RESONANTLY ENHANCED SAGNAC INTERFEROMETRY USING SURFACE
PLASMON POLARITONS
by
BRIAN K. HAKE
A DISSERTATION
Presented to the Department of Physics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
September 2019
DISSERTATION APPROVAL PAGE
Student: Brian K. Hake
Title: Resonantly Enhanced Sagnac Interferometry Using Surface Plasmon Polaritons
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Physics by:
Dr. Benjamin McMorran Chair
Dr. Miriam Deutsch Advisor
Dr. Raymond Frey Core Member
Dr. George Nazin Institutional Representative
and
Janet Woodruff-Borden Vice Provost and Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded September 2019
ii
©c 2019 Brian K. Hake
This work is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivs (United States) License.
iii
DISSERTATION ABSTRACT
Brian K. Hake
Doctor of Philosophy
Department of Physics
September 2019
Title: Resonantly Enhanced Sagnac Interferometry Using Surface Plasmon Polaritons
We present a theoretical model and experimental results for a Sagnac ring
interferometer resonantly enhanced via surface plasmon resonance (SPR). One
defining characteristic of a resonance is the π change in phase that occurs across
the resonance. For a narrow resonance, this results in a steep phase profile which
proves advantageous when utilized within a phase-based detection scheme, providing
a resonant enhancement that has the potential to greatly improve the sensitivity
of Sagnac interferometers. Furthermore, surface plasmon resonance serves as the
mechanism underlying surface plasmon sensing, one of the more commonly used
techniques for detection and characterization of a variety of interfacial events. This
method relies on the excitation of surface bound electromagnetic waves, known as
surface plasmon polaritons (SPPs), which propagate along the interface of a metal
and dielectric and exhibit a field profile that decays exponentially into both media.
Excitation of SPPs within an interferometric environment provides an avenue for
investigation into the potential benefits of this resonant enhancement. In addition, it
carries the capacity for insight into applications of SPR sensors using interferometric
techniques, some of which may be inaccessible to intensity based detection schemes.
iv
We begin this dissertation by providing necessary background information, such
as the interference equations describing the interferometer outputs and the SPP
dispersion relation. We then detail the features of SPR and explore modifications
to the resonance shape by variation of the system parameters. We fabricate and
characterize thin silver films, suitable for supporting SPPs, to be used in sensing
experiments. We design a theoretical model to compute the response of the resonantly
enhanced interferometer to various perturbations and calculate the corresponding
sensitivities. Finally, we construct a resonantly enhanced interferometer in the
laboratory, analyze the optical data collected from both interferometer output ports,
and comment on the promise of this and similar devices moving forward.
v
CURRICULUM VITAE
NAME OF AUTHOR: Brian K. Hake
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, Oregon
Point Loma Nazarene University, San Diego, California
DEGREES AWARDED:
Doctor of Philosophy, Physics, 2019, University of Oregon
Bachelor of Science, Engineering Physics, 2011, Point Loma Nazarene University
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, Department of Physics, University of Oregon,
Eugene, 2011-2013, 2015-2016, and 2018-2019
Graduate Research Assistant, Department of Physics, University of Oregon,
Eugene, 2012-2019
Science Literacy Teaching Fellow, University of Oregon, Eugene, 2016
PUBLICATIONS:
H. Grotewohl, B. Hake, and M. Deutsch, “Intensity and phase sensitivities in
metal/dielectric thin film systems exhibiting the coupling of surface plasmon
and waveguide modes,” Appl. Opt. 55, 8564–8570 (2016).
vi
ACKNOWLEDGEMENTS
I would like to start by thanking my advisor, Miriam Deutsch, for providing
me with the opportunity and resources needed to perform this research and for the
direction given and knowledge shared with me along the way. I would also like
to thank my committee chair, Ben McMorran, for the assurance and guidance he
provided to help me see this process through to completion, as well as the rest of
my committee, Ray Frey and George Nazin, for their participation in this process.
In addition, I would like to thank the many other professors and administrative staff
with whom I have interacted, for sharing their passions and expertise and providing
encouragement and assistance along the way.
I would also like to thank the students I have taken classes with or interacted
with as a teaching assistant for making this a fun and enjoyable place of learning.
Special thanks to my cohort of physics graduate students who were right there with
me through trying graduate level coursework and to all my colleagues in the Deutsch
Lab, especially Herbert Grotewohl and Andrea Goering, for the office camaraderie,
the tools and information passed on to me, and the insights gleaned through helpful
scientific discussions.
Thanks also are in order to my professors and fellow students from Point
Loma Nazarene University for nurturing my interest in physics and preparing me
for graduate school here at the University of Oregon.
I could not have accomplished this without the constant support and
encouragement of countless friends and family members throughout these years.
Thanks to my immediate and extended family in Phoenix, Arizona and around the
globe; those from Point Loma with whom I have maintained lasting friendships;
vii
and my faith family here at Eugene First Church of the Nazarene, among others,
for providing camaraderie, welcome distractions, and for taking part in my many
adventures both within and outside of physics. I could not possibly manage to list the
contributions of each one individually but there are a few I would like to highlight. To
the Smith-Gillespies, thank you for welcoming me as family and making the transition
to life in Eugene that much easier. To the Strunks, thank you for your supportive
friendship and for providing regular childcare to give me extra time to work. To my
siblings; Camille, Monica, Michael, Nathan, and Jonathan; you were my first and
best friends growing up and our visits together always give me something to look
forward to. To my parents, you have loved and cared for me since day one and I
aspire toward your example of joy, faithfulness, and humility. To Walter, you have
changed my life immeasurably since your arrival into this world. It is truly an honor
to be your dada and to have the privilege of watching you grow and learn; you have
taught me so much already. Finally, Alison, where do I begin? You believed in me
even when I expressed my doubts. You have been there with me at my best and at
my worst, and you continue to love me and motivate me toward bettering myself. I
am so blessed to be able to share this life with you. Thank you, and I love you.
To God be the glory.
This material is based upon work supported by the National Science Foundation
under Grant Nos. ECCS-1202182 and DMR-1404676. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the author
and do not necessarily reflect the views of the National Science Foundation.
viii
To my beloved grandparents, who I know are beaming with pride.
