be used for the

photonic crystal design. Unless noted otherwise, this would be the case for all subsequent

discussions.

Also, to preserve the mechanical performance of the photonic crystal slab, the slab is

designed to be suspended in air and to be made of silicon nitride, with a refractive index of nSiN ~

15

2.15. Silicon nitride is the dielectric material of choice due to its low loss at near-infrared

wavelengths and its excellent mechanical properties [2], [4], [8].

2.2 Structural Tuning

Constitutive and structural parameters of a photonic crystal slab, such as the slab thickness t,

hole diameter d, lattice constant a, and refractive index n (Fig. 2.1), have strong effects on

resultant guided resonances and spectral features [1], [2], [9], [10]. By adjusting such parameters,

the locations of the resonances on transmission and reflection spectra and their line shapes could

be tuned.

Figure 2.1. Relevant structural parameters of the photonic crystal slab designed in this section:

lattice constant a, hole diameter d and thickness t. These parameters are used to tune the slab to

produce guided resonances at the target locations and with the desired shapes.

a d t

16

To structurally tune the photonic crystal slab, we study the dependence of guided

resonances’ locations on the normalized thickness t/a of the slab [9]. 3D finite-difference time-

domain method (RSoft FullWAVE) is used to compute transmission spectra for different values

of t/a. The computational domain (Fig. 2.2) consists of a unit cell of the photonic crystal slab

with varied normalized thicknesses t/a and a fixed normalized hole diameter d/a. The top and

bottom boundaries are set to the Perfectly Matched Layer condition while the four surrounding

surfaces, perpendicular to the slab, are set to a Bloch periodic boundary condition. For the

excitation source, a plane wave is generated at normal incidence to the slab. A monitor point on

the other side of slab records field amplitudes in the time domain, from which transmission

spectra are obtained via Fourier transform. All calculated spectra are normalized to those

obtained with identical setup but without the slab.

17

Figure 2.2. Computational domain (cross-sectional view) for 3D finite-difference time-domain

calculation of the normal-incidence transmission spectrum of a photonic crystal slab. The

domain consists of a single unit cell of the photonic crystal slab. Perfectly matched layer

absorbing boundary condition is set for the top and bottom surfaces. Bloch periodic boundary

condition is set for the remaining four surfaces perpendicular to the slab. The excitation source is

a plane wave generated at normal incidence to the slab. Transmitted field amplitude is recorded

at a monitor point placed on the other side of the slab.

Single unit cell of photonic crystal

Plane wave source

Perfectly matched layer

Periodic boundary condition

Periodic boundary condition

Perfectly matched layer

× Monitor point

18

Computed results are shown in Fig. 2.3. As t/a decreases resonant frequencies are

observed to shift to higher values, as expected from previous reports [9]. With this effect

informing the tuning process, it is found that a photonic crystal slab with refractive index nSiN =

2.15, lattice constant a = 967 nm, slab thickness t = 50nm, and hole diameter d = 290 nm,

corresponding to t/a = 0.052 and d/a = 0.3, is able to produce a transmission dip near the target

freespace wavelength λ ~ 1064 nm. The calculated transmission spectrum of such slab is shown

in Fig. 2.3(bottom). It should be noted that the simulations are performed with measurement

units normalized to the lattice constant a. In choosing structural parameters in terms of absolute

values for our final design, we set the slab thickness at 50 nm due to practical reasons during the

fabrication process, to be discussed in the next chapter.

19

Figure 2.3. 3D finite-difference time-domain simulations for normal-incidence transmission

spectra of a photonic crystal slab with nSiN = 2.15 and d/a = 0.3 for: t/a = 0.3, 0.1, 0.052 (top to

bottom, respectively). When t is fixed at 50 nm, the bottom panel corresponds to a slab having a

= 967 nm, d = 290 nm, and producing a transmission dip (red dashed rectangular) at ~ 1064 nm.

