Applications of Micro/Nanoscale Optical

Applications of Micro/Nanoscale Optical Resonators: Plasmonic Photodetectors and Double-Disk Cavity Optomechanics - CaltechTHESIS
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Applications of Micro/Nanoscale Optical Resonators: Plasmonic Photodetectors and Double-Disk Cavity Optomechanics
Citation
Rosenberg, Jessie C.
(2010)
Applications of Micro/Nanoscale Optical Resonators: Plasmonic Photodetectors and Double-Disk Cavity Optomechanics.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/GA8B-F134.
Abstract
Optical resonators present the potential to serve vital purposes in many emergent technologies that require spectral filtering, high optical intensities, or optical delays. By scaling down the optical resonators to the micro or nanoscale, the relevant phenomena can increase significantly in magnitude, while the device geometries become suitable for chip-scale and integrated processing. In this thesis, research is presented on several valuable resonator geometries and implementations, beginning with a more standard all-optical design, and continuing on to investigate the novel phenomena and applications which are made possible when optical and mechanical structures can be synergistically combined.
First, the design and experimental implementation of a plasmonic photonic crystal spectral and polarization filtering element is presented. This resonator scheme, in addition to allowing for a tailorable frequency and polarization response for single detector pixels, also increases the absorption of a thin layer of detector material by utilizing the unique optical properties of metal to confine light more tightly within the detector active region. Demonstrated in the valuable mid-infrared regime, this method of producing pixel-integrated multispectral detectors could find application in biological sensing and spectroscopy, missile tracking and guidance, and night vision.
Following this discussion, progress is presented in the relatively new field of cavity optomechanics: utilizing mechanically compliant optical resonators to couple to, control, and read out mechanical motion via optical forces. The use of optical resonators allows the generally weak optical forces to be increased in strength by orders of magnitude due to the many passes light makes within the resonator, while miniaturizing optomechanical devices into a convenient form factor for on-chip applications. Using a fully silicon-compatible double-disk-geometry optomechanical resonator, extremely large optomechanical coupling and very high optical quality factors are shown, enabling the demonstration of regenerative mechanical amplification, high compression factor optomechanical cooling, coherent mechanical mode mixing, and wide-bandwidth all-optical wavelength routing. Applications to ground-state cooling of mesoscopic devices, tunable optical buffering, photonic-phononic quantum state transfer, channel routing/switching, pulse trapping/release, and tunable lasing are discussed.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
optomechanics; plasmonics; optical resonators; detectors; microcavities; photonics
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Painter, Oskar J.
Thesis Committee:
Vahala, Kerry J. (chair)
Schwab, Keith C.
Libbrecht, Kenneth George
Painter, Oskar J.
Defense Date:
8 December 2009
Record Number:
CaltechTHESIS:03072010-230109724
Persistent URL:
DOI:
10.7907/GA8B-F134
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
5581
Collection:
CaltechTHESIS
Deposited By:
Jessie Rosenberg
Deposited On:
22 Mar 2010 18:55
Last Modified:
08 Nov 2019 18:08
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Applications of Micro/Nanoscale Optical Resonators: Plasmonic
Photodetectors and Double-Disk Cavity Optomechanics

Thesis by

Jessie Rosenberg

In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California

2010
(Defended December 8, 2009)

c 2010
Jessie Rosenberg

iii

To my mother, father, and all of the friends, near and far, who have supported me.

Acknowledgments
First and foremost, I want to thank my advisor, Professor Oskar Painter. Your scientific brilliance,
creativity, and boundless energy will forever serve as an inspiration.
Thanks, also, to Qiang Lin. You have always been so generous with your knowledge, and never
balked at answering any question, seemingly trivial or otherwise. Our collaboration over these past
months has been an experience of immeasurable value for me.
Thanks to Raviv Perahia, for your companionship through five years and three different offices;
for all the conversations, and questions, and answers. It’s been a delight. Thanks, also, to my other
officemates, past and present: Orion Crisafulli, Qiang Lin, Thiago Alegre.
Thanks to the senior members of the group, Kartik Srinivasan, Paul Barclay, Matt Borselli, and
Tom Johnson, for the truly phenomenal amount of work you put in to get the group started, and for
sharing that knowledge and experience with those of us who followed. Thanks to Darrick Chang,
for so many fruitful discussions.
Thanks to the recently graduated and current members of the group, Raviv Perahia, Chris
Michael, Matt Eichenfield, Thiago Alegre, Ryan Camacho, Jasper Chan, Amir Safavi-Naeini, Jeff
Hill, Alex Krause, Daniel Chao, Chaitanya Rastogi, and Justin Cohen, for making the group such
a dynamic and vibrant scientific environment. Thanks to the old guard for sharing knowledge and
discoveries and stories along the way, and thanks to the new students for carrying everything on into
the future - and hopefully doing it all even better than we did.
Thanks to Professor Sanjay Krishna’s group at the University of New Mexico, for such a valuable collaboration, and for being so welcoming to a visitor. In particular, thanks to Rajeev Shenoi,
for the work we did together, and for being a great host.
Thanks to all the excellent staff at Caltech, who always were willing to contribute their time to
help out.
Thanks to all of my friends, Caltech students or otherwise, nearby or far away. I can’t even
begin to name everyone who had such an impact, but know that you are valued. In particular, a

heartfelt thanks to Eve Stenson, Neil Halelamien, Megan Nix, Jen Soto, and Jay Daigle. Thanks to
every person on the organizational team of the Caltech Ballroom Dance Club.
Thanks, above all, to my mother and father, for being the best and most supportive parents I
could have asked for.
Thanks, everyone.

Abstract
Optical resonators present the potential to serve vital purposes in many emergent technologies that
require spectral filtering, high optical intensities, or optical delays. By scaling down the optical resonators to the micro or nanoscale, the relevant phenomena can increase significantly in magnitude,
while the device geometries become suitable for chip-scale and integrated processing. In this thesis, research is presented on several valuable resonator geometries and implementations, beginning
with a more standard all-optical design, and continuing on to investigate the novel phenomena and
applications which are made possible when optical and mechanical structures can be synergistically
combined.
First, the design and experimental implementation of a plasmonic photonic crystal spectral and
polarization filtering element is presented. This resonator scheme, in addition to allowing for a tailorable frequency and polarization response for single detector pixels, also increases the absorption
of a thin layer of detector material by utilizing the unique optical properties of metal to confine light
more tightly within the detector active region. Demonstrated in the valuable mid-infrared regime,
this method of producing pixel-integrated multispectral detectors could find application in biological sensing and spectroscopy, missile tracking and guidance, and night vision.
Following this discussion, progress is presented in the relatively new field of cavity optomechanics: utilizing mechanically compliant optical resonators to couple to, control, and read out mechanical motion via optical forces. The use of optical resonators allows the generally weak optical
forces to be increased in strength by orders of magnitude due to the many passes light makes within
the resonator, while miniaturizing optomechanical devices into a convenient form factor for on-chip
applications. Using a fully silicon-compatible double-disk-geometry optomechanical resonator, extremely large optomechanical coupling and very high optical quality factors are shown, enabling the
demonstration of regenerative mechanical amplification, high compression factor optomechanical
cooling, coherent mechanical mode mixing, and wide-bandwidth all-optical wavelength routing.
Applications to ground-state cooling of mesoscopic devices, tunable optical buffering, photonic-

vii
phononic quantum state transfer, channel routing/switching, pulse trapping/release, and tunable lasing are discussed.

viii

Contents
Acknowledgments

Abstract

Preface

xiii

Plasmonic Resonators for Multispectral Mid-Infrared Detectors

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

Photonic Crystal Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

Single-Metal Plasmon Resonator Design . . . . . . . . . . . . . . . . . . . . . . .

1.4

Single-Metal Experimental Demonstration . . . . . . . . . . . . . . . . . . . . . .

1.5

Double-Metal Plasmon Resonator Design . . . . . . . . . . . . . . . . . . . . . .

1.6

Critical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Double Disk Optomechanical Resonators

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Optomechanical coupling and dynamic backaction . . . . . . . . . . . . . . . . .

2.3

Double-disk fabrication, optical, and mechanical design . . . . . . . . . . . . . . .

2.4

Optical and mechanical characterization . . . . . . . . . . . . . . . . . . . . . . .

2.5

Regenerative oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.1

Ambient pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.2

Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

Optomechanical cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Coherent Mechanical Mode Mixing in Optomechanical Nanocavities

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Zipper cavity and double-disk design, fabrication, and optical characterization . . .

3.3

Theory of optomechanical effects in the presence of mode mixing . . . . . . . . .

3.3.1

Intracavity field in the presence of optomechanical coupling . . . . . . . .

3.3.2

The power spectral density of the cavity transmission . . . . . . . . . . . .

3.3.3

The mechanical response with multiple excitation pathways . . . . . . . .

3.3.4

The mechanical response with external optical excitation . . . . . . . . . .

3.4

Mechanical mode renormalization in zipper cavities . . . . . . . . . . . . . . . . .

3.5

Coherent mechanical mode mixing in double-disks . . . . . . . . . . . . . . . . .

3.6

Coherent mechanical mode mixing in zipper cavities . . . . . . . . . . . . . . . .

3.7

Analogy to electromagnetically-induced transparency . . . . . . . . . . . . . . . .

3.8

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Mechanically Pliant Double Disk Resonators

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Spiderweb resonator design and optical characterization . . . . . . . . . . . . . . .

4.3

Static filter response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

Dynamic filter response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5

Discusssion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusion

List of Figures
1.1

Simulated bandstructure for a double-metal photonic crystal . . . . . . . . . . . . .

1.2

Simulated bandstructure for a square and rectangular lattice plasmonic photonic crystal

1.3

Schematic and bandstructure for single-metal detectors . . . . . . . . . . . . . . . .

1.4

Plasmonic photonic crystal detector simulations . . . . . . . . . . . . . . . . . . . .

1.5

Single-metal detector enhancement factor and active region absorption . . . . . . . .

1.6

Detector devices and measurement setup . . . . . . . . . . . . . . . . . . . . . . .

1.7

DWELL detector measurement results . . . . . . . . . . . . . . . . . . . . . . . . .

1.8

Waveguide thickness dispersion for double-metal waveguides . . . . . . . . . . . .

1.9

Field profiles for double-metal waveguides . . . . . . . . . . . . . . . . . . . . . .

1.10

Comparison of FDTD and group theory . . . . . . . . . . . . . . . . . . . . . . . .

1.11

Far-field profiles for stretched-lattice double metal resonators . . . . . . . . . . . . .

1.12

Schematic of double-metal detector loss mechanisms . . . . . . . . . . . . . . . . .

1.13

Vertical and substrate coupling vs. metal thickness and hole size for the single-metal
structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.14

Vertical coupling vs. metal thickness and hole size for the double-metal structure . .

1.15

Double-metal focal plane array schematic . . . . . . . . . . . . . . . . . . . . . . .

2.1

Double-slab waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Cavity Optomechanical Structures . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Double disk flapping mode displacement . . . . . . . . . . . . . . . . . . . . . . .

2.4

Fabrication and characterization of double-disk NOMS . . . . . . . . . . . . . . . .

2.5

Transmission scans for a double-disk microcavity . . . . . . . . . . . . . . . . . . .

2.6

Optical and mechanical mode spectroscopy . . . . . . . . . . . . . . . . . . . . . .

2.7

Regenerative oscillation in a double-disk microcavity . . . . . . . . . . . . . . . . .

2.8

Dynamical backaction: damping and amplification of mechanical motion . . . . . .

xi
3.1

Schematics and optical modes of two optomechanical systems . . . . . . . . . . . .

3.2

Mode mixing in zipper cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Mode mixing measurements in double disks . . . . . . . . . . . . . . . . . . . . . .

3.4

Zipper cavity mechanical mode mixing . . . . . . . . . . . . . . . . . . . . . . . .

3.5

Mechanical mode mixing analogues to optical systems and to EIT . . . . . . . . . .

4.1

Spiderweb microresonator images and simulations . . . . . . . . . . . . . . . . . .

4.2

Pump-probe experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

Static tuning of a spiderweb microresonator . . . . . . . . . . . . . . . . . . . . . .

4.4

Thermomechanical deflection of a spiderweb resonator . . . . . . . . . . . . . . . .

4.5

Dynamic response of a spiderweb microresonator . . . . . . . . . . . . . . . . . . .

xii

List of Tables
1.1

Point Group character tables for the square and rectangular lattice. . . . . . . . . . .

xiii

Preface
The startup of a research lab, especially one with such an extensive range of facilities, is a truly
overwhelming proposition requiring a daunting amount of effort. I was fortunate enough to join the
group at a rather fortuitous time, when the group was mature enough to have essentially all fully
functioning labs, but all of the original group members, with their tremendous experience born from
setting up everything in the lab, were still around and generously willing to share their knowledge.
I immediately began work on a project started by Kartik Srinivasan and Raviv Perahia, that I
imagined would occupy my entire graduate career: the patterning of photonic crystals in metal films,
to create spectrally sensitive pixels in mid-infrared detector material grown by our collaborators,
Prof. Sanjay Krishna’s group at the University of New Mexico. Though undoubtedly fraught with
unforeseeable complexities, the path to working devices seemed straightforward, and dovetailed
well with the many other photonic crystal efforts then under investigation in the lab.
My first major task, occupying nearly the entirety of my first summer in the lab, was to become
a fully functional user of our group cleanroom. We patterned some initial devices, and I also got
started learning about theory and doing some modeling in Matlab, as well as becoming familiar with
our group’s home-built finite difference time domain code. With some initial dubious successes on
the processing front, I also spent time working with Orion Crisafulli in doing a thorough investigation of the properties of metal-insulator-metal waveguides, which became quite useful later.
With a full plate of classwork interspersed, this occupied the first few years of my graduate
career, splitting my time between experiment and theory. We made several sets of photonic crystal
devices, but never saw clear resonant spectral enhancement correlating with the photonic crystal
patterns, which baffled us for quite some time. In the end, we finally realized that, due to the
particulars of the device fabrication, the apertures that we were patterning photonic crystals on
were, in fact, only a small fraction of the open area of the devices: there was a large outer area
of exposed detector material that had no photonic crystal patterning or covering of metal, and this
background scattering was completely overwhelming our signal.

xiv
After that discovery, and another round of device processing followed by my first visit to the
testing facilities at the University of New Mexico, we finally had our first clear success: spectrally
sensitive detector pixels with a resonance frequency having a direct correspondence to the designed
photonic crystal patterns. We celebrated, and I flew home to continue doing modeling. One more
processing run followed, on a new optimized detector material grown by the Krishna group to have
more quantum dot layers in a stable configuration, and we finally had data that we were satisfied
with.
Unfortunately, the simulations proved to be somewhat more difficult. The detector devices,
containing both metals and thick waveguide layers, required a very fine simulation mesh, proving a
strain on computational power. While I attempted to find methods to extract the numbers we wanted,
we also encountered processing difficulties in attempting to fabricate the double-metal detector
devices that we predicted would give us better results. In the end, although further progress in this
area seemed achievable with a fair amount of additional effort, we decided to focus on finishing up
what we already had accomplished.
As that project ended, then, I was freed up to begin something else. At the time, Qiang Lin
was intensely busy with the first demonstrations of the double-disk optomechanical resonators, so
the obvious thing for me to do was to join him to provide some additional manpower. With the
process optimization I had done working on the photonic crystal detectors, I was well-situated to
help optimize the double-disk processing, and I started to learn about cavity optomechanics at the
same time. The project started out as a crash course in fiber-optic testing for me, since the detector
testing I had done had all been free-space-based, but in the end we came out with some very nice
results and a world of possibilities to investigate.
The next task we tackled was modifying the double-disk geometry into a very flexible structure
that could achieve large static displacements with only a small applied optical force. Developing
the “spiderweb” geometry devices – double-ring structures with inner spokes, and rings to stabilize
the mechanics – went surprisingly quickly, with few fabrication setbacks, and we were able to begin
testing those devices with a minimum of delay. In the end, we had demonstrated a new type of
all-optical tunable resonator, with applications to a variety of fields in optical communications, as
well as nonlinear and quantum optics.
Emboldened by our successes in both of those endeavors, we returned to complete the work on
coherent mechanical mode mixing Qiang had been working on in the original double-disk geometry.
With the help of fruitful discussions with Darrick Chang, we completed the theoretical underpin-

xv
nings of the work, and expanded the original idea to encompass the potential for slow-light effects
relying on the long phononic timescale rather than the relatively quite short photonic one, as well as
the eventual possibility for quantum state transfer between phonons and photons.
Looking into the future, there is still a lot of work to be done: cavity optomechanics remains
a wide-open field. I see the possibilities presented by this and all the other innovative work being
done in the field, and I can only be excited for what will undoubtedly come next.
This thesis begins by presenting the work on plasmonic photonic crystal mid-infrared photodetectors, beginning with the theory and an experimental demonstration of single-metal waveguide
devices, and continuing to expand the theory to discuss double-metal devices. Continuing on, work
on dynamic backaction in double-disk resonators is presented, followed by the demonstration of
coherent mechanical mode mixing in the same device structure. Finally, the work on the flexible
double-ring “spiderweb” optomechanical resonators is discussed, finishing with several appendices
on the experimental and mathematical details of the optomechanical work presented in this thesis.

Chapter 1

Plasmonic Resonators for Multispectral
Mid-Infrared Detectors
1.1

Introduction

Optical sensors in the mid-infrared wavelength range are extremely important in a wide variety of
areas, such as night vision, missile guidance, and biological spectroscopy [1]. Currently, the best
mid-infrared detectors are based on mercury-cadmium-telluride (MCT). MCT detectors are very
efficient, but large-area focal-plane arrays are difficult and expensive to grow due to difficulties with
the epitaxial growth of mercury-based compounds [2, 3]. More recently, other detector materials
have become more common, but they have various limitations in the required direction of incoming
light, such as in quantum well detectors, or in detector efficiency, as in quantum dot or dots-in-awell (DWELL) detectors [4, 5]. With the use of a resonant cavity, it becomes possible to increase
the detector efficiency many times over by greatly extending the interaction length between the
incoming light and the active material. Instead of incoming light making only one pass through the
active region, in a resonant detector the light can make hundreds of passes.
The presence of a resonator can also make each pixel frequency and polarization specific [6–
8], allowing for a hyperspectral and hyperpolarization sensor without the need for any external
grating or prism. There are many applications for frequency and polarization-sensitive detectors. A
hyperspectral detector array could function as a spectrometer on a chip, filtering incoming signals
through the use of hundreds of highly sensitive detector pixels. A hyperpolarization detector could
be used in a camera to provide an additional layer of information which can be combined with
frequency and intensity data to better distinguish between different objects in an image.
The current dominant technologies in the field of multispectral imaging rely on the use of either

a broadband focal plane array (FPA) with a spinning filter wheel in front of it [9], or a bank of FPAs
with a dispersive element such as a grating or prism to separate light of different frequencies. These
methods are limited by the often high cost and complexity of such systems. However, when spectral
sensitivity is encoded at the pixel level within a single focal plane array, multi-spectral detection
becomes much more practical for use in a wide range of applications. In addition, the use of pixelintegrated resonators to provide spectral sensitivity can dramatically increase the efficiency of the
detector due to the many passes light makes within the resonator.
The resonator system we investigate here is composed of a photonic crystal cavity for in-plane
confinement, and a plasmonic waveguide [10, 11], composed of either a single or a double-layer of
metal, for the vertical confinement. This resonator design, combining the benefits of a plasmonic
waveguide and a photonic crystal cavity, has a number of advantages. The plasmonic waveguide
serves multiple purposes: it serves as a superior top contact (or in the double-metal case, top and
bottom contact) for the detector device providing enhanced extraction efficiency; it provides strong
vertical confinement (nearly total confinement, for the double-metal structure) of the resonator mode
within the active region; and it increases the index contrast in the photonic crystal, enhancing the inplane confinement of the resonator mode and enabling strong confinement even with a very shallow
photonic crystal etch extending only through the top metal layer [12, 13]. The photonic crystal
patterning also serves a dual purpose: it provides in-plane confinement to the resonator mode, serves
as a grating coupler to couple normal-incidence light into the in-plane direction of the detector, and
provides a mechanism for freely adjusting the polarization response of the detector pixel.
In the past, many promising schemes have been proposed and/or demonstrated illustrating various aspects of these concepts: optical resonators to provide spectral [14] or spectral and polarization
filtering [15–17], enhanced confinement of light to increase material absorption [14, 16–18], and
metallic gratings to enable strong confinement without the necessity for deep etching [15, 17, 18].
We have also demonstrated plasmonic photonic crystal designs with the maximum field intensity
at the top metal interface to allow for thinner devices and increase field overlap with the active region versus metallic Fabry Perot-based structures, in both deep-etched [19] and shallow-etched [13]
single-metal-layer implementations. Here we detail the design and experimental demonstration of
the shallow-etch single-metal resonators, and expand those design principles to propose a highlyefficient shallow-etch double-metal cavity design for hyperspectral and hyperpolarization, strongly
enhanced mid-infrared detection. Both single-metal and double-metal designs are detector material
agnostic, are easily incorporated into current FPA processing techniques, and do not involve the

damage or removal of any detector active region material, providing significantly increased flexibility and functionality with a minimal increase in complexity. This work was originally presented in
Refs. [13, 20].

1.2

Photonic Crystal Design

A significant obstacle to using resonant cavities to enhance detector absorption and provide spectral
and polarization sensitivity is achieving sufficient input coupling from free-space light. Commonly
such resonators, with their high confinement, have only very poor phase-matching to a normal incidence free-space beam such as that which we would ideally like to detect for imaging applications.
However, with suitable design and optimization of the plasmonic photonic crystal structure, it becomes possible to achieve significant free-space coupling, and indeed, even move towards achieving
critical coupling (as will be discussed in Section 1.6).
We used group theory to design a frequency and polarization sensitive photonic crystal structure suitable for coupling efficiently to normal incidence light. The simplest polarization-sensitive
resonator design would be a one-dimensional grating. However, it is beneficial to choose a fullyconnected photonic crystal design in order to take full advantage of the increased current extraction
efficiency from the plasmonic metal layer serving as the top contact of the detector device, as well
as allowing for continuous variation between polarization-sensitive and polarization-insensitive devices. Therefore, we analyze a square-lattice structure here, and describe how stretching the lattice
in one direction can split the degenerate modes of the structure and create a strong polarization
sensitivity for use in imaging applications.
The bandstructure of a square lattice photonic crystal is shown in Fig. 1.1(a), calculated using
plane wave expansion. The ratio of circular hole radius r to lattice spacing a used was r/a = 0.32,
and the index of the material was taken to be the effective index of the double metal plasmon waveguide, neff = 3.24. Details of the plasmon effective index calculation are discussed in Section 1.5.
There are no band gaps for this structure, however there are several flat-band regions. The group
velocity of these band-edge modes is close to zero, therefore the light travels very slowly and is
effectively confined within the patterned region. Band-edge modes are ideal for applications such
as detectors, as the mode volume is large, allowing more of the active region to be contained within
the resonator. There are a number of flat-band regions within the bandstructure in Fig. 1.1(a), but
we are interested in the modes at the Γ-point. The Γ-point corresponds to normal-incidence modulo

Normalized Frequency (a/λ0)

0.8

(a)

(b)

0.6
0.5

Γ3

0.4

(c)

Γ2

0.7

Γ0

Γ2

Γ1

Γ3

Γ0

0.3

X1 Γ1
G1

0.2
0.1

Γ4

Figure 1.1: (a) In-plane guided mode TM-like bandstructure plot of a square-lattice photonic crystal
(neff = 3.24) with r/a = 0.32. The light line is shown in yellow, and the modes of interest are circled.
(b) Reciprocal lattice for an unstretched photonic crystal. (c) Reciprocal lattice for a stretched
photonic crystal.

a reciprocal lattice vector, so the Γ-point modes are capable of coupling normal-incidence light into
the in-plane direction of the detector. In addition, the Γ-point modes are above the light line, and
therefore leak into the air, enabling them to couple more easily to an input free space beam. We
investigated the four lowest-order Γ-point modes, circled in Fig. 1.1(a), using group theory [21, 22].
The point group symmetry of the square photonic crystal lattice, with reciprocal lattice shown
in Fig. 1.1(b), is C4v . The in-plane field of the unperturbed waveguide is given by Ek? (r? ) =
ẑe

(k? ) r? , with k

? and r? representing the in-plane wavenumber and spatial position, respectively.

When the structure is patterned, coupling will occur between waveguide modes with similar unperturbed frequencies, and propagation constants that differ by a reciprocal lattice vector G.
There is one Γ-point within the first Brillouin zone (IBZ), at f(0, 0)kΓ g, with kΓ = 2π/a. Since
we are interested in modes with nonzero k-vectors in the in-plane direction, we will consider the
nearest Γ-points in the surrounding Brillouin zones, at (f (1, 0)kΓ , (0, 1)kΓ g). These points are
labeled in Fig. 1.1(b). The group of the wave vector, the symmetry group of a plane wave modulo
G, is C4v at the Γ-point. The character table of C4v is shown in Table 1.1.
The star of k (?k) at the Γ-point is the set of independent Γ-points within the region. In this
case, ?k is given, not uniquely, by fkΓ1 g. This will be our seed vector. We find the symmetry basis
for the modes at that satellite point by applying the symmetry operations of the group of the wave
vector to the seed vector. In this case, the basis is (EG1 , E G1 , EG2 , E G2 ). Projecting this symmetry

Normalized Frequency (a/λ0)

Table 1.1: Point Group character tables for the square and rectangular lattice.
C4v E C2 2C4 2σv 2σd C2v E C2 σx σy
A1 1 1
A1 1 1
-1
-1
A2 1 1 -1 -1
A2 1 1
B1 1 1
-1
-1
B1 1 -1 -1 1
B2 1 1
-1
-1
B2 1 -1 1 -1
2 -1

0.5 (a)

(b)

0.4

0.3

0.2

Figure 1.2: 2D bandstructure plots near the gamma point of (a) a square lattice and (b) a rectangular
lattice, stretched by 10%. The dipole-like modes are shown in bold.

basis onto the irreducible representation (IRREP) spaces of C4v , we find the modes:
EA1 = ẑ(cos(kG1 r) + cos(kG2 r)),
EB1 = ẑ(cos(kG1 r)

cos(kG2 r)),

(1.1)

EE,1 = ẑ(sin(kG2 r)),
EE,2 = ẑ(sin(kG1 r)),
where A1 , B1 , and E are IRREP spaces of C4v (see Table 1.1), and r has its origin at the center of the
air hole. Considering that modes with more electric field concentrated in areas with high dielectric
constant tend to have lower frequency than those with electric field concentrated in low dielectric
regions [23], we can order the modes by frequency. E is a two dimensional IRREP, so generates two
degenerate modes. We associate this pair of modes fEE,1 , EE,2 g with the second and third frequency
bands, which is in agreement with the bandstructure in Fig. 1.1(a). These degenerate modes, with
dipole-like symmetry and the spatial pattern given in Eqs. 1.1, radiate with a far-field pattern which
is uniform: in the case of a finite structure, a Gaussian-like far-field without anti-nodes.
In order to achieve polarization sensitivity, we need to split these two degenerate dipole-like
modes. To do this, we stretch the photonic crystal lattice (not the photonic crystal holes) in one
direction, giving the reciprocal lattice shown in Fig. 1.1(c). The effect of stretching the lattice on
the four lowest-order Γ-point modes is shown in Fig. 1.2. The symmetry group of this perturbation
is C2v ; the character table for C2v is shown in Table 1.1. Using the compatibility relations between
C4v and C2v , we find the new set of modes:
EA1 ,1 = ẑ(cos(kG1 r) + cos(kG2 r)),
EA1 ,2 = ẑ(cos(kG1 r)

cos(kG2 r)),

(1.2)

EB1 = ẑ(sin(kG2 r)),
EB2 = ẑ(sin(kG1 r)).
These modes are plotted in Fig. 1.10. The C4v two-dimensional representation E decomposes into
B1

B2 under C2v , therefore the dipole-like modes are no longer degenerate. This is in agreement

with what we see in the bandstructure of the stretched lattice, Fig. 1.2(b).

(a)

Ez Intensity of
Guided Mode

Refractive Index/Ez Intensity (a.u.)

0.4

(b)

(c)

Γ X

(d)

X2 M
Γ X1

Frequency (a/λ)

16
0.35

0.3

Active Region

Depth (µm)

0.25

XΓ

Figure 1.3: (a) A crossectional image of several lattice constants of the single-metal DWELL detector design. (b) Ez intensity profile of the fundamental plasmon waveguide mode (blue) and the real
part of the refractive index of the layers (red), with the detector active region highlighted. (c) FDTD
bandstructure for the unstretched single-metal photonic crystal structure shown in (a) in the region
between the Γ and X points. (d) FDTD bandstructure between the Γ and X1 points for a single-metal
photonic crystal structure stretched and compressed by 10% in the x and y directions, respectively.

1.3

Single-Metal Plasmon Resonator Design

In addition to the in-plane confinement provided by the photonic crystal pattern discussed in Section 1.2, it is necessary to confine the light in the vertical direction as well. We begin with a singlemetal design suitable for straightforward fabrication (as experimentally demonstrated in [13]), and
then expand the discussion to consider a double-metal design that mimics the top and bottom contact layers in detector focal plane arrays for easy integration. As we operate in the mid-infrared
frequency range, we are far from the plasmon resonance frequency of metals, typically in the ultraviolet; operating in this regime avoids the very high metal losses that occur at frequencies closer to
the plasmon frequency, and allows the mode to extend farther into the active region of the detector.
The single-metal resonant cavity consists of a single layer of metal with etched square holes in
a square lattice periodic array. A representation of several lattice constants of the device structure is
shown in Fig. 1.3(a). The plasmonic layer provides the vertical confinement, confining the optical
mode with a maximum at the surface of the metal (Fig. 1.4(b,c)), while the etched air-holes create a
PC pattern to confine the light in-plane. Combined together, this resonator design provides full 3D
confinement, significantly increasing the amount of time light spends within the detector active region, and therefore enhancing the probability of detection. Due to the strong index contrast between
the surface plasmon [10, 11, 24] mode beneath the metal regions and the dielectric-confined mode
beneath the air holes, this plasmonic PC grating is strong enough to generate an in-plane confined
resonant mode without etching into the detector active material [12, 25], allowing a resonator to be
fabricated without damaging or removing active material. The numerical and symmetry analysis
presented in Sec. 1.2 and Ref. 20 shows that the two degenerate dipole-like in-plane modes of the
structure (Fig. 1.4(b-d)) couple most easily from free space. Further improvements in the free-space
coupling efficiency were performed by optimizing the top metal thickness and hole size. In addition,
as the two dipole-like modes couple to orthogonal polarizations of incoming light, a stretch of the
PC lattice breaks the degeneracy of the two modes, splitting their resonance frequencies and thus
achieving high polarization selectivity [6, 16].
The single-metal resonant DWELL detector structure we study is shown in Fig. 1.3(a), along
with the 1D Ez intensity profile of the fundamental plasmon waveguide mode in Fig. 1.3(b) (not
including the effects of the photonic crystal holes). The vertical confinement factor of this mode
within the active region of the detector is η = 91%, in strong contrast to the generally much lower
confinement factor of purely dielectric waveguides. This extremely high confinement, even for a

(a)

x10-3

(b)

0.5

z (µm)

-2 active region

-1

-8

-1
x10-3
1 (d)

(c)

z (µm)

-2 active region

-4

-0.5-0.5

-8

-1

y (µm)

0.5

x (µm)

0.5

x10-3
0.5

-6

-0.5

0.5

y (µm)

0.5

-1

-0.5

-6

-4

air

-1

semiconductor
-0.5
0.5
x (µm)

-0.5
-1

Figure 1.4: (a) A diagram of the unstretched PC structure showing relevant dimensions. The expanded plots show the Ez mode profile for one lattice constant of one of the two dipole modes for an
unstretched PC lattice in (b) the x-z plane along the hole edge, (c) the y-z plane along the hole edge,
and (d) the x-y plane just beneath the metal-semiconductor interface, for a structure with lattice
constant a = 2.38 µm, W̄ = 0.6, and metal thickness t = 150 nm.