ix
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Sagnac Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . 2
Surface Plasmon Resonance Sensors . . . . . . . . . . . . . . . . . 5
Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
II. SAGNAC INTERFEROMETRY . . . . . . . . . . . . . . . . . . . . . . . 10
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Interference Equations . . . . . . . . . . . . . . . . . . . . . . . . . 15
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
III. SURFACE PLASMON RESONANCE . . . . . . . . . . . . . . . . . . . . 20
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Surface Plasmon Polariton Dispersion . . . . . . . . . . . . . . . . 20
Discussion of Materials . . . . . . . . . . . . . . . . . . . . . . . . 24
Coupling to Surface Plasmon Polaritons . . . . . . . . . . . . . . . 27
Surface Plasmon Resonance Features . . . . . . . . . . . . . . . . . 28
x
Chapter Page
Sample Preparation and Characterization . . . . . . . . . . . . . . 32
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
IV. RESONANTLY ENHANCED SAGNAC INTERFEROMETRY . . . . . . 39
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Inclusion of a Resonance . . . . . . . . . . . . . . . . . . . . . . . . 40
Perturbation Detection . . . . . . . . . . . . . . . . . . . . . . . . 42
Dark Port Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Dark Port Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 49
Bright Port Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Bright Port Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 56
Angular Detuning . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
V. EXPERIMENTAL IMPLEMENTATION AND RESULTS . . . . . . . . . 65
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Configuration and Alignment . . . . . . . . . . . . . . . . . . . . . 65
Perturbation Implementation . . . . . . . . . . . . . . . . . . . . . 72
Reciprocal Ohmic Heating in the Bright Port . . . . . . . . . . . . 76
Nonreciprocal Plasmon Drag in the Dark Port . . . . . . . . . . . . 81
Simultaneous Detection . . . . . . . . . . . . . . . . . . . . . . . . 94
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
xi
Chapter Page
VI. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
xii
LIST OF FIGURES
Figure Page
1. The interferometer configuration used by Georges Sagnac. . . . . . . . . . 3
2. Early surface plasmon sensing data. . . . . . . . . . . . . . . . . . . . . . 6
3. Schematic of a Sagnac interferometer. . . . . . . . . . . . . . . . . . . . . 11
4. Schematic diagram of a field incident on an interface. . . . . . . . . . . . 12
5. The reverse time picture of a field incident on an interface. . . . . . . . . 13
6. Illustration of the Stokes relations using time reversal invariance. . . . . . 14
7. Impact of a beamsplitter split ratio imbalance. . . . . . . . . . . . . . . . 18
8. Single interface schematic for calculation of SPP dispersion relation. . . . 22
9. Spectral permittivity of a few noble metals. . . . . . . . . . . . . . . . . . 25
10. Electromagnetic field of SPPs. . . . . . . . . . . . . . . . . . . . . . . . . 26
11. Mode dispersion of SPPs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
12. Kretschmann configuration for coupling to SPPs. . . . . . . . . . . . . . . 28
13. SPR angular reflectance and phase. . . . . . . . . . . . . . . . . . . . . . 29
14. SPR angular reflectance for varied silver film thicknesses. . . . . . . . . . 30
15. Angular dependence of SPR phase with varied film thickness. . . . . . . . 31
16. Schematic for angle conversion. . . . . . . . . . . . . . . . . . . . . . . . . 33
17. Angular reflectance experimental data with best fit functions. . . . . . . . 34
18. Angular reflectance data with fit for a near-critical thickness film. . . . . 36
19. Visualization of silver film topography created from AFM data. . . . . . . 37
20. AFM step height analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . 38
21. Resonantly enhanced Sagnac interferometer using SPPs. . . . . . . . . . . 41
22. Differential phase magnitude with variation in metal film thickness. . . . 42
xiii
Figure Page
23. SPR reflectance and phase in the refractive index domain. . . . . . . . . . 43
24. Phase difference produced by a nonreciprocal perturbation. . . . . . . . . 45
25. Dark port response to nonreciprocal perturbations. . . . . . . . . . . . . . 47
26. Dark port perturbation response. . . . . . . . . . . . . . . . . . . . . . . . 48
27. Dark port response to small perturbations. . . . . . . . . . . . . . . . . . 49
28. Nonreciprocal and reciprocal sensitivity in the dark port. . . . . . . . . . 51
29. Dark port sensitivity for simultaneous detection. . . . . . . . . . . . . . . 52
30. Dark port sensitivity for supercritical thickness. . . . . . . . . . . . . . . 53
31. Bright port response to reciprocal perturbations. . . . . . . . . . . . . . . 54
32. Bright port perturbation response. . . . . . . . . . . . . . . . . . . . . . . 56
33. Bright port response to small perturbations. . . . . . . . . . . . . . . . . 57
34. Reciprocal and nonreciprocal sensitivity in the bright port. . . . . . . . . 58
35. Bright port sensitivity for simultaneous detection. . . . . . . . . . . . . . 59
36. Bright port sensitivity for supercritical thickness. . . . . . . . . . . . . . . 60
37. Angle detuned bright port intensity. . . . . . . . . . . . . . . . . . . . . . 61
38. Angle detuned dark port intensity. . . . . . . . . . . . . . . . . . . . . . . 62
39. Initial alignment of the interferometer . . . . . . . . . . . . . . . . . . . . 67
40. A symmetric configuration for alignment at the desired angle. . . . . . . . 68
41. An asymmetric configuration for alignment at the desired angle. . . . . . 69
42. Schematic of full experimental configuration. . . . . . . . . . . . . . . . . 72
43. Experimental configuration as constructed in the laboratory. . . . . . . . 73
44. Diagram of the SPR prism mount used by L. J. Davis. . . . . . . . . . . . 75
45. Angles of incidence used in bright port data collection. . . . . . . . . . . . 77
46. Bright port response to reciprocal ohmic heating for θ =42.92◦. . . . . . . 78
47. Bright port response to reciprocal ohmic heating for θ =43.12◦. . . . . . . 80
xiv
Figure Page
48. Dark port response to nonreciprocal plasmon drag. . . . . . . . . . . . . . 84
49. Dark port model of plasmon drag. . . . . . . . . . . . . . . . . . . . . . . 86
50. Dark port nonreciprocal response at the resonance minimum. . . . . . . . 88
51. Dark port nonreciprocal response at an angle less than resonance. . . . . 90
52. Photo of apparatus with film oriented vertically. . . . . . . . . . . . . . . 91
53. Dark port response for varied current with vertically oriented film. . . . . 92
54. Scaling of modulation amplitude. . . . . . . . . . . . . . . . . . . . . . . . 93
xv
LIST OF TABLES
Table Page
1. Best fit parameter values from optical film characterization. . . . . . . . . 35
2. Interferometer response to reciprocal and nonreciprocal effects. . . . . . . 63
3. Decay constants of the dark port signal at the resonance minimum. . . . . 89
4. Decay constants for an angle less than the resonance minimum. . . . . . . 90
xvi
CHAPTER I
INTRODUCTION
Overview
This dissertation explores the functions and capabilities of a Sagnac ring
interferometer resonantly enhanced via surface plasmon polaritons (SPPs). Both
SPPs and Sagnac interferometers have undergone ample scientific investigation for
their use in sensing applications. Sagnac interferometers are most ubiquitous in
gyroscopes and for measurement of rotation; however, their sensing capabilities
encompass a much broader class of phenomena which break the symmetry between
the interferometer paths. In an effort to improve the detection limits achievable
through these techniques, some within the field have begun to explore the inclusion
of a resonance within the framework of this geometry with the intent to capitalize
on the resonant phase behavior inherent to these systems. The second overarching
topic of importance within this dissertation, surface plasmon sensing, is an established
technique well suited for the investigation of interfacial phenomena across a variety
of areas including biochemistry, biomedical, and environmental science applications.