20

2.3 Band Structure and Mode Shapes

To further investigate the properties of the photonic crystal design, band structure of the slab is

computed using the plane wave expansion method (RSoft BandSOLVE). At the Γ point on the

band structure (Fig. 2.4), a leaky mode is found at normalized frequency 0.896(2πc/a) [mode A

in Fig. 2.4(b)]. This leaky mode indeed corresponds to the resonant frequency found previously

on the transmission spectrum in the red dashed region in Fig. 2.3(bottom). It is also evident from

the band structure that not all modes at Γ point manifest themselves on the normal-incidence

transmission spectrum. To examine this behavior, electric field intensity distribution of some

leaky modes are computed and shown in Fig. 2.4(b) along with their locations. Modes that are

doubly degenerate, such as mode A in Fig. 2.4(b), appears on the normal-incidence transmission

spectrum. On the other hand, singly degenerate modes, such as mode B in Fig. 2.4(b), are unable

to couple to a plane wave propagating normal to the slab due to symmetry mismatch and

therefore do not appear on the transmission spectrum [1], [2], [11]. Such analysis proves useful

later on when non-normal illumination is introduced during optical measurement and able to

produce additional transmission dips that are otherwise not present under normal-incidence plane

wave excitation.

21

Figure 2.4. (a) Computed band structure for the designed photonic crystal slab. Dotted line

denotes the light line, above which modes are leaky. The red dashed region contains the leaky

mode at Γ point that is responsible for the dip on the transmission spectrum at the target

wavelength, shown earlier in Fig. 2.3(bottom). (b) Close-up view of band structure within the red

dashed region and calculated electric field intensity distributions of the leaky modes at Γ point

within this region: doubly degenerate mode A at 0.896 (2πc/a) and singly degenerate mode B at

0.886 (2πc/a). Mode A appears in Fig. 2.3(bottom) as the transmission dip.

22

Bibliography



[3] A. Miroshnichenko, S. Flach, and Y. Kivshar, “Fano resonances in nanoscale structures,” Reviews of Modern Physics, vol. 82, no. 3, pp. 2257–2298, Aug. 2010.






23

[9] T. Prasad, V. L. Colvin, and D. M. Mittleman, “Dependence of guided resonances on the structural parameters of terahertz photonic crystal slabs,” Journal of the Optical Society of America B, vol. 25, no. 4, p. 633, Mar. 2008.

[10] N. Ou, J. H. Shyu, H. M. Lee, and J. C. Wu, “Diameter-dependent guided resonance of dielectric hole-array membrane,” Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures, vol. 27, no. 6, p. 3183, 2009.

[11] T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Physical Review B, vol. 63, no. 12, Mar. 2001.

24

Chapter 3

Fabrication

In this chapter, photonic crystal slabs are fabricated according to structural parameters described

in the previous chapter. For our samples, we start with commercially available low-stress SiNx

membranes (Norcada Inc.). These membranes, according to manufacturer’s specifications, have

a 1 mm × 1 mm square surface with a thickness of 50 nm, and are suspended from 200 µm thick

silicon frames (Fig 3.1). Membranes of this particular model are identical to those used in

references [1]. Those are expected to have excellent fabrication quality in terms smoothness,

flatness and cleanliness and have demonstrated remarkably high mechanical Q factors in excess

of 1 × 106 at room temperature [2].

Figure 3.1. Commercial 1 mm × 1 mm × 50 nm SiNx membrane suspended from a 200 µm thick

silicon frame (Norcada Inc.).

25

Photonic crystal structures are patterned in the suspended SiNx membranes by electron-

beam lithography, with PMMA as resist, and reactive ion etching (RIE). The fabrication process

flow is shown in Fig 3.2. For compatibility with spin-coating and vacuum processes, the silicon

frame supporting the membrane is first temporarily attached to a separate silicon substrate using

resist as an adhesive [Fig. 3.2(I)]. This silicon substrate is pre-etched with small venting trenches

to promote pressure equalization between both sides of the suspended SiNx membrane. A6

PMMA is spin-coated on the membrane to a thickness of ~ 300 nm. As the membrane support

frame has a square shape, resist thickness uniformity is expected to be poor in areas close to the

edges. Contact profilometer measurement, however, verifies good uniformity in the area

surrounding the membrane. After baking (at 180 oC), the membrane is loaded into a scanning

electron microscope (FEI Company) fitted with the Nanometer Pattern Generation System (JC

Nabity Lithography Systems) for electron-beam lithography. After the hole array pattern is

written, the sample is developed in a MIBK:IPA solution and then rinsed in IPA [Fig 3.2 (II)].