10
waveguide utilizing only a single layer of metal, immediately showcases the benefits of choosing a
plasmon-based photonic crystal design.
Figure 1.4(a) shows relevant dimensions of the structure; we define the normalized hole width
as W̄ = 2W /(ax + ay ). Field profiles along different planes for the square lattice dipole-like mode
are plotted in Fig. 1.4(b-d) over a single unit cell; the other dipole mode has the same field pattern,
rotated by 90 degrees in the x-y plane. The simulated active region absorption corresponding to
this mode is 10.9%, given an approximate 2% single-pass absorption in the DWELL material, and
corresponds to an expected resonant responsivity enhancement of 5-6 times that of a control sample
with no plasmonic layer or PC patterning.
The TM square lattice single-metal plasmonic photonic crystal bandstructure for the region
near the Γ-point is shown in Fig. 1.3(c), calculated using finite difference time domain (FDTD)
methods with metal material properties from Ref. 26. The simulated structure has a lattice constant
of a = 2.939 µm, a hole width W vs. lattice constant ratio of W̄ = 0.567, and a metal thickness
of tm = 150 nm, and shows Γ-point modes which are in good agreement with the group theory
predictions in the previous section. Figure 1.3(d) shows the same bandstructure region for a lattice
stretched and compressed by 10% along the x̂- and ŷ- axes respectively, with the two dipole modes
showing a significant frequency splitting, also as predicted. These two 3D FDTD simulations match
up well with the 2D plane-wave expansion bandstructure predictions shown in Fig. 1.2, with the
addition of visible higher-order vertical modes from the single-metal plasmon waveguide. The
double-metal waveguide structure can be designed such that these higher-order vertical modes are
eliminated.
The overall quality factor of the two degenerate dipole modes in the unstretched-lattice case (for
resonator parameters given in the caption to Fig. 1.5) is calculated to be Qper = 48 for a perfectly
periodic structure, not including the in-plane quality factor, Qxy , which can be increased indefinitely
by adding more lattice constants to the resonator structure. By simulating the structure as excited
by an incoming, normal incidence plane wave, we can measure the percentage of the incoming light
absorbed within the active region. The simulated DWELL active region absorption corresponding
to this mode is At = 11.5% (Fig. 1.5(a)), with material parameters specified so as to reproduce
the approximate DWELL material single-pass absorption, ADWELL = 2%. This corresponds to an
expected responsivity enhancement factor of E

At /ADWELL = 5.75 at the resonant wavelength,

versus a sample with no top patterned plasmonic metal layer at the same wavelength (Fig. 1.5).
Note that this enhancement factor is not normalized to the area of the holes in the plasmonic metal,

WG1

WG2 P10

P11

P01
Wavelength (µm)

P00

0.04

WG1

0.02

-2

-0.02

-6
-8

0.01

0.005

-2

-4

-0.04

-1 0 1
y (µm)

0.015

P01

-4

-0.005

-6
-8

-0.01
-0.015

-1 0 1
y (µm)

-0.01

-6
-8

0.02
0.01

z (µm)

(b)

z (µm)

P00

z (µm)

Enhancement Factor, E

(a)

Active Region Absorption (%)

-0.02

-1 0 1
y (µm)

Figure 1.5: (a) FDTD simulated enhancement factor and active region absorption vs. wavelength,
based on a 2% single-pass absorption, using a structure with lattice constant a = 2.939 µm, W̄ =
0.567, and metal thickness tm = 150 nm. (b) Ez mode profiles in the y-z plane at the hole edge
for the three longer-wavelength peaks in (a), for one lattice constant. The three shorter-wavelength
peaks have similar vertical field profiles, but are higher-order in the x-y plane.

as is typically done in the case of ’extraordinary’ transmission through thin metal layers [24]; the
absorption values are compared over the same physical region of detector material.
The simulated active region absorption, At , and enhancement factor, E, are plotted versus frequency in Fig. 1.5(a), showing the fundamental plasmon mode at 9.6 µm and a series of higher-order
modes at shorter wavelengths. The mode Pxy corresponds to the xth -order in-plane and yth -order
vertical plasmon waveguide mode. Thus P00 corresponds to the fundamental plasmon mode as discussed above. W G1 and W G2 are the first and second-order TM waveguide modes of the structure,
and Fig. 1.5(b) shows that they have a lower overlap with the metal surface than the plasmon modes.
Thus their Q per is higher than the plasmon modes for the perfectly periodic structure simulated, as
can be seen in Fig. 1.5(a), but their in-plane Q-factor, Qxy , will be very small due to their minimal
interaction with the photonic crystal grating. As the overall absorption enhancement factor is lower
for these modes even in the infinite-structure limit, we conclude that the response of this resonator
structure will be dominated by the surface plasmon-guided modes.

plasmon metal

(a)

plasmon metal

(b)

top metal
DWELL active region

AlGaAs
bottom metal

(c)

5 µm

GaAs substrate

(d)
LN2 Dewar
Blackbody

Chopper
Controller

5 µm

Trigger

ESA

Current Amplifier

Figure 1.6: (a) Optical image of the fabricated device indicating the top, bottom and plasmon metallizations. (b) Cross-sectional SEM of the fabricated square-lattice device indicating the plasmon
metal and device layers. (c) SEM image of the fabricated square-lattice PC pattern on the plasmon
metallization. (d) Schematic of the setup used for measuring responsivity and detectivity of the
fabricated devices. Responsivity measurements were performed by illuminating the sample with
a calibrated Mikron M365 blackbody at T = 800 K. The blackbody radiation was modulated at a
frequency of 400 Hz using a chopper and this signal was used as a trigger for the SRS 760 fast
Fourier transform (FFT) spectrum analyzer. The photocurrent was amplified using a SRS 570 low
noise amplifier and then measured in the spectrum analyzer.

a=1.90µm
a=1.97µm
a=2.04µm
a=2.10µm
a=2.17µm
a=2.24µm

unpolarized
90º
polarized
0º
polarized

a=2.31µm

9 10

Wavelength (µm)

5 x 10-7
unpatterned
response

5.5

6.5

Wavelength (µm)

normalized

0.05

0.00

(d)

-6
control
divided 2

0.5

a=2.38µm

Spectral Response (a.u.)

a=1.83µm

Spectral Response (a.u.)

0.5

0.10

9 10

Wavelength (µm)

-4

1010

-2

Voltage (V)

109
108

(e)

107-6

-4

-2

Voltage (V)

Responsivity Enhancement, E

unprocessed 1x 10-6

(c)

unpatterned

Responsivity (A/W)

(b)

unpatterned

Detectivity (cm Hz1/2/W)

(a)

(f )

Sample A −5V bias
Sample A 5V bias
Sample B −5V bias
Sample B 5V bias

5.5 6 6.5 7 7.5 8 8.5

Wavelength (µm)

Figure 1.7: (a) Normalized spectral response from square lattice devices at a bias of 5 V, indicating
tuning of peak wavelength with the lattice constant. (b) Normalized spectral response from a rectangular lattice device (ax = 1.65 µm, ay = 2.02 µm) at a bias of -5 V. The response to unpolarized light
is shown in green, and beneath is the response to light polarized at 0 degrees (red) and 90 degrees
(blue) relative to the shorter lattice constant dimension of the lattice. (c) Data processing of the
a = 2.38 µm device response. The unprocessed data (green) is divided by the unpatterned DWELL
response to show the resonances independently of the base detector response. The background
scattering from other regions of the sample (yellow) is then subtracted from the control divided
data (blue) and normalized, to give the final spectral response (red) that is plotted in Fig. 1.7(a,b).
(d) Measured responsivity and (e) detectivity of a stretched lattice device with ax = 1.78 µm and
ay = 2.16 µm (red), and a control device (black). (f) Measured peak responsivity enhancement E
versus resonant device wavelength for two samples at positive and negative bias.

1.4

Single-Metal Experimental Demonstration

To test these predictions, we fabricated two detector samples: sample A with a square lattice PC,
and sample B with a rectangular lattice PC having a lattice constant stretching ratio ay /ax = 1.2.
Figure 1.6(a-c) shows representative images of the fabricated devices. All of the PC patterns had the
same normalized hole width (W̄

0.6), but different lattice constant values, and therefore different

resonant wavelengths determined by the scaling of the pattern. The spectral response of the surfaceplasmon resonant detectors was measured at 30 K using a Nicolet 870 Fourier transform infrared
spectrometer (FTIR) with the patterned detector sample used in place of the standard FTIR detector.
Responsivity and detectivity measurements were performed at 77 K using the experimental setup
shown in Fig. 1.6(d). To separate out the background scattering and the frequency response of
the DWELL material from the resonant enhancement, we perform data processing as shown for a
typical spectral response measurement in Fig. 1.7(c).
As predicted, by varying the lattice constant and symmetry of the patterned grating, we are
able to tailor the wavelength and polarization response of each detector pixel. Figure 1.7(a) shows
the resonant spectral response from a set of representative detector pixels on sample A, varying

14
the peak wavelength response from 5.5 µm to 7.2 µm by choosing PC lattice constants in a range
from 1.83 µm to 2.38 µm. The linewidth of these resonances is

0.9 µm, providing strong spectral

sensitivity within the broad background DWELL response which covers more than 5 µm. In addition
to the fundamental surface plasmon resonant mode, we also observe a higher-order plasmon mode
as predicted by the FDTD simulations in Fig. 1.4(b), at a wavelength in good agreement with the
theory. In order to generate a polarization-sensitive response, we stretch the lattice constant in one
direction (sample B), splitting the resonant detector response into two well-separated peaks as shown
in the green curve of Fig. 1.7(b). By varying the polarization of the light incident on the detector,
we show that these two peaks correspond to orthogonal linear polarization directions of incoming
light, as represented by the blue and red curves of Fig. 1.7(b). The high polarization extinction
between the two curves indicates clearly the strong polarization dependence in our device. The
experimentally measured spectral peaks are broadened relative to the FDTD simulated values in
Fig. 1.4(b) due to the finite extent of the PC pattern ( 50 µm in diameter), and therefore the limited
in-plane confinement.
To characterize the efficiency of the detector response and the resonant enhancement, we define
and measure the responsivity and detectivity of samples A and B as follows. The peak responsivity
was computed using the expression
Rp = R λ

λ1

I0
RN (λ)Le (λ, T )As Ad tFr2F dλ

(1.3)

where RN (λ), I0 , Le , T , As , and Ad are the normalized spectral response, measured photocurrent, the
black body spectral excitance, the black body source temperature, the area of the source, and the area
of the detector, and r, t, FF are the distance between the source and the detector, the transmission of
the window and geometrical form factor, respectively. The lower and upper wavelength bounds of
the detector response are given by λ1 and λ2 . The detectivity D is then
Ad ∆ f
Rp
D =
in

(1.4)

where Ad is the detector area, ∆ f is the noise equivalent bandwidth ofour measurement, and in is
the noise current.
Compared to a control (unpatterned) sample, the plasmonic PC patterned devices provide a
strong enhancement of responsivity and a corresponding increase in detectivity across an applied

15
0.35

(a)

0.3

Effective Index, neff

(b)

0.25
0.2
0.15
0.1
0.05

200

400

600

800

Thickness, t (nm)

1000

200

400

600

800

Thickness, t (nm)

1000

Figure 1.8: The (a) real (red) and (b) imaginary (blue) parts of the dielectric constant dispersion
relation of an Ag/GaAs/Ag waveguide vs. waveguide thickness t, for a free-space wavelength of
λ = 10 µm. The GaAs core index is indicated by a dotted black line.

bias range from -5 V to 5 V, as shown in Fig. 1.7(d) and (e). The enhancement factor E is defined
as E = R(λi )/Rc (λi ), where R(λi ) is the responsivity of the patterned detector at the resonant wavelength and Rc (λi ) is the responsivity of the control sample at the same wavelength. In Fig. 1.7(f),
we show enhancement factors across a range of wavelengths reaching as high as 5X for sample A,
and 4X for sample B.

1.5

Double-Metal Plasmon Resonator Design

Though we have shown, theoretically and experimentally, that a single-metal plasmonic device
can have high active-region confinement and reasonable Q-factors, we can move to a double-metal
design to increase both of these quantities even further. The double-metal structure brackets the
active region with a thin layer of plasmonic metal on either side, and the photonic crystal holes are
etched only into the top metal layer, as before. All of the advantages of the single-metal device
are preserved, while the substrate loss can be essentially eliminated and the detector active region
vertical confinement can approach 100%. These are achieved at the price of higher plasmonic metal
loss, but in the mid-infrared region, this loss is not prohibitive.
In choosing the ideal waveguide thickness, t, for the double-metal (or metal-insulator-metal,
MIM) plasmonic waveguide, there are several considerations. Figure 1.8 shows the variation of
mode effective index neff with waveguide thickness for a Ag/GaAs/Ag waveguide at a free-space
wavelength of λ = 10 µm. As the waveguide thickness decreases and more energy moves into the
metal regions, both the real (Fig. 1.8(a)) and the imaginary (Fig. 1.8(b)) parts of the effective index

16
x 105

(a)
abs(E) (N/C)

abs(E) (N/C)

neff = 7.64+0.2584i

(b)

-100

-50

z (nm)

100

150

neff = 3.44+0.0112i

x 103

(c)

x 104

abs(E) (N/C)

neff = 3.30+0.0005i

200

z (nm)

400

600

z (µm)

Figure 1.9: The field profile for Ag/GaAs/Ag plasmon waveguides with a thickness t of (a) 10 nm,
(b) 500 nm, and (c) 50 µm are shown, for a free-space wavelength of λ = 10 µm. The effective index
neff for each plasmon waveguide is also given.

increase. A high real part of the effective index is beneficial, because it increases the index contrast
of the photonic crystal by increasing the contrast between the photonic crystal holes and the metalcovered regions (the double-metal waveguide). A higher index contrast increases the strength of the
photonic crystal perturbation, improving the in-plane confinement Qxy of the resonator. If the index
contrast is high enough, the photonic crystal holes can be etched only into the top metal, without
removing material from the detector active region. In the double-metal case, this condition is even
easier to achieve than in the single-metal case. However, the imaginary part of the effective index
is proportional to the loss in the waveguide, and must be minimized. Therefore a waveguide width
must be chosen to balance the competing factors of index contrast and loss.
The field profiles of three Ag/GaAs/Ag plasmon waveguides are shown in Fig. 1.9, with the
calculated effective index neff , for a free-space wavelength of λ = 10 µm. Though the effective index
values shown here would seem to generate only a low index contrast with the core dielectric material
(nGaAs = 3.3) for reasonable waveguide thicknesses, it must be considered that these simulations do
not take into account the effect of the photonic crystal holes etched in the metal. In fact, it can be
shown [12, 25] that the presence of holes in a metal layer has the effect of lowering the effective
plasmon frequency of that layer without significantly raising losses, increasing the real but not the
imaginary part of the waveguide effective index. Thus, the actual combined photonic crystal and
plasmon structure will have a considerably higher index contrast than would be expected from these
effective index values. This is demonstrated via simulations of the full 3D structure which show that,
even with the photonic crystal holes etched only into the top metal layer, we still achieve significant

(a)

(b)

(c)

(d)

Figure 1.10: A comparison of the FDTD simulated and group theory predicted Ez field profiles for
the four lowest gamma-point modes of a 20% stretched lattice, in order of increasing frequency. The
modes, labeled according to C2v designations, are (a) A1,1 , (b) B2 , (c) B1 , (d) A1,2 . The group theory
predictions are shown on the left, while the FDTD results, including the effects of the photonic
crystal air hole (overlaid white square), are shown on the right. The FDTD fields shown are 2D
slices of the full simulations taken just below the top metal layer, inside the active region.

in-plane confinement and vertical coupling.
Full structure simulations (photonic crystal plus plasmon waveguide) were performed using the
FDTD method on a perfectly periodic lattice (as before, due to computational constraints), with
a lattice stretched by 20%. These FDTD simulations confirm the results of our separate photonic
crystal and plasmon waveguide simulations. Figure 1.10 shows the FDTD field plots (left) in comparison with the group theory mode plots (right). Though the presence of the air-hole distorts the
shape of the modes in the center of the FDTD images, at the outside of the simulation region it can
be seen that the simple group theory calculations have accurately predicted the mode shapes given
by the more complex FDTD simulations.
We have also investigated the far-field profiles of the four Γ-point modes, through examining
the spatial fourier transform of Ez . Due to time-reversal symmetry, the field profile of a mode that
can be coupled into the resonator is equivalent to the field profile of the resonator mode propagated

(a)

Py/Px > 109

-100

-50

y (µm)

-150

(b)

Px/Py > 109

-100

-50

y (µm)

-150

100

150

-100 -50

x (µm)

50 100

150

-100 -50

x (µm)

50 100

Figure 1.11: Far-field plots at z = 90 µm of the two Γ-point dipole modes in a stretched-lattice structure with W̄ = 0.5309 and ay /ax = 1.2. (a) B2 mode power density, with dominant ŷ-polarization.
(b) B1 mode power density, with dominant x̂-polarization. For both B1 and B2 modes, the polarization selectivity is calculated to be greater than 109 , limited entirely by error in the numerical
simulation.

out into the far-field, therefore these far-field plots indicate the mode-shapes and polarizations that
couple most strongly from free space to the resonator mode. Far-field plots of the two fundamental
stretched-lattice dipole modes, B1 and B2 , are shown in Fig. 1.11, generated from an 10X10 tiled
array of the FDTD simulated field profile (itself one lattice constant in size) and apodized using
a Gaussian function with a standard deviation of two lattice constants (ax and ay , respectively)
in the x and y directions. We can see from Fig. 1.11(a,b) that the B1 and B2 modes are wellsuited for coupling to incident free-space light, since the far-field profile has a single lobe at normal
incidence and does not contain any anti-nodes, in agreement with the group theory predictions from
Section 1.2.

1.6

Critical Coupling

After optimizing the large-scale resonator design and choosing photonic crystal modes which have
the largest coupling to normal-incident light, it still remains to find the best values for the design
parameters to increase detector absorption, and to determine the fundamental limits on absorption
enhancement for these two (single-metal and double-metal) resonator designs. We find that, for the
double-metal resonator, the absorption is greatest at the point of critical input coupling, whereas we
find a different optimal point for the single-metal resonator, as the parasitic substrate loss increases
along with the detector absorption as the input coupling is increased.

Qe
Qmetal

-V

Qxy

Figure 1.12: The dominant loss mechanisms within a double-metal plasmonic photonic crystal
resonant detector.

Critical coupling occurs when the external coupling to the resonator (the resonator “coupling
loss”) is equal to the total internal cavity loss from all other loss mechanisms. When that condition
occurs, the reflection coefficient goes to zero, and all of the incident light at the resonance frequency
is absorbed in the resonator [27]. Figure 1.12 shows the various scattering and absorption processes
that are involved in near normal incidence resonant detection. The reflection from the cavity is given
by
R=
where ∆ = ω

γ0 γe 2
2 ,
∆2 + γ2t

∆2 +

(1.5)

ω0 is the frequency detuning from the resonance frequency ω0 , γ0 is the intrinsic

cavity loss rate, γe is the vertical coupling rate to free space, and γt = γ0 + γe . The loss rates γ are
related to the Q-factors given previously by γ = nλQ/2πc. It is clear from Eq. 1.5 that, when the
cavity is excited on-resonance (∆ = 0), the reflection goes to zero when γ0 = γe , when the rate at
which the cavity can be fed from free space is equal to the sum of all the internal cavity loss rates.
This is the critical coupling condition.
From the expression for the reflection in Eq. 1.5, we can write the power dropped into the cavity

20
(not only the absorbed power, but all of the power not reflected):
Pd = Pin (1

R) = Pin

γ0 γe
∆2 +

γt 2

(1.6)

where Pin represents the power incident on the cavity. Therefore the fractional absorption efficiency
into the i-th loss channel is
pi =

γi Pd
γi γe
.
γ0 Pin ∆ + γt 2

(1.7)

We can enumerate the loss mechanisms in Fig. 1.12, such that γ0 = γD + γmetal + γxy + γsub , corresponding to the (beneficial) detector absorption, the metal absorption, the in-plane loss, and the
substrate loss, respectively. The in-plane loss can always be made negligible, by adding more lattice constants to the photonic crystal patterning region to increase the in-plane confinement strength
relative to the other loss mechanisms. In the single-metal case, we can consider γsub = mγe , representing a mode coupling into the substrate that is m times larger than that into the air due to the
higher substrate refractive index; in the double-metal case, m

0 due to the thick bottom layer of

plasmon metal.
The fractional absorption into the DWELL detector material is then, at resonance, given by
pD =

4γD γe
[(m + 1)γe + γmetal + γD ]2

(1.8)

From this expression, we see that the maximum fractional absorption occurs at
γe =

γD + γmetal
1+m

(1.9)

As the input coupling γe can be adjusted by varying resonator parameters, this maximal condition
should be readily achievable, corresponding to a fractional absorption into the detector material of
pD,max =
For the double-metal case in which m

γD
(1 + m)(γD + γmetal )

(1.10)

0, the maximal detector absorption occurs at the pure critical

coupling condition, γe = γD + γmetal . In this case, the fractional power absorbed is primarily limited
by the relatively small metal losses in the mid-infrared region. For the silver plasmon waveguides
simulated in this work, we find a metal loss quality factor of Qmetal = 149, in comparison to the

105

(a)
Quality Factor

Quality Factor

104

103

102

101

100

200

300

400

Metal Thickness, tm (nm)

500

(b)

104
103
102
101

0.4

0.5

0.6

0.7

0.8

Figure 1.13: (a) Variation of external coupling and substrate loss quality factors, Qe and Qsub , with
metal thickness tm , and (b) with W̄ for the fundamental (blue) and higher-order (red) modes of
the unstretched single-metal photonic crystal lattice. Open circles represent Qe and filled triangles
represent Qsub . A dotted line marks the value of the parameter held constant in the opposing plot.

estimated DWELL detector absorption quality factor of QD = 188. This indicates that 55.8% of
the incoming light will be absorbed in the active material for the optimal external coupling quality
factor of Qe = 83.
There are many free parameters in this resonator structure which can be optimized in order to
achieve the optimal input coupling value. Choosing two of the most significant, the normalized hole
width, W̄ , and the top metal thickness, tm , we investigate their effect on Qe for both the single-metal
unstretched lattice (Fig. 1.13) and double-metal stretched-lattice (Fig. 1.14, with a lattice stretching
ratio of 1.2) structures. With variation of W̄ , shown in Figs. 1.13(b) for single-metal and 1.14(b)
for double-metal, the overall trend for both structures is the same, showing a curve most likely
due to a combination of factors: the increased hole size provides a larger aperture through which
light can escape, decreasing the Qe ; but the larger air hole also distorts the shape of the mode,
causing it to generate a less pure far-field profile which does not match as well with a free-space
beam. In contrast, the variations in quality factor with changes in the top metal thickness, shown in
Figs. 1.13(a) for single-metal and 1.14(a) for double-metal, illustrate vertical quality factors Qe of
both dipole-like modes increasing monotonically as the top metal becomes thicker. Varying the top
metal thickness tm is an effective way to change Qe to better match the internal loss, and thus more
closely approach critical coupling, without changing the mode frequency.

(a)

8000

(b)

500

6000

400

4000

300

2000
00

600

External Quality Factor, Qe

10000

200

200 400 600 800 1000 1200

Metal Thickness, tm (nm)

100 0.54

0.58

0.62

0.66

0.7

Figure 1.14: Variation of vertical coupling quality factor Qe of the B1 (blue square) and B2 (green
circle) modes of the double-metal photonic crystal lattice with changing (a) tm and (b) W̄ .

Though the behavior as hole size and metal thickness are varied is similar for both singlemetal and double-metal structures, the double-metal structure has a lower achievable Qe , indicating
more favorable external coupling conditions; the stretching of the photonic crystal lattice does not
significantly decrease Qe .

1.7

Conclusion

We have designed a plasmonic photonic crystal resonator utilizing either a single-metal or doublemetal plasmon waveguide for use in mid-infrared photodetectors, and experimentally demonstrated
single-metal devices with responsivity enhancement of up to 5X. This resonator design shows good
frequency and polarization selectivity for use in hyperspectral and hyperpolarization detectors. We
theoretically analyzed the conditions for optimal detector absorption enhancement, and by varying
the photonic crystal hole size and top metal thickness, we adjusted the vertical coupling efficiency
to more closely match the resonator loss, moving towards achieving critical coupling. Additional
increases in coupling efficiency or reductions in loss could bring the system to near 100% absorption
in the detector. This resonator can be optimized for use at any wavelength from the terahertz to the
visible with suitable scaling of the photonic crystal holes and waveguide width, and can easily be
modified to suit any detector material, since no photonic crystal holes are etched into the active
region itself.

Infrared Light In

Photonic Crystal Resonator
Highly Doped Layer

Metal
Silicon Nitride
Metal

Active Region

Highly Doped Layer

Indium Bump

Epoxy Underfill

Figure 1.15: A design schematic for a resonant double-metal plasmonic photonic crystal FPA.

The flip-chip bonding method of focal plane array (FPA) fabrication naturally lends itself to
use with a double-metal resonant cavity, with only the top metal photonic crystal lithography step
differing from standard process techniques. DWELL FPAs have already been demonstrated with
hybridization to a readout integrated circuit [4, 28]. In Fig. 1.15, a proposed FPA schematic is
shown, illustrating the ease with which double-metal plasmonic photonic crystal resonators can
be incorporated into current FPA designs and presenting the possibility to achieve highly sensitive
mid-infrared spectral and polarization imaging at low cost.

Chapter 2

Double Disk Optomechanical
Resonators
2.1

Introduction

Many precision position measurement devices involve the coupling of mechanical degress of freedom to an electromagnetic interferometer or cavity [29, 30]. Today, cavity-mechanical systems span
a wide range of geometries and scales, from multi-kilometer long gravitational-wave detectors [31]
to coupled nanomechanical-microwave circuits [32]. For the sensitive detection and actuation of
mechanical motion, each of these systems depend upon “dynamical backaction” [33, 34] resulting
from the position-dependent feedback of electromagnetic wave momentum. Recent work in the
optical domain has used the scattering radiation pressure force to both excite and dampen oscillations of a micro-mechanical resonator [35–41], with the intriguing possibility of self-cooling the
mechanical system down to its quantum ground-state. As has been recently proposed [42, 43] and
demonstrated [44, 45], the optical gradient force within guided-wave nanostructures can be ordersof-magnitude larger than the scattering force. In this work we combine the large per-photon optical
gradient force with the sensitive feedback of a high quality factor whispering-gallery microcavity.
The cavity geometry, consisting of a pair of silica disks separated by a nanoscale gap, shows extremely strong dynamical backaction, powerful enough to excite giant coherent oscillations even
under heavily damped conditions (mechanical Q

4). In vacuum, the threshold for regenerative

mechanical oscillation is lowered to an optical input power of only 270 nanoWatts, or roughly 1000
stored cavity photons, and efficient cooling of the mechanical motion is obtained with a temperature
compression factor of 13 dB for 4 microWatt of dropped optical input power. These properties of
the double-disk resonator make it interesting for a broad range of applications from sensitive force

25
and mass detection in viscous environments such as those found in biology [46, 47], to quantum
cavity-optomechanics in which a versatile, chip-scale platform for studying the quantum properties
of the system may be envisioned. This work was initially presented in Ref. 48.

2.2

Optomechanical coupling and dynamic backaction

The per photon force exerted on a mechanical object coupled to the optical field within a resonant
cavity is given by ~gOM , where gOM

dωc /dx is a coefficient characterizing the dispersive nature

of the cavity with respect to mechanical displacement, x. In a Fabry-Perot (Fig. 2.2a) or microtoroid
resonator (Fig. 2.2b), the optical force manifests itself as a so-called scattering radiation pressure
due to direct momentum transfer from the reflection of photons at the cavity boundary [49, 50]. As
the momentum change of a photon per round trip is fixed inside such cavities, while the round-trip
time increases linearly with the cavity length, the radiation pressure per photon scales inversely with
the cavity size. In contrast, for the gradient optical force the cavity length and the optomechanical
coupling can be decoupled, allowing for photon momentum to be transfered over a length scale approaching the wavelength of light [42, 43]. This method was recently employed in a silicon photonic
circuit to manipulate a suspended waveguide [44]. However, without the feedback provided by an
optical cavity or interferometer, the optical force only provides a static mechanical displacement.
In the case of a cavity optomechanical system, dynamical backaction can be quantified by considering the magnitude of the damping/amplification that an input laser has on the mechanical motion. For a fixed absorbed optical input power in the bad-cavity limit (κ
is given by
Γm,opt

Ωm ), the maximum rate

!
3 3g2OM
Pd ,
(1 + K)2
κ3i ωc mx

(2.1)

where ωc is the optical cavity resonance frequency, mx is the motional mass of the optomechanical
system, Pd is the optical power dropped (absorbed) within the cavity, and K

κe /κi is a cavity

loading parameter (κi , the intrinsic energy loss rate of the optical cavity; κe , the energy coupling
rate between external laser and internal cavity fields). The effectiveness of the coupling between
the optical and mechanical degrees of freedom can thus be described by a back-action parameter,
B = g2OM / κ3i ωc mx , which depends upon the motional mass, the per-photon force, and the optical
cavity Q-factor.

2.3

Double-disk fabrication, optical, and mechanical design

Here we describe the design, fabrication, and characterization of a nano-optomechanical system
(NOMS) consisting of a pair of optically thin disks separated by a nanoscale gap. The doubledisk structure (Fig. 2.2c) supports high-Q whispering-gallery resonances, and provides back-action
several orders of magnitude larger than in previously demonstrated gradient force optomechanical
systems [44, 45] (very recent work [51] involving the versatile coupling of external nanomechanical
elements to the near-field of a high-Finesse microtoroid has realized very strong dynamical backaction, although still roughly two-orders of magnitude smaller than in our integrated device).
Fabrication of the double-disk whispering-gallery resonator began with initial deposition of
the cavity layers. The two silica disk layers and the sandwiched amorphous silicon (α-Si) layer
were deposited on a (100) silicon substrate by plasma-enhanced chemical vapor deposition, with
a thickness of 340

4 nm and 158

3 nm for the silica and α-Si layers, respectively. The wafer

was then thermally annealed in a nitrogen environment at a temperature of T = 1050 K for 6 hours
to drive out water and hydrogen in the film, improving the optical quality of the material. The
disk pattern was created using electron beam lithography followed by an optimized C4 F8 -SF6 gas
chemistry reactive ion etch. Release of the double-disk structure was accomplished using a SF6
chemical plasma etch which selectively (30, 000 : 1) attacks the intermediate α-Si layer and the
underlying Si substrate, resulting in a uniform undercut region between the disks extending radially
inwards 6 µm from the disk perimeter. Simultaneously, the underlying silicon support pedestal
is formed. The final gap size between the disks was measured to be 138

8 nm (shrinkage having

occurred during the anneal step). Two nanoforks were also fabricated near the double-disk resonator
to mechanically stabilize and support the fiber taper during optical coupling; the geometry was
optimized such that the forks introduce a total insertion loss of only

8%.

The final double-disk structure, shown in Fig. 2.4, consists of 340-nm-thick silica disks separated by a

140 nm air gap extending approximately 6 µm in from the disk perimeter (the undercut

region). Two different sized cavities are studied here, one large (D = 90 µm; Sample I) and one
small (D = 54 µm; Sample II) in diameter. The small diameter cavity structure represents a minimal
cavity size, beyond which radiation loss becomes appreciable (Qr

108 ).

Finite element method (FEM) simulations of the whispering-gallery optical modes of the doubledisk structure shows substantial splitting of the cavity modes into even and odd parity bonded and
anti-bonded modes (Fig. 2.2(e-f)). Due to its substantial field intensity within the air gap, the bonded

27
mode tunes rapidly with changing gap size as shown in the inset to Fig. 2.2(g). As the mode confinement in a double-disk NOMS is primarily provided by the transverse boundaries formed by the two
disks, the double-disk structure can be well approximated by a symmetric double-slab waveguide
shown in Fig. 2.1.
For the bonding mode polarized along the êy direction, the tangential component of the electric
field is given by:
Ae γx ,
B cos κx +C sin κx,
Ey =
D cosh γx,
B cos κx C sin κx,
Aeγx ,

x > h + x0 /2
x0 /2 < x < h + x0 /2
(2.2)

x0 /2 < x < x0 /2
x0 /2 > x > h
x< h

x0 /2

where κ is the transverse component of the propagation constant inside the slabs and γ is the field
decay constant in the surrounding area. They are given by the following expressions:
κ2 = k02 n2c

β2 ,

γ2 = β2

k02 n2s ,

(2.3)

where k0 = ω0 /c is the propagation constant in vacuum and β = k0 neff is the longitudinal component
of the propagation constant of the bonding mode. neff is the effective refractive index for the guided
mode. Accordingly, the tangential component of the magnetic field can be obtained through Hz =
i ∂Ey
µω0 ∂x . The continuity of Ey and Hz across the boundaries requires κ and γ to satisfy the following

equation:
κγ [1 + tanh(γx0 /2)] = κ2

γ2 tanh(γx0 /2) tan κh,

(2.4)

which reduces to tan κh = γ/κ when x0 ! 0, as expected.
The circular geometry of the double disk forms the whispering-gallery mode, in which the resonance condition requires the longitudinal component of the propagation constant, β, to be fixed as
2πRβ = 2mπ, where R is the mode radius and m is an integer. Thus, any variation on the disk spacing x0 transfers to a variation on the resonance frequency ω0 through Eqs. (2.3) and (2.4), indicating
that ω0 becomes a function of x0 . By using these two equations, we find that the optomechanical

28
ns

Figure 2.1: Schematic of a symmetric double-slab waveguide. h and x0 are the slab thickness and
the slab spacing, respectively. nc and ns are the refractive indices for the slab and surrounding area,
respectively.

coupling coefficient, gOM = dω
dx0 , is given by the general form

gOM (x0 ) =

where χ

4(n2c

cχγ2
2 γx0
k0 sech
n2s ) tan κh + n2s x0 χsech2 γx20 + 2ξ (n2c γh csc2 κh + 2n2s ) tan κh + nsγκ

κ + γ tan κh and ξ

n2c γ

i(2.5)

1 + tanh( γx20 ).