These sensors rely primarily on intensity-based detection techniques; however, with
aspirations to lower the limit of detection, there has been increased interest within
the field toward the employment of phase-based detection techniques. The efforts
focused both toward resonance inclusion within the Sagnac interferometry community
and concurrently toward phase based detection for surface plasmon sensing share an
area of overlap and suggest that the marriage of Sagnac interferometry with surface
plasmon sensing is an intersection worthy of exploration. Insights gleaned from
1
research in this domain have the potential to both provide insight to the broader
field of resonantly enhanced Sagnac interferometry and expand upon the capabilities
of traditional surface plasmon sensing techniques.
In this chapter, we provide a cursory overview of both Sagnac interferometry
and surface plasmon sensors. This discussion will include aspects ranging from
early historical details up to, and including, modern applications. In addition, we
introduce the intention of utilizing the two in tandem which serves as the fundamental
architecture employed throughout this dissertation. We begin now with further details
regarding Sagnac interferometry.
Sagnac Interferometry
Interferometers come in a wide variety of shapes and sizes, but all rely on the
singular phenomenon of interference. Two interfering waves are superimposed to
form a resultant wave which can exhibit constructive or destructive interference
depending on the alignment of the waves with respect to one another, referred to
as the relative phase. In optical interferometry, the two are often electromagnetic
waves which originated from the same source, were separated at a beamsplitter, and
made to recombine at some later time. Studying the interference of these waves
and the corresponding response to perturbations enacted upon the system allows us
to glean information about the intensity or relative phase of the waves and often
provides insight regarding the nature of the perturbation itself. The configuration of
interest in this work is the Sagnac interferometer, distinguished by two beams which
share a common path but propagate around that path in opposite directions before
recombining at the same beamsplitter used to separate the paths initially.
2
The Sagnac interferometer bears the name of French physicist Georges Sagnac,
who in the early 1900s designed experiments using a rotating interferometer, in hopes
to support the existence and study the behavior of a luminiferous ether [1–3]. The
original configuration used by Sagnac, shown in Figure 1, consists of a source lamp
(labeled 0), a beamsplitter (J), four mirrors (M1-M4), and a photographic plate (pp’)
all mounted upon a rotating disc. Modern iterations of this class of interferometer
may utilize a different number of mirrors in a variety of configurations or dispel with
free space optics entirely and be constituted of fiber optics, but all operate under the
same general set of guiding principles [4].
FIGURE 1. The interferometer configuration used by Georges Sagnac.
As depicted in his 1913 paper published in Comptes Rendus [3], this configuration
consists of a source lamp (labeled 0), a beamsplitter (J), four mirrors (M1-M4), and
a photographic plate (pp’) all mounted upon a rotating disc.
In a 1913 paper published by Sagnac[3], he related the angular rotation of the
interferometer, Ω~ , to the fringe phase shift, δφ, by the following equation
8πΩ~ · A~
δφ = (1.1)
λv
3
where A~ is the area vector of the region enclosed by the interferometer loop, λ
is the wavelength of light used, and v is the speed of light [5]. Sagnac was able
to experimentally measure the expected phase shift based on the rotation of his
apparatus, and used this result to argue in favor of an immobile rather than a dragging
aether. Although the argument of an immobile aether eventually proved inaccurate;
the results can instead be explained by the theory of relativity [6]; this relation
between the phase shift and angular rotation has stood the test of time and has been
termed the Sagnac effect. Furthermore, interferometers of this type, with two beams
that traverse a common path in opposite directions, have come to be referred to as
Sagnac interferometers. Other terms such as loop or ring interferometers are also
commonplace.
The foremost advantages of this type of interferometer as compared with others
such as the Mach-Zehnder or Michelson interferometer rely on the commonality
of the paths. The interferometer is naturally aligned at the zero path difference
fringe thereby reducing constraints on the necessary coherence length of the light
source [7]. Furthermore, because both beams traverse the same path, there is an
inherently reduced sensitivity to low frequency phase noise including many thermal
and environmental sources [8, 9]. The elimination of numerous potential noise sources
allows for improved detection of perturbations which break the symmetry of these
paths. A common application of ring interferometers is their use as detectors of
rotation [10–13]. As the interferometer rotates, one path travels with the rotation
while the other propagates against the direction of rotation, breaking the symmetry of
the optical path lengths. Other applications involve the use of this device in the study
of magnetic effects [14, 15], acoustics [16, 17], temperature [18, 19], humidity [20],
4
biosensing [21], and gravity waves in the case of the proposed Laser Interferometer
Space Antenna (LISA) [22, 23].
Sagnac interferometers which incorporate a resonance, such as that of
electromagnetically induced transparency (EIT), have been posited to improve the
rotational sensitivity of such an instrument by as many as five to eight orders of
magnitude [24]. The types of resonances explored in pursuit of obtaining such a
drastic improvement in sensitivity include atomic resonances [24–26], ring resonators
[27–29], and coupled sequences of micro-resonators [30–33]. The pertinent feature
here is the large differential phase found on resonance. This phase behavior allows
for a detectable interferometric response to a particularly small detuning in some
parameter relevant to the particular resonant system. It is this resonant enhancement
of the sensitivity that we aim to exploit in this dissertation.
Surface Plasmon Resonance Sensors
A sensor is a device in which an output or measurement of the system responds
to stimuli from its surrounding environment in a calculable manner. Surface sensitive
sensing techniques preferentially respond to phenomena which occur in the near
vicinity of an interface between two materials. One such technique utilizes surface
plasmon resonance (SPR). In this technique, SPPs, coupled modes of charge-density
waves and photons, are excited along the interface between a metal and dielectric and
display an electromagnetic field profile which decays exponentially into both media.
These surface waves exhibit subwavelength confinement of light creating increased
field intensity and, thus, particularly high sensitivity to interfacial events. Detailed
review articles regarding the concepts and technologies of SPR sensors have been
authored by Jǐŕı Homola [34, 35].
5
The use of surface plasmon resonance as an optical biosensor was first realized
in the early 1980s by Claes Nylander and Bo Liedberg [36–38]. They recognized the
concept as conducive to gas detection and performed an experimental investigation
utilizing a gas cavity as part of the Kretschmann configuration [39]. They were
successful in measuring both the shift in resonance angle, ∆θ, and the shift in
reflectivity at a fixed angle of incidence, ∆R, in response to the introduction of
differing concentrations of gas. Their results are shown in Figure 2. The differing
gas concentrations manifest as a change in refractive index as experienced by the
propagating surface plasmon, modifying the resonance. This general principle persists
as the applications have expanded beyond gas concentration detection to others
including film characterization [40, 41], liquid identification [42, 43], temperature
sensing [44, 45], and chemical sensors [46, 47]. Additionally, techniques have expanded
to encompass those using photonic crystals [48–50], fiber optics [51–53] and even
interfacing with smart phones [54, 55].
FIGURE 2. Early surface plasmon sensing data.
Published by Claes Nylander and Bo Liedberg in 1982 [36], this figure shows
measurement of gas concentration as indicated by both a shift in the resonance
angle, ∆θ, and a shift in reflectivity, ∆R, at a fixed angle.