The pattern is then transferred to the SiNx layer in a CF4/O2 RIE etcher (Technics) using a recipe

(75 W power, 100 mTorr pressure) optimized for a PMMA:SiNx selectivity of 1:1 [Fig. 3.2 (III)].

Finally, PMMA is removed along with the temporary holding substrate with acetone [Fig. 3.2

(IV)]. Subsequent O2 plasma ensures cleanliness.

SEM images (Fig 3.3) demonstrate the fabrication quality. The hole array is positioned at

the center of the membrane and consists of 250 × 250 holes. The photonic crystal area is

minimized to only ~ 6% of the total membrane area to preserve its mechanical properties while

remains large enough to cover the 1/e2 diameter of the Nd:YAG laser’ TEM00 cavity mode (90

µm [1]).

26

Figure 3.2. Fabrication process: (I) Membrane frame attached to substrate; (II) PMMA spin-

coating and e-beam lithography; (III) Pattern transfer using RIE etch; and (IV) PMMA and

substrate removal.

27

Figure 3.3. SEM images showing part of the fabricated hole array at the center of the membrane.

Inset shows close-up view of air holes.

28

Bibliography



29

Chapter 4

Measurements and Discussions

4.1 Mechanical Ringdown Measurements and Discussions

To be used as the mechanical element in the membrane-in-the-middle optomechanical platform,

it is crucial for the SiNx membrane to preserve its high mechanical Q factors in the presence of

the photonic crystal structures. To verify Q factors of the patterned membranes, mechanical

ringdown measurements are performed using method similar to that in reference [1].

The membrane is mounted on a piezoelectric actuator under a 10-6 Torr vacuum at room

temperature and driven at its mechanical resonances. The vibrational modes are identified and

indexed as νi,j where i and j are positive integers relating the eigenfrequencies as

2 2, 1,1 ( ) / 2i jv v i j= + , similar to a stressed square membrane. The Q factor of each mode is

obtained by rapidly switching off the piezoelectric actuator and monitoring the mechanical

ringdown using a low-finesse interferometer operated at a wavelength of 635 nm. Mode

assignments are further confirmed by sweeping the laser across the surface of the membrane and

noting the expected locations of vibrational nodes. The modes are carefully driven to avoid the

nonlinear Duffing oscillator regime [2]. Measured resonant frequencies and associated Q factors

for several vibrational modes are plotted in Fig. 4.1(a). A typical ringdown is shown in Fig.

4.1(b). The results show Q factors to remain at high values of up to ~ 5 × 106, similar to those

reported for unpatterned membranes [1], [3], [4], thus establishing that introducing the hole array

indeed does not appreciably degrade the membrane’s high Q factors.

30

Figure 4.1. (a) Measured Q factors and resonant frequencies at 10-6 Torr and room temperature

of various mechanical modes of the SiNx membrane after being patterned with a 250 × 250

photonic crystal hole array at the center. Modes are indexed as (i,j) where i and j are positive

integers relating the eigenfrequencies as 2 2, 1,1 ( ) / 2i jv v i j= + . (b) A typical mechanical ringdown

of the photonic crystal membrane, shown here for mode (3,2) at ν3,2 = 498 kHz with ~ 3.2 s 1/e

lifetime. Solid line is the exponential fit.

31

4.2 Optical Transmission Measurements and Discussions

Broadband transmission measurements are performed for the fabricated photonic crystal

membranes. The experimental setup is shown in Fig 4.2. Light from a 900 nm - 1200 nm

broadband source (BL) coupled to a multimode fiber is collected by an objective lens (MO) and

illuminates the membrane at the photonic crystal site. Transmitted light is collected by a second

object lens (MO) and coupled into a single mode fiber connected to an optical spectrum analyzer

(OSA). A linear polarizer (PL) is placed before the second objective lens with adjustable

polarizer angle θ. Incident angle of illumination could be varied by rotating the sample in the

direction shown in Fig. 4.2 (red arrow). For alignment, imaging, instead of focusing, method is

used to minimize the beam angular spread. The transmission signal is at least 10 dB higher than

the optical spectrum analyzer noise floor. All transmission measurements are normalized to the

same measurements with no sample in place. Transmission is measured at normal incidence,

varied polarizer angles, and varied incidence angles.