When x0 ! 0, Eq. (2.5) leads to the maximum optomechanical coupling of
gOM (0) =

ω0 γ3
2β2 + 2k02 n2c γh

(2.6)

In analogy to Fabry-Perot cavities and microtoroids, the magnitude of the optomechanical coupling
can be characterized by an effective length, LOM , defined such that gOM

ω0
LOM . Equation (2.6) infers

a minimum effective length
2 + k hn2 n2
n2s
λ0 eff
eff
1 + 20 (n2s + n2c γh) =
L0 =
3/2
n2
n2
eff

(2.7)

which is approximately on the order of the optical wavelength λ0 .
Physically, as the two slabs are coupled through the evanescent field between them with amplitude decaying exponentially with slab spacing at a rate γ [see Eq. (2.2)], the resulting optomechani-

29
cal coupling can be well approximated by an exponential function
gOM (x0 )

gOM (0)e γx0 ,

(2.8)

where gOM (0) is given by Eq. (2.6). As indicated by the red curve in Fig. 2.2(g), Eq. (2.8) provides
an excellent approximation for the optomechanical coupling coefficient in a double-disk NOMS.
Therefore, the approximate effective length, LOM

ω0
γx0
gOM (0) e , agrees well with the results simu-

lated by the finite element method, shown in Fig. 2.2(g), and the effective length decreases roughly
exponentially with decreasing disk spacing, reaching a minimum value of 3.8 µm at a resonance
optical wavelength of λc

1.5 µm. For the air gap of 138 nm used in this work, the optomechanical

coupling is estimated to be gOM /2π = 33 GHz/nm (LOM = 5.8 µm), equivalent to 22 fN/photon.
The double-disk structure also supports a number of different micro-mechanical resonances,
ranging from radial breathing modes to whispering-gallery-like vibrations of the disk perimeter.
The most strongly coupled mechanical resonance is that of the symmetric (i.e., azimuthal mode
number, m = 0) flapping motion of the disks. With a clamped inner edge and a free outer edge,
the mechanical displacement of a double disk exhibiting a flapping mode is generally a function
of radius (Fig. 2.3). What matters for the optomechanical effect, however, is the disk spacing at
the place where the whispering-gallery mode is located, as that determines the magnitude of the
splitting between the bonding and antibonding cavity modes.
As the mechanical displacement actuated by the gradient force is generally small compared with
the original disk spacing x0 , we can assume it is uniform in the region of the whispering-gallery
mode and define the effective disk spacing xm (r0 ) at the mode center, where r0 is the radius of the
whispering-gallery mode. The effective mechanical displacement is then given by xeff = xm (r0 ) x0 ,
2 /2, where m is the
corresponding to an effective mechanical potential energy of E p = mx Ω2m xeff

corresponding effective motional mass and Ωm is the resonance frequency of the flapping mode.
Note that xeff is twice the real displacement at the mode center for a single disk, xeff = 2d(r0 ). E p
reaches its maximum value when the double disk is at rest at its maximum displacement, at which
point all of the mechanical energy is stored in the strain energy Us . Therefore, E p = Us and the
effective motional mass is given by
mx =

2Us
Ωm [xm (r0 )

x0 ]2

Us
2Ωm d 2 (r0 )

(2.9)

gOM= -ω0/LC

gOM= ω0/LOM

SiO2
α-Si

SiO2

gOM= -ω0/R

Resonance frequency (THz)

208

-1

Height (µm)

-2

-1
-2

42
44
46
Radius (µm)

40
204

gOM
2π (GHz/nm)

200
196

194
400
800
Disk spacing x (nm)

10
200
400
600
Disk spacing x (nm)

800

Effective length LOM (µm)

Height (µm)

Figure 2.2: Schematic of the corresponding (a) Fabry-Parot and (b) microtoroid optomechanical
cavities. (c) Schematic of the double-disk NOMS structure, showing the mechanical flapping motion of the disks. FEM-simulated optical mode profiles of the radial component of the electric field
for the (d) bonded mode at λ = 1520 nm and (e) antibonded mode at λ = 1297.3 nm. (f) FEMsimulated tuning curve of the bonded mode. (g) Optomechanical coupling coefficient and effective
length (blue curves) for the bonded mode. gOM and LOM are both well-approximated by exponential
functions (red curves).

xm(r0)
d(r)

x0
r0
rb
whispering-gallery mode
Figure 2.3: Illustration of the disk displacement. x0 is the disk spacing in the absence of the optical
field. r0 is the radius of the whispering-gallery mode. xm (r0 ) corresponds to the effective disk
spacing at the mode center. ra and rb are the inner and outer radii of the disk region involved in the
flapping motion. d(r) is the mechanical displacement at radius r.

where both Us and d(r0 ) can be obtained from the mechanical simulations by the finite element
method.
The relationship between the effective mass and the physical mass of the double-disk NOMS
can be found by examining the mechanical potential energy. With a mechanical displacement d(r)
for each single disk [Fig. 2.3], we can find the total mechanical potential energy by integrating over
the disk regions involved in the flapping motion:
Z rb

Ep =
ra

Ω2m d 2 (r)ζ2πrhdr,

(2.10)

where ζ is the material density, h is the thickness for a single disk, and ra and rb are the inner and
outer radii of the disk region involved in the flapping motion (see Fig. 2.3). Note that E p is the
total potential energy for the two disks, which is simply two times that of a single one because of
the symmetry between the two disks. As the physical mass of a single disk region involved in the
flapping motion is given by m p = πζh(rb2

ra2 ), using Eq. (2.10), we find that the effective mass is

related to the physical mass through the following expression:
mx =

Z rb

4m p
rb2

ra2 [xm (r0 )

x0 ]2

rd (r)dr =
ra

rb2

mp
ra2 d 2 (r0 )

Z rb

rd 2 (r)dr.

(2.11)

As the whispering-gallery mode is generally located close to the disk edge (i.e., the mode radius
r0 = 44 µm in a double disk with rb = 45 µm), d 2 (r)/d 2 (r0 )

1 for most of the region between

32
ra and rb , and Eq. (2.11) shows that mx

m p /2. Therefore, the effective mass is significantly

less than half the physical mass of a single disk region. In practice, the effective mass is much
smaller than this value because of the real displacement function d(r). For the 90-µm device used
in our experiment, with an 6 µm undercut air gap region involved in the flapping motion (Fig. 2.2c
and Fig. 2.4), the effective mass is 0.264 nanogram, only about one fifth of the physical mass of a
single disk region m p = 1.18 nanogram. The effective mass decreases to 0.145 nanogram for the
54-µm device, due to the decrease in the disk radius. Note that both these values are more than two
orders of magnitude smaller than commonly used micromirrors and microtoroids [36–41, 49], and
in combination with the large per-photon force, provide a significant enhancement to the dynamic
back-action parameter which scales as g2OM /mx .

2.4

Optical and mechanical characterization

Optical and mechanical measurements were initially performed at room temperature in a one atmosphere nitrogen environment. Fig. 2.6(a) shows the wavelength scan of a large diameter double-disk
cavity (Sample I). Several radial-order whispering-gallery modes are evident in the spectrum, all of
them of TE-like polarization and bonded mode character. The fundamental TE-like bonded optical
mode at λ = 1518.57 nm is shown in the Fig. 2.6(a) inset, from which an intrinsic optical Q-factor
of 1.75

106 is inferred, taking into account the mechanical perturbations.

Unlike other microcavities in which the linear transmission is determined only by the cavity loss
and dispersion, for the double-disk NOMS, even the small thermal Brownian motions of the flapping
mode introduce significant perturbations to the cavity resonance due to the large optomechanical
coupling, leading to considerably broadened cavity transmission. Figure 2.5(a) shows an example
of the cavity transmission of Sample I. With a small input power of 5.8 µW well below the oscillation
threshold, the cavity transmission exhibits intense fluctuations when the laser frequency is scanned
across the cavity resonance. As a result, the averaged spectrum of the cavity transmission (red
curve) is significantly broader than the real cavity resonance. A correct description of the cavity
transmission requires an appropriate inclusion of the optomechanical effect, which is developed in
the following.
When the optical power is well below the oscillation threshold and the flapping mode of the
double disk is dominantly driven by thermal fluctuations, the mechanical motion can be described

VOA
1495-1565 nm
Tunable Laser
10 : 90
splitter

vacuum chamber

Polarization
Controller

Reference
Detector 1
MZ I
High-speed
Detector

Reference
Detector 2

Oscilloscope

TBPF

EDFA

double disk

VOA

fiber taper

10 : 90
splitter

1 µm

nanoforks

pedestal
undercut

20 µm
Figure 2.4: (a) Schematic of the experimental setup for optical testing of the double-disk cavity. The
cavity input and transmission are both transported through a single-mode silica fiber taper, which is
supported by two nanoforks for stable operation. A tunable laser source is used to optically probe
and actuate the double-disk structure, with input power controlled by a variable optical attenuator
(VOA) and wavelength calibrated by a Mach-Zehnder interferometer (MZI). For experiments performed in a nitrogen environment, the cavity transmission is sent directly to the photodetectors,
while it is first amplified by an erbium-doped fiber amplifier (EDFA) for the experiments performed
in vacuum. (b,c) Scanning electron microscope images of the 54-µm double-disk NOMS. False
color is used to indicate different relevant regions of the device.

34
Normalized Transmission

0.8

0.6

0.4
(a)
−10

−5

Normalized Transmission

0.8
0.6

0.98

0.4

0.96
0.94

0.2

Qi = 1.75 106
−30 −20 −10 0 10 20 30
[λ−1518.57nm] (pm)

1516

1520

Wavelength Detuning (pm)

1524

1528

(b)
1532

Wavelength (nm)

Figure 2.5: (a) The cavity transmission of Sample I in a nitrogen environment, when the laser is
scanned across the cavity resonance at 1518.57 nm with an input power of 5.8 µW. The blue curve is
the instantaneous signal collected by the high-speed detector and the red curve is the average signal
collected by the slow reference detector 2. The slight asymmetry in the transmission spectrum is
due to the static component of mechanical actuation when the laser is scanned from blue to red. The
dashed line indicates the laser frequency detuning used to record the power spectral density shown
in the top panel of Fig. 2.6(b). (b) Linear scan of the averaged cavity transmission of Sample I at an
input power of 2.9 µW. The inset shows a detailed scan for the bonding mode at 1518.57 nm, with
the experimental data in blue and the theoretical fitting in red.

by the following equation:
d2x
FT (t)
dx
+ Γm + Ω2m x =
dt
dt
mx

(2.12)

where Ωm , Γm , and mx are the resonance frequency, damping constant, and effective mass of the
flapping mode, respectively. FT is the Langevin force driving the mechanical Brownian motion, a
Markovin process with the following correlation function:
hFT (t)FT (t + τ)i = 2mx Γm kB T δ(τ),

(2.13)

where T is the temperature and kB is Boltzmann’s constant. It can be shown easily from Eqs. (2.12)
and (2.13) that the Brownian motion of the flapping mode is also a Markovin process with a spectral
correlation given by he
x(Ω1 )e
x (Ω2 )i = 2πSx (Ω1 )δ(Ω1

Ω2 ), where xe(Ω) is the Fourier transform

of the mechanical displacement x(t) defined as xe(Ω) =

R +∞

iΩt
∞ x(t)e dt, and Sx (Ω) is the spectral

intensity for the thermal mechanical displacement with the following form:
Sx (Ω) =

(Ω2m

2Γm kB T /mx
Ω2 )2 + (ΩΓm )2

(2.14)

35
The time correlation of the mechanical displacement is thus given by
hx(t)x(t + τ)i =
2π

Z +∞

hx2 iρ(τ)

Sx (Ω)e iΩτ dτ

hx2 ie Γm jτj/2 cos Ωm τ,

(2.15)

where hx2 i = kB T /(mx Ω2m ) is the variance of the thermal mechanical displacement and ρ(τ) is the
normalized autocorrelation function for the mechanical displacement.
To be general, we consider a doublet resonance in which two optical fields, one forward and
the other backward propagating, circulate inside the microcavity and couple via Rayleigh scattering
from the surface roughness. The optical fields inside the cavity satisfy the following equations:
da f
= (i∆0
dt
dab
= (i∆0
dt

κ/2

igOM x)a f + iηab + i κe Ain ,

(2.16)

κ/2

igOM x)ab + iηa f ,

(2.17)

where a f and ab are the forward and backward whispering-gallery modes (WGMs), normalized
such that U j = ja j j2 ( j = f , b) represents the mode energy. Ain is the input optical wave, normalized
such that Pin = jAin j2 represents the input power. κ is the photon decay rate for the loaded cavity,
and κe is the photon escape rate associated with the external coupling. ∆0 = ω

ω0 is the frequency

detuning from the input wave to the cavity resonance and η is the mode coupling coefficient. In the
case of a continuous-wave input, Eqs. (2.16) and (2.17) provide a formal solution of the forward
WGM:
a f (t) = i κe Ain
where f (τ)

Z +∞

Rτ

cos(ητ) f (τ)e igOM 0 x(t τ )dτ dτ,

(2.18)

e(i∆0 κ/2)τ represents the cavity response. Using Eq. (2.15), we find that the statisti-

cally averaged intracavity field is given as:
ha f (t)i = i κe Ain
where ε

Z +∞

cos(ητ) f (τ)e 2 h(τ) dτ,

(2.19)

g2OM hx2 i and h(τ) is defined as
ZZ τ

h(τ)

ρ(τ1

τ2 )dτ1 dτ2 .

(2.20)

36
Similarly, we can find the averaged energy for the forward WGM as:
hU f (t)i = κe Pin

ZZ +∞

f (τ1 ) f (τ2 ) cos(ητ1 ) cos(ητ2 )e 2 h(jτ1 τ2 j) dτ1 dτ2

κe Pin κ iη
2κ κ 2iη
where f j (τ)

Z +∞

e 2 h(τ) [ fc (τ) + fs (τ)] dτ + c.c.,

e(i∆ j κ/2)τ ( j = c, s), with ∆c = ∆0 + η and ∆s = ∆0

(2.21)

η. c.c. denotes complex conju-

gate.
As the transmitted power from the double disk is given by
p
PT (t) = Pin + κeU f (t) + i κe Ain a f (t)
the averaged cavity transmission, hT i
hT i = 1

κe κi
2κ

Ain a f (t) ,

(2.22)

hPT i/Pin , thus takes the form

iηκe
κi (κ 2iη)

Z +∞

2 h(τ)

[ fc (τ) + fs (τ)] dτ + c.c. .

(2.23)

In the case of a singlet resonance, η = 0 and Eq. (2.23) reduces to the simple form expression
hT i = 1

κe κi

Z +∞

e 2 h(τ) [ f (τ) + f (τ)] dτ.

(2.24)

In the absence of opto-mechanical coupling, gOM = 0 and Eq. (2.24) reduces to the conventional
form of
T =1

κe κi
∆0 + (κ/2)2

(2.25)

as expected.
Using the theory developed above and fitting the experimental averaged cavity transmission
spectrum, we obtain the optical Q factor of the resonance, as shown in Fig. 2.5(b) for Sample I. The
same approach is used to describe the cavity transmission of Sample II, given in Fig. 2.6(a).
The radio-frequency (RF) power spectrum of the optical signal transmitted through the cavity
(Fig. 2.6(b), top panel) exhibits three clear frequency components at 8.30, 13.6, and 27.9 MHz
corresponding to thermally-actuated resonances of the double-disk structure. These values agree
well with FEM simulations of the differential flapping mode (7.95 MHz), and the first (14.2 MHz)
and second (28.7 MHz) order radial breathing modes (Fig. 2.6c). The strong dynamic back-action
of the flapping mode (under thermal excitation) also produces a broadband spectral background in

37
the RF spectrum with a shoulder at the second harmonic frequency. A correct description of the
power spectrum (Fig. 2.6(b), red curve) shows that the flapping mode has a 3-dB linewidth of 2.1
MHz (mechanical Q-factor, QM = 3.95), limited by the squeeze-film process of the nitrogen gas
between the disks [52].
We can describe the power spectral density of the cavity transmission in the presence of mechanical Brownian motion using a linear-perturbation approximation when the optomechanical effects
are small, and a non-perturbation theory, accurate for arbitrarily strong optomechanical effects,
when the effects are larger. Both analyses are presented here.
If the induced optomechanical perturbations are small, Eq. (2.18) can be approximated as
a f (t)

i κe Ain

Z +∞

cos(ητ) f (τ) 1

Z τ

igOM

x(t

τ0 )dτ0 dτ.

(2.26)

In this case, the transmitted optical field can be written as AT (t) = Ain + i κe a f (t)

A0 + δA(t),

where A0 is the transmitted field in the absence of the optomechanical effect and δA is the induced
perturbation. They take the following forms:
A0 = Ain 1

Z +∞

κe

cos(ητ) f (τ)dτ

δA(t) = igOM κe Ain

dτ cos(ητ) f (τ)

Z +∞

Ain Â0 ,

Z τ

x(t

τ0 )dτ0 .

(2.27)
(2.28)

The transmitted power then becomes P(t) = jAT (t)j2

jA0 j2 + A0 δA(t) + A0 δA (t). It is easy to

show that hδA(t)i = 0 and hPT (t)i = jA0 j2 . As a result, the power fluctuations, δP(t)

PT (t)

hPT (t)i, become
Z +∞

δP(t)
where u(τ)

iκe cos(ητ)[Â0 f (τ)

gOM Pin

Z τ

dτu(τ)

x(t

τ0 )dτ0 ,

(2.29)

Â0 f (τ)]. By using Eq. (2.15), we find the autocorrelation func-

tion for the power fluctuation to be
hδP(t)δP(t + t0 )i

εPin2

ZZ +∞

dτ1 dτ2 u(τ1 )u(τ2 )ψ(t0 , τ1 , τ2 ),

(2.30)

where ψ(t0 , τ1 , τ2 ) is defined as
Z τ1

ψ(t0 , τ1 , τ2 )

dτ01

Z τ2

dτ02 ρ(t0 + τ01

τ02 ).

(2.31)

38
Taking the Fourier transform of Eq. (2.30), we obtain the power spectral density SP (Ω) of the cavity
transmission to be
SP (Ω)

g2OM Pin2 H(Ω)Sx (Ω),

(2.32)

where Sx (Ω) is the spectral intensity of the mechanical displacement given in Eq. (2.14) and H(Ω)
is the cavity transfer function given by
H(Ω) =

Z +∞

u(τ)(eiΩτ

1)dτ .

(2.33)

In the case of a singlet resonance, the cavity transfer function takes the form:
4∆20 (κ2i + Ω2 )
κ2e
H(Ω) =
∆20 + (κ/2)2 [(∆0 + Ω) + (κ/2) ] [(∆0 Ω) + (κ/2) ]

(2.34)

In most cases, the photon decay rate inside the cavity is much larger than the mechanical damping rate, κ

Γm . For a specific mechanical mode at the frequency Ωm , the cavity transfer function

can be well approximated by H(Ω)

H(Ωm ). In particular, in the sideband-unresolved regime, the

cavity transfer function is given by a simple form of
4κ2e κ2i ∆20
H=
4 .
∆20 + (κ/2)2

(2.35)

Therefore, Eq. (2.32) shows clearly that, if the optomechanical effect is small, the power spectral
density of the cavity transmission is directly proportional to the spectral intensity of the mechanical
displacement.
The situation becomes quite complicated when the optomechanical effects are large. From
Eq. (2.22), the autocorrelation function for the power fluctuation of the cavity transmission, δP(t)
PT (t)

hPT i, is given by
hδP(t1 )δP(t2 )i = κ2e hU f 1U f 2 i κe h Ain a f 1 Ain a f 1 Ain a f 2 Ain a f 2 i

3/2
+ iκe hU f 1 Ain a f 2 Ain a f 2 i + hU f 2 Ain a f 1 Ain a f 1 i
2
κe hU f i + i κe Ain ha f i Ain ha f i ,

(2.36)

where U f j = U f (t j ) and a f j = a f (t j ) ( j = 1, 2). Equation (2.36) shows that the autocorrelation
function involves various correlations between the intracavity energy and field, all of which can

39
be found using Eqs. (2.15) and (2.18). For example, we can find the following correlation for the
intracavity field:
h Ain a f 1

Ain a f 1

κe Pin2

ZZ +∞

Ain a f 2

Ain a f 2 i

dτ1 dτ2C1C2 e 2 (h1 +h2 ) f1 f2 e εψ + f1 f2 eεψ + c.c. ,

(2.37)

where, in the integrand, C j = cos(ητ j ), h j = h(τ j ), f j = f (τ j ) (with j = 1, 2), and ψ = ψ(t2
t1 , τ1 , τ2 ). h(τ) and ψ(t2

t1 , τ1 , τ2 ) are given by Eqs. (2.20) and (2.31), respectively.

Equations (2.20) and (2.31) show that h(τ) and ψ(t2

t1 , τ1 , τ2 ) vary with time on time scales

of 1/Ωm and 1/Γm . However, in the sideband-unresolved regime, κ

Γm and κ

Ωm . As the

cavity response function f (τ) decays exponentially with time at a rate of κ/2, the integrand in
Eq. (2.37) becomes negligible when τ1

2/κ or τ2

2/κ. Therefore, ψ(t2

t1 , τ1 , τ2 ) can be well

approximated as

ψ(t2

Similarly, h(τ)

t1 , τ1 , τ2 ) =

(2.38)

τ2 , since h(τ) = ψ(0, τ, τ). Therefore, Eq. (2.37) becomes
h Ain a f 1

where ∆t = t2

Z +∞
iΩτ
Sx (Ω) iΩ(t2 t1 )
iΩτ1
dΩ
2πhx2 i ∞ Ω2
Z +∞
τ1 τ2
Sx (Ω)e iΩ(t2 t1 ) dΩ = τ1 τ2 ρ(t2 t1 ).
2πhx2 i ∞

Ain a f 1

Ain a f 2

Ain a f 2 i

κe Pin2 Φ(∆t,C1C2 ),

(2.39)

t1 and Φ(∆t,C1C2 ) is defined as
ZZ +∞

Φ(∆t,C1C2 )

2
dτ1 dτ2C1C2 e 2 (τ1 +τ2 ) f1 f2 e ετ1 τ2 ρ + f1 f2 eετ1 τ2 ρ + c.c. ,

(2.40)

with ρ = ρ(∆t). Following a similar approach, we can find the other correlation terms in Eq. (2.36).
Using these terms in Eq. (2.36), we find that the autocorrelation function of the power fluctuations
is given by
hδP(t1 )δP(t2 )i

κ2e Pin2 Φ(∆t, σ1 σ2 )

κe hU f i + i κe Ain ha f i

2
Ain ha f i ,

(2.41)

40
where σ j = σ(τ j ) ( j = 1, 2) and σ(τ) is defined as

σ(τ)

κe (κ2 + 2η2 )
ηκe
sin(ητ).
cos(ητ) + 2
κ(κ + 4η )
κ + 4η2

(2.42)

Moreover, Eq. (2.19) and (2.21) show that, in the sideband-unresolved regime, ha f i and hU f i
are well approximated by
ha f (t)i

i κe Ain

hU f (t)i

κe Pin

Z +∞

ZZ +∞

ε 2

cos(ητ) f (τ)e 2 τ dτ,

(2.43)

f (τ1 ) f (τ2 ) cos(ητ1 ) cos(ητ2 )e 2 (τ1 τ2 ) dτ1 dτ2 .

(2.44)

Therefore, we obtain the final term in Eq. (2.41) as
κe hU f i + i κe Ain ha f i

Ain ha f i

Z +∞

κe Pin

ε 2

σ(τ) [ f (τ) + f (τ)] e 2 τ dτ.

(2.45)

Using this term in Eq. (2.41), we obtain the final form for the autocorrelation of the power fluctuations:
hδP(t1 )δP(t2 )i

κ2e Pin2 [Φ(∆t, σ1 σ2 )

Φ(∞, σ1 σ2 )] .

(2.46)

It can be further simplified if we notice that the exponential function e ετ1 τ2 ρ(∆t) in Eq. (2.40) can
be expanded in a Taylor series as
+∞

( ετ1 τ2 )n n
ρ (∆t).
n!
n=0

e ετ1 τ2 ρ(∆t) = ∑

(2.47)

Substituting this expression into Eq. (2.40) and using it in Eq. (2.46), we obtain the autocorrelation
function for the power fluctuation in the following form
+∞

hδP(t)δP(t + t0 )i

εn ρn (t0 )
jGn + ( 1)n Gn j2 ,
n!
n=1

κ2e Pin2 ∑

(2.48)

where Gn is defined as
Z +∞

ε 2

τn σ(τ) f (τ)e 2 τ dτ.

(2.49)

In the case of a singlet resonance, η = 0 and σ(τ) simplifies considerably to σ = κi /κ. The autocorrelation function for the power fluctuation is still described by Eq. (2.48).
In general, the power spectral density of the cavity transmission is given by the Fourier transform

41
of Eq. (2.48):
+∞

εn Sn (Ω)
jGn + ( 1)n Gn j2 ,
n!
n=1

S p (Ω) = κ2e Pin2 ∑

(2.50)

where Sn (Ω) is defined as
Z +∞

Sn (Ω) =

ρn (τ)eiΩτ dτ.

(2.51)

Eq. (2.14) shows that the spectral intensity of the mechanical displacement can be approximated
by a Lorentzian function, resulting in an approximated ρ(τ) given as ρ(τ)

e Γm jτj/2 cos Ωm τ [see

Eq. (2.15)]. As a result, Eq. (2.51) becomes
Sn (Ω)

1 n
n!
nΓm
∑ k!(n k)! (nΓm /2)2 + [(2k n)Ωm + Ω]2 .
2n k=0

(2.52)

Combining Eq. (2.50) and (2.52), we can see that, if the optomechanical coupling is significant, the
thermal mechanical motion creates spectral components around the harmonics of the mechanical
frequency with broader linewidths. As shown clearly in Fig. 2.6(b), the second harmonic is clearly
visible. In particular, if the fundamental mechanical linewidth is broad, various frequency components on the power spectrum would smear out, producing a broadband spectral background, as
shown in the top panel of Fig. 2.6(b) for Sample I. This phenomenon is similar to the random-fieldinduced spectral broadening in nuclear magnetic resonance [53] and atomic resonance fluorescence
[54].
This theory can be extended easily for the case with multiple mechanical frequencies. In this
case, the power spectrum only only exhibits harmonics of each mechanical frequency, but also
their frequency sums and differences. As shown in the bottom panel of Fig. 2.6(b), the frequency
components near 0 MHz are the differential frequencies and those near 18-20 MHz are the second
harmonic and sum frequencies.

2.5

Regenerative oscillation

2.5.1

Ambient pressure

Despite the near-unity mechanical quality factor of the flapping mode, the powerful dynamic backaction in the double-disk structure provides sufficient compensation of mechanical loss to excite
regenerative mechanical oscillation. As shown in Fig. 2.6(d), with an input optical power of 760 µW
launched at the blue detuned side of the resonance, the induced parametric mechanical instability

Power Spectral Density (dBm/Hz)

Normalized Transmission

0.8
0.6

0.98

0.4

0.96
0.94

0.2

1516

Qi = 1.75 106
−30 −20 −10 0 10 20 30
[λ−1518.57nm] (pm)

1520

1524

1528

Wavelength (nm)

1532

-90
-100

Qm = 3.95

-110

3 4

-120
-130

Qm = 4.07 103

-90

-80
-90
-100

-100

-110
8.2 8.6 9 9.4 9.8
Frequency (MHz)

-110

Frequency (MHz)

-25
0.5
0.5
-50

0.2

0.4

0.6

Time (µs)

0.8

100

150

Time (ns)

200

Mechanical displacement (pm)

Normalized transmission

Figure 2.6: (a) Optical transmission spectrum of a large diameter (D = 90 µm; Sample I) double-disk
cavity. The inset shows the fundamental TE-like bonded mode at λ = 1518.57 nm. (b) Upper panel:
optical transmission power spectral density (PSD) of a Sample I double-disk in the 1 atm. nitrogen
environment for Pi = 5.8 µW. Experimental data in blue, theoretical modeling in red, and detector
noise background in yellow. Lower panel: transmission PSD of a small diameter (D = 54 µm;
Sample II) double-disk cavity in vacuum for Pi = 44 nW. The inset shows a zoom-in of the spectrum
around the fundamental flapping mode frequency. (c) FEM simulated mechanical modes indicated
in (b). (d) Recorded transmission waveform of Sample I for Pi = 0.76 mW. (e) Comparison of
experimental (blue curve) and simulated (red curve) waveforms, with the corresponding simulated
mechanical displacement (green curve).

43
causes the cavity transmission to oscillate over the entire coupling depth with a fundamental frequency of 13.97 MHz (this value is about 68% larger than the intrinsic mechanical frequency due
to the optical spring effect [55]). A zoom-in of the recorded time waveform (Fig. 2.6e) agrees well
with our numerical simulation which shows that the gradient force actuates an extremely large
(50 pm) mechanical displacement amplitude, dragging the cavity resonance over more than 10
cavity-linewidths and leaving distinctive features of the Lorentzian cavity transfer function. In particular, two sequential passes of the cavity resonance across the laser frequency can be seen, along
with an overshoot and oscillation of the transmitted optical power resulting from the quick release
of Doppler shifted photons from the cavity.
The optomechanical oscillations are simulated through the following coupled equations governing the intracavity optical field and mechanical motions, respectively:
da
= (i∆0
igOM x)a + i κe Ain ,
dt
dx
d2x
FT (t) Fo (t)
+ Γm + Ω2m x =
dt 2
dt
mx
mx

(2.53)
(2.54)

where we have counted in both the thermal Langevin force FT and the optical gradient force Fo =
gOM jaj2
for actuating mechanical motions.
ω0

The threshold for regenerative oscillation depends sensitively upon the optical input power and
the average laser-cavity resonance detuning, a map of which can be used to quantify the strength of
the dynamic back-action. An estimate of the threshold detuning (∆th ), for a given input power, can
be determined from the abrupt kink in the cavity transmission that marks the onset of regenerative
oscillation (Fig. 2.8(a) and Fig. 2.7).
Figure 2.7 shows an example of the cavity transmission of Sample I. The mechanical flapping
mode starts to oscillate when the input laser frequency is scanned across a certain detuning. Within
this detuning value, the same magnitude of optomechanical oscillation is excited over a broad range
of laser blue detuning. The intense transmission oscillations cover the entire coupling depth, leaving
an abrupt kink on the transmission spectrum. The coupling depth at the kink point, ∆Tth , corresponds
to the threshold coupling at the given power level, from which we can obtain the threshold frequency
detuning ∆th .
The detuning dependence of the optomechanical amplification coefficient can be lumped into a

Normalized Transmission

1.0

0.8

0.6

0.4

0.2

−30

−20

−10

Wavelength Detuning (pm)

Figure 2.7: Scan of the cavity transmission of Sample I at an input power of 0.76 mW, with the instantaneous and averaged signals shown in blue and red, respectively. The dashed line indicated the
laser frequency detuning used to record the time-dependent cavity transmission given in Fig. 2.6(d).

single detuning function,
f (∆)

∆2 + (κ/2)2
κκe κ3i ∆

κ 2
(∆ + Ωm ) +
(∆

Ωm ) +

κ 2

(2.55)

where κ = κi + κe is the total photon decay rate of the loaded cavity. The right panel of Fig. 2.8(b)
shows a map of f (∆th ) versus optical input power for the 90 µm diameter double-disk cavity in
the heavily damped nitrogen environment. The data in Fig. 2.8(b), as expected, shows a linear
dependence of f (∆th ) on input power, and is well described in the unresolved sideband regime [50]
by
f (∆th ) =

2g2OM Pi
ωc mx Γm κ3i

2B
Pi ,
Γm

(2.56)

where Γm = 2.1 MHz is the bare mechanical damping rate of the flapping mode. Fitting of eq.
(2.56) to the data in Fig. 2.8(b) yields a dynamic back-action parameter of B = 0.061 MHz/µW,
corresponding to an optomechanical coupling factor of gOM /2π = 33.8

0.4 GHz/nm, in good

agreement with the simulated result of 33 GHz/nm.