6
In the last decade, there has been much discussion regarding the lower detection
limits achievable using SPR sensors, with many of the best devices capable of
detecting a refractive index changes as small as 10−7 RIU [56]. It has been argued that
phase sensitive detection techniques have the ability to lower this limit of detection
by an order of magnitude, down to 10−8 RIU [57–59]. Others have argued against
this, stating that the drawbacks of the lower intensity at the resonance minimum,
where the differential phase is maximized, outweigh the advantages of phase based
detection [60]. One potential method of combating this counterpoint is through the
use of a stacked metal and dielectric layer configuration, which creates a geometric
Fano resonance, and alters the reflectance and phase profiles [61–64]. This allows for
increased intensity in regions with significant phase change, thus minimizing concerns
regarding detection noise. This work utilizes an interferometric surface plasmon
sensing technique, contributing to the knowledge base regarding the capabilities of
phase based probes.
Chapter Outline
This dissertation will explore the integration of surface plasmon sensing
techniques within the construct of a Sagnac interferometer as a means of improved
sensing capabilities and resonant enhancement of the sensitivity.
The second chapter will focus entirely on Sagnac interferometry and delve into
some of the mathematical formalism. The optical components which comprise the
interferometer and the general configuration in which they will be used will be
introduced with particular attention given to the beamsplitter. As a part of this
chapter, we will present the interference equations describing the normalized intensity
7
at the interferometer outputs. These equations will act as a foundational building
block upon which much of the remaining discussion will be based.
The focus of this dissertation will then shift back to surface plasmon resonance
in the third chapter. This chapter will include a derivation of the surface plasmon
dispersion relation for wave propagation along the interface of two semi-infinite media.
A discussion of the parameters necessary to satisfy this equation will follow, along
with proposed materials that fit these requirements. Next, we will address coupling
schemes for excitation of surface plasmons and introduce the configuration used for
these experiments. Some features of the resulting resonance, and their importance
to sensing techniques, will be highlighted. Finally, the chapter will conclude with
experimental details regarding both the preparation of samples as well as the optical
interrogation techniques utilized to characterize them.
The fourth chapter describes the theoretical model used and the results obtained
by employing it. We begin with some logistical details regarding the model and
methods for calculation of the outputs. We identify and differentiate two distinct
types of perturbations, reciprocal and nonreciprocal, and delve into the response at
both interferometer outputs to reciprocal perturbations, nonreciprocal perturbations,
and the simultaneous existence of both. Included in this chapter is an examination
of the sensitivity at each output and how well the results withstand variation
to parameters of the system. The results of this investigation are summarized;
highlighted by the observation that, unlike a typical Sagnac, the resonantly enhanced
interferometer is in fact sensitive to reciprocal effects. This unlocks the potential for
simultaneous detection of both reciprocal and nonreciprocal perturbations through
use of a single source beam and monitoring of the two output channels.
8
In the fifth chapter, we will document the construction of the optical system
while describing the components used and the procedures for alignment. We will
then discuss the experimental implementation of both reciprocal and nonreciprocal
effects by introduction of an electrical current to the metal film of the surface plasmon
configuration. This will introduce ohmic heating as the reciprocal perturbation and
plasmon drag as the nonreciprocal perturbation. Data collected from both outputs
of the interferometer will be presented and discussed in attempt to support the
expected outcomes based on the theoretical modeling contained in chapter four. A
few comments on simultaneous detection will conclude this chapter.
The final chapter will highlight the conclusions drawn from this dissertation as
well as present additional avenues for further exploration and closing remarks.
9
CHAPTER II
SAGNAC INTERFEROMETRY
Overview
In this chapter we take a more detailed and mathematical approach to Sagnac
interferometry. We begin by presenting the generic configuration used in this
dissertation and discussing the components used, with particular attention to the
beamsplitter. We then derive the interference equations that describe the normalized
intensity at the outputs of the interferometer and simplify them with use of the Stokes
relations. Finally, we introduce a beamsplitter imbalance parameter and explore the
effects of an uneven split ratio on the outputs.
Configuration
A Sagnac interferometer is the name given a closed loop interferometer consisting
of two counterpropagating beams which traverse a common path. The commonality
of the paths yields increased stability to laser fluctuations, environmental noise, or
other noise sources which affect both paths equivalently. Consequently, this type
of interferometer has proven useful in detection of non-reciprocal phenomena which
break the inherent symmetry of the two paths. This detection is primarily done via
a single output of the interferometer; however, there exists another interferometer
output directed back along the input beam path. In this work, we explore the
response of both interferometer outputs in search of any additional information
that may be gleaned through observation of both output ports. We begin that
discussion by describing the particular configuration used in this dissertation and
10
working through the mathematical formalism which describes the intensity at the
interferometer outputs.
FIGURE 3. Schematic of a Sagnac interferometer.
This Sagnac configuration consists of a beamsplitter (BS) and three mirrors (M).
The locations marked A and B act as both inputs and outputs of the system.
A schematic of the basic Sagnac interferometer configuration that will be
discussed in this dissertation is portrayed in Figure 3. This type of interferometer
consists of a single beamsplitter and two or more mirrors, three in the case of this
work. For an input laser beam incident at A as marked in the figure, the outputs
exit the interferometer at the locations marked A and B. In a perfectly balanced
interferometer, with equivalent phase accumulation in the two paths, the output that
exits the beamsplitter at the laser input exhibits perfectly constructive interference.
In this same scenario, the complementary interferometer output displays destructive
interference. Furthermore, assuming an ideal 50/50 reflection/transmission split
at the beamsplitter, the signal at the destructive interference output is identically
zero. These two outputs, which exhibit constructive and destructive interference, will
11
henceforth be referred to in this dissertation as the bright and dark interferometer
output ports, respectively. Although the nature of the interference at each output is
subject to change with the introduction of a phase difference to the propagating
beams, the naming convention will continue to refer to that of the balanced
interferometer configuration.
Beamsplitter
The lynchpin of the entire interferometer configuration is the beamsplitter itself.
This component acts both to split the optical source into two beams and to recombine
them, thereby producing the wave interference. As we will soon see in working out
the full interference equations, the beamsplitter will introduce multiple reflection
and transmission coefficients as the beam navigates the interferometer. The Stokes
relations will help to simplify the resulting terms and elucidate the interpretation
of the interferometer output equations. These Stokes relations follow from time
reversal invariance of electromagnetism in non-absorbing partial reflectors, such as
the beamsplitter used in the Sagnac interferometer.
FIGURE 4. Schematic diagram of a field incident on an interface.
The incident field is partially reflected and partially transmitted with coefficients of
reflection and transmission labeled r and t, respectively.
12
In the forward time picture, shown in Figure 4, an incident field results in
both a transmitted and reflected field, with coefficients of reflection and transmission
denoted here by r and t, respectively. In the time reversed picture of this scenario, in
which phase conjugation has reversed the direction of propagation, the reflected and
transmitted fields are now incident upon the interface. In order to satisfy time reversal
invariance, the resulting outgoing fields must be both the time reversed counterpart
of the above incident field as well as a null field. This time reversed scenario is
diagrammed in Figure 5.
FIGURE 5. The reverse time picture of a field incident on an interface.