32

Figure 4.2. Setup for transmission spectrum measurement: BL - halogen broadband source, MO -

objective lenses, PL - linear polarizer with adjustable polarizer angle θ, OSA - optical spectrum

analyzer. Membrane is illuminated at the photonic crystal site. Incident angle of illumination

could be varied by rotating the sample in the direction shown (red arrow). All measurements are

normalized to the same measurements with no sample in place.

Measured normal-incidence transmission spectrum is shown in Fig 4.3 along with finite-

difference time-domain simulation. A pronounced transmission dip is observed at ~ 1035 nm,

confirming the guided resonance. The slight wavelength deviation from the target wavelength of

1064 nm, predicted by finite-difference time-domain simulation, could be explained by

fabrication-related disorder or the fact that the membrane thickness of 50 nm given by the

manufacturer is not exact. It should also be noted that the suppression of transmission could be

made broadband, thus lessening the requirement of having a guided resonance at exactly 1064

nm. This could be accomplished via the dependence of resonance linewidth on the diameter of

the air hole [5], an effect that has been previously exploited to create guided-mode mirrors with

strong reflection over a spectral band as broad as ~ 300 nm [6].

33

The normal-incidence transmission measurement is then repeated for different polarizer

angles θ ranging from 0o to 90o. The results are shown in Fig 4.4. In all cases, near 1035 nm, we

observe transmission dips in the range of 43% to 59% with linewidths varying from 3.1 nm to

6.5 nm [Fig 4.4(b)]. The spectra are found to be relatively polarization-insensitive due to the

rotational symmetry of the circular-hole square lattice, similar to previous reports [6–8].

Figure 4.3. Measured normal-incidence transmission spectrum of the fabricated photonic crystal

membrane, shown with finite-difference time-domain simulation.

34

Figure 4.4. (a) Measured transmission spectra at varied polarizer angles (θ ~ 0o to 90o). (b)

Magnitude and linewidth of the transmission dip associated with each polarizer angle.

(b)

(a)

35

Transmission spectra are also measured at different angles of incidence with fixed

polarization (Fig. 4.5). When the angle of incidence is varied from 0o to ~ 5o, the dip at ~ 1035

nm become less pronounced and another dip appears at ~ 1070 nm. This represents the coupling

of non-normal components of illumination to a singly degenerate mode, specifically to mode B

in the band structure shown previously in Fig. 2.4(b) in chapter 2 [5], [9], [10].

Figure 4.5. Measured transmission spectra at varied incidence angles of illumination (0o to ~ 5o,

in arbitrary steps). Note the coupling to the singly degenerate mode as incidence angle moves

away from surface-normal direction.

36

In principles, near-zero transmission could be achieved. The performance observed here

is most likely limited by fabrication-related disorder. Since various types of structural disorder

affect the guided resonances’ spectra differently [6], [11], we instead quantify losses from

disorder by using a theoretical model for a single guided resonance, first described in reference

[5] and modified in reference [12] to account for losses. We use the term τ/τloss in such model to

quantify losses, where 1/τ is the decay rate of the guided resonance and 1/τloss is the extra decay

term associated with all losses caused by fabrication-related disorder. We estimate τ/τloss to be ~

1 for the spectral performance observed here. For quantum effects to be detectable, membrane

reflectivity needs to approach ~ 0.99, which is estimated to correspond to τ/τloss approaching ~

0.01. Although challenging, as fabrication becomes optimized, this requirement should be

achievable.