2.5.2

Vacuum

In order to eliminate the squeeze-film damping of the nitrogen environment, measurements were
also performed in vacuum (P < 5

10 4 Torr). The significantly reduced mechanical linewidth

in vacuum shows that the flapping mode consists of a small cluster of modes (Fig. 2.6(b), bottom

0.8
0.6
0.05

f = 8.53 MHz

-0.05

10-27

-10

-6

-8

-4

-2

Wavelength detuning (pm)

10-29

10-28

10-31

10-32

10-29

Frequency (MHz)

sing
increa

ca
p ti

10-30
6.8

vacuum

air

60
50
40
30
20
10
00

0.5

1.5

200 400 600 800

Input power (µW)

100

10-30

Displacement spectral density (m2/Hz)

-0.6

f = 9.63 MHz

Spectral density (m2/Hz)

0.2
-0.2

Threshold detuning function f(∆th)

Normalized transmission

III I

Effective temperature (K)

7.2

7.4

7.6

7.8

Frequency (MHz)

8.2

8.4

8.6

0.1

Input power (µW)

Figure 2.8: (a) Top panel: Normalized cavity transmission for Sample II in vacuum and Pi = 11
µW. Blue and red traces show the instantaneous and low-pass-filtered signals, respectively. Middle
panel: the transduction amplitude of the frequency component at 8.53 MHz and its higher-order
harmonics. Bottom panel: the transduction amplitude of the frequency component at 9.63 MHz and
its higher-order harmonics. (b) f (∆th ) as a function of optical input power. Right panel: Sample
I in a 1 atm. nitrogen environment. Left panel: Sample II in vacuum (inset shows the minimum
achievable threshold (green arrow)). (c) Spectral intensity of the thermally-driven fundamental
flapping mode at various input powers, recorded for Sample II in vacuum, with a laser detuning
of ∆ = −1.45(κ/2) (inset shows the displacement sensitivity at the highest input power with the
second optical attenuator removed), and (d) the corresponding effective temperature. In (d), the red
curve is a fit to the data, the solid green (dashed black) curve is a theoretical curve obtained using
the estimated B-parameter
from theleft panel of (b) and the experimental (optimal) detuning of
∆ = −1.45(κ/2) ∆ = −(κ/2)/ 5 .

46
panel). These modes are a mixture of the lower-lying azimuthal modes, coupled together due to
deviations in circularity of the undercut region and support pedestal.
Because of the extremely short round-trip time of the cavity mode, the optical wave is sensitive
only to the variations of averaged disk spacing around the whole disk. As a result, the optomechanical coupling for the fundamental flapping mode, which has a flapping amplitude uniformly
distributed around the disk perimeter, is maximum, but it is nearly zero for flapping modes with
higher-order azimuthal mode numbers. However, due to the asymmetry in practical devices, the net
variations in the average disk spacing induced by the higher-order flapping modes (with azimuthal
mode number

1) is not zero, and their thermal motion is visible in the transmission power spec-

trum. In general, their optomechanical coupling is weak and does not provide efficient dynamic
back action.
Measurements of the optical spring effect indicates that the optical field renormalizes the cluster
of modes, with the lowest-frequency mode at 8.53 MHz transforming into the fundamental flapping
mode with uniformly distributed displacement along the disk perimeter (the rest of the modes decouple from the light field). With an in-vacuum QM = 4070 (Fig. 2.6(b), inset), the fundamental
flapping mode has an extremely low threshold input power for regenerative oscillation. Figure 2.8(a)
shows a transmission spectrum when the laser is scanned across the cavity resonance. Three different regimes can be clearly seen: (I) transduction of thermal motion, (II) onset of optically-driven
oscillation, and (III) optically damped motion. The onset of regenerative oscillation coincides with
a frequency shift in the fundamental flapping mode to 9.63 MHz as shown in the bottom two panels
of Fig. 2.8(a). The left panel of Figure 2.8(b) shows a plot of the in-vacuum f (∆th ) versus input
power, with a measured minimum threshold power of Pi = 267 nW. Extrapolation of the experimental data using Eqs. (2.55) and (2.56) to the optimal detuning point shows a minimum threshold
power of only 40 nW.

2.6

Optomechanical cooling

In general, the optomechanical effect is governed by Eqs. (2.53) and (2.54). However, the optomechanical effect during mechanical cooling is well described by linear perturbation theory since
the thermal mechanical motion is significantly suppressed. The intracavity field can thus be approximated as a(t)

a0 (t) + δa(t), where a0 is the cavity field in the absence of optomechanical

coupling and δa is the perturbation induced by the thermal mechanical motion. From Eq. (2.53),

47
they are found to satisfy the following equations:
da0
= (i∆0
dt
dδa
= (i∆0
dt

κ/2)a0 + i κe Ain ,

(2.57)

igOM xa0 .

(2.58)

κ/2)δa

In the case of a continuous-wave input, Eq. (2.57) gives a steady-state value given as:
i κe Ain
a0 =
κ/2 i∆0

(2.59)

and Eq. (2.58) provides the spectral response for the perturbed field amplitude,
δe
a(Ω) =

igOM a0 xe(Ω)
i(∆0 + Ω) κ/2

where δe
a(Ω) is the Fourier transform of δa(t) defined as δe
a(Ω) =

(2.60)
R +∞

iΩt
e(Ω)
∞ δa(t)e dt. Similarly, x

is the Fourier transform of x(t).
The optical gradient force, Fo =
Fo (t) =

gOM jaj2
ω0 , is given by

gOM 2
ja0 j + a0 δa(t) + a0 δa (t) .
ω0

(2.61)

The first term is a static term which only affects the equilibrium position of the mechanical motion,
and can be removed simply by shifting the zero-point of the mechanical displacement to the new
equilibrium position. Therefore, we neglect this term in the following discussion. The second and
third terms provide the dynamic optomechanical coupling. From Eq. (2.60), the gradient force is
given by the following equation in the frequency domain:
Feo (Ω) =

2g2OM ja0 j2 ∆0 xe(Ω)
∆20 Ω2 + (κ/2)2 + iκΩ
ω0
[(∆0 + Ω)2 + (κ/2)2 ] [(∆0 Ω)2 + (κ/2)2 ]

(2.62)

As expected, the gradient force is linearly proportional to the thermal mechanical displacement.
Equation (2.54) can be solved easily in the frequency domain, which becomes
(Ω2m

Ω2

iΓm Ω)e
x=

FeT
Feo
+ .
mx mx

(2.63)

Equation (2.63) together with (2.62) provides the simple form for the thermal mechanical displace-

48
ment,
xe(Ω) =

FeT
mx (Ω0m )2

Ω2

iΓ0m Ω

(2.64)

where Ω0m and Γ0m are defined as
(Ω0m )2

Γ0m

2g2OM ja0 j2 ∆0
∆20 Ω2 + (κ/2)2
mx ω0
[(∆0 + Ω)2 + (κ/2)2 ] [(∆0 Ω)2 + (κ/2)2 ]
2g2 ja0 j2 ∆0
∆20 Ω2m + (κ/2)2
Ω2m + OM
mx ω0
[(∆0 + Ωm )2 + (κ/2)2 ] [(∆0 Ωm )2 + (κ/2)2 ]
2g2OM ja0 j2 κ∆0
Γm
mx ω0
[(∆0 + Ω) + (κ/2) ] [(∆0 Ω)2 + (κ/2)2 ]
2g2OM ja0 j2 κ∆0
Γm
mx ω0
[(∆0 + Ωm )2 + (κ/2)2 ] [(∆0 Ωm )2 + (κ/2)2 ]

Ω2m +

(2.65)

(2.66)

Equations (2.64)-(2.66) show clearly that the primary effect of the optical gradient force on the
mechanical motion is primarily to change its mechanical frequency (the so-called optical spring
effect) and energy decay rate to the new values given by Eqs. (2.65) and (2.66). The efficiency of
optomechanical control is determined by the figure of merit g2OM /mx . On the red detuned side, the
optical wave damps the thermal mechanical motion and thus increases the energy decay rate. At the
same time, the mechanical frequency is modified, decreasing with increased cavity energy in the
sideband-unresolved regime.
Using Eqs. (2.13) and (2.64), we find that the spectral intensity of the thermal displacement is
given by a form similar to Eq. (2.14):
Sx (Ω) =

2Γm kB T /mx
Ω2 ]2 + (ΩΓ0m )2

[(Ω0m )2

(2.67)

m kB T
which has a maximum value Sx (Ω0m ) = mx2Γ
. The variance of the thermal mechanical displace(Ω0 Γ0 )2
m m

ment is equal to the area under the spectrum,
h(δx)2 i =

2π

Z +∞

Sx (Ω)dΩ =

kB T Γm
mx (Ω0m )2 Γ0m

(2.68)

Cooling the mechanical motion reduces the spectral magnitude and the variance of thermal displacement.
The large mechanical amplification of the double-disk NOMS implies a correspondingly efficient cooling of mechanical motion on the red-detuned side of the cavity resonance. As shown in
Fig. 2.8(c) for Sample II in vacuum, the spectral intensity of the fundamental flapping mode de-

49
creases dramatically with increased input power, accompanied by a significant broadening of the
mechanical linewidth. Even for the strongest damping levels, the inset to Fig. 2.8(c) shows good
signal to noise for the transduced motion due to the high displacement sensitivity of the double-disk
(7

10 17 m/Hz1/2 , as limited by the background level).
A measure of the optical cooling can be determined from the integrated area under the displace-

ment spectrum [56]. For a mechanical mode in thermal equilibrium, the effective temperature can
be inferred from the thermal mechanical energy using the equipartition theorem:
kB Teff = mx (Ω0m )2 h(δx)2 i.

(2.69)

The area under the displacement spectrum thus provides an accurate measure of the effective temperature. In practice, fluctuations on the laser frequency detuning may cause the mechanical frequency and damping rate to fluctuate over a certain small range [Eq. (2.65) and (2.66)], with a
probability density function of p(Ω0m ). As a result, the experimentally recorded displacement spectrum is given by the averaged spectrum

Sx (Ω) =

Sx (Ω)p(Ω0m )dΩ0m ,

where Sx (Ω) is given by Eq. (2.67) and we have assumed

(2.70)

p(Ω0m )dΩ0m = 1. The experimentally

measured spectral area is thus
2π

Z +∞

Sx (Ω)dΩ =

h(δx)2 ip(Ω0m )dΩ0m

h(δx)2 i.

(2.71)

Therefore, the integrated spectral area obtained from the experimental spectrum is the averaged
variance of thermal mechanical displacement, from which, according to the equipartition theorem,
we obtain the effective average temperature

kB T eff = mx (Ωm )2 h(δx)2 i,

where Ωm

(2.72)

Ω0m p(Ω0m )dΩ0m is the center frequency of the measured displacement spectrum Sx (Ω).

Compared with the room temperature, the effective temperature is thus given by

T eff (Ωm )2 h(δx)2 i
= 2
T0
Ω h(δx)2 i

(2.73)

50
where h(δx)2 i0 is the displacement variance at room temperature, given by the spectral area at T0 .
Figure 2.8(d) plots the inferred temperaure, Teff , which drops down to 12.5 K for a maximum
input power of Pi = 11 µW (Pd = 4.4 µW). In principle, the effective temperature is related to the
optical damping rate (Γm,opt ) through the relation T0 /Teff = 1 + Γm /Γm,opt , where T0 = 300 K is the
bath temperature. In Fig. 2.8d the red curve is a fit of the measured cooling curve using the relation
T0 /Teff = 1 + αPi , whereas the green curve represents the expected cooling curve for the dynamic
back-action parameter (B = 0.032 MHz/µW) determined from the threshold plot in the right panel
of Fig. 2.8(b) and the experimental laser-cavity detuning (∆ =

1.45(κ/2)). For comparison, we

have also plotted (dashed black line) the theoretical cooling curve in the case of optimal laser-cavity
detuning (∆ = (κ/2)/ 5). The difference between the two theoretical curves and the measured
data, along with the limited range of optical input power studied, can largely be attributed to issues
associated with the limited bandwidth and range of our current cavity locking scheme (a problem
exacerbated by the very large transduction of even the Brownian motion of the disks). As the
dashed black curve indicates, technical improvements in the cavity locking position and stability
should enable temperature compression factors of 20 dB for less than 1 µW of dropped power.

2.7

Discussion

The large dynamic back-action of the double-disk cavity, primarily a result of the large per-photon
force and small motional mass of the structure, opens up several areas of application outside the
realm of more conventional ultra-high-Q cavity geometries. This can be seen by considering not
only the efficiency of the cooling/amplification process, but also the maximum rate of effective
cooling/amplification, the scale of which is set by the optical cavity decay rate [57, 58]. In the
double-disk cavities presented here, the dynamic back-action parameter is B
cavity decay rate of κ/2π

0.06 MHz/µW for a

100 MHz. The combination allows for higher mechanical frequencies

of operation, where the bare damping is expected to scale with frequency, and makes possible enormous temperature compression ratios. A quantum mechanical analysis of the optical self-cooling
process [57, 58], indicates that the sideband resolved regime (κ . 32Ωm ) is necessary to reduce
the phonon occupancy below unity. Having already achieved optical Q-factors in excess of 106 , and
planar silica microdisks having already been demonstrated with Q > 107 [59], we expect that further
optimization of the double-disk NOMS will be able to extend its operation well into the sideband
resolved regime. The combination of large dynamic back-action parameter and large maximum

51
amplification rate also present intriguing possibilites for sensitive, high temporal resolution force
detection [60], particularly in heavily damped environments such as fluids for biological applications [46, 47]. Other application areas enabled by the chip-scale format of these devices include
tunable photonics [42–44], optical wavelength conversion [61], and RF-over-optical communication.

Chapter 3

Coherent Mechanical Mode Mixing in
Optomechanical Nanocavities
3.1

Introduction

The coherent mixing of multiple excitation pathways provides the underlying mechanism for many
physical phenomena. Well-known examples include the Fano resonance [62] and electromagnetically induced transparency (EIT) [63], arising from the interference between excitations of discrete states and/or a continuum background. In the past few decades, Fano-like or EIT-like resonances have been discovered in a variety of physical systems, such as electron transport in quantum wells/dots [64, 65], phonon interactions in solids [66, 67], inversion-free lasers [68, 69], coupled photonic microcavities [70–73], and plasmonic metamaterials [74]. Here we report a new
class of coherent excitation mixing which appears in the mechanical degree of freedom of nanooptomechanical systems (NOMS). We use two canonical systems, coupled microdisks and coupled photonic-crystal nanobeams, to show that the large optical stiffening introduced by the optical
gradient force actuates significant coherent mixing of mechanical excitations, not only leading to
renormalization of the mechanical modes, but also producing Fano-like and EIT-like optomechanical interference, both of which are fully tunable by optical means. The demonstrated phenomena
introduce the possibility for classical/quantum information processing via optomechanical systems,
providing an on-chip platform for tunable optical buffering, storage, and photonic-phononic quantum state transfer. This work was initially presented in Ref. 75.
Optical forces within micromechanical systems have attracted considerable interest of late due
to the demonstration of all-optical amplification and self-cooling of mesoscopic mechanical resonators [35–39]. This technique for sensing and control of mechanical motion relies on the radia-

53
tion pressure forces that build up in a mechanically compliant, high-Finesse optical cavity, resulting
in strong dynamical back-action between the cavity field and mechanical motion. More recently
[44, 45, 48, 76–78], it has been realized that guided wave nanostructures can also be used to generate extremely large per-photon optical forces via the gradient optical force [79]. The combination of
tailorable mechanical geometry, small motional mass, and large per-photon force in such nanostructures results in a regime of operation in which the dynamic response of the coupled optomechanical
system can significantly differ from that of the bare mechanical structure. In particular, the mechanical motion can be renormalized by the optical spring effect [29, 33, 45, 55, 80, 81], creating a
highly anistropic, intensity-dependent effective elastic modulus of the optomechanical structure.

3.2

Zipper cavity and double-disk design, fabrication, and optical characterization

We have focused on two specific implementations of nanoscale cavity optomechanical systems,
shown in Fig. 3.1, in which dynamical back-action effects are particularly strong. The first system
consists of two patterned nanobeams in the near-field of each other, forming what has been termed a
zipper cavity [45, 82]. In this cavity structure the patterning of the nanobeams localizes light through
Bragg-scattering, resulting in a series of high Finesse (F

104 ), near-infrared (λ

1550 nm)

optical supermodes of the beam pair. Clamping to the substrate at either end of the suspended
beams results in a fundamental in-plane mechanical beam resonance of frequency

8 MHz. The

second cavity optomechanical system is based upon the whispering-gallery microdisk optical cavity
structure presented in Chapter 2 and Ref. 48. By creating a pair of microdisks, one on top of the
other with a nanoscale gap in between, strong optical gradient forces may be generated between
the microdisks while maintaining the benefits of the low-loss, high-Q (Q

106 ) character of the

whispering-gallery cavity. As shown schematically in Fig. 3.1(a), the double-disk structure [48]
is supported and pinned at its center, allowing the perimeter of the disks to vibrate in myriad of
different ways.
The zipper cavity is formed from a thin-film (400 nm) of tensile-stressed, stoichiometric Si3 N4
deposited by low-pressure chemical vapor deposition on a silicon substrate. Electron beam-lithography,
followed by a series of plasma and wet chemical etches, are used to form the released nanobeam
structure. The double-disk structure is formed from a 158 nm sacrifical amorphous silicon layer
sandwiched in between two 340 nm thick silica glass layers, all of which are deposited via plasma-

1 µm

Figure 3.1: (a) Schematic and (b) zoomed-in scanning electron microscopic (SEM) image of the
double-disk NOMS. (c) FEM-simulated electric field intensity of a transverse-electric (TE) polarized, bonded (even parity) whispering-gallery supermode between the two microdisks (shown in
cross-section and for resonance wavelength λc 1550 nm). The double-disk bonded supermode
has an optomechanical coupling coefficient of gOM /2π 33 GHz/nm. The device studied here has
a measured resonance wavelength of λc = 1538 nm and an intrinsic and loaded quality (Q) factor
of 1.07 106 and 0.7 106 , respectively. (d) Schematic, (e) SEM image, and (f) FEM-simulated
bonded (even parity) optical supermode of the zipper cavity. The zipper cavity bonded supermode
has an optomechanical coupling coefficient of gOM /2π 68 GHz/nm, a measured resonance wavelength of λc = 1545 nm, and an intrinsic and loaded Q-factor of 3.0 104 and 2.8 104 , respectively.
Additional details for both devices are in Refs. 45, 48.

55
enhanced chemical vapor deposition. A high temperature (1050 K) thermal anneal is used to improve the optical quality of the as-deposited silica layers. The microdisk pattern was fabricated by
reactive ion etching, and the sandwiched α-Si layer was undercut by 6 µm from the disk edge using
a sulfur hexafluoride dry release etch. This etch simultaneously undercuts the silicon substrate to
form the underlying silicon pedestal. The final air-gap between the silica disks size is measured to
be 138 nm due to shrinkage of the amorphous silicon layer during annealing.
A fiber-taper optical coupling technique is used to in-couple and out-couple light from the zipper
and double-disk cavities. The fiber taper, with extremely low-loss (88% transmission efficiency),
is put in contact with the substrate near the cavities in order to mechanically anchor it during all
measurements (thus avoiding power-dependent movement of the taper due to thermal and/or optical
forces). An optical fiber polarization controller, consisting of a series of circular loops of fiber, is
used to selectively excite the transverse-electric polarized optical modes of both cavities.
RF spectra are measured by direct detection of the optical power transmitted through the cavities
using a 125 MHz bandwidth photoreceiver (noise-equivalent-power NEP= 2.5 pW/Hz1/2 from 010 MHz and 22.5 pW/Hz1/2 from 10-200 MHz, responsivity R = 1 A/W, transimpedance gain
G=4

104 V/A) and a high-speed oscilloscope (2 Gs/s sampling rate and 1 GHz bandwidth). A

pair of “dueling” calibrated optical attenuators are used before and after the cavities in order to vary
the input power to the cavity while keeping the detected optical power level constant. The measured
electrical noise floor is set by the circuit noise of the photodetector for the optical power levels
considered in this work, corresponding to

3.3

125 dBm/Hz near 10 MHz.

Theory of optomechanical effects in the presence of mode mixing

Of particular interest in both the zipper and double-disk systems are two types of motion: the differential motion of the nanobeams or disks, in which the changing gap between the elements creates
a large dispersive shift in the internally propagating cavity light field; and the common motion, in
which both nanobeams or disks move together, and the gap remains approximately constant, resulting in mechanical motion that is decoupled from the light field. Due to the strong light-field coupling
and dynamical backaction of the differential mode, and the correspondingly weak coupling of the
common mode, we term these two motional states optically-bright and optically-dark, respectively.
The theory for gradient-force optomechanical systems in which there is coupling between these two
types of mechanical excitations is presented in the next sections.

3.3.1

Intracavity eld in the presence of optomechanical coupling

In the presence of optomechanical coupling, the optical field inside the cavity satisfies the following
equation:
da
= (i∆0
dt

Γt /2

igom xb )a + i Γe Ain ,

(3.1)

where a is the optical field of the cavity mode, normalized such that U = jaj2 represents the mode
energy, and Ain is the input optical wave, normalized such that Pin = jAin j2 represents the input
power. Γt is the photon decay rate for the loaded cavity and Γe is the photon escape rate associated
with the external coupling. ∆0 = ω

ω0 is the frequency detuning from the input wave to the cavity

resonance. gom is the optomechanical coupling coefficient associated with the optically bright mode,
with a mechanical displacement given by xb . In Eq. (3.1), we have neglected the optomechanical
coupling to the optically dark mode because of its negligible magnitude.
Well below the threshold of mechanical oscillation, the mechanical motion is generally small,
and its impact on the intracavity optical field can be treated as a small perturbation. As a result, the
intracavity field can be written as a(t)

a0 (t) + δa(t), where a0 is the cavity field in the absence of

optomechanical coupling and δa is the perturbation induced by the mechanical motion. They satisfy
the following two equations:
da0
= (i∆0
dt
dδa
= (i∆0
dt

Γe Ain ,

(3.2)

igom xb a0 .

(3.3)

Γt /2)a0 + i
Γt /2)δa

In the case of a continuous-wave input, Eq. (3.2) leads to a steady state given by
i Γe Ain
a0 =
Γt /2 i∆0

(3.4)

and Eq. (3.3) provides a spectral response for the perturbed field amplitude of
δe
a(Ω) =

igom a0 xeb (Ω)
i(∆0 + Ω) Γt /2

where δe
a(Ω) is the Fourier transform of δa(t) defined as δe
a(Ω) =
is the Fourier transform of xb (t).

(3.5)
R +∞

iΩt
eb (Ω)
∞ δa(t)e dt. Similarly, x

3.3.2

The power spectral density of the cavity transmission

From the discussion in the previous section, the transmitted optical power from the cavity is given
by
PT = Ain + i Γe a

jA0 j2 + i

Γe (A0 δa

A0 δa ) ,

(3.6)

where A0 is the steady-state cavity transmission in the absence of optomechanical coupling. It is
given by
A0 = Ain

(Γ0

Γe )/2 i∆0
Γt /2 i∆0

(3.7)

where Γ0 is the photon decay rate of the intrinsic cavity. It is easy to show that the averaged cavity
transmission is given by hPT i = jA0 j2 , as expected. By using Eqs. (3.5), (3.6), and (3.7), we find the
power fluctuations, δPT (t)

PT (t)

hPT i, are given in the frequency domain by

iΓ
(Ω)
(Γ0 Γe )/2 + i∆0
in
om
δPeT (Ω) =
Γt /2 i(∆0 + Ω)
(Γt /2) + ∆0

(Γ0 Γe )/2
Γt /2 + i(∆0

i∆0
Ω)

(3.8)

where δPeT (Ω) is the Fourier transform of δPT (t). By using Eq. (3.8), we obtain a power spectral
density (PSD) for the cavity transmission of
SP (Ω) = g2om Pin2 Sxb (Ω)H(Ω),

(3.9)

where Sxb (Ω) is the spectral intensity of the mechanical displacement for the optically bright mode
which will be discussed in detail in the following sections. H(Ω) is the cavity transfer function
defined as
H(Ω)

4∆20 (Γ20 + Ω2 )
Γ2e
2
2
∆0 + (Γt /2)2 [(∆0 + Ω) + (Γt /2) ] [(∆0 Ω) + (Γt /2) ]

(3.10)

In general, when compared with Sxb (Ω), H(Ω) is a slowly varying function of Ω and can be well
approximated by its value at the mechanical resonance: H(Ω)

H(Ω0mb ). Clearly then, the power

spectral density of the cavity transmission is linearly proportional to the spectral intensity of the
mechanical displacement of the optically bright mode.

3.3.3

The mechanical response with multiple excitation pathways

When the optically bright mode is coupled to an optically dark mode, the Hamiltonian for the
coupled mechanical system is given by the general form:

Hm =

p2b
p2
+ kb xb2 + d + kd xd2 + κxb xd ,
2mb 2
2md 2

(3.11)

where x j , p j , k j , and m j ( j = b, d) are the mechanical displacement, kinetic momentum, the spring
constant, and the effective motional mass for the jth mechanical mode, respectively, and κ represents the mechanical coupling between the bright and dark modes. The subscripts b and d denote
the optically bright and optically dark modes, respectively. With this system Hamiltonian, including
the optical gradient force on the optically bright mode and counting in the mechanical dissipation
induced by the thermal mechanical reservoir, we obtain the equations of motion for the two mechanical modes:
dxb
Fb
Fo
d 2 xb
+ Γmb
+ Ω2mb xb + xd =
+ ,
dt 2
dt
mb
mb mb
d 2 xd
dxd
Fd
+ Γmd
+ Ω2md xd +
xb =
dt
dt
md
md
where Ω2m j

(3.12)
(3.13)

kj
th
m j is the mechanical frequency for the j mode. Fj ( j = b, d) represents the Langevin

forces from the thermal reservoir actuating the Brownian motion, with the following statistical properties in the frequency domain:
hFei (Ωu )Fej (Ωv )i = 2mi Γmi kB T δi j 2πδ(Ωu

Ωv ),

(3.14)

where i, j = b, d, T is the temperature and kB is the Boltzmann constant. Fei (Ω) is the Fourier
transform of Fi (t).
In Eq. (3.12), Fo =

gom jaj2
ω0 represents the optical gradient force. From the previous section, we

find that it is given by
Fo (t) =

gom 2
ja0 j + a0 δa(t) + a0 δa (t) .
ω0

(3.15)

The first term is a static term which only changes the equilibrium position of the mechanical motion. It can be removed simply by shifting the mechanical displacement to be centered at the new
equilibrium position. Therefore, we neglect this term in the following discussion. The second and

59
third terms provide the dynamic optomechanical coupling. From Eq. (3.5), the gradient force is
found to be given in the frequency domain by
Feo (Ω)

fo (Ω)e
xb (Ω) =

∆20 Ω2 + (Γt /2)2 + iΓt Ω
2g2om ja0 j2 ∆0 xeb (Ω)
, (3.16)
ω0
[(∆0 + Ω)2 + (Γt /2)2 ] [(∆0 Ω)2 + (Γt /2)2 ]

which is linearly proportional to the mechanical displacement of the optically bright mode.
Equations (3.12) and (3.13) can be solved easily in the frequency domain, in which the two
equations become
Feb
Feo
xed =
+ ,
mb
mb mb
Fed
Ld (Ω)e
xd +
xeb =
md
md
Lb (Ω)e
xb +

where L j (Ω)

Ω2m j

Ω2

(3.17)
(3.18)

iΓm j Ω ( j = b, d). Substituting Eq. (3.16) into Eq. (3.17), we find that

Eq. (3.17) can be written in the simple form,
Lb (Ω)e
xb +

Feb
xed =
mb
mb

(3.19)

where Lb (Ω) is now defined with a new mechanical frequency Ω0mb and energy decay rate Γ0mb as
Lb (Ω) = Ω2mb

Ω2

iΓmb Ω

fo (Ω)
mb

(Ω0mb )2

Ω2

iΓ0mb Ω,

(3.20)

and the new Ω0mS and Γ0mS are given by
(Ω0mb )2

Γ0mb

∆20 Ω2 + (Γt /2)2
2g2om ja0 j2 ∆0
mb ω0
[(∆0 + Ω)2 + (Γt /2)2 ] [(∆0 Ω)2 + (Γt /2)2 ]
∆20 Ω2mb + (Γt /2)2
2g2 ja0 j2 ∆0
Ω2mb + om
mb ω0
[(∆0 + Ωmb )2 + (Γt /2)2 ] [(∆0 Ωmb )2 + (Γt /2)2 ]
2g2om ja0 j2 Γt ∆0
Γmb
m b ω0
[(∆0 + Ω) + (Γt /2) ] [(∆0 Ω)2 + (Γt /2)2 ]
2g2om ja0 j2 Γt ∆0
Γmb
m b ω0
[(∆0 + Ωmb )2 + (Γt /2)2 ] [(∆0 Ωmb )2 + (Γt /2)2 ]
Ω2mb +

(3.21)

(3.22)

Clearly, the effect of the optical gradient force on the optically bright mode is primarily to change
its mechanical frequency (the optical spring effect) and energy decay rate (mechanical amplification
or damping).

60
Equations (3.18) and (3.19) can be solved easily to obtain the solution for the optically bright
mode,
xeb (Ω) =
where η4

Feb (Ω)
mb Ld (Ω)

κ Fed (Ω)
mb md
Lb (Ω)Ld (Ω) η4

(3.23)

κ2
mb md represents the mechanical coupling coefficient. By using Eq. (3.14) and (3.23),

we obtain the spectral intensity of the mechanical displacement for the optically bright mode,
Sxb (Ω) =

2kB T η4 Γmd + Γmb jLd (Ω)j2
mb jLb (Ω)Ld (Ω) η4 j2

(3.24)

where Lb (Ω) is given by Eq. (3.20). The mechanical response given by Eq. (3.24) is very similar to
the atomic response in EIT.

3.3.4

The mechanical response with external optical excitation

The previous section focuses on the case in which the mechanical excitations are primarily introduced by the thermal perturbations from the environmental reservoir. However, the mechanical
motion can be excited more intensely through the optical force by modulating the incident optical
wave. In this case, the input optical wave is composed of an intense CW beam together with a small
modulation: Ain = Ain0 + δA(t). As a result, Eq. (3.3) now becomes
dδa
= (i∆0
dt

Γt /2)δa

igom xb a0 + i Γe δA.

(3.25)

This equation leads to the intracavity field modulation given in the frequency domain as:
igom a0 xeb (Ω) i Γe δA(Ω)
δe
a(Ω) =
i(∆0 + Ω) Γt /2

(3.26)

where δA(Ω)
is the Fourier transform of δA(t). By use of this solution together with Eq. (3.15), the
gradient force now becomes
Feo (Ω) = fo (Ω)e
xb (Ω) + Fee (Ω),

(3.27)

where fo (Ω) is given by Eq. (3.16) and Fee (Ω) represents the force component introduced by the
input modulation. It is given by the following form:
e ( Ω)
A(Ω)
om
Fee (Ω) =
ω0
i(∆ + Ω) Γt /2 i(∆ Ω) + Γt /2

(3.28)

61
In particular, in the sideband-unresolved regime, Eq. (3.28) can be well approximated by
Fee (Ω)

i Γe gom h
e ( Ω) .
a0 δA(Ω)
+ a0 δA
ω0 (i∆ Γt /2)

(3.29)

In the case that the mechanical excitation is dominated by the external optical modulation, the
thermal excitation from the reservoir is negligible and Eqs. (3.12) and (3.13) become
dxb
Fo
d 2 xb
+ Γmb
+ Ω2mb xb + xd =
dt
dt
mb
mb
d 2 xd
dxd
+ Γmd
+ Ω2md xd +
xb = 0.
dt
dt
md

(3.30)
(3.31)

Using Eqs. (3.26) and (3.27), following a similar procedure as the previous section, we find that the
mechanical displacement for the optically bright mode is now given by
xeb (Ω) =

Fee (Ω)
Ld (Ω)
mb Lb (Ω)Ld (Ω)

η4

(3.32)

where Lb (Ω) and Ld (Ω) are given in the previous section. Clearly, the mechanical response given
in Eq. (3.32) is directly analogous to the atomic response in EIT systems [83].