Here the time reversed counterparts of the reflected and transmitted fields from
above are incident onto the interface. For time reversal invariance to be satisfied, the
resulting field must be the time reversed counterpart of the original incident field.
But for any field impingent upon an interface, we can write the reflected and
transmitted components of that field in terms of the coefficients of reflection and
transmission. This, then, also applies to the time reversed reflected and transmitted
fields: when incident on the surface, each results in their own transmitted and reflected
fields. The time reversed form of the reflected field results in transmitted and reflected
fields as illustrated in the leftmost diagram of Figure 6. Similarly, the time reversed
form of the transmitted field has its own reflected and transmitted fields as shown in
the middle diagram of Figure 6. These field reflection and transmission coefficients,
13
r′ and t′, are denoted with a prime as they are incident from below and presumably
may differ from r and t associated with incidence from above. When these resulting
fields are added together, as is depicted in Figure 6, they should recover the time
reversed picture of Figure 5.
FIGURE 6. Illustration of the Stokes relations using time reversal invariance.
The time reversed counterparts of the original reflected and transmitted fields, each
with their own partial reflections and transmissions. These two diagrams, when
combined, must recover the time reversed incident field and a complementary null
field in order to satisfy the requirement of time reversal invariance. The equations
resulting from the outgoing fields of each diagram uncover the Stokes relations.
With this in mind, we can now write two equations that must hold: the first coming
from the outgoing fields in the upper-left quadrant of each diagram and the second
coming from the outgoing fields in the lower-left.
rr∗E∗ + t′t∗E∗ ∗0 0 = E0 (2.1)
tr∗E∗ ′0 + r t
∗E∗0 = 0 (2.2)
leading to the familiar Stokes relations
rr∗ + t′t∗ = 1 (2.3)
tr∗ + r′t∗ = 0 (2.4)
14
With these tools and the overall configuration laid out, we are now equipped to derive
the interference equations which describe the bright and dark outputs of the Sagnac
interferometer.
Interference Equations
The normalized output intensities of the bright and dark ports can be calculated
by keeping track of the electric field as it propagates around the interferometer. To
calculate the output intensities at A and B, first suppose an input field of E eiφ00
impingent upon the beamsplitter. The corresponding transmitted and reflected
fields travel opposite directions around the interferometer. Upon again reaching the
beamsplitter they will have accumulated additional phase φ1 and φ2 respectively and
√ √
their amplitude will be modified by the factor T1 and T2 respectively where T1
and T2 are the round trip intensity coefficients for the two paths. The transmitted
field is incident again on the same face of the beamsplitter, resulting in the terms
√ √
rt T i(φ0+φ1) i(φ0+φ1)1E0e and tt T1E0e . Upon return to the beamsplitter the reflected
√
field is incident on the opposite face resulting in the terms t′r T2E e
i(φ0+φ2)
0 and
r′
√
r T E ei(φ0+φ2)2 0 , where the prime denotes incidence from the opposite face of the
beamsplitter. By superposition, the field at the bright output port is as follows.
( √ √ )
E+ = E e
iφ0 rt T eiφ1 + t′0 1 r T
iφ2
2e (2.5)
Similarly, addition of the fields in the complementary port results in the following
equation for the field at the dark output port of the interferometer.
( √ √ )
E = E eiφ0 tt T eiφ1 + r′r T eiφ2− 0 1 2 (2.6)
15
Since intensity is proportional to |E|2 the normalized output intensity at the bright
port can be drawn from the electric field equation at that output.
√ √
I iφ1 ′ iφ2 2+ = I0|rt T1e + t r T2e | (2.7)
where I = |E eiφ00 0 |2. Or equivalently,
[ √ ( )]
I = I |r|2|t|2T + |t′|2|r|2T + T T |r|2 t∗t′ei(φ2−φ1) + t′∗te−i(φ2−φ1)+ 0 1 2 1 2 (2.8)
Making use of the stokes relation, rr∗ + t′t∗ = 1, the resulting intensity may be
written as
[ √
′ ( ) ]I+ = I |r|2|t|2T + |t |2 20 1 |r| T2 + 2 T 2 21T2|r| 1− |r| cos (φ2 − φ1) (2.9)
In the same manner, the intensity at the dark output port may be computed using
the respective field equation.
√ √
I = I |tt T eiφ1 + r′r T eiφ2 2− 0 1 2 | (2.10)
Which may be equivalently expressed as
[ √ ( ]
I = I |t|4T + |r′|2|r|2T + T T t∗t∗r′rei(φ2−φ1) + r′∗r∗
)
tte−i(φ2−φ1)− 0 1 2 1 2 (2.11)
Then, by applying a second stokes relation, tr∗+ r′t∗ = 0, the dark port intensity can
be written as
[ √ ]
I = I |t|4T + |r′|2− 0 1 |r|2T2 − 2 T1T |t|2|r|22 cos (φ2 − φ1) (2.12)
16
As a sanity check, we first assume the round trip transmission coefficients and
phase accumulation are equivalent for both directions of travel around the loop,
meaning T1 = T2 and φ1 = φ2. We further assume the beamsplitter maintains an ideal
50/50 split ratio for incidence from both sides, such that |r|2 = |r′|2 = |t|2 = |t′|2 = 1 .
2
Under this scenario, the bright port intensity reduces to I+ = TI0 and the dark port
intensity goes to zero, displaying complete constructive and destructive interference,
respectively.
Split Ratio Imbalance
In practical applications, the transmission coefficient may depend on the
direction the beam travels around the path, and the beamsplitter may not provide an
ideal 50/50 split. A directionally dependent transmission coefficient can be accounted
for by a difference in T1 and T2, which are not required to be equivalent. To describe
how an uneven beamsplitter may modify the output intensities, a split ratio imbalance
parameter, ε, can be defined such that
|r|2 = |r′|2 1= (1 + ε) (2.13)
2
|t|2 = |t′|2 1= (1− ε) (2.14)
2
Written in terms of this parameter, the normalized intensity outputs at the bright
and dark ports are as follows
1 ( ) [ √ ]
[ I+ = I 20 1− ε T1 + T2 +( 2 T1)T√2cos (φ2 − φ1) (2.15)41 ]
I− = I0 (1− ε)2 T + (1 + ε)21 T2 − 2 1− ε2 T1T2cos (φ2 − φ1) (2.16)
4
17
(a) Bright port. (b) Dark port.
FIGURE 7. Impact of a beamsplitter split ratio imbalance.
Bright and dark port normalized output intensities as a function of ε for an
interferometer with T1 = T2 and φ1 − φ2 = 0; ε of 0.1 represents a 55/45 split ratio.
The normalized output intensities as a function of ε for an interferometer with
T1 = T2 and φ1 − φ2 = 0 are shown in Figure 7. From this, we can see that for ε up
to 0.1, which represents a 55/45 split ratio, the normalized intensities vary by up to
0.01. This is not too significant a concern with respect to the bright port, where this
variation is riding atop a large magnitude signal to begin with. However, in the dark
port this is not the case. Using a laser line, non-polarizing beamsplitter optimized for
632.8 nm we expect to measure a value of ε below 0.1; how far below will set a limit
on the lowest intensity theoretically achievable at the dark port of our interferometer.