4.3 Optomechanical Performance

In the context of optomechanics, these results translate to a membrane of enhanced reflectivity

|rm|2 of up to 57 % at the target wavelength, assuming a negligible absorption of the ultrathin

SiNx membrane [3], [13]. The background of the spectrum in Fig. 4.3 corresponds to the

reflectivity of an unpatterned SiNx membrane [5] and is ~ 15 %, similar to reported values [3],

[14]. This work thus represents a ~ 4 × increase in reflectivity. To quantify the improvement on

optomechanical performance, we calculate and plot in Fig. 4.6 the expected cavity detuning for a

cavity containing such membrane as a function of membrane position x, given by reference [15]

−−+= )/π4(1cos|m|1cos)/( /)(cav r λφω xrLcLcx

37

where the wavelength λ = 1064 nm and cavity length L = 6.313 cm with a free spectral range

FSR of 2.374 GHz. rφ is the complex phase of rm. As mentioned previously, optomechanical

coupling strength increases with the curvature of ωcav(x). For a patterned membrane of |rm|2 =

57%, the detuning curve achieves π2/)(cav xω ′′ = 108 kHz nm-2 at its extrema, a marked

improvement over π2/)(cav xω ′′ = 42 kHz nm-2 for an unpatterned membrane of |rm|2 = 15%. Such

enhanced cavity detuning (Fig. 4.6) also shows an improvement in the traditional linear coupling

regime when the membrane is not placed at a node or antinode of the intracavity mode.

38

Figure 4.6. Computed optomechanical detuning curves of the membrane-in-the-middle platform,

at normal and enhanced membrane reflectivity. ωcav(x) is given in units of 2π×FSR where FSR =

2.374 GHz.

39

Bibliography


[2] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer-Verlag, 1983.




[6] O. Kilic, S. Kim, W. Suh, Y.-A. Peter, A. S. Sudbø, M. F. Yanik, S. Fan, and O. Solgaard, “Photonic crystal slabs demonstrating strong broadband suppression of transmission in the presence of disorders,” Optics Letters, vol. 29, no. 23, p. 2782, 2004.

[7] V. Lousse, W. Suh, O. Kilic, S. Kim, O. Solgaard, and S. Fan, “Angular and polarization properties of a photonic crystal slab mirror,” Optics Express, vol. 12, no. 8, p. 1575, 2004.

[8] Z. Qiang, H. Yang, L. Chen, H. Pang, Z. Ma, and W. Zhou, “Fano filters based on transferred silicon nanomembranes on plastic substrates,” Applied Physics Letters, vol. 93, no. 6, p. 061106, 2008.


40

[10] T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Physical Review B, vol. 63, no. 12, Mar. 2001.

[11] T. Prasad, V. L. Colvin, and D. M. Mittleman, “The effect of structural disorder on guided resonances in photonic crystal slabs studied with terahertz time-domain spectroscopy,” Optics Express, vol. 15, no. 25, p. 16954, 2007.

[12] W. Suh, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Applied Physics Letters, vol. 82, no. 13, p. 1999, 2003.


[14] J. C. Sankey, A. M. Jayich, B. M. Zwickl, C. Yang, and J. G. E. Harris, “Improved ‘Position Squared’ Readout Using Degenerate Cavity Modes,” in Proceedings of the XXI International Conference on Atomic Physics, Storrs, CT, USA, 2008, pp. 131–149.


41

Chapter 5

Summary

In this thesis, we demonstrate enhanced reflectivity and preserved high mechanical Q factors,

typically in excess of 1 × 106 with the (Qi,j × νi,j) product reaching 3.3 × 1012 Hz, of SiNx

membranes. The enhanced reflectivity, along with the low eigenmode mass and small optical

loss of the membranes, should allow significantly improved optomechanical coupling via

increased radiation pressure per photon, presenting a path for membrane-in-the-middle type

optomechanical systems to achieve motional quantum ground state cooling and observations of

related quantum effects.

Improvements for photonic crystal membranes described in this work would involve

further refinement of the fabrication process. Current process allows a direct modification of a

freestanding SiNx membrane. However, it also means that during the etching step, the etch sites

are without a bulk silicon substrate, which puts them at a disadvantageous situation in terms of

heat dissipation. Potentially inadequate cooling during etching might lead to a reduction in etch

quality. One solution to this risk is to perform lithography and etching before the releasing the

membrane.

Another further consideration involves lessening the requirement of producing a guided

resonance at a specific target central wavelength by making the reflection broadband. As

mentioned previously, this could be accomplished by a design with a larger hole diameter.

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