3.4

Mechanical mode renormalization in zipper cavities

We begin with an analysis of the zipper cavity, in which the strong optically-induced rigidity associated with differential in-plane motion of the nanobeams results in a dressing of the mechanical
motion by the light field. Optical excitation provides both a means to transduce mechanical motion
(which is imparted on the transmitted light field through phase and intensity modulation) and to
apply an optical-intensity-dependent mechanical rigidity via the strong optical gradient force. By
fitting a Lorentzian to the two lowest-order in-plane mechanical resonances in the radio-frequency
(RF) optical transmission spectrum, we display in Fig. 3.2(a) and (b) the resonance frequency and
resonance linewidth, respectively, of the two coupled mechanical modes of the nanobeam pair
as a function of laser-cavity detuning. At large detuning (low intra-cavity photon number) the
nanobeams’ motion is transduced without inducing significant optical rigidity, and the measured
mechanical resonances are split by

200 kHz, with similar linewidths (damping) and transduced

amplitudes (Fig. 3.2(c)). As the laser is tuned into resonance from the blue-side of the cavity, and
the intra-cavity photon number increases (to

7000), the higher frequency resonance is seen to

Power Spectral Density (dBm/Hz)

Mechanical Frequency (MHz)

62
8.4
8.2

-90

Nphoton

-100

7.8
7.6
7.4
0.25 b

Power Spectral Density (dBm/Hz)

Mechanical Linewidth (MHz)

-110

0.2
0.15
0.1
0.05

-2

-1.5

-1

-0.5

0.5

Normalized Detuning (∆ο/Γt)

1.5

-60

Nphoton

-70

-80

7.2

7.6

8.4

Frequency (MHz)

8.8

Figure 3.2: (a) Mechanical frequency and (b) linewidth of the fundamental in-plane mechanical
resonances of the zipper cavity’s coupled nanobeams as a function of laser frequency detuning. The
input power for these measurements is 127 µW, corresponding to a maximum cavity photon number
of 7000 on resonance. The circles show the experimental data and the solid curves correspond to
a fit to the data using Eq. (3.37). Optically-transduced RF spectrum at a laser-cavity detuning of (c)
∆0 /Γt = 2.1 and (d) ∆0 /Γt = 0.32. The two nanobeams vibrate independently when the laser-cavity
detuning is large, but are renormalized to the cooperative (e) differential and (f) common motions
near resonance.
significantly increase in frequency while the lower frequency mode tunes to the average of the independent beam frequencies with its transduced amplitude significantly weaker. The linewidth of
the high frequency resonance also tends to increase, while that of the lower frequency mode drops.
Tuning from the red-side of the cavity resonance reverses the sign of the frequency shifts and the
roles of the high and low frequency modes.
A qualitative understanding of the light-induced tuning and damping of the zipper cavity nanobeam
motion emerges if one considers the effects of squeeze-film damping [84]. Squeeze-film effects, a
result of trapped gas in-between the beams (measurements were performed in 1 atm. of nitrogen),
tend to strongly dampen differential motion of the beams and should be negligible for common motion of the beams. Similarly, the optical gradient force acts most strongly on the differential beam
motion and negligibly on the common-mode motion. The sign of the resulting optical spring is positive for blue detuning and negative for red detuning from the cavity resonance. Putting all of this

63
together, a consistent picture emerges from the data in Fig. 3.2 in which the nanobeams start out at
large detuning moving independently with similar damping (the frequency splitting of

200 kHz is

attributable to fabrication assymetries in the beams). As the detuning is reduced, and approaches the
cavity half-linewidth, the motion of the nanobeams is dressed by the internal cavity field into differential motion with a large additional optical spring constant (either positive or negative) and large
squeeze-film damping component, and common motion with reduced squeeze-film damping and
minimal coupling to the light field. Due to the strong light-field coupling of the differential mode
and the correspondingly weak coupling of the common mode, we term these dressed motional states
optically-bright and optically-dark, respectively.
In general, the motion of individual disks or nanobeams satisfies the following equations:
Fq
d 2 x1
F1
Fo
dx1
+ Γm1
+ Ω2m1 x1 =
+ ,
dt 2
dt
m1 m1 m1
Fq
dx2
d 2 x2
F2
Fo
+ Γm2
+ Ω2m2 x2 =
dt
dt
m2 m2 m2

(3.33)
(3.34)

where Fq is the viscous force from the squeeze film damping, and m j , x j , Ωm j , Γm j , Fj ( j = 1, 2)
are the effective mass, the mechanical displacement, resonance frequency, damping rate, and the
Langevin force for individual disks (or beams), respectively.
The optically bright mechanical mode corresponds to the differential motion of the two disks/beams,
with a mechanical displacement given by xb

x2 . By transferring Eqs. (3.33) and (3.34) into

the frequency domain, it is easy to find that the mechanical displacement of the optically bright
mode is given by
Fe1 (Ω)
xeb (Ω) =
m1 L1 (Ω)
where L j (Ω) = Ω2m j

Ω2

h
Fe2 (Ω)
Feq (Ω) + Feo (Ω) ,
m2 L2 (Ω)
m1 L1 (Ω) m2 L2 (Ω)

(3.35)

iΓm j Ω ( j = 1, 2). The squeeze-film effect is produced by the pressure

differential between the gap and the outer region introduced by the differential mechanical motion,
and thus has a magnitude linearly proportional to the differential displacement. In general, it can be
described by Feq (Ω) = fq (Ω)e
xb (Ω), where fq (Ω) represents the spectral response of the squeeze gas
film [84]. Using this form together with Eq. (3.16) in Eq. (3.35), we obtain the spectral intensity of

64
the optically bright mode displacement,
2kB T

Sxb (Ω) =
L1 (Ω)L2 (Ω)

Γm2
Γm1
m1 jL2 (Ω)j + m2 jL1 (Ω)j

[ fo (Ω) + fq (Ω)]

L1 (Ω)
L2 (Ω)
m2 + m1

i 2.

(3.36)

As the squeeze-film effect primarily damps the differential motion, its spectral response can be
approximated as fq (Ω)

iαq Ω. Moreover, since the two disks or nanobeams generally have only

slight asymmetry due to fabrication imperfections, they generally have quite close effective masses
and energy damping rates: m1

m2 = 2mb and Γm1

Γm2

Γm , where we have used the fact

that the effective motional mass of the differential motion is given by mb = m1 m2 /(m1 + m2 ). As a
result, Eq. (3.36) can be well approximated by
Sxb (Ω)

where Γq

kB T Γm
mb L1 (Ω)L2 (Ω)

jL1 (Ω)j2 + jL2 (Ω)j2
2 [ f o (Ω)/mb + iΓq Ω] [L1 (Ω) + L2 (Ω)]

(3.37)

αq /mb represents the damping rate introduced by the squeeze gas film, and the spectral

response of the gradient force fo (Ω) is given by Eq. (3.16).
The intrinsic mechanical frequencies of 7.790 and 7.995 MHz for the two individual nanobeams
are measured from the experimental recorded PSD with a large laser-cavity detuning. The optomechanical coupling coefficient is 68 GHz/nm and the effective mass is 10.75 pg for the fundamental
differential mode, both obtained from FEM simulations (note that these values are different than
those quoted in Ref. 45 due to the different definition of mode amplitude for xb ). The intrinsic and
loaded optical Q factors are 3.0

104 and 2.8

104 , respectively, obtained from optical charac-

terization of the cavity resonance. By using these values in Eqs. (3.37) and (3.16), we can easily
find the mechanical frequencies and linewidths for the two renormalized modes, where we treat
the intrinsic mechanical damping rate Γm and the squeeze-film-induced damping rate Γq as fitting
parameters. As shown in Fig. 3.2, this theoretical model provides an accurate description of the
mechanical mode renormalization, with a fitted intrinsic mechanical and squeeze-film damping rate
of 0.03 and 0.2 MHz, respectively.
Similarly, we can obtain the spectral intensity of xd

x1 + x2 for the optically-dark mechanical

65
mode, which is given by the following form:

Sxd (Ω) = 2kB T

Γm2
m2

L1 (Ω)

m1 [ f o (Ω) + f q (Ω)]

L1 (Ω)L2 (Ω)

+ Γmm11 L2 (Ω) m22 [ fo (Ω) + fq (Ω)]
i2
L2 (Ω)
[ fo (Ω) + fq (Ω)] L1m(Ω)
m1
m2 = 2mb and Γm1

Similar to the optically-bright mode, with m1

Γm2

(3.38)

Γm , Eq. (3.38) can be

well approximated by
Sxd (Ω)

kB T Γm jL1 (Ω) h(Ω)j2 + jL2 (Ω) h(Ω)j2
md L1 (Ω)L2 (Ω) 1 h(Ω) [L1 (Ω) + L2 (Ω)] 2

where md = m/2 is the effective mass of the common mode and h(Ω)

(3.39)

[ fo (Ω)/mb + iΓq Ω] repre-

sents the total spectral response of the optical gradient force and squeeze film damping. In particular, when the optical-spring-induced frequency shift is much larger than the intrinsic mechanical
frequency splitting, the spectral intensities of these two modes reduce to
Sxb (Ω)

2kB T Γm /mb
jLo (Ω)

where L0 (Ω) = (Ωm1 + Ωm2 )2 /4

Ω2

h(Ω)j

Sxd (Ω)

2kB T Γm /md
jLo (Ω)j2

(3.40)

iΓm Ω. Equation (3.40) indicates that the optically bright

and dark modes reduce to pure differential and common modes, respectively.

3.5

Coherent mechanical mode mixing in double-disks

A similar optically-induced renormalization mechanism applies to the double-disk cavity structure
shown in Fig. 3.1(a-c). In this case, the large optical spring effect for the differential motion of
the two microdisks excites another, more intriguing form of coherent optomechanical mixing with
the optically dark common mode of the disks. Unlike in the zipper cavity, FEM modeling of the
mechanics of the double-disk structure indicates a significant frequency splitting between the differential and common modes of motion of the double disk (shown in Fig. 3.3(b) and (c)), primarily due
to the difference in the extent of the undercut between the disk layers and the extent of the central
pedestal which pins the two disk layers. The result is that the differential, or “flapping” motion,
of the undercut disk region has a lower frequency of 7.95 MHz, whereas the common motion of
the disks results in a higher frequency (14.2 MHz) “breathing” motion of the entire double-disk
structure.

66
The RF-spectrum of the transmitted optical intensity through a double-disk cavity, measured
using the same fiber probing technique as for the zipper cavity, is shown in Fig. 3.2(c) versus lasercavity detuning. For the largest detuning (in which the optical spring is negligible) the spectrum
shows a broad (2.1 MHz) resonance at 8.3 MHz and a much narrower (0.11 MHz) resonance at
13.6 MHz, in good corresponce with the expected frequencies of the flapping and breathing modes,
respectively. The difference in damping between the two resonances can be attributed to the strong
squeeze-film damping of the differential flapping motion of the disks. As shown in Fig. 3.2(c), the
flapping mode can be tuned in frequency via the optical spring effect from its bare value of 8.3
MHz all the way out to 15.7 MHz (optical input power of Pi = 315 µW). In the process, the flapping
mode is tuned across the breathing mode at 13.6 MHz. Although the optically-dark breathing mode
is barely visible in the tranduced spectrum at large laser-cavity detunings, its spectral amplitude is
considerably enhanced as the optically-bright flapping mode is tuned into resonance. In addition,
a strong Fano-like lineshape, with

13 dB anti-resonance, appears in the power spectrum near

resonance of the two modes (Fig. 3.3(f-h)).
As shown schematically in Fig. 3.5(a), the Fano-like interference in the optically-bright power
spectral density can be attributed to an internal mechanical coupling between the flapping and
breathing mechanical modes. This is quite similar to the phonon-phonon interaction during the
structural phase transition in solids [66, 85–89], in which the internal coupling between phonon
modes produces Fano-like resonances in the Raman-scattering spectra.
The power spectral density (PSD) of the cavity transmission is linearly proportional to Eq. (3.24).
Equation (3.24) together with (3.9) is used to find the theoretical PSD shown in Fig. 3.3, by using an
optomechanical coupling coefficient of gom /2π = 33 GHz/nm and an effective mass of mb = 264 pg
for the flapping mode, both obtained from FEM simulations. The intrinsic and loaded optical quality factors of 1.07

106 and 0.7

106 are obtained from optical characterization of the cavity

resonance, and are also given in the caption of Fig. 3.1. The intrinsic mechanical frequencies and
damping rates of the two modes (Ωmb , Ωmd , Γmb , and Γmd ) are obtained from the experimentally
recorded PSD of cavity transmission with a large laser-cavity detuning, as given in the caption of
Fig. 3.3. The mechanical coupling coefficient η is treated as a fitting parameter. Fitting of the PSDs
results in η = 3.32 MHz, indicating a strong internal coupling between the two mechanical modes.
As shown clearly in Fig. 3.3(d, f-h), our theory provides an excellent description of the observed
phenomena.

Power Spectral Density (20 dB/div)

10
12
14
16
Frequency (MHz)

-75

01 2 3
∆0/Γt

PSD (dBm/Hz)

-80
-85

-90
-95

-100

Frequency (MHz)

Figure 3.3: (a,b) FEM simulated mechanical motion of the differential flapping mode (a) and
the common breathing mode (b), with simulated frequencies of 7.95 and 14.2 MHz. The color
map indicates the relative magnitude (exaggerated) of the mechanical displacement. (c) Recorded
power spectral density (PSD) of the cavity transmission for the double-disk, with an input power of
315 µW. Each curve corresponds to a normalized laser-cavity frequency detuning, ∆0 /Γt indicated
in (e). For display purposes, each curve is relatively shifted by 10 dB in the vertical axis. (d) The
corresponding theoretical PSD. (f-h) Detailed PSD at three frequency detunings indicated by the
arrows in (e), with the experimental and theoretical spectra in blue and red, respectively.

3.6

Coherent mechanical mode mixing in zipper cavities

The coherent mixing of mechanical excitation is universal to gradient-force-based NOMS with a
giant optical spring effect. Similar phenomena to that presented for double-disks were also observed
in the zipper cavity. However, due to the device geometry, the coupled nanobeams have more
complex mechanical mode families in which all the even-order mechanical modes are optically
dark, because they exhibit a mechanical node at the beam center where the optical mode is located.
As the same-order common and differential motions of the two beams have similar mechanical
frequencies, they can simultaneously couple to the same optically bright mode, leading to multiple
excitation interferences on the mechanical response.
In the case when the optically bright mode is coupled to two optically dark modes, the Hamiltonian for the mechanical system is given by the following general form:

Hm =

i=b,1,2

p2i
+ ki xi2 + κ1 xb x1 + κ2 xb x2 ,
2mi 2

(3.41)

where i = b, 1, 2 corresponds to the optically bright mode and optically dark modes 1 and 2, respectively. With this Hamiltonian, counting in both the optical gradient force and the Langevin forces
from the thermal reservoir, we obtain the equations of motions for the three modes:
d 2 xb
κ1
κ2
Fb
Fo
dxb
+ Γmb
+ Ω2mb xb + x1 + x2 =
+ ,
dt 2
dt
mb
mb
mb mb
dx1
d 2 x1
κ1
F1
+ Γm1
+ Ω2m1 x2 + xb =
dt
dt
m1
m1
dx2
d 2 x2
κ2
F2
+ Γm2
+ Ω2m2 x3 + xb =
dt 2
dt
m2
m2

(3.42)
(3.43)
(3.44)

where the gradient force Fo is given by Eq. (3.15), and the statistical properties of the Langevin
forces are given by Eq. (3.14). Following the same analysis as in Section 3.3.3, we can obtain the
spectral intensity for the mechanical displacement of the optically bright mode as
Sxb (Ω) =

2kB T η41 Γm1 jL2 (Ω)j2 + η42 Γm2 jL1 (Ω)j2 + Γmb jL1 (Ω)L2 (Ω)j2
mb
Lb (Ω)L1 (Ω)L2 (Ω) η4 L2 (Ω) η4 L1 (Ω)

where η4j

(3.45)

κ2j
mb m j ( j = 1, 2) represents the mechanical coupling coefficient. L j (Ω) = Ωm j

Ω2

iΓm j Ω ( j = 1, 2) and Lb (Ω) is given by Eq. (3.20) with Ω0mb and Γ0mb given in Eqs. (3.21) and (3.22),
respectively. As the optical wave is coupled to the optically bright mode only, the power spectral

69
density of the cavity transmission is still given by Eq. (3.9), with the mechanical response Sxb given
in Eq. (3.45).
Figure 3.4 shows the PSD of the cavity transmission by launching a continuous wave into a resonance of the coupled nanobeams with an intrinsic and loaded Q factor of 3.0

104 and 2.8

104 ,

respectively. Three mechanical modes are clearly visible, where mode I is the fundamental differential mode [Fig. 3.4(h)I], and mode II and III correspond to the second-order common and
differential modes [Fig. 3.4(h)II and III], respectively. Similar to the double-disk NOMS, the gigantic optical spring effect shifts the frequency of the optically bright mode I from its intrinsic
value of 8.06 MHz to 19 MHz, crossing over both optically dark modes II and III closely located
at 16.54 and 17.04 MHz and resulting in complex interferences on the power spectra [Fig. 3.4(a)].
Equation (3.45) provides an accurate description of the observed phenomena, as shown clearly in
Fig. 3.4(b), (d)-(f). Fitting of the PSD results in mechanical coupling coefficients of η1 = 3.45 MHz
and η2 = 3.48 MHz, implying that the two optically dark modes couple to the fundamental optically
bright mode with a similar magnitude.

3.7

Analogy to electromagnetically-induced transparency

The mechanical response given by Eq. (3.24) is directly analogous to the atomic response in EIT [83].
Just as in EIT, one can understand the resulting Fano lineshape in two different ways. The first
perspective considers the interference associated with multiple excitation pathways. In the optomechanical system, the mechanical motion of the flapping mode is thermally excited along two different pathways, either directly into the broadband (lossy) flapping mode, or indirectly, through the
flapping mode, into the long-lived breathing mode, and then back again into the flapping mode. The
two excitation pathways interfere with each other, resulting in the Fano-like resonance in the spectral response of the optically bright flapping mode. An alternative, but perfectly equivalent view of
the coupled optomechanical system considers the dressed states resulting from the internal mechanical coupling. In this picture the internal mechanical coupling renormalizes the broadband flapping
mode and the narrowband breathing mode into two dressed mechanical modes, both broadband and
optically-bright. In particular, when the flapping and breathing mechanical frequencies coincide,
the two dressed modes are excited with equal amplitude and opposite phase at the center frequency
between the split dressed states. Destructive interference results, suppressing excitation of the mechanical system at the line center. Consequently, the mechanical motion becomes purely a trapped

ΙΙ

ΙΙΙ

(c)

(b)

Power Spectral Density (20 dB/div)

(a)

14
16
18
Frequency (MHz)

∆0/Γt

PSD (dBm/Hz)

-80

-90

-100

-110

(e)

(d)
14

Frequency (MHz)

(f )
Frequency (MHz)

Frequency (MHz)

(h)

ΙΙ

ΙΙΙ

Figure 3.4: (a) Experimentally recorded power spectral densities of the cavity transmission for the
zipper cavity of Fig. 3.1(d-f), with an input power of 5.1 mW. Each curve corresponds to a laser
frequency detuning indicated in (c). Each curve is relatively shifted by 5 dB in the vertical axis for
a better vision of the mechanical frequency tuning and the induced mechanical interference. The
optically dark mode II and III have a full-width at half maximum (FWHM) of 0.16 and 0.15 MHz,
respectively. The optically bright mode I has an intrinsic FWHM of 0.30 MHz. (b) The corresponding theoretical spectra of the power spectral density. (d)-(f) The detailed spectra of the power
spectral density at three frequency detunings indicated by the three arrows in (c). The blue and red
curves show the experimental and theoretical spectra, respectively. (h) FEM simulated mechanical
motions for the fundamental differential mode (I), the second-order common (II) and differential
(III) modes, whose frequencies are indicated by the arrows in (a). The color map indicates the
relative magnitude (exaggerated) of the mechanical displacement.

input

output

input

coupling

k12

|+〉

|2〉

pump

γ2

|−〉

control

pump

probe
γ3

|1〉

anti-Stokes

|3〉

Γ2
|1〉

Stokes

RF/microwave
photon

|2〉
coupling
|3〉

Γ3

Figure 3.5: (a) Schematic of an equivalent Fabry-Perot cavity system showing mechanical mode
mixing. The mechanical motion of the cavity mirror (m1 , equivalent to the optically-bright flapping
mode) is primarily actuated by the spring k1 and the optical force. It is internally coupled to a second
mass-spring system (m2 , equivalent to the breathing mode) actuated by the spring k2 which is decoupled from the optical wave. The two masses are internally coupled via spring k12 . (b) A photonic
analogue to the optomechanical system involving coupled resonators. Microcavity 1 is directly coupled to the external optical waveguide (equivalent to the optically-bright flapping mode) and also
internally coupled the narrowband cavity 2 (equivalent to the optically-dark breathing mode). (c)
State diagram of an EIT-like medium. The excited state (j2i) is split by the optical control beam into
two broadband dressed states (j+i and j−i). The dipole transition between ground-states j1i and
j3i is forbidden. (d) The state diagram corresponding to the optomechanical system of (a) where
j1i is the phonon vacuum state, and j2i and j3i correspond to the flapping and breathing modes,
respectively.

72
mechanically-dark state, transparent to external excitation. As shown in Fig. 3.3(c), this induced
mechanical transparency is a direct analogue to EIT in atomic systems [63, 83, 90, 91], in which
the quantum interference between the transition pathways to the dressed states of the excited electronic state, through either j1i $ j+i or j1i $ j i, leads to an induced spectral window of optical
transparency.
Despite the intriguing similarities between the optomechanical system studied here and EIT in
atomic media, there are some important, subtle differences. For instance, in the optomechanical
system, rather than the linear dipole transition of EIT, the interaction corresponds to a second-order
transition. The dynamic backaction between the cavity field and mechanical motion creates Stokes
and anti-Stokes optical sidebands, whose beating with the fundamental optical wave resonates with
the mechanical motion to create/annihilate phonons (see Fig. 3.3(d)). Functionally, this is like
coherent Stokes and anti-Stokes Raman scattering, albeit with unbalanced scattering amplitudes
resulting from the coloring of the electromagnetic density of states by the optical cavity.
The system Hamiltonian of an optomechanical cavity is given by the following general form:

H = ~ω0 a† a + ~Ωm b† b + ~gom xb a† a,

(3.46)

where a and b are the annihilation operators for photon and phonon, respectively, normalized such
that a† a and b† b represent the operators for photon and phonon number. xb is the mechanical
displacement for the optically bright mode, related to b by
xb =

b + b† .
2mb Ωmb

(3.47)

Therefore, the interaction Hamiltonian between the optical wave and the mechanical motion is given
by

Hi = hga† a b + b† ,

where the factor g

g2om ~3
2mb Ωmb

1/2

(3.48)

The mechanical motion modulates the intracavity field to create two optical sidebands. As a
result, the optical field can be written as
a = a p + as e iΩmbt + ai eiΩmbt ,

(3.49)

73
where a p is the field amplitude of the fundamental wave, and as and ai are those of the generated
Stokes and anti-Stokes wave, respectively. As the magnitudes of the Stokes and anti-Stokes sidebands are much smaller than the fundamental wave, when we substitute Eq. (3.49) into Eq. (3.48)
and leave only the first-order terms of as and ai , under the rotating-wave approximation, the interaction Hamiltonian becomes

Hi = ~g b + b† a†p a p + ~gb† a†s a p + a†p ai + ~gb a†p as + a†i a p .

(3.50)

In Eq. (3.50), the first term describes the static mechanical actuation, which changes only the equilibrium position of mechanical motion and is neglected in the current analysis, as discussed previously. The second and third terms show clearly that the process corresponds directly to coherent
Stokes and anti-Stokes Raman scattering as shown in Fig. 3.5(d).
Therefore, in analogy to EIT, it is the modulation signal carried by the incident optical wave
(radio-frequency or microwave photons) that fundamentally probes/excites the mechanical motion
and to which the trapped mechanically-dark state becomes transparent. Moreover, rather than tuning
the Rabi-splitting through the intensity of a control beam resonant with the j3i $ j2i electronic
transition (Fig. 3.3(c)), this optically-induced mechanical transparency is controlled via optical
spring tuning of the resonance frequency of the optically bright flapping mechanical mode. Perhaps
the most apt analogy to the optomechanical system can be made to the photonic resonator system
shown in Fig. 3.3(d). The interference in this case is between the two optical pathways composed of
the waveguide-coupled low-Q optical resonator 1, and the waveguide-decoupled high-Q resonator 2.
This interference again leads to a Fano-like resonance, or what has been termed coupled-resonatorinduced transparency, in the optical cavity transmission [70–73].

3.8

Discussion

Although the studies considered here involve thermal excitation of the optomechanical system, the
same phenomena can be excited more efficiently, and with greater control, using external optical
means (Sec. 3.3.4). As such, beyond the interesting physics of these devices, exciting application
in RF/microwave photonics and quantum optomechanics exist. Similar to the information storage
realized through EIT [83, 92, 93], optical information can be stored and buffered in the dark mechanical degree of freedom in the demonstrated NOMS. This can be realized through a procedure

74
similar to that recently proposed for coupled optical resonators [94, 95] in which dynamic, adiabatic
tuning of optical resonances are used to slow, store, and retrieve optical pulses. The corresponding optomechanical system would consist of an array of double-disk resonators, all coupled to a
common optical bus waveguide into which an optical signal carrying RF/microwave information
would be launched. In this scheme, a second control optical beam would adiabatically tune the
frequency of the optically-bright flapping mode of each resonator, allowing for the RF/microwave
signal to be coherently stored in (released from) the long-lived breathing mode through adiabatic
compression (expansion) of the mechanical bandwidth [94, 95]. In comparison to the all-photonic
system, optomechanical systems have several advantages, primarily related to the attainable lifetime of the dark mechanical state. For example, the radial breathing mechanical mode of a similar
whispering-gallery cavity has been shown to exhibit a lifetime of more than 2 ms [96], a timescale
more than seven orders of magnitude longer than that in demonstrated photonic-coupled-resonator
systems [97] and comparable with EIT media [92, 93]. Moreover, mechanical lifetimes of more
than one second have recently been demonstrated using stressed silicon nitride nanobeam [98] and
nanomembrane [40] mechanical resonators operating in the MHz frequency regime. In the quantum
realm, such a system operating in the good-cavity or sideband-resolved regime (by increasing either
the optical Q factor [59] or the mechanical frequency), would reduce the simultaneous creation and
annihilation of Stokes and anti-Stokes photons, enabling efficient information storage and retrieval
at the single-quanta-level suitable for quantum state transfer.

Chapter 4

Mechanically Pliant Double Disk
Resonators
4.1

Introduction

Optical information processing in photonic interconnects relies critically on the capability for wavelength management [99, 100]. The underlying essential functionalities are optical filtering and
wavelength routing, which allow for precise selection and flexible switching of optical channels
at high speeds over a broad bandwidth [99, 101–103]. In the past two decades, a variety of technologies have been developed for this purpose [104–106]; those based on micro/nano-resonators
are particularly attractive because of their great potential for future on-chip integrated photonic applications [107–119]. In general, reconfigurable tuning of cavity resonances is realized through
thermo-optic [109, 110, 117, 119], electro-optic [112, 118, 119], photochemical [111], optofluidic
[115], or microelectricalmechanical approaches [108, 114, 120]. However, all of these tuning mechanisms have intrinsic limitations on their tuning speed [108–111, 114, 115, 117], tuning bandwidth
[112, 118, 119], routing efficiency [108, 109, 118, 119], and/or routing quality [112, 118, 119].
Here we propose and demonstrate an all-optical wavelength-routing approach which combines the
advantages of various approaches into one nanophotonic device. By using a tuning mechanism
based upon the optical gradient forces in a specially-designed nano-optomechanical system, we are
able to realize seamless wavelength routing over a range about 3000 times the channel intrinsic
linewidth, with a tuning efficiency of 309 GHz/mW, a switching time of less than 200 ns, and 100%
channel-quality preservation over the entire tuning range. The demonstrated approach and device
geometry indicates great prospects for a variety of applications such as channel routing/switching,
buffering, dispersion compensation, pulse trapping/release, and tunable lasing, with easy on-chip

76
integration on a silicon-compatible platform. This work was initially presented in Ref. 78.
The physics of electromagnetic forces within mechanically-compliant resonant cavities is by
now well established, with some of the early experimental considerations being related to the
quantum-limited measurement of weak, classical forces [29]. In the optical domain, experiments
involving optical Fabry-Perot “pendulum cavities” were first explored [121], with more recent
studies having measured radiation pressure forces in micro- and nano-mechanical structures [35–
38, 40, 44, 45, 48, 51]. In each of these systems, whether it be gravitational wave observatory
[31] or photonic crystal nanomechanical cavity [45], the same fundamental physics applies. A narrowband laser input to the system, of fixed frequency, results in a “dynamical back-action” [33]
between mechanical fluctuations and the internal electromagnetic field. This dynamical back-action
modifies both the real and imaginary parts of the frequency of the mechanical motion, yielding an
optically-controllable, dynamic mechanical susceptibility. A separate effect occurs when the laser
frequency is swept across the cavity resonance, pushing on the mechanical system as the internal
light field builds up near cavity resonance. The more compliant the mechanical system, the larger
the static displacement and the larger the tuning of the optical cavity. Here we utilize both the static
and dynamic mechanical susceptibilities of a coupled opto-mechanical system to realize a chipbased optical filter technology in which wideband tuning and fast switching can be simultaneously
accomplished.

4.2

Spiderweb resonator design and optical characterization

The optomechanical system we consider here is a simple modification to the common microring
whispering-gallery cavity that has found widespread application in microphotonics. As shown in
Fig. 4.1(d), it consists of a pair of planar microrings, one stacked on top of the other [122, 123]. The
resulting near-field modal coupling forms a “super-cavity,”, with a resonance frequency ω0 strongly
dependent on the vertical cavity spacing, x.
Fabrication of the spiderweb whispering-gallery resonator began with initial deposition of the
cavity layers. The two silica web layers and the sandwiched amorphous silicon (α-Si) layer were
deposited on a (100) silicon substrate by plasma-enhanced chemical vapor deposition, with a thickness of 400

4 nm and 150

3 nm for the silica and α-Si layers, respectively. The wafer was

then thermally annealed in a nitrogen environment at a temperature of T = 1050 K for 10 hours to
drive out water and hydrogen in the film, improving the optical quality of the material. The spider-

1 µm

gap

25 µm

50
40

gOM/2π (GHz/nm)

SiO2

α-Si

SiO2

Height (µm)

20
1460

Radius (µm)

0 200 400 600 800

15
10

-1
42

1500

Ring spacing (nm)

1540

-2

1580

Per-photon force (fN)

Wavelength (nm)

150

20 µm

200

400

600

Ring spacing (nm)

800

1000

Figure 4.1: Scanning electron microscope images of (a) the between-ring gap, (b) the 54 µm spiderweb resonator, and (c) the 90 µm spiderweb resonator. (d) Schematic of a cross-section of the
resonator, showing the bending of the two silica rings under the influence of the optical force. (e)
Mechanical FEM simulation of the bending of the 90 µm spiderweb resonator. The outward bending motion is shown for ease of viewing, and is exaggerated for clarity. (f) FEM simulation of the
radial component of the electric field for the fundamental TE bonding mode of the 90 µm spiderweb structure. (g) The theoretical wavelength tunability, per-photon force, and wavelength (inset)
of the spiderweb cavity as a function of the ring spacing. The vertical dashed lines represent the
experimentally-realized ring spacing of 150 nm.

78
web pattern was created using electron beam lithography followed by an optimized C4 F8 -SF6 gas
chemistry reactive ion etch. Release of the web structure was accomplished using a SF6 chemical
plasma etch which selectively (30, 000 : 1) attacks the intermediate α-Si layer and the underlying Si
substrate, resulting in a uniform undercut region which extends radially inwards 4 µm on all boundaries, fully releasing the web. Simultaneously, the underlying silicon support pedestal is formed.
Two nanoforks were also fabricated near the double-disk resonator to mechanically stabilize and
support the fiber taper during optical coupling; the geometry was optimized such that the forks
introduce a total insertion loss of only

4%.