Summary
This chapter provided further background information regarding Sagnac
interferometry that will be utilized throughout the remainder of this dissertation.
We first presented the basic Sagnac configuration that will be further expanded upon
later on in this dissertation, but that serves as an informative starting point. Next,
we derived the interference equations which characterize the intensity outputs of the
interferometer. These will prove useful in the theoretical modeling contained within
18
the following chapters. Lastly, we introduced a split ratio imbalance parameter to
explore the impact of an imbalanced beamsplitter. With an introductory discussion
of Sagnac interferometry complete, we now shift gears and focus our attention on the
concept of surface plasmon resonance.
19
CHAPTER III
SURFACE PLASMON RESONANCE
Overview
This chapter contains theoretical background information and experimental
outcomes regarding the concept of surface plasmon resonance (SPR), as a detailed
understanding of this phenomenon is vital before moving forward to incorporate
it within the Sagnac geometry. We begin with a conceptual description of SPR
and accompanying surface plasmon polaritons (SPPs). We then exploit Maxwell’s
equations of electromagnetism to arrive at the surface plasmon dispersion relation.
Next, we consider the requirements that must be satisfied by the material parameters
involved in this equation and explore some suitable materials for doing so. Armed
with this knowledge, we introduce a configuration of these materials that allows
for coupling to the surface plasmon mode. Furthermore, we explore the features
of the resonance and the tunability of the resonance shape through variation of
material parameters. Lastly, we detail experimental techniques for creation, optical
interrogation, and independent characterization of samples exhibiting these resonance
features.
Surface Plasmon Polariton Dispersion
SPR spectroscopy is a popular surface-sensitive characterization technique which
provides a repeatable and quantifiable optical response to some interfacial event. This
may include the binding of proteins to the surface, a change in temperature of the
one of the constituent materials, or the presence of a biomarker within a biological
20
sample, among other stimuli [46]. This technique is facilitated by the existence of
coherent electron oscillations known as surface plasmons. The oscillations may be
localized to a small particle or propagate along an interface. The latter, also known
as SPPs, are coupled modes of charge density waves and photons that propagate
along the boundary between metal and dielectric materials. They are characterized
by an electromagnetic field profile that peaks sharply at the interface and decays
exponentially away from the interface. In order to gain a more mathematical
understanding of the propagation of SPPs it is instructive to uncover the dispersion
relation.
The dispersion relation for SPPs can be found by application of Maxwell’s
equations. To solve, consider an interface between two semi-infinite media and begin
with an electric field of the following form that propagates along the interface, here
designated as the x axis, and decays exponentially into the two media, along the z
direction.  ik→− ‖x−k1z−iωt(x̂E 1x
+ ẑE1z) e z > 0
E =  (3.1)(x̂E2x + ẑE ik‖x+k2z−iωt2z) e z < 0
Where subscripts 1 and 2 refer to the materials above and below the x axis according
to the schematic shown in Figure 8; Ex and Ez are the field amplitude coefficients in
the x and z directions; k‖ is the wave vector along the direction of propagation; k1 and
k2 are the decay coefficients into the two materials, both taken to be positive; and
ω is the frequency of the wave. We first impose boundary conditions, namely that
the component of the electric field parallel to the interface, Ex, must be continuous
across the surface z = 0. In doing so, we can relate the amplitude coefficients in the
following manner.
E1x = E2x (3.2)
21
FIGURE 8. Single interface schematic for calculation of SPP dispersion relation.
Orientation of axes is as shown and 1 and 2 represent the permittivities of the
materials used. The curved lines at the interface are meant to evoke the exponential
decay of electric field into the two media.
Secondly, in the absence of surface charge, we apply the boundary condition that
the component of the electric displacement perpendicular to the interface, Dz, must
also be continuous across the surface z = 0. Doing so gives a similar relation,
1E1z = 2E2z (3.3)
where 1 and 2 are the relative permittivities of the material. Then, apply Maxwell’s
equation for the electric displacement, first by taking the divergence as follows
 ( ) ik x−k z−iωt
→− →− 1 ik‖E1x − k ‖ 11E1z e z > 0
∇ · D =  ( ) (3.4)2 ik E + k E eik‖x+k2z−iωt‖ 2x 2 2z z < 0
22
In the absence of a source term, the divergence of the electric displacement is equal
to zero. By imposing this condition, we arrive at the following two equations.
ik‖E1x − k1E1z = 0 (3.5)
ik‖E2x + k2E2z = 0 (3.6)
Combining these with Equations 3.2 and 3.3 results in the statement below.
k12 + k21 = 0 (3.7)
Based on the condition that k1 and k2 are assumed positive, this requires that 
change sign across the interface. As will be discussed in more detail later on, this
can be accomplished if one of the materials is a metal. Here, we will take the first
medium to be a typical dielectric material with 1 > 0 and the second to be a metal
with negative permittivity, 2 < 0.
To arrive at the dispersion relation for SPPs, we begin with the wave equation
→− 2→− µ ∂2 →−∇ E − E = 0 (3.8)
c2 ∂t2
and apply it to Equation 3.1, which yields the following two equations.
− 2 2 1k‖ + k1 + ω2 = 0 (3.9)c2
−k2 + k2 2+ ω2‖ 2 = 0 (3.10)c2
23
These may be combined with Equation 3.7 to obtain the surface plasmon dispersion
relation √
ω 12
k‖ = (3.11)
c 1 + 2
Although this relation looks linear in ω, there is a frequency dependence of 2 which
makes the relationship nonlinear. Additionally, the lossy nature of metals give a
complex permittivity, 2, which will be accounted for here by a complex wave vector,
′ ′′ ′ ′′
k‖. Assuming ω and 1 real, and using 2 = 2 + i2 where |2| > 2 the real and
imaginary components of the wave vector can be written, to lowest order, as
√
′
′ ω 1
k 2‖(= c 1′ ) ′
(3.12)
+ 2
3/2 ( ′′′′ ω 12 k‖ = 2′ )2 (3.13)c 1 +  ′2 2 2
′
Recalling that the second material is chosen to be a metal with 2 < 0, here we see
′ ′
that in order to arrive at a real solution for k‖ we further require that |2| > 1.
′′
The imaginary component of the wave vector, k‖ , describes the absorption of the
propagating wave and determines the SPP propagation length. The field penetration
depths into the respective materials are set by k1 and k2 which can be solved from
Equations 3.9 and 3.10.
Discussion of Materials
All of the necessary requirements for 2 are satisfied by a number of metals such as
silver, gold, and aluminum, leading to their recurring use in plasmonics applications.