The optomechanical coupling coefficient, gOM

dω0 /dx, determines both the tunability and

per-photon optical force [43, 76]. Finite-element-method (FEM) simulation shows that, for two
400-nm-thick planar silica whispering-gallery microcavities placed 150 nm apart (Fig. 4.1(f-g)),
the resonance tunability is as large as gOM /2π = 31 GHz/nm (corresponding to a 21 fN/photon
force). The corresponding static mechanical displacement for N photons stored inside the cavity is
∆xstatic = N~gOM /k, where k is the intrinsic spring constant of the mechanical structure. The overall
magnitude of the cavity resonance tuning is then,
∆ω0 = gOM ∆xstatic =

N~g2OM g2OM Pd
kω0 Γ0

(4.1)

where Pd is the power dropped into the cavity and Γ0 is the intrinsic photon decay rate, inversely
proportional to the optical quality factor.
As the optical gradient force stems from the evanescent field coupling between the two nearfield-spaced cavities, it is completely independent of the round-trip length of the cavity. This feature
enables independent control of the optical and mechanical properties, allowing us to freely engineer the intrinsic mechanical rigidity through the scalability of the structure without changing the
per-photon force. In order to minimize the mechanical stiffness while also providing mechanical
stability, we utilize a spiderweb-like support structure consisting of an arrangement of spokes and
inner rings [96]. The zeroth-order spiderweb cavity (Fig. 4.1(b)) has a 54 µm diameter outer ring
supported by five spokes, while the first-order structure (Fig. 4.1(c)) has a 90 µm outer diameter
ring with six spokes and one supporting inner ring. FEM simulations show that these structures
have spring constants of 9.25 N/m and 1.63 N/m for the smaller and larger resonators, respectively.
In addition to the favorable mechanical properties, the whispering-gallery nature of the spiderweb resonator provides for high-Q optical resonances. Optical spectroscopy of the devices is

EDFA

BPF

VOA

Pump Laser

Polarization
Controller

MUX
Probe Laser

Reference
Detector 1

Polarization
Controller

DC Bias

RF Drive

50 : 50
splitter

VOA

EOM

MZ I

Oscilloscope

Reference
Detector 2

OSA

Network Analyzer

High-speed
Detector

fiber taper
VOA
10 : 90
splitter

DEMUX

10 : 90
splitter

Reference
Detector 3

Figure 4.2: The pump and probe lasers are coupled to the spiderweb resonator via a single-mode
silica fiber taper stabilized by two nanoforks fabricated near the device. The pump laser power is
boosted by an erbium-doped fiber amplifier (EDFA) and passed through a band-pass filter (BPF).
The two lasers are split into separate wavelength channels using a mux/demux system (providing
greater than 120 dB pump-probe isolation). For modulation experiments, the pump laser wavelength
is modulated using an electro-optic modulator (EOM) driven by a network analyzer. The laser
power levels are controlled by several variable optical attenuators (VOAs), the probe wavelength
is calibrated by a Mach-Zehnder interferometer (MZI), and the pump wavelength is monitored by
an optical spectrum analyzer (OSA). The spiderweb device itself is contained within a nitrogen
environment at atmospheric pressure.

80
performed using the experimental set-up shown in Fig. 4.2. Figure 4.3(a) shows the low power, inplane polarized, wavelength scan of a 54-µm diameter resonator. The excited family of resonances,
corresponding to the fundamental transverse-electric-like (TE-like) modes, has a free-spectral range
(FSR) of 9.7 nm, with resonances at λ = 1529 nm and λ = 1549 nm exhibiting intrinsic quality factors of Qi = 1.04

106 and Qi = 0.90

106 , respectively.

The extremely small intrinsic spring constant of the spiderweb resonator leads to significant
thermal Brownian mechanical motion and introduces considerable fluctuations on the cavity transmission spectrum, as shown in Fig. 4.5(a). This makes it difficult to measure the optical Q factor
of a cavity resonance. As discussed previously, the thermal Brownian mechanical motion can be
significantly suppressed through the optical spring effect. This feature provides an elegant way to
accurately characterize the optical Q of a cavity mode, by launching a relatively intense wave at a
different resonance to suppress the perturbations induced by the thermal mechanical motion. Moreover, a complete theory developed previously in Sec. 2.4 and Ref. 48 was used to describe the linear
cavity transmission with the inclusion of the optomechanical effect.

4.3

Static lter response

The combination of high cavity Q-factor, large gOM , and floppy spiderweb structure result in the
large optomechanical bistability shown in Fig. 4.3(b). With a power of 1.7 mW dropped into the
cavity, the cavity resonance initially at λ = 1549 nm is shifted by 4.4 nm (a little more than 0.5
THz), corresponding to a static mechanical displacement of ∆xstatic = 17.7 nm. We observed similar
performance from the larger 90 µm spiderweb structures, although device yield (20%) and a slow
change in device properties over time (despite devices being tested in a nitrogen environment to
avoid water adsorption), indicate that further mechanical design optimization may be necessary for
the larger structures. By comparison, the smaller 54-µm diameter structures had near-100% yield
and maintained their properties over the entire period of testing.
As the mechanical displacement is universally experienced by all double-ring cavity modes,
the displacement actuated by one cavity mode can be used to control the wavelength routing of
an entire mode family, indicating a great potential for broad waveband translation and switching
in the wavelength-division multiplexing configuration. This is demonstrated in Fig. 4.3(d), where
the mechanical displacement actuated by the “pump” mode at λ = 1549 nm is used to control the
wavelength of a “probe” mode initially located at λ = 1529 nm. With increased dropped pump

Normalized transmission

0.8
0.6

probe

pump

0.8
0.6

1520

1530

1540

0.5

1530

1531

1532

1533

Probe wavelength (nm)

0.4

1550

200

14 GHz/mW
0.5

1.5

0.8

0.5

100

1552 1553

Wavelength (nm)

400
300

1549 1550 1551

1560

Probe tuning (GHz)

500

Detuning (pm)

Wavelength (nm)

1.5

1529

-8

low power

[ lp - 1531] (nm)

1510

1500

Detuning (pm)

-8

0.5

0.2

pow
er

0.6

0.4

Pump dropped power (mW)

high

0.8

0.6

0.4

0.2

-0.5

0.1

0.2

Pump dropped power (mW)

0.3

0.4

0.5

Figure 4.3: (a) Broadband optical transmission spectrum of the 54 µm spiderweb cavity. Inset:
fine frequency scan of pump (probe) mode, highlighed in red (green), with Lorentzian fit to the
lineshape. (b) Overcoupled pump-mode transmission spectrum at Pd = 275 nW (blue) and Pd =
1.7 mW (red). (c) Probe-mode transmission curves for a selection of dropped powers in d, with
Pd indicated by the baseline of each transmission curve. (d) Measured (blue circles) and linear
fit (red curve) to the probe resonance wavelength tuning versus Pd . Green curve corresponds to
thermo-optic component of tuning. (e) Intensity image of the optical transmission spectrum near
the anticrossing of two TE- and TM-like probe modes.

82
power, the probe wavelength is tuned linearly and continuously by 4.2 nm, approximately 3000
times the probe resonance intrinsic channel linewidth (or 500 times the loaded linewidth). This
factor is at least one order of magnitude larger than any other conventional approach previously
reported [108–115, 117–119]. The tuning range shown in Fig. 4.3(d) is about 43% of the FSR. In
principle, it is possible to tune over the entire free-spectral range with a moderate dropped pump
power of only 4 mW. Importantly, this wavelength-routing approach is purely dispersive in nature
and completely preserves the channel quality during the wavelength routing process as can be clearly
seen in Fig. 4.3(c). This is in contrast to other tuning mechanisms such as the electro-optic approach
via carrier injection [112, 118, 119], in which the accompanying carrier absorption degrades the
quality of the switched channel and thus limits the ultimate tuning bandwidth.
A linear fit to the probe resonance tuning data data in Fig. 4.3(d) gives a tuning efficiency of
309 GHz/mW. This value agrees reasonably well with the theoretically predicted value of 393 GHz/mW,
inferred from optical and mechanical FEM simulations and the measured optical Q-factor (see
eq. (4.1)). Independent measurements show that the thermo-optic effect contributes only a small
component to the overall tuning rate (13.8 GHz/mW; green curve in Fig. 4.3(d)), and FEM simulations indicate a negligible thermo-mechanical component ( 0.06%).
The thermo-optical effect on the resonance tuning was calibrated by using another identical device on the same sample. To isolate the thermo-optic effect from the optomechanical effect, we
caused the two rings to stick together through the van der Waals force, so the flapping mechanical
motion was completely eliminated. Testing was performed on a cavity mode at 1552 nm using exactly the same conditions as for the wavelength routing measurements. A power of 2.1 mW dropped
into the cavity introduces a maximum resonance red tuning by 0.23 nm, corresponding to a tuning
rate of 0.11 nm/mW (13.8 GHz/mW), about 4% of the total tuning rate recorded experimentally.
The thermo-optic resonance tuning indicates a maximum temperature change of 21 K in the
resonator. FEM simulations show that such a temperature variation of the resonator introduces
a ring-gap change by only about 10 pm, shown by the differential displacement of the top and
bottom rings in Fig. 4.4. Therefore, thermally induced static mechanical deformation has only a
negligible contribution of 0.06% of the experimentally recorded wavelength tuning. The negligible
contributions of both thermo-optic and thermo-mechanical effects are confirmed by the pump-probe
modulation spectra shown in Fig. 4.5(d) and (e). This, and properties of the dynamical response of
the system (see below), show that the wavelength routing is indeed a result of the optical gradient
force. The difference between theoretical and experimental optical force tuning rates (

25%) can

-10
-20

z-displacement (pm)

Figure 4.4: FEM simulation illustrating the z-displacement of a 54 µm spiderweb resonator under
the 21 K temperature differential between substrate and ring induced by 2.1 mW dropped optical
power.

likely be attributed to the uncertaintity in the Young’s modulus of the annealed PECVD silica used
to form the spiderweb structure.
In addition to the TE-like modes, the spiderweb double-ring resonator also supports a family
of high-Q transverse-magnetic-like (TM-like) modes with a FSR of 10 nm. FEM simulations show
that the per-photon force is slightly larger for the TM modes (26.5 fN/photon, or a 59% larger tuning
efficiency), due primarily to the enhanced electric field strength in the nanoscale gap between the
rings for polarization normal to the plane of the rings. Figure 4.3(e) shows the mode hybridization
between a pair of TE and TM-like modes (the slight angle in the outer sidewall of the two rings
breaks the vertical symmetry, allowing for mode-mixing) induced by the optical force tuning of the
two mode families.
The anti-crossing between the two probe modes when they approach each other is primarily due
to the internal coupling between the two cavity modes, which can be described by a simple theory
as follows. Assume two cavity resonances located at ω01 and ω02 . For an input probe wave at ω,
the two cavity modes are excited through the following equations:
da1
= (i∆1
dt
da2
= (i∆2
dt
where ∆ j = ω

Γt1
)a1 + iβa2 + i Γe1 Ain ,
Γt2
)a2 + iβa1 + i Γe2 Ain ,

(4.2)
(4.3)

ω0j represents the cavity detuning of the jth mode, and β is the optical coupling co-

efficient between the two cavity modes. With a continuous-wave input, the steady state of Eqs. (4.2)

84
and (4.3) is given by the following solution
p
iAin (i∆2 Γt2 /2) Γe1 iβ Γe2
(i∆1 Γt1 /2)(i∆2 Γt2 /2) + β2
iAin (i∆1 Γt1 /2) Γe2 iβ Γe1
(i∆1 Γt1 /2)(i∆2 Γt2 /2) + β2

a1 =
a2 =

(4.4)
(4.5)

As the transmitted field from the cavity is given by AT = Ain + i Γe1 a1 + i Γe2 a2 , the cavity
transmission thus has the following equation

(i∆1
jAT j2
jAin j

Γ01 Γe1
)(i∆2

(i∆1

i Γe1 Γe2 )2
Γt2 /2) + β2

Γ02 Γe2
) + (β

Γt1 /2)(i∆2

(4.6)

The experimental observation agrees well with this simple theory (dashed curve in Fig. 4.3(e)),
giving a tuning efficiency for the TM modes which is 42% larger than that of the TE modes. This
precisely tunable channel coupling may find applications in polarization switching/multiplexing/demultiplexing
in optical signal processing, or carrier-sideband filtering in microwave photonics [124].

4.4

Dynamic lter response

In addition to the static mechanical actuation of the spiderweb structure, the optical gradient force
also introduces dynamical back action which alters the dynamic response of the mechanical motion
[33, 50, 125]. The in-phase component of the optical force leads to a modified mechanical resonance
frequency and effective dynamical spring constant of

k0 = k +
where ∆ = ωl

2g2OM Pd ∆
ω0 Γ0 [∆2 + (Γt /2)2 ]

(4.7)

ωo is the detuning of the input laser (ωl ) from the cavity resonance (ωo ) frequency

and Γt is the photon decay rate of the loaded cavity. As the intrinsic spring constant of the spiderweb
resonator is small (9.5 N/m), the dynamical spring can be greatly modified optically. The alteration
of the effective dynamic spring is clearly seen in the resonance spectra of the cavity resonances
(left panel of Fig. 4.5(a)). For the floppy spiderweb structure, thermal Brownian motion introduces
significant fluctuations in the cavity resonances. As pump power is dropped into the cavity, however,
the dynamic spring stiffens and strongly suppresses the magnitude of the thermal fluctuations (right
panel of Fig. 4.5(a)).

0.8
0.6
0.4
0.2

1529.08 1529.12

0.6
0.8
0.6
0.8
0.6

1529.28 1529.32

0.1

0.2

20
15
10
-5
-10
-15
-20

Frequency (MHz)

0.3

Time(µs)

pump

0.6
0.4
0.2
0.8

probe

0.6
0.4
0.2

Modulation spectrum (dB)

Wavelength (nm)

-25

0.8

Fractional Modulation

Normalized transmission

0.4

0.5

0.2

0.4

0.6

0.8

Time(µs)

1.2

78 dB

-20
-40
-60
-80

-100

200

400

600

800

1000

Frequency (MHz)

1200

1400

Figure 4.5: (a) Undercoupled probe transmission spectra recorded at low (Pd = 0 mW (blue)) and
high (Pd = 0.20 mW (red)) pump power (time-averaged trace in black). (b) Time waveforms of the
probe transmission for sinusoidally modulated (22.3 MHz; vertical dashed line in (d)) pump mode
with modulation depths of 1.9% (blue), 14.9% (green), and 20.5% (red) at average Pd = 0.85 mW.
(c) Pulsed modulation of the pump (top) and corresponding probe response (bottom). The fractional
modulation for the pump is defined relative to the average dropped power, while that for the probe is
defined relative to the on-resonance probe-mode coupling depth. (d) Normalized probe modulation
spectra for Pd = f14, 110, 210, 430, 850g µW. The dashed black curves show the corresponding
model. (e) Probe modulation spectrum (Pd = 0.85 mW) shown over a wide frequency span. The
green curve shows the modeled response including only the dominant flapping mechanical mode
(the red curve includes other, breathing-like, mechanical resonances). The orange curve shows the
measured noise floor.

86
Optical control of the dynamic response is most clearly demonstrated through the pump-probe
modulation response of the spiderweb structure. In general, the pump wave inside the cavity satisfies
the following equation:
da p
= (i∆ p
dt

Γtp
)a p

igOM xa p + iγja p j2 a p + i Γep Ap ,

(4.8)

where a p and A p are the intracavity and input field of the pump wave, respectively, normalized such
that Up

ja p j2 and Pp

jA p j2 represent the intracavity energy and input power. ∆ p = ω p

ω0p

represents the detuning of pump frequency ω p to the cavity resonance ω0p and Γtp is the photon
decay rate of the loaded cavity for the pump mode. In Eq. (4.8), the third term represents the back
action of mechanical motion on the cavity resonance, where gOM is the optomechanical coupling
coefficient and x(t) is the mechanical displacement of the cavity structure. The fourth term describes
the self-phase modulation introduced by the Kerr nonlinearity, where the nonlinear parameter γ =
cω p n2
, n2 = 2.6
n20Veff

10 20 m2 /W is the Kerr nonlinear coefficient of silica, n0 = 1.44 is the silica

refractive index, and Veff = 370 µm2 (from FEM simulation) is the effective mode volume [126–
128]. However, compared with the dominant optomechanical effect, the self-phase modulation on
the pump wave is negligible in the spiderweb ring resonator. The final term in Eq. (4.8) represents
the external field coupling with a photon escape rate of Γep .
Assume that the input pump wave consists of an intense continuous wave together with a small
time-varying modulation, A p = Ap0 + δA p (t). The intracavity field can be written as a p = ap0 +
δa p (t), governed by the following equations:
dap0
= (i∆ p
dt
dδa p
= (i∆ p
dt

Γtp
)ap0 + i Γep Ap0 ,
Γtp
)δa p igOM xap0 + i Γep δAp ,

(4.9)
(4.10)

where we have neglected the negligible self-phase modulation for the pump wave. Equation (4.9)
provides a steady-state solution of
i Γep Ap0
ap0 =
Γtp /2 i∆ p

(4.11)

from which we obtain the average pump power dropped into the cavity, Ppd , given by
Ppd =

Pp0 Γ0p Γep
∆ p + (Γtp /2)2

(4.12)

87
where Pp0 = jAp0 j2 is the averaged input pump power and Γ0p is the intrinsic photon decay rate of
the pump mode. Clearly, to the zeroth order, the relative magnitude of the dropped pump power
modulation is directly equal to that of the input modulation:
δPpd (t) δPp (t)
Ppd
Pp0

(4.13)

where δPp = Ap0 δA p + Ap0 δA p is the time-varying component of the input pump power.
Eq. (4.10) leads to a pump-field modulation in the frequency domain of
ep (Ω)
igOM ap0 xe(Ω) i Γep δA
δe
a p (Ω) =
i(∆ p + Ω) Γtp /2

(4.14)

ep (Ω) are Fourier transforms of δa p (t), x(t), and δA p (t), respectively,
where δe
a p (Ω), xe(Ω), and δA
defined as B(Ω)

R +∞

∞ B(t)e

iΩt dt. Physically, the first term in Eq. (4.14) represents the perturba-

tion induced by the mechanical motion, while the second term represents the effect of direct input
modulation.
The optical gradient force is linearly proportional to the cavity energy as Fo =

gOMU p
ω p . With

modulation of the pump energy, the gradient force thus consists of two terms, Fo = Fo0 + δFo (t),
where Fo0 =

gOMUp0
is the static force component introduced by the averaged pump energy Up0 =
ωp

jap0 j2 , and δFo (t) is the dynamic component related to the pump energy modulation δUp (t), given
by
gOM δUp
ωp

δFo (t) =

gOM
ap0 δa p (t) + ap0 δa p (t) .
ωp

(4.15)

Substituting Eq. (4.14) into Eq. (4.15), we find the force modulation is described by this general
form in the frequency domain:

δFe0 (Ω) = fo (Ω)e
x(Ω) +

ep (Ω)
ep ( Ω)
ap0 δA
ap0 δA
Γep gOM
ωp
i(∆ p + Ω) Γtp /2 i(∆ p Ω) + Γtp /2

(4.16)

where the first term represents the back action introduced by the mechanical motion, with a spectral
response f0 (Ω) given by
fo (Ω)

∆2p Ω2 + (Γtp /2)2 + iΓtp Ω
2g2OM jap0 j2 ∆ p

.
ωp
(∆ p + Ω)2 + (Γtp /2)2 (∆ p Ω)2 + (Γtp /2)2

(4.17)

Figure 4.5(d) shows the spectral response of a probe resonance to small-signal sinusoidal pump

88
modulation for several different (average) pump dropped powers. When the pump dropped power
is low, the pump back-action on mechanical motion is negligible and the probe response is given
by a combination of the intrinsic mechanical stiffness and the squeeze-film effect [84] of trapped
gas in between the rings. When the pump power is increased, however, the mechanical resonance
frequency increases correspondingly, reaching a value of 22.3 MHz at a dropped power of 0.85 mW.
This value is about 32 times larger than the intrinsic mechanical frequency, and implies a dynamical
stiffness more than 1000 times that of the silica rings.
The spiderweb ring resonators are separated by a 150 nm gap, which is only about 2.2 times
the mean free path in a nitrogen environment ( 68 nm). As the ring is

6.3 µm wide, much

larger than the ring gap, the nitrogen gas sandwiched in the gap is highly confined by the two silica
layers and cannot move freely during the flapping motion of the two rings. The resulting significant
pressure differential between the internal and external regions of the paired silica rings functions as a
viscous force to damp the mechanical motion. This phenomenon is well-known as the squeeze-film
effect, which has a profound impact on the dynamic response of micro/nano-mechanical systems
[84]. Apart from the optical gradient force, the squeeze-film effect is the dominant mechanism
responsible for the dynamic mechanical response of our devices. The associated damping force
can be described by a general form of Fesq (Ω) = fsq (Ω)e
x(Ω), where fsq (Ω) represents the spectral
response of the squeeze film.
In general, the squeeze-film effect is typically described by two theories which work in quite
different regimes, depending on the Knudsen number Kn characterizing the ratio between the meanfree path and the gap [84]. In the classical regime with Kn

1 where the gas can be considered

a continuum, the squeeze-film viscous force for a rectangular plate is well described by fsq (Ω) =
ke (Ω) + iCd (Ω), where ke and Cd represent the spring constant and damping, respectively, induced
by the squeeze film. They are given by the following equations [129]
ke (Ω) =

64σ2 Pa L0W0
∑ m2 n2 [(m2 + (n/η)2 )2 + σ2 /π4 ] ,
π8 h0
m,n odd

(4.18)

Cd (Ω) =

64σPa L0W0
m2 + (n/η)2
∑ 22 2
2 2
2 4
π6 h0
m,n odd m n [(m + (n/η) ) + σ /π ]

(4.19)

where Pa is the ambient gas pressure, W0 and L0 are the width and length of the plate, h0 is the gap,

89
η = L0 /W0 is the aspect ratio of the plate, and σ is the squeeze number given by
σ(Ω) =

12µeffW02 Ω
Pa h20

(4.20)

where µeff = µ/(1 + 9.638Kn1.159 ) is the effective value of the viscosity coefficient µ [130]. Under
this model, the squeeze film functions primarily as a damping (or elastic) force when the modulation
frequency is below (or above) the cutoff frequency given by
π2 Pa h20
Ωc =
12µeff
In contrast, in the free-molecule regime with Kn

+ 2
W0 L0

(4.21)

1 where the interaction between gas molecules

is negligible, the squeeze film approximately behaves like a damping force, fsq (Ω) = iCr Ω, with Cr
given by the following equation [131, 132]
Cr =

16πh0

4Pa L0W0

2Mm
πR T

(4.22)

where Mm is the molar mass of gas, T is the temperature, R is the ideal gas constant, S is the
perimeter length of the gap region.
However, our devices have a Knudsen number of Kn = 0.45, falling in the crossover regime
where neither theory adequately describes the squeeze-film effect [133]. As the device works in
the regime between the continuum and free-molecule limit, we heuristically propose that the damping/elastic force of the squeeze film is effectively described by a composite of the two theories:
fsq (Ω) = ke (Ω) + iCd (Ω) + iηrCr Ω,

(4.23)

with a modified effective coefficient of viscosity µ0eff = ηµ µeff , where ηr and ηµ are parameters used
for a best description of the squeeze-film response in our devices. Detailed analysis shows that
ηµ = 0.7 and ηr = 0.03 provides the best fit for our devices. As our devices have a spiderweb
geometry, we approximate it with an equivalent rectangular shape with W0 given by the ring width,
L0 given by the circumference at the ring center, and S

2L0 . As shown by the experimental results

and theoretical fits, this model provides an accurate description of the squeeze-film effect in our
devices.
Although the intrinsic mechanical frequency of the 54 µm spiderweb structure is 694 kHz (in-

90
dicated by FEM simulation), Fig. 4.5(d) shows a minimum dynamic frequency response of 6 MHz,
dominated by the squeeze-film damping. Interestingly, although squeeze-film damping is generally
detrimental in other micro/nanomechanical systems [84, 134], it is beneficial in this case, as it helps
to extend the modulation bandwidth for wavelength routing.
With the optical gradient force and the squeeze-film damping force, the mechanical motion of
the cavity satisfies the following equation:
d2x
dx
+ Γm + Ω2m x =
(Fo + Fsq + FT ) =
(Fo0 + δFo + Fsq + FT ),
dt
dt
meff
meff

(4.24)

where meff is the effective motional mass of the flapping mechanical mode, and Ωm and Γm are
intrinsic mechanical frequency and damping rate, respectively. FT is the thermal Langevin force
responsible for the thermal Brownian motion, a Markovin process with the following correlation
function:
hFT (t)FT (t + τ)i = 2meff Γm kB T δ(τ),

(4.25)

where kB is Boltzmann’s constant.
As the squeeze-film viscous force is zero at Ω = 0, the squeeze gas film impacts only the dynamic response of mechanical motion. Equation (4.24) shows clearly that the static mechanical
displacement is actuated only by the static component of the optical force given by
x0 =

gomUp0
gom Ppd
jFo0 j
meff Ω2m
km ω p
km ω p Γ0p

(4.26)

where km = meff Ω2m is the intrinsic spring constant of the spiderweb structure. With a specifically
designed extremely small spring constant, x0 can be quite significant for a given dropped power.
As a result, the cavity resonance can be tuned by a significant magnitude of gOM x0 . This is the
primary mechanism responsible for the resonance tuning. On the other hand, this static mechanical
displacement primarily changes the equilibrium position of the mechanical motion. It is convenient
to remove this component in Eq. (4.24) by defining x0 = x x0 , since both the squeeze-film damping
force and dynamic component of the optical force affect only the dynamics of x0 .
Substituting Eqs. (4.16), (4.17), (4.23) into Eq. (4.24) in the frequency domain, we find that the
squeeze-film damping force and the backaction term of the optical force primarily change the values

91
of the resonant frequency and damping rate of the mechanical motions. Defining

L (Ω)

Ω2m

Ω2

iΓm Ω

fo (Ω)
meff

fsq (Ω)
meff

(4.27)

the mechanical displacement is thus given by
ep (Ω)
ep ( Ω)
ap0 δA
ap0 δA
i Γep gOM
FeT (Ω)
xe(Ω) =
meff L (Ω) meff ω p L (Ω) i(∆ p + Ω) Γtp /2 i(∆ p Ω) + Γtp /2

(4.28)

where we have dropped the prime notation of x0 for simplicity.
The first term in Eq. (4.28) represents the thermal Brownian motion while the second term
describes the motions actuated by the pump modulation. In the absence of pump modulation, the
mechanical motion is dominated by the Brownian motion. By using Eq. (4.25), we find the spectral
density of thermal mechanical displacement has the form

Sx (Ω) =

2Γm kB T
meff jL (Ω)j2

(4.29)

Equations (4.16), (4.17), and (4.27) show that one dominant effect of the pump energy inside the
cavity is to increase the mechanical rigidity, the so-called optical spring effect. In most cases, L (Ω)
can be well approximated by L (Ω)

(Ω0m )2

Ω2

iΓ0m Ω with a new mechanical resonance Ω0m

and damping rate Γ0m affected by the optical force. Equation (4.29) thus leads to a variance of the
thermal mechanical displacement given by
h(δx) i =
2π

Z +∞

k T Γm
km Γ0m

Sx (Ω)dΩ = B0

kB T
km

(4.30)

0 = m (Ω0 )2 is the effective spring constant and the approximation in the final term aswhere km
eff

sumes a negligible change in the mechanical linewidth. Clearly, the increase of the mechanical
resonance frequency through the optical spring effect dramatically suppresses the magnitude of
the thermal mechanical displacement and its perturbation of the cavity resonance, as shown in
Fig. 4.5(a). In the presence of pump modulation, the mechanical motion is primarily dominated
by the dynamic optical force rather than the actuation from the thermal Langevin force, and the
first term is negligible compared with the second term in Eq. (4.28). Thus, we neglect the thermal
Brownian term in the following discussion.

92
The probe wave inside the cavity is governed by a dynamic equation similar to Eq. (4.8):
das
= (i∆s
dt

Γts
)as

igOM xas + 2iγja p j2 as + i

Γes As ,

(4.31)

except that the Kerr-nonlinear term now describes the cross-phase modulation from the pump wave.
With the perturbations induced by the pump modulation, similar to the previous discussion of the
pump wave, the intracavity probe field can be written as as = as0 +δas (t), governed by the following
equations:
das0
= (i∆s
dt
dδas
= (i∆s
dt

Γts
)as0 + 2iγUp0 as0 + i Γes As ,
Γts
)δas + 2iγUp0 δas igOM xas0 + 2iγδUp as0 ,

(4.32)
(4.33)

where we have assumed the probe input is a continuous wave with a power of Ps = jAs j2 . The
second terms of Eqs. (4.32) and (4.33) represent the static cavity tuning introduced by cross-phase
modulation, which can be included in the cavity tuning term ∆s for simplicity. In general, it is
negligible compared with the cavity linewidth at the power level used for exciting optomechanical
effects, leading to 2γUp0

Γtp , Γts .

Equation (4.32) provides a steady-state solution of
i Γes As
as0 =
Γts /2 i∆s

(4.34)

and Eq. (4.33) results in a probe-field modulation in the frequency domain of

δe
as (Ω) =

ias0 gOM xe(Ω)
i(∆s + Ω)

ep (Ω)
2γδU
Γts /2

(4.35)

ep (Ω) is the Fourier transform of δUp (t). As the transmitted field of the probe is given by
where δU
ATs = As + i Γes as , the modulation of the transmitted probe power thus takes the form
δPTs = i Γes (A0s δas

A0s δas ),

(4.36)

where A0s = As + i Γes as0 is the transmitted probe wave in the absence of modulation. By use
of Eqs. (4.14), (4.28), (4.34) and (4.35), we find that the power spectrum of the transmitted probe

93
modulation is given by the following equation:
2 2
Ppd jδPepd (Ω)j2 4Γ2es Γ20s ∆2s
g2OM
jδPeTs (Ω)j2
2γ
Ps2
meff ω p L (Ω)
Γ20p
Ppd
[∆2s + (Γts /2)2 ]4

(4.37)

where Γ0s is the intrinsic photon decay rate of the probe mode, and δPeTs (Ω) and δPepd (Ω) are
the Fourier transforms of δPTs (t) and δPpd (t), respectively. To obtain Eq. (4.37), we have used
Eq. (4.13) to relate the dropped pump power to the input, and have also taken into account the
fact that the Kerr effect is relatively small, such that 2γUp0
the sideband-unresolved regime with Ωm

Γtp . We also assume the cavity is in

Γtp , Γts . The modulation spectra given in Fig. 4.5 are

defined as
ρ(Ω)

jδPeTs (Ω)j2 /Ps2
jδPepd (Ω)j2 /P2

(4.38)

For a better comparison of the dynamic-backaction induced variations on the probe modulation, the
modulation spectra shown in Fig. 4.5(d) are normalized by a factor corresponding to the ratio of
the dropped power for each curve relative to the maximum dropped power. Therefore, the plotted
modulation spectra are given by
ρ0 (Ω)

ρ(Ω)

Ppd0
Ppd

jδPeTs (Ω)j2 /Ps2
jδPepd (Ω)j2 /P2

(4.39)

pd0

where Ppd0 = 0.85 mW is the maximum drop power used in Fig. 4.5(d).
The derivations above take into account only the flapping mechanical mode, since it is most
strongly actuated by the optical gradient force. In general, there are many mechanical resonances
for the spiderweb resonators, but weakly coupled to the optical waves inside the cavity. In this
case, following the same procedure above, it is easy to show that the spectral response of probe
modulation now becomes

ρ(Ω) = 2γ + ∑

g2j
mj ω p L j (Ω)

Ppd

4Γ2es Γ20s ∆2s

Γ20p [∆2s + (Γts /2)2 ]4

(4.40)

where g j , m j , and L j (Ω) are optomechanical coupling coefficient, effective motional mass, and
the spectral response of mechanical motions, respectively, for the jth mechanical mode. For those
weakly actuated mechanical modes, L j (Ω) = Ω2mj

Ω2

iΓmj Ω where Ωmj and Γmj are the reso-

nance frequency and damping rate of the jth mechanical mode. Equation (4.40) was used to describe

94
the modulation spectrum shown in Fig. 4.5(d).
As shown in the dashed curves in Fig. 4.5(d), eq. (4.37) provides an accurate description of
the pump-probe modulation response (the Fano-like resonance seen at Pd = 0.85 mW is due to
intrinsic mechanical coupling between different types of motion, and is discussed in Chapter 3). In
general, the small modulation of the pump wave which actuates the mechanical oscillation is greatly
magnified on the probe resonance. Figure 4.5(d) shows that for Pd = 0.85 mW there is a resonant
modulation “gain” of greater than 20 dB. This can also be seen in the time waveform of the probe in
Fig. 4.5(b), where a 1.9% modulation of the pump power is large enough to introduce considerable
fractional modulation in the probe time waveform (top panel). Increasing the pump modulation to
14.9% (Fig. 4.5(b), middle panel) results in a probe modulation of larger than a half-linewidth (full
contrast modulation). Further increase in the pump modulation depth actuates flapping mechanical
motion so intense it begins to excite a second mechanical mode (the Fano-like feature in Fig. 4.5(d)),
resulting in a beat signal with a period of 0.36 µs on the probe time waveform.
One metric for characterizing the response time of the spiderweb optomechanical cavity is the
resonant oscillation period [120]. Figure 4.5(d) shows that the optical spring effect enables a modulation time as fast as 44.8 ns. This can be further enhanced by using the transduction “gain” to push
the probe modulation into the nonlinear regime, where in the lower panel of Fig. 4.5(b) the probe
wavelength (10%–90%) on-off switching time is reduced to 7 ns, roughly 3 orders of magnitude
faster than modulation schemes based upon thermo-optic, optofluidic, photochemical, or microelectricalmechancial approaches [108–111, 113–115, 117, 120]. For many switching applications,
however, one is more interested in the impulse response of the system. The pulsed response of the
probe is shown in Fig. 4.5(c). As is common in micro/nanomechanical systems [114, 120], the resonant response causes ringing during switching, with a settling time determined by the mechanical
linewidth. The measured settling time constant of the probe response is 196 ns, consistent with the
mechanical linewidth of

2-3 MHz (see Fig. 4.5(d)).

In addition to the optomechanical nonlinearity, other optical (material, etc.) nonlinearities
can also contribute to the probe modulation. As shown in the expanded modulation spectrum of
Fig. 4.5(e), the resonant optomechanical nonlinearity is dominant out to a frequency of 500 MHz,
after which the response plateaus due to the ultrafast Kerr nonlinearity of silica. The Kerr nonlinearity is measured to be 78 dB below the resonant optomechanical response. This ratio agrees well
2
g2
with the theoretical value of 81 dB given by 2γmeff ωOM
, where Ω0m and Γ0m are the effective
pΩ Γ
m m

mechanical resonance frequency and damping rate, respectively. The Kerr nonlinearity in silica has

95
been extensively studied over more than three decades for optical signal processing [126, 135, 136],
and the excellent agreement between the theoretical and experimental spectra provides yet another
indication that the optical gradient force is the dominant tuning mechanism in the spiderweb cavity
structure.

4.5

Discusssion

The versatility of the gradient optical force tuning approach described here provides considerable
room for future improvement of device performance. An increase in the tuning range and efficiency
(actuation power) can be expected with further engineering of the mechanical stability of the spiderweb structure. For example, the 90 µm diameter first-order spiderweb cavities should allow for
a six-fold increase in tuning efficiency to approximately 15 nm/mW. There are also many wellestablished methods for managing the dynamical response, in particular the ringing, of resonant
micro- and nano-mechanical systems [114, 120]. In contrast to cavity-optomechanical applications
such as cooling and amplification of mechanical motion [50, 125], a reduction in the mechanical Qfactor, which can be obtained through elevated gas pressure or incorporation of damping materials,
is sought to improve the switching time. Given the similarity of the double-ring spiderweb stucture
to other more conventional planar microring technologies, one can also incorporate other chip-based
optical components such as waveguides, lasers, and modulators to enable full control and functionality of the optomechanics. One example technology would be an on-chip reconfigurable optical
add/drop multiplexer or wavelength selective switch/crossconnect, which could be accomplished
by integrating an array of double-ring cavities into a parallel or cascaded configuration. In addition
to the demonstrated wavelength routing, other prospective applications for optomechanical devices
include tunable optical buffering [137], dispersion compensation [138], tunable lasers [139], and
nonlinear signal processing [126].

Chapter 5

Conclusion
Here has been presented work on several optical resonator systems: the single and double-metal
plasmonic photonic crystal resonator, the double-disk whispering-gallery cavity, and the doublering spiderweb cavity. Each of these resonator designs was developed and optimized for a particular
range of applications, and each has been shown to be effective in at least initial demonstrations.
Multispectral mid-infrared resonant detectors were demonstrated with enhanced responsivity
and detectivity, and tailorable polarization and spectral sensitivity. These devices were fabricated
using a very simple single-etch process, and are detector agnostic, with design principles that are
easily transferrable to any other detector material or frequency range with a minimum of difficulty.
Expanding to a double-metal device structure, this method could be used to easily and inexpensively
impart frequency and polarization selectivity, as well as absorption enhancement, to current detector
focal plane array processing.
A novel optomechanical device structure has also been presented, consisting of two stacked
microdisks with an optically narrow gap between them. This device has a very large optomechanical
coupling and a high quality factor, giving rise to extremely large dynamical backaction in the form
of both regenerative mechanical oscillation and optomechanical cooling. Due to the large optical
spring effect in these structures, we also demonstrate tunable coherent mechanical mode mixing
with an analogy to electromagnetically induced transparency, showing the possibility for slow-light
effects on the very long phononic timescale, and the potential for phonon-photon quantum state
transfer.
Finally, an extremely flexible double-ring optomechanical device is shown, demonstrating alloptical wavelength routing with unprecedented range and efficiency, and 100% channel quality
preservation. As this device can be easily integrated on-chip, it shows great promise for optical
communications applications, as well as for more fundamental physics-based applications such as

97
in cavity quantum electrodynamics or for dispersion compensation in nonlinear optics.

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Applications of Micro/Nanoscale Optical Resonators: Plasmonic
Photodetectors and Double-Disk Cavity Optomechanics

Thesis by

Jessie Rosenberg

In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California

2010
(Defended December 8, 2009)

c 2010
Jessie Rosenberg

iii

To my mother, father, and all of the friends, near and far, who have supported me.

vii
phononic quantum state transfer, channel routing/switching, pulse trapping/release, and tunable lasing are discussed.

viii

Contents
Acknowledgments

Abstract

Preface

xiii

Plasmonic Resonators for Multispectral Mid-Infrared Detectors

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

Photonic Crystal Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

Single-Metal Plasmon Resonator Design . . . . . . . . . . . . . . . . . . . . . . .

1.4

Single-Metal Experimental Demonstration . . . . . . . . . . . . . . . . . . . . . .

1.5

Double-Metal Plasmon Resonator Design . . . . . . . . . . . . . . . . . . . . . .

1.6

Critical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Double Disk Optomechanical Resonators

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Optomechanical coupling and dynamic backaction . . . . . . . . . . . . . . . . .

2.3

Double-disk fabrication, optical, and mechanical design . . . . . . . . . . . . . . .

2.4

Optical and mechanical characterization . . . . . . . . . . . . . . . . . . . . . . .

2.5

Regenerative oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.1

Ambient pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.2

Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

Optomechanical cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Coherent Mechanical Mode Mixing in Optomechanical Nanocavities

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Zipper cavity and double-disk design, fabrication, and optical characterization . . .

3.3

Theory of optomechanical effects in the presence of mode mixing . . . . . . . . .

3.3.1

Intracavity field in the presence of optomechanical coupling . . . . . . . .

3.3.2

The power spectral density of the cavity transmission . . . . . . . . . . . .

3.3.3

The mechanical response with multiple excitation pathways . . . . . . . .

3.3.4

The mechanical response with external optical excitation . . . . . . . . . .

3.4

Mechanical mode renormalization in zipper cavities . . . . . . . . . . . . . . . . .

3.5

Coherent mechanical mode mixing in double-disks . . . . . . . . . . . . . . . . .

3.6

Coherent mechanical mode mixing in zipper cavities . . . . . . . . . . . . . . . .

3.7

Analogy to electromagnetically-induced transparency . . . . . . . . . . . . . . . .

3.8

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Mechanically Pliant Double Disk Resonators

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Spiderweb resonator design and optical characterization . . . . . . . . . . . . . . .

4.3

Static filter response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

Dynamic filter response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5

Discusssion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusion

List of Figures
1.1

Simulated bandstructure for a double-metal photonic crystal . . . . . . . . . . . . .

1.2

Simulated bandstructure for a square and rectangular lattice plasmonic photonic crystal

1.3

Schematic and bandstructure for single-metal detectors . . . . . . . . . . . . . . . .

1.4

Plasmonic photonic crystal detector simulations . . . . . . . . . . . . . . . . . . . .

1.5

Single-metal detector enhancement factor and active region absorption . . . . . . . .

1.6

Detector devices and measurement setup . . . . . . . . . . . . . . . . . . . . . . .

1.7

DWELL detector measurement results . . . . . . . . . . . . . . . . . . . . . . . . .

1.8

Waveguide thickness dispersion for double-metal waveguides . . . . . . . . . . . .

1.9

Field profiles for double-metal waveguides . . . . . . . . . . . . . . . . . . . . . .

1.10

Comparison of FDTD and group theory . . . . . . . . . . . . . . . . . . . . . . . .

1.11

Far-field profiles for stretched-lattice double metal resonators . . . . . . . . . . . . .

1.12

Schematic of double-metal detector loss mechanisms . . . . . . . . . . . . . . . . .

1.13

Vertical and substrate coupling vs. metal thickness and hole size for the single-metal
structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.14

Vertical coupling vs. metal thickness and hole size for the double-metal structure . .

1.15

Double-metal focal plane array schematic . . . . . . . . . . . . . . . . . . . . . . .

2.1

Double-slab waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Cavity Optomechanical Structures . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Double disk flapping mode displacement . . . . . . . . . . . . . . . . . . . . . . .

2.4

Fabrication and characterization of double-disk NOMS . . . . . . . . . . . . . . . .

2.5

Transmission scans for a double-disk microcavity . . . . . . . . . . . . . . . . . . .

2.6

Optical and mechanical mode spectroscopy . . . . . . . . . . . . . . . . . . . . . .

2.7

Regenerative oscillation in a double-disk microcavity . . . . . . . . . . . . . . . . .

2.8

Dynamical backaction: damping and amplification of mechanical motion . . . . . .

xi
3.1

Schematics and optical modes of two optomechanical systems . . . . . . . . . . . .

3.2

Mode mixing in zipper cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Mode mixing measurements in double disks . . . . . . . . . . . . . . . . . . . . . .

3.4

Zipper cavity mechanical mode mixing . . . . . . . . . . . . . . . . . . . . . . . .

3.5

Mechanical mode mixing analogues to optical systems and to EIT . . . . . . . . . .

4.1

Spiderweb microresonator images and simulations . . . . . . . . . . . . . . . . . .

4.2

Pump-probe experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

Static tuning of a spiderweb microresonator . . . . . . . . . . . . . . . . . . . . . .

4.4

Thermomechanical deflection of a spiderweb resonator . . . . . . . . . . . . . . . .

4.5

Dynamic response of a spiderweb microresonator . . . . . . . . . . . . . . . . . . .

xii

List of Tables
1.1

Point Group character tables for the square and rectangular lattice. . . . . . . . . . .

xiii

Chapter 1

Plasmonic Resonators for Multispectral
Mid-Infrared Detectors
1.1

Introduction

1.2

Photonic Crystal Design

Normalized Frequency (a/λ0)

0.8

(a)

(b)

0.6
0.5

Γ3

0.4

(c)

Γ2

0.7

Γ0

Γ2

Γ1

Γ3

Γ0

0.3

X1 Γ1
G1

0.2
0.1

Γ4

(k? ) r? , with k

? and r? representing the in-plane wavenumber and spatial position, respectively.

Normalized Frequency (a/λ0)

0.5 (a)

(b)

0.4

0.3

0.2

Figure 1.2: 2D bandstructure plots near the gamma point of (a) a square lattice and (b) a rectangular
lattice, stretched by 10%. The dipole-like modes are shown in bold.

basis onto the irreducible representation (IRREP) spaces of C4v , we find the modes:
EA1 = ẑ(cos(kG1 r) + cos(kG2 r)),
EB1 = ẑ(cos(kG1 r)

cos(kG2 r)),

(1.1)

cos(kG2 r)),

(1.2)

EB1 = ẑ(sin(kG2 r)),
EB2 = ẑ(sin(kG1 r)).
These modes are plotted in Fig. 1.10. The C4v two-dimensional representation E decomposes into
B1

B2 under C2v , therefore the dipole-like modes are no longer degenerate. This is in agreement

with what we see in the bandstructure of the stretched lattice, Fig. 1.2(b).

(a)

Ez Intensity of
Guided Mode

Refractive Index/Ez Intensity (a.u.)

0.4

(b)

(c)

Γ X

(d)

X2 M
Γ X1

Frequency (a/λ)

16
0.35

0.3

Active Region

Depth (µm)

0.25

XΓ

1.3

Single-Metal Plasmon Resonator Design

(a)

x10-3

(b)

0.5

z (µm)

-2 active region

-1

-8

-1
x10-3
1 (d)

(c)

z (µm)

-2 active region

-4

-0.5-0.5

-8

-1

y (µm)

0.5

x (µm)

0.5

x10-3
0.5

-6

-0.5

0.5

y (µm)

0.5

-1

-0.5

-6

-4

air

-1

semiconductor
-0.5
0.5
x (µm)

-0.5
-1

At /ADWELL = 5.75 at the resonant wavelength,

versus a sample with no top patterned plasmonic metal layer at the same wavelength (Fig. 1.5).
Note that this enhancement factor is not normalized to the area of the holes in the plasmonic metal,

WG1

WG2 P10

P11

P01
Wavelength (µm)

P00

0.04

WG1

0.02

-2

-0.02

-6
-8

0.01

0.005

-2

-4

-0.04

-1 0 1
y (µm)

0.015

P01

-4

-0.005

-6
-8

-0.01
-0.015

-1 0 1
y (µm)

-0.01

-6
-8

0.02
0.01

z (µm)

(b)

z (µm)

P00

z (µm)

Enhancement Factor, E

(a)

Active Region Absorption (%)

-0.02

-1 0 1
y (µm)

plasmon metal

(a)

plasmon metal

(b)

top metal
DWELL active region

AlGaAs
bottom metal

(c)

5 µm

GaAs substrate

(d)
LN2 Dewar
Blackbody

Chopper
Controller

5 µm

Trigger

ESA

Current Amplifier

a=1.90µm
a=1.97µm
a=2.04µm
a=2.10µm
a=2.17µm
a=2.24µm

unpolarized
90º
polarized
0º
polarized

a=2.31µm

9 10

Wavelength (µm)

5 x 10-7
unpatterned
response

5.5

6.5

Wavelength (µm)

normalized

0.05

0.00

(d)

-6
control
divided 2

0.5

a=2.38µm

Spectral Response (a.u.)

a=1.83µm

Spectral Response (a.u.)

0.5

0.10

9 10

Wavelength (µm)

-4

1010

-2

Voltage (V)

109
108

(e)

107-6

-4

-2

Voltage (V)

Responsivity Enhancement, E

unprocessed 1x 10-6

(c)

unpatterned

Responsivity (A/W)

(b)

unpatterned

Detectivity (cm Hz1/2/W)

(a)

(f )

Sample A −5V bias
Sample A 5V bias
Sample B −5V bias
Sample B 5V bias

5.5 6 6.5 7 7.5 8 8.5

Wavelength (µm)

1.4

Single-Metal Experimental Demonstration

0.6), but different lattice constant values, and therefore different

14
the peak wavelength response from 5.5 µm to 7.2 µm by choosing PC lattice constants in a range
from 1.83 µm to 2.38 µm. The linewidth of these resonances is

0.9 µm, providing strong spectral

λ1

I0
RN (λ)Le (λ, T )As Ad tFr2F dλ

(1.3)

(1.4)

15
0.35

(a)

0.3

Effective Index, neff

(b)

0.25
0.2
0.15
0.1
0.05

200

400

600

800

Thickness, t (nm)

1000

200

400

600

800

Thickness, t (nm)

1000

1.5

Double-Metal Plasmon Resonator Design

16
x 105

(a)
abs(E) (N/C)

abs(E) (N/C)

neff = 7.64+0.2584i

(b)

-100

-50

z (nm)

100

150

neff = 3.44+0.0112i

x 103

(c)

x 104

abs(E) (N/C)

neff = 3.30+0.0005i

200

z (nm)

400

600

z (µm)

(a)

(b)

(c)

(d)

(a)

Py/Px > 109

-100

-50

y (µm)

-150

(b)

Px/Py > 109

-100

-50

y (µm)

-150

100

150

-100 -50

x (µm)

50 100

150

-100 -50

x (µm)

50 100

1.6

Critical Coupling

Qe
Qmetal

-V

Qxy

Figure 1.12: The dominant loss mechanisms within a double-metal plasmonic photonic crystal
resonant detector.

γ0 γe 2
2 ,
∆2 + γ2t

∆2 +

(1.5)

ω0 is the frequency detuning from the resonance frequency ω0 , γ0 is the intrinsic

20
(not only the absorbed power, but all of the power not reflected):
Pd = Pin (1

R) = Pin

γ0 γe
∆2 +

γt 2

(1.6)

where Pin represents the power incident on the cavity. Therefore the fractional absorption efficiency
into the i-th loss channel is
pi =

γi Pd
γi γe
.
γ0 Pin ∆ + γt 2

(1.7)

0 due to the thick bottom layer of

plasmon metal.
The fractional absorption into the DWELL detector material is then, at resonance, given by
pD =

4γD γe
[(m + 1)γe + γmetal + γD ]2

(1.8)

From this expression, we see that the maximum fractional absorption occurs at
γe =

γD + γmetal
1+m

(1.9)

γD
(1 + m)(γD + γmetal )

(1.10)

0, the maximal detector absorption occurs at the pure critical

105

(a)
Quality Factor

Quality Factor

104

103

102

101

100

200

300

400

Metal Thickness, tm (nm)

500

(b)

104
103
102
101

0.4

0.5

0.6

0.7

0.8

(a)

8000

(b)

500

6000

400

4000

300

2000
00

600

External Quality Factor, Qe

10000

200

200 400 600 800 1000 1200

Metal Thickness, tm (nm)

100 0.54

0.58

0.62

0.66

0.7

Figure 1.14: Variation of vertical coupling quality factor Qe of the B1 (blue square) and B2 (green
circle) modes of the double-metal photonic crystal lattice with changing (a) tm and (b) W̄ .

1.7

Conclusion

Infrared Light In

Photonic Crystal Resonator
Highly Doped Layer

Metal
Silicon Nitride
Metal

Active Region

Highly Doped Layer

Indium Bump

Epoxy Underfill

Figure 1.15: A design schematic for a resonant double-metal plasmonic photonic crystal FPA.

Chapter 2

Double Disk Optomechanical
Resonators
2.1

Introduction

4). In vacuum, the threshold for regenerative

2.2

Optomechanical coupling and dynamic backaction

The per photon force exerted on a mechanical object coupled to the optical field within a resonant
cavity is given by ~gOM , where gOM

dωc /dx is a coefficient characterizing the dispersive nature

Ωm ), the maximum rate

!
3 3g2OM
Pd ,
(1 + K)2
κ3i ωc mx

(2.1)

where ωc is the optical cavity resonance frequency, mx is the motional mass of the optomechanical
system, Pd is the optical power dropped (absorbed) within the cavity, and K

κe /κi is a cavity

2.3

Double-disk fabrication, optical, and mechanical design

4 nm and 158

3 nm for the silica and α-Si layers, respectively. The wafer

8 nm (shrinkage having

8%.

The final double-disk structure, shown in Fig. 2.4, consists of 340-nm-thick silica disks separated by a

140 nm air gap extending approximately 6 µm in from the disk perimeter (the undercut

108 ).

x > h + x0 /2
x0 /2 < x < h + x0 /2
(2.2)

x0 /2 < x < x0 /2
x0 /2 > x > h
x< h

x0 /2

β2 ,

γ2 = β2

k02 n2s ,

(2.3)

equation:
κγ [1 + tanh(γx0 /2)] = κ2

γ2 tanh(γx0 /2) tan κh,

(2.4)

28
ns

coupling coefficient, gOM = dω
dx0 , is given by the general form

gOM (x0 ) =

where χ

4(n2c

cχγ2
2 γx0
k0 sech
n2s ) tan κh + n2s x0 χsech2 γx20 + 2ξ (n2c γh csc2 κh + 2n2s ) tan κh + nsγκ

κ + γ tan κh and ξ

n2c γ

i(2.5)

1 + tanh( γx20 ).

When x0 ! 0, Eq. (2.5) leads to the maximum optomechanical coupling of
gOM (0) =

ω0 γ3
2β2 + 2k02 n2c γh

(2.6)

In analogy to Fabry-Perot cavities and microtoroids, the magnitude of the optomechanical coupling
can be characterized by an effective length, LOM , defined such that gOM

ω0
LOM . Equation (2.6) infers

a minimum effective length
2 + k hn2 n2
n2s
λ0 eff
eff
1 + 20 (n2s + n2c γh) =
L0 =
3/2
n2
n2
eff

(2.7)

29
cal coupling can be well approximated by an exponential function
gOM (x0 )

gOM (0)e γx0 ,

(2.8)

ω0
γx0
gOM (0) e , agrees well with the results simu-

1.5 µm. For the air gap of 138 nm used in this work, the optomechanical

2Us
Ωm [xm (r0 )

x0 ]2

Us
2Ωm d 2 (r0 )

(2.9)

gOM= -ω0/LC

gOM= ω0/LOM

SiO2
α-Si

SiO2

gOM= -ω0/R

Resonance frequency (THz)

208

-1

Height (µm)

-2

-1
-2

42
44
46
Radius (µm)

40
204

gOM
2π (GHz/nm)

200
196

194
400
800
Disk spacing x (nm)

10
200
400
600
Disk spacing x (nm)

800

Effective length LOM (µm)

Height (µm)

xm(r0)
d(r)

Ep =
ra

Ω2m d 2 (r)ζ2πrhdr,

(2.10)

ra2 ), using Eq. (2.10), we find that the effective mass is

related to the physical mass through the following expression:
mx =

Z rb

4m p
rb2

ra2 [xm (r0 )

x0 ]2

rd (r)dr =
ra

rb2

mp
ra2 d 2 (r0 )

Z rb

rd 2 (r)dr.

(2.11)

As the whispering-gallery mode is generally located close to the disk edge (i.e., the mode radius
r0 = 44 µm in a double disk with rb = 45 µm), d 2 (r)/d 2 (r0 )

1 for most of the region between

32
ra and rb , and Eq. (2.11) shows that mx

m p /2. Therefore, the effective mass is significantly

2.4

Optical and mechanical characterization

106 is inferred, taking into account the mechanical perturbations.

VOA
1495-1565 nm
Tunable Laser
10 : 90
splitter

vacuum chamber

Polarization
Controller

Reference
Detector 1
MZ I
High-speed
Detector

Reference
Detector 2

Oscilloscope

TBPF

EDFA

double disk

VOA

fiber taper

10 : 90
splitter

1 µm

nanoforks

pedestal
undercut

34
Normalized Transmission

0.8

0.6

0.4
(a)
−10

−5

Normalized Transmission

0.8
0.6

0.98

0.4

0.96
0.94

0.2

Qi = 1.75 106
−30 −20 −10 0 10 20 30
[λ−1518.57nm] (pm)

1516

1520

Wavelength Detuning (pm)

1524

1528

(b)
1532

Wavelength (nm)

by the following equation:
d2x
FT (t)
dx
+ Γm + Ω2m x =
dt
dt
mx

(2.12)

(2.13)

Ω2 ), where xe(Ω) is the Fourier transform

of the mechanical displacement x(t) defined as xe(Ω) =

R +∞

iΩt
∞ x(t)e dt, and Sx (Ω) is the spectral

intensity for the thermal mechanical displacement with the following form:
Sx (Ω) =

(Ω2m

2Γm kB T /mx
Ω2 )2 + (ΩΓm )2

(2.14)

35
The time correlation of the mechanical displacement is thus given by
hx(t)x(t + τ)i =
2π

Z +∞

hx2 iρ(τ)

Sx (Ω)e iΩτ dτ

hx2 ie Γm jτj/2 cos Ωm τ,

(2.15)

κ/2

igOM x)a f + iηab + i κe Ain ,

(2.16)

κ/2

igOM x)ab + iηa f ,

(2.17)

ω0 is the frequency

Z +∞

Rτ

cos(ητ) f (τ)e igOM 0 x(t τ )dτ dτ,

(2.18)

e(i∆0 κ/2)τ represents the cavity response. Using Eq. (2.15), we find that the statisti-

cally averaged intracavity field is given as:
ha f (t)i = i κe Ain
where ε

Z +∞

cos(ητ) f (τ)e 2 h(τ) dτ,

(2.19)

g2OM hx2 i and h(τ) is defined as
ZZ τ

h(τ)

ρ(τ1

τ2 )dτ1 dτ2 .

(2.20)

36
Similarly, we can find the averaged energy for the forward WGM as:
hU f (t)i = κe Pin

ZZ +∞

f (τ1 ) f (τ2 ) cos(ητ1 ) cos(ητ2 )e 2 h(jτ1 τ2 j) dτ1 dτ2

κe Pin κ iη
2κ κ 2iη
where f j (τ)

Z +∞

e 2 h(τ) [ fc (τ) + fs (τ)] dτ + c.c.,

e(i∆ j κ/2)τ ( j = c, s), with ∆c = ∆0 + η and ∆s = ∆0

(2.21)

η. c.c. denotes complex conju-

gate.
As the transmitted power from the double disk is given by
p
PT (t) = Pin + κeU f (t) + i κe Ain a f (t)
the averaged cavity transmission, hT i
hT i = 1

κe κi
2κ

Ain a f (t) ,

(2.22)

hPT i/Pin , thus takes the form

iηκe
κi (κ 2iη)

Z +∞

2 h(τ)

[ fc (τ) + fs (τ)] dτ + c.c. .

(2.23)

In the case of a singlet resonance, η = 0 and Eq. (2.23) reduces to the simple form expression
hT i = 1

κe κi

Z +∞

e 2 h(τ) [ f (τ) + f (τ)] dτ.

(2.24)

In the absence of opto-mechanical coupling, gOM = 0 and Eq. (2.24) reduces to the conventional
form of
T =1

κe κi
∆0 + (κ/2)2

(2.25)

i κe Ain

Z +∞

cos(ητ) f (τ) 1

Z τ

igOM

x(t

τ0 )dτ0 dτ.

(2.26)

In this case, the transmitted optical field can be written as AT (t) = Ain + i κe a f (t)

A0 + δA(t),

where A0 is the transmitted field in the absence of the optomechanical effect and δA is the induced
perturbation. They take the following forms:
A0 = Ain 1

Z +∞

κe

cos(ητ) f (τ)dτ

δA(t) = igOM κe Ain

dτ cos(ητ) f (τ)

Z +∞

Ain Â0 ,

Z τ

x(t

τ0 )dτ0 .

(2.27)
(2.28)

The transmitted power then becomes P(t) = jAT (t)j2

jA0 j2 + A0 δA(t) + A0 δA (t). It is easy to

show that hδA(t)i = 0 and hPT (t)i = jA0 j2 . As a result, the power fluctuations, δP(t)

PT (t)

hPT (t)i, become
Z +∞

δP(t)
where u(τ)

iκe cos(ητ)[Â0 f (τ)

gOM Pin

Z τ

dτu(τ)

x(t

τ0 )dτ0 ,

(2.29)

Â0 f (τ)]. By using Eq. (2.15), we find the autocorrelation func-

tion for the power fluctuation to be
hδP(t)δP(t + t0 )i

εPin2

ZZ +∞

dτ1 dτ2 u(τ1 )u(τ2 )ψ(t0 , τ1 , τ2 ),

(2.30)

where ψ(t0 , τ1 , τ2 ) is defined as
Z τ1

ψ(t0 , τ1 , τ2 )

dτ01

Z τ2

dτ02 ρ(t0 + τ01

τ02 ).

(2.31)

38
Taking the Fourier transform of Eq. (2.30), we obtain the power spectral density SP (Ω) of the cavity
transmission to be
SP (Ω)

g2OM Pin2 H(Ω)Sx (Ω),

(2.32)

where Sx (Ω) is the spectral intensity of the mechanical displacement given in Eq. (2.14) and H(Ω)
is the cavity transfer function given by
H(Ω) =

Z +∞

u(τ)(eiΩτ

1)dτ .

(2.33)

In the case of a singlet resonance, the cavity transfer function takes the form:
4∆20 (κ2i + Ω2 )
κ2e
H(Ω) =
∆20 + (κ/2)2 [(∆0 + Ω) + (κ/2) ] [(∆0 Ω) + (κ/2) ]

(2.34)

In most cases, the photon decay rate inside the cavity is much larger than the mechanical damping rate, κ

Γm . For a specific mechanical mode at the frequency Ωm , the cavity transfer function

can be well approximated by H(Ω)

H(Ωm ). In particular, in the sideband-unresolved regime, the

cavity transfer function is given by a simple form of
4κ2e κ2i ∆20
H=
4 .
∆20 + (κ/2)2

(2.35)

(2.36)

39
be found using Eqs. (2.15) and (2.18). For example, we can find the following correlation for the
intracavity field:
h Ain a f 1

Ain a f 1

κe Pin2

ZZ +∞

Ain a f 2

Ain a f 2 i

dτ1 dτ2C1C2 e 2 (h1 +h2 ) f1 f2 e εψ + f1 f2 eεψ + c.c. ,

(2.37)

where, in the integrand, C j = cos(ητ j ), h j = h(τ j ), f j = f (τ j ) (with j = 1, 2), and ψ = ψ(t2
t1 , τ1 , τ2 ). h(τ) and ψ(t2

t1 , τ1 , τ2 ) are given by Eqs. (2.20) and (2.31), respectively.

Equations (2.20) and (2.31) show that h(τ) and ψ(t2

t1 , τ1 , τ2 ) vary with time on time scales

of 1/Ωm and 1/Γm . However, in the sideband-unresolved regime, κ

Γm and κ

Ωm . As the

cavity response function f (τ) decays exponentially with time at a rate of κ/2, the integrand in
Eq. (2.37) becomes negligible when τ1

2/κ or τ2

2/κ. Therefore, ψ(t2

t1 , τ1 , τ2 ) can be well

approximated as

ψ(t2

Similarly, h(τ)

t1 , τ1 , τ2 ) =

(2.38)

τ2 , since h(τ) = ψ(0, τ, τ). Therefore, Eq. (2.37) becomes
h Ain a f 1

where ∆t = t2

Z +∞
iΩτ
Sx (Ω) iΩ(t2 t1 )
iΩτ1
dΩ
2πhx2 i ∞ Ω2
Z +∞
τ1 τ2
Sx (Ω)e iΩ(t2 t1 ) dΩ = τ1 τ2 ρ(t2 t1 ).
2πhx2 i ∞

Ain a f 1

Ain a f 2

Ain a f 2 i

κe Pin2 Φ(∆t,C1C2 ),

(2.39)

t1 and Φ(∆t,C1C2 ) is defined as
ZZ +∞

Φ(∆t,C1C2 )

2
dτ1 dτ2C1C2 e 2 (τ1 +τ2 ) f1 f2 e ετ1 τ2 ρ + f1 f2 eετ1 τ2 ρ + c.c. ,

(2.40)

κ2e Pin2 Φ(∆t, σ1 σ2 )

κe hU f i + i κe Ain ha f i

2
Ain ha f i ,

(2.41)

40
where σ j = σ(τ j ) ( j = 1, 2) and σ(τ) is defined as

σ(τ)

κe (κ2 + 2η2 )
ηκe
sin(ητ).
cos(ητ) + 2
κ(κ + 4η )
κ + 4η2

(2.42)

Moreover, Eq. (2.19) and (2.21) show that, in the sideband-unresolved regime, ha f i and hU f i
are well approximated by
ha f (t)i

i κe Ain

hU f (t)i

κe Pin

Z +∞

ZZ +∞

ε 2

cos(ητ) f (τ)e 2 τ dτ,

(2.43)

f (τ1 ) f (τ2 ) cos(ητ1 ) cos(ητ2 )e 2 (τ1 τ2 ) dτ1 dτ2 .

(2.44)

Therefore, we obtain the final term in Eq. (2.41) as
κe hU f i + i κe Ain ha f i

Ain ha f i

Z +∞

κe Pin

ε 2

σ(τ) [ f (τ) + f (τ)] e 2 τ dτ.

(2.45)

Using this term in Eq. (2.41), we obtain the final form for the autocorrelation of the power fluctuations:
hδP(t1 )δP(t2 )i

κ2e Pin2 [Φ(∆t, σ1 σ2 )

Φ(∞, σ1 σ2 )] .

(2.46)

It can be further simplified if we notice that the exponential function e ετ1 τ2 ρ(∆t) in Eq. (2.40) can
be expanded in a Taylor series as
+∞

( ετ1 τ2 )n n
ρ (∆t).
n!
n=0

e ετ1 τ2 ρ(∆t) = ∑

(2.47)

Substituting this expression into Eq. (2.40) and using it in Eq. (2.46), we obtain the autocorrelation
function for the power fluctuation in the following form
+∞

hδP(t)δP(t + t0 )i

εn ρn (t0 )
jGn + ( 1)n Gn j2 ,
n!
n=1

κ2e Pin2 ∑

(2.48)

where Gn is defined as
Z +∞

ε 2

τn σ(τ) f (τ)e 2 τ dτ.