Each have a negative permittivity across a wide spread of wavelengths from the
ultraviolet through the infrared as can be seen in Figure 9a, which uses Palik data
24
to interpolate the permittivity as a function of wavelength [65]. The magnitude of
′
the permittivity is large enough to satisfy the requirement that |2| > 1 for many
gases and liquids that may be used as the dielectric material 1. In addition, the
imaginary components, also interpolated from Palik data and displayed in Figure 9b,
′ ′′
are small enough to satisfy the condition |2| > 2 . Of the metals pictured, silver is the
least lossy and therefore will provide the narrowest resonance. A narrower resonance
results in a higher sensitivity and it is for this reason that silver is the metallic
material of choice for all experiments discussed throughout this dissertation. However,
this choice of silver is not without drawbacks; the greatest being the propensity for
oxidation when exposed to ambient conditions, thus altering the material properties
and degrading the quality of the resonance [66–68]. This effect can be mitigated, in
part, by either applying a protective coating layer such as gold [69, 70] or oxide layers
[71] atop the silver or by controlling the environment to which the surface of the film
is exposed.
(a) (b)
FIGURE 9. Spectral permittivity of a few noble metals.
Real and imaginary components of the complex relative permittivity as a function
of wavelength for silver (Ag), gold (Au), and aluminum (Al). Curves are
interpolated using optical data from Palik [65].
For the experiments presented in chapter five of this dissertation, the dielectric
used adjacent to the silver will be room temperature air. To prevent oxidation, the
25
samples will be stored under nitrogen until ready for use. Using the configuration of
Figure 8 with air and silver, the field distribution of the SPP is shown in Figure 10a.
Here the magnetic field is plotted as it is transverse to the direction of propagation and
creates a more helpful visualization. This figure also illustrates the SPP wavelength
which is very near, but slightly less than, the free space wavelength of 632.8 nm, as
expected [72]. The approximate penetration depth of the field into the two media
is observed more clearly by plotting the field amplitude, as is shown in Figure 10b.
For this particular configuration, the 1/e optical penetration depths are found to be
400nm and 24 nm, respectively. It is this region, where the field is strongest, that
will be probed be the surface plasmon sensing techniques.
(a) (b)
FIGURE 10. Electromagnetic field of SPPs.
(a) SPP magnetic field for the configuration shown in Figure 8 with air and silver as
the two materials and for wavelength of 632.8 nm. Magnetic, rather than electric
field is shown because it is transverse to the propagation. (b) Normalized
electromagnetic field amplitude as a function of the distance for the same
configuration. This illustrates the exponential decay of the field away from the
surface; 1/e optical penetration depths into the dielectric and metal are found to be
400nm and 24 nm, respectively.
26
Coupling to Surface Plasmon Polaritons
In order to utilize propagating surface plasmons for sensing applications, the
issue of exciting SPPs must first be resolved. The difficultly lies in the fact that
the SPP mode is inaccessible by light incident from the dielectric itself. For a given
frequency, the wave vector of the light in the dielectric medium must be increased by
some amount, ∆k in order to match that of the SPP. This can be seen in Figure 11
which plots the SPP dispersion relation for a silver/vacuum interface along with the
light line in vacuum; notice that the SPP mode lies entirely to the right of the light
line, and thus cannot be coupled to directly from vacuum.
FIGURE 11. Mode dispersion of SPPs.
Dispersion relation of SPPs at an Ag/vacuum interface and light lines in both
vacuum and glass to show that use of a glass prism enables coupling to SPPs.
Frequencies are normalized to the plasma frequency, ωp, and the real part of k‖ is
plotted after normalization to kp = ωp/c. The HeNe frequency, the frequency used
in the calculation of the SPP dispersion, is indicated in the figure as a reference.
To excite SPPs, the momentum mismatch must be overcome through some
coupling scheme. This can be accomplished via grating coupling, in which the
higher order diffracting beams compensate the additional momentum, or attenuated
total reflection, also termed prism coupling [73]. In this dissertation we focus on
27
the coupling scheme of attenuated total reflection, which involves use of a prism of
higher index material than the SPP dielectric in order to shift the light line to higher
wave vector – further to the right as seen in Figure 11. From the prism, the SPP is
accessible by incident light frequencies below the intersection of the mode dispersion
with the light line in the prism. At a particular frequency, the momentum of the
incident light parallel to the interface is matched to that of the SPP by variation of
the incident angle of the light within the prism. This coupling scheme is referred to
as attenuated total reflection, because the angle which accomplishes this is beyond
the angle of total internal reflection. The prism can be implemented on either side
of the interface, replacing either of the two semi-infinite media with a finite thickness
film. The more common configuration, and the one that will be employed in this
dissertation, is the Kretschmann configuration [39], in which a finite thickness metal
film is sandwiched by prism and dielectric material as shown in Figure 12.
FIGURE 12. Kretschmann configuration for coupling to SPPs.
Light incident on the silver from the glass at an angle θ excites SPPs on the
opposite surface of the silver, along the interface with the adjacent dielectric as
indicated with the cartoon of the field amplitude profile.
Surface Plasmon Resonance Features
Using the Kretschmann configuration described above, it is possible to excite
surface plasmons on the metal/dielectric interface. The incident light must be
28
FIGURE 13. SPR angular reflectance and phase.
Calculated for a glass prism, silver, and air in the Kretschmann configuration using
Palik data for the silver permittivity[65]. The thickness of the silver film is
optimized to give zero reflectance at the resonance minimum and exhibits a π jump
in phase at the angle of minimum reflectance.
polarized with p, or transverse magnetic, polarization so that the electric field
oscillates within the plane of incidence and penetrates the metal film. Variation of the
angle of incidence affects the projection of the momentum along the surface according
√
to k = ω‖  sin(θ). By varying the incident angle of the beam upon the surface andc
measuring the output at the specular beam, we can map out the angular reflectance
spectrum. Above the angle for total internal reflection, there will be a dip in the
reflectance, shown in Figure 13 along with the accompanying phase profile, which
corresponds to the excitation of SPPs. This profile is referred to as the resonance
lineshape. It can be observed not only in the angular domain, but also with variation
in wavelength of the incident light or permittivity of the dielectric adjoining the metal
film. The resonance is slightly asymmetric, with the larger magnitude slope occurring
on the side of the resonance nearer the angle of total internal reflection. The width
and depth of this resonance depend on the parameters involved; they can be tuned,
for example, through variation of the thickness of the metal film.
29
For a particular wavelength of incident light and set of prism, metal, and
dielectric materials, there is a critical thickness of the metal film for which the
reflectance drops to zero at the minimum of the resonance. This occurs when the
intrinsic damping is equal to the radiative damping [74]. As the thickness of the
metal film is detuned from this critical value in either direction, the resonance becomes
shallower. For thicknesses below this critical thickness, the shallower resonance is also
broadened and shifted to higher angle, whereas for thicknesses above this critical value
the resonance narrows and the resonance minimum trends toward a lesser angle. This
behavior is portrayed in Figure 14 for film thicknesses ranging from 25 nm to 75 nm.
FIGURE 14. SPR angular reflectance for varied silver film thicknesses.
Computed for a glass prism, silver, and air in the Kretschmann configuration with
silver film thickness varied as indicated. Changes in thickness of the film alters both
the location of the minimum and the depth of the resonance curve.