(2.49)

41
of Eq. (2.48):
+∞

εn Sn (Ω)
jGn + ( 1)n Gn j2 ,
n!
n=1

S p (Ω) = κ2e Pin2 ∑

(2.50)

where Sn (Ω) is defined as
Z +∞

Sn (Ω) =

ρn (τ)eiΩτ dτ.

(2.51)

Eq. (2.14) shows that the spectral intensity of the mechanical displacement can be approximated
by a Lorentzian function, resulting in an approximated ρ(τ) given as ρ(τ)

e Γm jτj/2 cos Ωm τ [see

Eq. (2.15)]. As a result, Eq. (2.51) becomes
Sn (Ω)

1 n
n!
nΓm
∑ k!(n k)! (nΓm /2)2 + [(2k n)Ωm + Ω]2 .
2n k=0

(2.52)

2.5

Regenerative oscillation

2.5.1

Ambient pressure

Power Spectral Density (dBm/Hz)

Normalized Transmission

0.8
0.6

0.98

0.4

0.96
0.94

0.2

1516

Qi = 1.75 106
−30 −20 −10 0 10 20 30
[λ−1518.57nm] (pm)

1520

1524

1528

Wavelength (nm)

1532

-90
-100

Qm = 3.95

-110

3 4

-120
-130

Qm = 4.07 103

-90

-80
-90
-100

-100

-110
8.2 8.6 9 9.4 9.8
Frequency (MHz)

-110

Frequency (MHz)

-25
0.5
0.5
-50

0.2

0.4

0.6

Time (µs)

0.8

100

150

Time (ns)

200

Mechanical displacement (pm)

Normalized transmission

(2.53)
(2.54)

where we have counted in both the thermal Langevin force FT and the optical gradient force Fo =
gOM jaj2
for actuating mechanical motions.
ω0

Normalized Transmission

1.0

0.8

0.6

0.4

0.2

−30

−20

−10

Wavelength Detuning (pm)

single detuning function,
f (∆)

∆2 + (κ/2)2
κκe κ3i ∆

κ 2
(∆ + Ωm ) +
(∆

Ωm ) +

κ 2

(2.55)

2g2OM Pi
ωc mx Γm κ3i

2B
Pi ,
Γm

(2.56)

0.4 GHz/nm, in good

agreement with the simulated result of 33 GHz/nm.

2.5.2

Vacuum

In order to eliminate the squeeze-film damping of the nitrogen environment, measurements were
also performed in vacuum (P < 5

10 4 Torr). The significantly reduced mechanical linewidth

in vacuum shows that the flapping mode consists of a small cluster of modes (Fig. 2.6(b), bottom

0.8
0.6
0.05

f = 8.53 MHz

-0.05

10-27

-10

-6

-8

-4

-2

Wavelength detuning (pm)

10-29

10-28

10-31

10-32

10-29

Frequency (MHz)

sing
increa

ca
p ti

10-30
6.8

vacuum

air

60
50
40
30
20
10
00

0.5

1.5

200 400 600 800

Input power (µW)

100

10-30

Displacement spectral density (m2/Hz)

-0.6

f = 9.63 MHz

Spectral density (m2/Hz)

0.2
-0.2

Threshold detuning function f(∆th)

Normalized transmission

III I

Effective temperature (K)

7.2

7.4

7.6

7.8

Frequency (MHz)

8.2

8.4

8.6

0.1

Input power (µW)

1) is not zero, and their thermal motion is visible in the transmission power spec-

2.6

Optomechanical cooling

a0 (t) + δa(t), where a0 is the cavity field in the absence of optomechanical

coupling and δa is the perturbation induced by the thermal mechanical motion. From Eq. (2.53),

47
they are found to satisfy the following equations:
da0
= (i∆0
dt
dδa
= (i∆0
dt

κ/2)a0 + i κe Ain ,

(2.57)

igOM xa0 .

(2.58)

κ/2)δa

In the case of a continuous-wave input, Eq. (2.57) gives a steady-state value given as:
i κe Ain
a0 =
κ/2 i∆0

(2.59)

and Eq. (2.58) provides the spectral response for the perturbed field amplitude,
δe
a(Ω) =

igOM a0 xe(Ω)
i(∆0 + Ω) κ/2

where δe
a(Ω) is the Fourier transform of δa(t) defined as δe
a(Ω) =

(2.60)
R +∞

iΩt
e(Ω)
∞ δa(t)e dt. Similarly, x

is the Fourier transform of x(t).
The optical gradient force, Fo =
Fo (t) =

gOM jaj2
ω0 , is given by

gOM 2
ja0 j + a0 δa(t) + a0 δa (t) .
ω0

(2.61)

2g2OM ja0 j2 ∆0 xe(Ω)
∆20 Ω2 + (κ/2)2 + iκΩ
ω0
[(∆0 + Ω)2 + (κ/2)2 ] [(∆0 Ω)2 + (κ/2)2 ]

(2.62)

As expected, the gradient force is linearly proportional to the thermal mechanical displacement.
Equation (2.54) can be solved easily in the frequency domain, which becomes
(Ω2m

Ω2

iΓm Ω)e
x=

FeT
Feo
+ .
mx mx

(2.63)

Equation (2.63) together with (2.62) provides the simple form for the thermal mechanical displace-

48
ment,
xe(Ω) =

FeT
mx (Ω0m )2

Ω2

iΓ0m Ω

(2.64)

where Ω0m and Γ0m are defined as
(Ω0m )2

Γ0m

Ω2m +

(2.65)

(2.66)

2Γm kB T /mx
Ω2 ]2 + (ΩΓ0m )2

[(Ω0m )2

(2.67)

m kB T
which has a maximum value Sx (Ω0m ) = mx2Γ
. The variance of the thermal mechanical displace(Ω0 Γ0 )2
m m

ment is equal to the area under the spectrum,
h(δx)2 i =

2π

Z +∞

Sx (Ω)dΩ =

kB T Γm
mx (Ω0m )2 Γ0m

(2.68)

10 17 m/Hz1/2 , as limited by the background level).
A measure of the optical cooling can be determined from the integrated area under the displace-

(2.69)

Sx (Ω) =

Sx (Ω)p(Ω0m )dΩ0m ,

where Sx (Ω) is given by Eq. (2.67) and we have assumed

(2.70)

p(Ω0m )dΩ0m = 1. The experimentally

measured spectral area is thus
2π

Z +∞

Sx (Ω)dΩ =

h(δx)2 ip(Ω0m )dΩ0m

h(δx)2 i.

(2.71)

kB T eff = mx (Ωm )2 h(δx)2 i,

where Ωm

(2.72)

Ω0m p(Ω0m )dΩ0m is the center frequency of the measured displacement spectrum Sx (Ω).

Compared with the room temperature, the effective temperature is thus given by

T eff (Ωm )2 h(δx)2 i
= 2
T0
Ω h(δx)2 i

(2.73)

1.45(κ/2)). For comparison, we

2.7

Discussion

0.06 MHz/µW for a

100 MHz. The combination allows for higher mechanical frequencies

Chapter 3

Coherent Mechanical Mode Mixing in
Optomechanical Nanocavities
3.1

Introduction

3.2

Zipper cavity and double-disk design, fabrication, and optical characterization

104 ), near-infrared (λ

1550 nm)

optical supermodes of the beam pair. Clamping to the substrate at either end of the suspended
beams results in a fundamental in-plane mechanical beam resonance of frequency

8 MHz. The

106 ) character of the

1 µm

104 V/A) and a high-speed oscilloscope (2 Gs/s sampling rate and 1 GHz bandwidth). A

3.3

125 dBm/Hz near 10 MHz.

Theory of optomechanical effects in the presence of mode mixing

3.3.1

Intracavity eld in the presence of optomechanical coupling

In the presence of optomechanical coupling, the optical field inside the cavity satisfies the following
equation:
da
= (i∆0
dt

Γt /2

igom xb )a + i Γe Ain ,

(3.1)

ω0 is the frequency detuning from the input wave to the cavity

a0 (t) + δa(t), where a0 is the cavity field in the absence of

optomechanical coupling and δa is the perturbation induced by the mechanical motion. They satisfy
the following two equations:
da0
= (i∆0
dt
dδa
= (i∆0
dt

Γe Ain ,

(3.2)

igom xb a0 .

(3.3)

Γt /2)a0 + i
Γt /2)δa

In the case of a continuous-wave input, Eq. (3.2) leads to a steady state given by
i Γe Ain
a0 =
Γt /2 i∆0

(3.4)

and Eq. (3.3) provides a spectral response for the perturbed field amplitude of
δe
a(Ω) =

igom a0 xeb (Ω)
i(∆0 + Ω) Γt /2

where δe
a(Ω) is the Fourier transform of δa(t) defined as δe
a(Ω) =
is the Fourier transform of xb (t).

(3.5)
R +∞

iΩt
eb (Ω)
∞ δa(t)e dt. Similarly, x

3.3.2

The power spectral density of the cavity transmission

From the discussion in the previous section, the transmitted optical power from the cavity is given
by
PT = Ain + i Γe a

jA0 j2 + i

Γe (A0 δa

A0 δa ) ,

(3.6)

where A0 is the steady-state cavity transmission in the absence of optomechanical coupling. It is
given by
A0 = Ain

(Γ0

Γe )/2 i∆0
Γt /2 i∆0

(3.7)

PT (t)

hPT i, are given in the frequency domain by

iΓ
(Ω)
(Γ0 Γe )/2 + i∆0
in
om
δPeT (Ω) =
Γt /2 i(∆0 + Ω)
(Γt /2) + ∆0

(Γ0 Γe )/2
Γt /2 + i(∆0

i∆0
Ω)

(3.8)

where δPeT (Ω) is the Fourier transform of δPT (t). By using Eq. (3.8), we obtain a power spectral
density (PSD) for the cavity transmission of
SP (Ω) = g2om Pin2 Sxb (Ω)H(Ω),

(3.9)

4∆20 (Γ20 + Ω2 )
Γ2e
2
2
∆0 + (Γt /2)2 [(∆0 + Ω) + (Γt /2) ] [(∆0 Ω) + (Γt /2) ]

(3.10)

In general, when compared with Sxb (Ω), H(Ω) is a slowly varying function of Ω and can be well
approximated by its value at the mechanical resonance: H(Ω)

H(Ω0mb ). Clearly then, the power

spectral density of the cavity transmission is linearly proportional to the spectral intensity of the
mechanical displacement of the optically bright mode.

3.3.3

The mechanical response with multiple excitation pathways

When the optically bright mode is coupled to an optically dark mode, the Hamiltonian for the
coupled mechanical system is given by the general form:

Hm =

p2b
p2
+ kb xb2 + d + kd xd2 + κxb xd ,
2mb 2
2md 2

(3.11)

(3.12)
(3.13)

kj
th
m j is the mechanical frequency for the j mode. Fj ( j = b, d) represents the Langevin

forces from the thermal reservoir actuating the Brownian motion, with the following statistical properties in the frequency domain:
hFei (Ωu )Fej (Ωv )i = 2mi Γmi kB T δi j 2πδ(Ωu

Ωv ),

(3.14)

where i, j = b, d, T is the temperature and kB is the Boltzmann constant. Fei (Ω) is the Fourier
transform of Fi (t).
In Eq. (3.12), Fo =

gom jaj2
ω0 represents the optical gradient force. From the previous section, we

find that it is given by
Fo (t) =

gom 2
ja0 j + a0 δa(t) + a0 δa (t) .
ω0

(3.15)

59
third terms provide the dynamic optomechanical coupling. From Eq. (3.5), the gradient force is
found to be given in the frequency domain by
Feo (Ω)

fo (Ω)e
xb (Ω) =

∆20 Ω2 + (Γt /2)2 + iΓt Ω
2g2om ja0 j2 ∆0 xeb (Ω)
, (3.16)
ω0
[(∆0 + Ω)2 + (Γt /2)2 ] [(∆0 Ω)2 + (Γt /2)2 ]

where L j (Ω)

Ω2m j

Ω2

(3.17)
(3.18)

iΓm j Ω ( j = b, d). Substituting Eq. (3.16) into Eq. (3.17), we find that

Eq. (3.17) can be written in the simple form,
Lb (Ω)e
xb +

Feb
xed =
mb
mb

(3.19)

where Lb (Ω) is now defined with a new mechanical frequency Ω0mb and energy decay rate Γ0mb as
Lb (Ω) = Ω2mb

Ω2

iΓmb Ω

fo (Ω)
mb

(Ω0mb )2

Ω2

iΓ0mb Ω,

(3.20)

and the new Ω0mS and Γ0mS are given by
(Ω0mb )2

Γ0mb

(3.21)

(3.22)

60
Equations (3.18) and (3.19) can be solved easily to obtain the solution for the optically bright
mode,
xeb (Ω) =
where η4

Feb (Ω)
mb Ld (Ω)

κ Fed (Ω)
mb md
Lb (Ω)Ld (Ω) η4

(3.23)

κ2
mb md represents the mechanical coupling coefficient. By using Eq. (3.14) and (3.23),

we obtain the spectral intensity of the mechanical displacement for the optically bright mode,
Sxb (Ω) =

2kB T η4 Γmd + Γmb jLd (Ω)j2
mb jLb (Ω)Ld (Ω) η4 j2

(3.24)

where Lb (Ω) is given by Eq. (3.20). The mechanical response given by Eq. (3.24) is very similar to
the atomic response in EIT.

3.3.4

The mechanical response with external optical excitation

Γt /2)δa

igom xb a0 + i Γe δA.

(3.25)

This equation leads to the intracavity field modulation given in the frequency domain as:
igom a0 xeb (Ω) i Γe δA(Ω)
δe
a(Ω) =
i(∆0 + Ω) Γt /2

(3.26)

where δA(Ω)
is the Fourier transform of δA(t). By use of this solution together with Eq. (3.15), the
gradient force now becomes
Feo (Ω) = fo (Ω)e
xb (Ω) + Fee (Ω),

(3.27)

(3.28)

61
In particular, in the sideband-unresolved regime, Eq. (3.28) can be well approximated by
Fee (Ω)

i Γe gom h
e ( Ω) .
a0 δA(Ω)
+ a0 δA
ω0 (i∆ Γt /2)

(3.29)

(3.30)
(3.31)

Using Eqs. (3.26) and (3.27), following a similar procedure as the previous section, we find that the
mechanical displacement for the optically bright mode is now given by
xeb (Ω) =

Fee (Ω)
Ld (Ω)
mb Lb (Ω)Ld (Ω)

η4

(3.32)

where Lb (Ω) and Ld (Ω) are given in the previous section. Clearly, the mechanical response given
in Eq. (3.32) is directly analogous to the atomic response in EIT systems [83].

3.4

Mechanical mode renormalization in zipper cavities

200 kHz, with similar linewidths (damping) and transduced

amplitudes (Fig. 3.2(c)). As the laser is tuned into resonance from the blue-side of the cavity, and
the intra-cavity photon number increases (to

7000), the higher frequency resonance is seen to

Power Spectral Density (dBm/Hz)

Mechanical Frequency (MHz)

62
8.4
8.2

-90

Nphoton

-100

7.8
7.6
7.4
0.25 b

Power Spectral Density (dBm/Hz)

Mechanical Linewidth (MHz)

-110

0.2
0.15
0.1
0.05

-2

-1.5

-1

-0.5

0.5

Normalized Detuning (∆ο/Γt)

1.5

-60

Nphoton

-70

-80

7.2

7.6

8.4

Frequency (MHz)

8.8

63
together, a consistent picture emerges from the data in Fig. 3.2 in which the nanobeams start out at
large detuning moving independently with similar damping (the frequency splitting of

200 kHz is

(3.33)
(3.34)

x2 . By transferring Eqs. (3.33) and (3.34) into

the frequency domain, it is easy to find that the mechanical displacement of the optically bright
mode is given by
Fe1 (Ω)
xeb (Ω) =
m1 L1 (Ω)
where L j (Ω) = Ω2m j

Ω2

h
Fe2 (Ω)
Feq (Ω) + Feo (Ω) ,
m2 L2 (Ω)
m1 L1 (Ω) m2 L2 (Ω)

(3.35)

iΓm j Ω ( j = 1, 2). The squeeze-film effect is produced by the pressure

64
the optically bright mode displacement,
2kB T

Sxb (Ω) =
L1 (Ω)L2 (Ω)

Γm2
Γm1
m1 jL2 (Ω)j + m2 jL1 (Ω)j

[ fo (Ω) + fq (Ω)]

L1 (Ω)
L2 (Ω)
m2 + m1

i 2.

(3.36)

As the squeeze-film effect primarily damps the differential motion, its spectral response can be
approximated as fq (Ω)

iαq Ω. Moreover, since the two disks or nanobeams generally have only

slight asymmetry due to fabrication imperfections, they generally have quite close effective masses
and energy damping rates: m1

m2 = 2mb and Γm1

Γm2

Γm , where we have used the fact

that the effective motional mass of the differential motion is given by mb = m1 m2 /(m1 + m2 ). As a
result, Eq. (3.36) can be well approximated by
Sxb (Ω)

where Γq

kB T Γm
mb L1 (Ω)L2 (Ω)

jL1 (Ω)j2 + jL2 (Ω)j2
2 [ f o (Ω)/mb + iΓq Ω] [L1 (Ω) + L2 (Ω)]

(3.37)

αq /mb represents the damping rate introduced by the squeeze gas film, and the spectral

104 and 2.8

104 , respectively, obtained from optical charac-

x1 + x2 for the optically-dark mechanical

65
mode, which is given by the following form:

Sxd (Ω) = 2kB T

Γm2
m2

L1 (Ω)

m1 [ f o (Ω) + f q (Ω)]

L1 (Ω)L2 (Ω)

+ Γmm11 L2 (Ω) m22 [ fo (Ω) + fq (Ω)]
i2
L2 (Ω)
[ fo (Ω) + fq (Ω)] L1m(Ω)
m1
m2 = 2mb and Γm1

Similar to the optically-bright mode, with m1

Γm2

(3.38)

Γm , Eq. (3.38) can be

well approximated by
Sxd (Ω)

kB T Γm jL1 (Ω) h(Ω)j2 + jL2 (Ω) h(Ω)j2
md L1 (Ω)L2 (Ω) 1 h(Ω) [L1 (Ω) + L2 (Ω)] 2

where md = m/2 is the effective mass of the common mode and h(Ω)

(3.39)

[ fo (Ω)/mb + iΓq Ω] repre-

2kB T Γm /mb
jLo (Ω)

where L0 (Ω) = (Ωm1 + Ωm2 )2 /4

Ω2

h(Ω)j

Sxd (Ω)

2kB T Γm /md
jLo (Ω)j2

(3.40)

iΓm Ω. Equation (3.40) indicates that the optically bright

and dark modes reduce to pure differential and common modes, respectively.

3.5

Coherent mechanical mode mixing in double-disks

13 dB anti-resonance, appears in the power spectrum near

106 and 0.7

106 are obtained from optical characterization of the cavity

Power Spectral Density (20 dB/div)

10
12
14
16
Frequency (MHz)

-75

01 2 3
∆0/Γt

PSD (dBm/Hz)

-80
-85

-90
-95

-100

Frequency (MHz)

3.6

Coherent mechanical mode mixing in zipper cavities

Hm =

i=b,1,2

p2i
+ ki xi2 + κ1 xb x1 + κ2 xb x2 ,
2mi 2

(3.41)

(3.42)
(3.43)
(3.44)

2kB T η41 Γm1 jL2 (Ω)j2 + η42 Γm2 jL1 (Ω)j2 + Γmb jL1 (Ω)L2 (Ω)j2
mb
Lb (Ω)L1 (Ω)L2 (Ω) η4 L2 (Ω) η4 L1 (Ω)

where η4j

(3.45)

κ2j
mb m j ( j = 1, 2) represents the mechanical coupling coefficient. L j (Ω) = Ωm j

Ω2

104 and 2.8

104 ,

3.7

Analogy to electromagnetically-induced transparency

ΙΙ

ΙΙΙ

(c)

(b)

Power Spectral Density (20 dB/div)

(a)

14
16
18
Frequency (MHz)

∆0/Γt

PSD (dBm/Hz)

-80

-90

-100

-110

(e)

(d)
14

Frequency (MHz)

(f )
Frequency (MHz)

Frequency (MHz)

(h)

ΙΙ

ΙΙΙ

input

output

input

coupling

k12

|+〉

|2〉

pump

γ2

|−〉

control

pump

probe
γ3

|1〉

anti-Stokes

|3〉

Γ2
|1〉

Stokes

RF/microwave
photon

|2〉
coupling
|3〉

Γ3

H = ~ω0 a† a + ~Ωm b† b + ~gom xb a† a,

(3.46)

b + b† .
2mb Ωmb

(3.47)

Therefore, the interaction Hamiltonian between the optical wave and the mechanical motion is given
by

Hi = hga† a b + b† ,

where the factor g

g2om ~3
2mb Ωmb

1/2

(3.48)

The mechanical motion modulates the intracavity field to create two optical sidebands. As a
result, the optical field can be written as
a = a p + as e iΩmbt + ai eiΩmbt ,

(3.49)

Hi = ~g b + b† a†p a p + ~gb† a†s a p + a†p ai + ~gb a†p as + a†i a p .

(3.50)

3.8

Discussion

Chapter 4

Mechanically Pliant Double Disk
Resonators
4.1

Introduction

4.2

Spiderweb resonator design and optical characterization

4 nm and 150

3 nm for the silica and α-Si layers, respectively. The wafer was

then thermally annealed in a nitrogen environment at a temperature of T = 1050 K for 10 hours to
drive out water and hydrogen in the film, improving the optical quality of the material. The spider-

1 µm

gap

25 µm

50
40

gOM/2π (GHz/nm)

SiO2

α-Si

SiO2

Height (µm)

20
1460

Radius (µm)

0 200 400 600 800

15
10

-1
42

1500

Ring spacing (nm)

1540

-2

1580

Per-photon force (fN)

Wavelength (nm)

150

20 µm

200

400

600

Ring spacing (nm)

800

1000

4%.

The optomechanical coupling coefficient, gOM

dω0 /dx, determines both the tunability and

N~g2OM g2OM Pd
kω0 Γ0

(4.1)

EDFA

BPF

VOA

Pump Laser

Polarization
Controller

MUX
Probe Laser

Reference
Detector 1

Polarization
Controller

DC Bias

RF Drive

50 : 50
splitter

VOA

EOM

MZ I

Oscilloscope

Reference
Detector 2

OSA

Network Analyzer

High-speed
Detector

fiber taper
VOA
10 : 90
splitter

DEMUX

10 : 90
splitter

Reference
Detector 3

106 and Qi = 0.90

106 , respectively.

4.3

Static lter response

Normalized transmission

0.8
0.6

probe

pump

0.8
0.6

1520

1530

1540

0.5

1530

1531

1532

1533

Probe wavelength (nm)

0.4

1550

200

14 GHz/mW
0.5

1.5

0.8

0.5

100

1552 1553

Wavelength (nm)

400
300

1549 1550 1551

1560

Probe tuning (GHz)

500

Detuning (pm)

Wavelength (nm)

1.5

1529

-8

low power

[ lp - 1531] (nm)

1510

1500

Detuning (pm)

-8

0.5

0.2

pow
er

0.6

0.4

Pump dropped power (mW)

high

0.8

0.6

0.4

0.2

-0.5

0.1

0.2

Pump dropped power (mW)

0.3

0.4

0.5

25%) can

-10
-20

z-displacement (pm)

Figure 4.4: FEM simulation illustrating the z-displacement of a 54 µm spiderweb resonator under
the 21 K temperature differential between substrate and ring induced by 2.1 mW dropped optical
power.

Γt1
)a1 + iβa2 + i Γe1 Ain ,
Γt2
)a2 + iβa1 + i Γe2 Ain ,

(4.2)
(4.3)

ω0j represents the cavity detuning of the jth mode, and β is the optical coupling co-

efficient between the two cavity modes. With a continuous-wave input, the steady state of Eqs. (4.2)

84
and (4.3) is given by the following solution
p
iAin (i∆2 Γt2 /2) Γe1 iβ Γe2
(i∆1 Γt1 /2)(i∆2 Γt2 /2) + β2
iAin (i∆1 Γt1 /2) Γe2 iβ Γe1
(i∆1 Γt1 /2)(i∆2 Γt2 /2) + β2

a1 =
a2 =

(4.4)
(4.5)

As the transmitted field from the cavity is given by AT = Ain + i Γe1 a1 + i Γe2 a2 , the cavity
transmission thus has the following equation

(i∆1
jAT j2
jAin j

Γ01 Γe1
)(i∆2

(i∆1

i Γe1 Γe2 )2
Γt2 /2) + β2

Γ02 Γe2
) + (β

Γt1 /2)(i∆2

(4.6)

4.4

Dynamic lter response

k0 = k +
where ∆ = ωl

2g2OM Pd ∆
ω0 Γ0 [∆2 + (Γt /2)2 ]

(4.7)

ωo is the detuning of the input laser (ωl ) from the cavity resonance (ωo ) frequency

0.8
0.6
0.4
0.2

1529.08 1529.12

0.6
0.8
0.6
0.8
0.6

1529.28 1529.32

0.1

0.2

20
15
10
-5
-10
-15
-20

Frequency (MHz)

0.3

Time(µs)

pump

0.6
0.4
0.2
0.8

probe

0.6
0.4
0.2

Modulation spectrum (dB)

Wavelength (nm)

-25

0.8

Fractional Modulation

Normalized transmission

0.4

0.5

0.2

0.4

0.6

0.8

Time(µs)

1.2

78 dB

-20
-40
-60
-80

-100

200

400

600

800

1000

Frequency (MHz)

1200

1400

Γtp
)a p

igOM xa p + iγja p j2 a p + i Γep Ap ,

(4.8)

where a p and A p are the intracavity and input field of the pump wave, respectively, normalized such
that Up

ja p j2 and Pp

jA p j2 represent the intracavity energy and input power. ∆ p = ω p

ω0p

10 20 m2 /W is the Kerr nonlinear coefficient of silica, n0 = 1.44 is the silica

Γtp
)ap0 + i Γep Ap0 ,
Γtp
)δa p igOM xap0 + i Γep δAp ,

(4.9)
(4.10)

where we have neglected the negligible self-phase modulation for the pump wave. Equation (4.9)
provides a steady-state solution of
i Γep Ap0
ap0 =
Γtp /2 i∆ p

(4.11)

from which we obtain the average pump power dropped into the cavity, Ppd , given by
Ppd =

Pp0 Γ0p Γep
∆ p + (Γtp /2)2

(4.12)

(4.13)

(4.14)

ep (Ω) are Fourier transforms of δa p (t), x(t), and δA p (t), respectively,
where δe
a p (Ω), xe(Ω), and δA
defined as B(Ω)

R +∞

∞ B(t)e

iΩt dt. Physically, the first term in Eq. (4.14) represents the perturba-

tion induced by the mechanical motion, while the second term represents the effect of direct input
modulation.
The optical gradient force is linearly proportional to the cavity energy as Fo =

gOMU p
ω p . With

modulation of the pump energy, the gradient force thus consists of two terms, Fo = Fo0 + δFo (t),
where Fo0 =

gOMUp0
is the static force component introduced by the averaged pump energy Up0 =
ωp

jap0 j2 , and δFo (t) is the dynamic component related to the pump energy modulation δUp (t), given
by
gOM δUp
ωp

δFo (t) =

gOM
ap0 δa p (t) + ap0 δa p (t) .
ωp

(4.15)

Substituting Eq. (4.14) into Eq. (4.15), we find the force modulation is described by this general
form in the frequency domain:

δFe0 (Ω) = fo (Ω)e
x(Ω) +

ep (Ω)
ep ( Ω)
ap0 δA
ap0 δA
Γep gOM
ωp
i(∆ p + Ω) Γtp /2 i(∆ p Ω) + Γtp /2

(4.16)

where the first term represents the back action introduced by the mechanical motion, with a spectral
response f0 (Ω) given by
fo (Ω)

∆2p Ω2 + (Γtp /2)2 + iΓtp Ω
2g2OM jap0 j2 ∆ p

.
ωp
(∆ p + Ω)2 + (Γtp /2)2 (∆ p Ω)2 + (Γtp /2)2

(4.17)

Figure 4.5(d) shows the spectral response of a probe resonance to small-signal sinusoidal pump

6.3 µm wide, much

1 where the gas can be considered

64σ2 Pa L0W0
∑ m2 n2 [(m2 + (n/η)2 )2 + σ2 /π4 ] ,
π8 h0
m,n odd

(4.18)

Cd (Ω) =

64σPa L0W0
m2 + (n/η)2
∑ 22 2
2 2
2 4
π6 h0
m,n odd m n [(m + (n/η) ) + σ /π ]

(4.19)

where Pa is the ambient gas pressure, W0 and L0 are the width and length of the plate, h0 is the gap,

89
η = L0 /W0 is the aspect ratio of the plate, and σ is the squeeze number given by
σ(Ω) =

12µeffW02 Ω
Pa h20

(4.20)

+ 2
W0 L0

(4.21)

1 where the interaction between gas molecules

is negligible, the squeeze film approximately behaves like a damping force, fsq (Ω) = iCr Ω, with Cr
given by the following equation [131, 132]
Cr =

16πh0

4Pa L0W0

2Mm
πR T

(4.22)

(4.23)

2L0 . As shown by the experimental results

(4.24)

(4.25)

gomUp0
gom Ppd
jFo0 j
meff Ω2m
km ω p
km ω p Γ0p

(4.26)

91
of the resonant frequency and damping rate of the mechanical motions. Defining

L (Ω)

Ω2m

Ω2

iΓm Ω

fo (Ω)
meff

fsq (Ω)
meff

(4.27)

the mechanical displacement is thus given by
ep (Ω)
ep ( Ω)
ap0 δA
ap0 δA
i Γep gOM
FeT (Ω)
xe(Ω) =
meff L (Ω) meff ω p L (Ω) i(∆ p + Ω) Γtp /2 i(∆ p Ω) + Γtp /2

(4.28)

Sx (Ω) =

2Γm kB T
meff jL (Ω)j2

(4.29)

(Ω0m )2

Ω2

iΓ0m Ω with a new mechanical resonance Ω0m

and damping rate Γ0m affected by the optical force. Equation (4.29) thus leads to a variance of the
thermal mechanical displacement given by
h(δx) i =
2π

Z +∞

k T Γm
km Γ0m

Sx (Ω)dΩ = B0

kB T
km

(4.30)

0 = m (Ω0 )2 is the effective spring constant and the approximation in the final term aswhere km
eff

92
The probe wave inside the cavity is governed by a dynamic equation similar to Eq. (4.8):
das
= (i∆s
dt

Γts
)as

igOM xas + 2iγja p j2 as + i

Γes As ,

(4.31)

Γts
)as0 + 2iγUp0 as0 + i Γes As ,
Γts
)δas + 2iγUp0 δas igOM xas0 + 2iγδUp as0 ,

(4.32)
(4.33)

Γtp , Γts .

Equation (4.32) provides a steady-state solution of
i Γes As
as0 =
Γts /2 i∆s

(4.34)

and Eq. (4.33) results in a probe-field modulation in the frequency domain of

δe
as (Ω) =

ias0 gOM xe(Ω)
i(∆s + Ω)

ep (Ω)
2γδU
Γts /2

(4.35)

A0s δas ),

(4.36)

where A0s = As + i Γes as0 is the transmitted probe wave in the absence of modulation. By use
of Eqs. (4.14), (4.28), (4.34) and (4.35), we find that the power spectrum of the transmitted probe

93
modulation is given by the following equation:
2 2
Ppd jδPepd (Ω)j2 4Γ2es Γ20s ∆2s
g2OM
jδPeTs (Ω)j2
2γ
Ps2
meff ω p L (Ω)
Γ20p
Ppd
[∆2s + (Γts /2)2 ]4

(4.37)

Γtp . We also assume the cavity is in

Γtp , Γts . The modulation spectra given in Fig. 4.5 are

defined as
ρ(Ω)

jδPeTs (Ω)j2 /Ps2
jδPepd (Ω)j2 /P2

(4.38)

ρ(Ω)

Ppd0
Ppd

jδPeTs (Ω)j2 /Ps2
jδPepd (Ω)j2 /P2

(4.39)

pd0

ρ(Ω) = 2γ + ∑

g2j
mj ω p L j (Ω)

Ppd

4Γ2es Γ20s ∆2s

Γ20p [∆2s + (Γts /2)2 ]4

(4.40)

Ω2

iΓmj Ω where Ωmj and Γmj are the reso-

nance frequency and damping rate of the jth mechanical mode. Equation (4.40) was used to describe

2-3 MHz (see Fig. 4.5(d)).

mechanical resonance frequency and damping rate, respectively. The Kerr nonlinearity in silica has

4.5

Discusssion

Chapter 5

97
in cavity quantum electrodynamics or for dispersion compensation in nonlinear optics.