In addition to the zero in reflectance, the critical thickness metal film is also
marked by a singularity in the differential phase and a π phase jump on resonance
[57]. The magnitude of the phase slope on resonance decreases as the film thickness
departs from this critical value. In addition, the sign of the slope flips across this
critical thickness; it is negative for thinner films and positive on resonance for thicker
30
films. This behavior is shown in Figure 15, which displays the phase profiles for silver
film thicknesses ranging from 25 nm to 75 nm.
FIGURE 15. Angular dependence of SPR phase with varied film thickness.
Computed for a glass prism, silver, and air in the Kretschmann configuration with
silver film thickness varied as indicated. Steeper phase profiles are accomplished by
tuning the thickness nearer the critical value. The slope of the phase on resonance is
negative for thicknesses below this critical value and positive for thicknesses above.
These features of a narrow intensity resonance and large differential phase are
advantageous when employed in sensing applications. For intensity based sensors,
which rely on measurement of the reflectance, it is the slope of the reflectance which
is crucial. For this reason it is advantageous to maximize the slope, which occurs
at a position slightly detuned from the minimum of the resonance. When operated
in this way, a small change in the material parameter to be detected will induce the
largest possible change in intensity. By the same argument, intensity based detection
schemes tend to operate near the angle which minimizes the intensity, which exhibits
the maximum slope in phase.
31
Sample Preparation and Characterization
The type of samples desired are smooth silver films with uniform thickness of
approximately 50 nm. We choose to deposit the silver onto plain Corning soda-
lime glass microscope slides which can then be easily adhered to, and subsequently
removed from, the hypotenuse of a right angle prism using index matching liquids.
The glass slides are cut to match the height of the prism using a score and snap
technique, followed by a smoothing of the cut edges with lapping film. Smaller glass
substrates, less than 1 cm on a side, were also cut and deposited on, to be used
later for characterization by atomic force microscopy (AFM). The slides are cleaned
before deposition with a drop and drag technique using acetone and ethanol. Plasma
cleaning and piranha cleaning methods were also tested for comparison but made no
noticeable difference in the quality of the deposited films.
Once the substrates have been prepared, sample fabrication is performed in the
Center for Advanced Materials Characterization in Oregon (CAMCOR) facility at the
University of Oregon. The film is thermally evaporated using the Amod deposition
system from Angstrom Engineering with the substrates mounted on a rotating plate
within the chamber to facilitate uniform deposition. We use a source material of
99.99% pure silver pellets from Kurt J. Lesker company resting in a molybdenum
boat. The silver is deposited under vacuum conditions of approximately 10−6 torr
with deposition rates ranging from 0.1-15 Å/s onto glass substrates trimmed to the
desired size and cleaned as described above. Deposition rates are primarily chosen
from the higher end of this range in order to deposit higher quality films [75]. The
thickness of the Ag films is controlled both by deposition rate and by variation of the
deposition time and is monitored by a quartz crystal microbalance (QCM) mounted
in the deposition chamber. Because of their differing locations, a tooling factor is
32
calculated to related the QCM thickness to the thickness of the film deposited on the
substrate. Upon completion of the fabrication procedure, samples are stored under
nitrogen to preserve their integrity until ready for use.
FIGURE 16. Schematic for angle conversion.
Displays the geometry needed to relate the incidence angle within the prism, θ, to
the angle of the motorized rotation stage, θm, required to achieve this incidence.
To optically interrogate a sample, it is removed from the dry box and promptly
mounted, with the silver film facing outward, onto the hypotenuse of a Thorlabs 25
mm right angle BK7 glass prism using Cargille refractive index liquid. The index
matching liquid is applied so as to eliminate any air gaps between the prism and glass
slide and to provide a constant refractive index environment to remove any disruption
of the incoming beam. The prism, with the silver film now attached, is mounted onto
a motorized rotation stage, enabling variation of the angle of incidence of the source
beam onto the sample. The motor angle is set to zero for 45 degree incidence on
the back of the prism. The angle of the motorized stage can then be related to the
angle of incidence within the prism through the use of basic geometry and Snell’s law
yielding (
n ( ))a π
θm = arcsin sin θ − (3.14)
np 4
where na and np and the refractive indices of the air and prism respectively, and
θm and θ are the angles of the motorized stage and incidence within the prism as
33
indicated in Figure 16. The beam spot on the back of the sample is aligned as closely
as possible over the center of rotation of the motorized stage so that the same region
of the sample is interrogated as the prism is rotated and the specularly reflected beam
emanates from the same location over a range of angles. Two inch diameter optics
are used to collimate the output and focus it onto the detector allowing an angular
scan over nearly ten degrees to be completed without repositioning of the detector.
Detection is performed with a silicon photodiode through a lock-in amplifier designed
to pick out the optical signal modulated by an optical chopper wheel at 911 Hz.
The motorized rotation and signal collection is controlled via GPIB interface with a
computer running LabVIEW software.
FIGURE 17. Angular reflectance experimental data with best fit functions.
Silver films of different thickness were deposited by variation of deposition time and
characterized optically. The dots are the experimental optical data points and the
solid lines are a least squares fit to that optical data. The thickness values indicated
in the legend are those output by the fitting procedure.
The optical data collected is then analyzed in Mathematica. Reflectance from
the SPR configuration is computed by R = |r|2 with reflectivity, r, of a single thin
34
film system given by [76]
r12 + r
−2iδ
23e
r = (3.15)
1 + r −2iδ12r23e
where rij is the p wave reflectivity at the interface between two materials, i and j,
with relative permittivities i and j and angles of incidence θi and θj according to
√
i cos(θj)−
√
j cos(θi)
rij = √ √ (3.16)
i cos(θj) + j cos(θi)
and where the parameter δ is proportional to the film thickness, d, and is given by
2πd√
δ = 2 cos(θ2) (3.17)
λ
Thickness characterization of the samples is performed via a three parameter fit to
the reflectance data with the three parameters being the real and imaginary parts of
the permittivity and the thickness of the silver film. The optical data, along with
the best fit curves, are shown in Figure 17 for a set of four samples fabricated with
different deposition times. The determined film thicknesses are displayed in the plot
legend and the permittivity values for silver are consistent with the literature [65, 77].
TABLE 1. Best fit parameter values from optical film characterization.
Fit to Independent
Property Source
Optical Data Verification
-15.9 + 1.07 i Palik [65]
 -17.3 + 0.96 i
-18.3 + 0.48 i Johnson & Christy [77]
d 51.6 nm 50.5 nm AFM
Using these thicknesses as calibration, the deposition time can be optimized to
produce silver films with a resonance minimum even closer to zero than any of the
initial samples. Figure 18 shows optical data in which the normalized reflectance dips
35
just below 0.05 at its minimum. Performing the fitting procedure as outlined above
yields a film thickness of 51.6 nm and relative permittivity of −17.3 + 0.97i. Table
1 shows that this value falls between the values for the silver permittivity at this
wavelength from Palik and Johnson and Christy.
FIGURE 18. Angular reflectance data with fit for a near-critical thickness film.
A near-critical thickness film was achieved based on the calibration from Figure 17
and then characterized optically. The dots represent the measured reflectance values
and the solid line a fit to the optical data. The fitting procedure yields a relative
permittivity of  = −17.3 + 0.97i and thickness of 51.6 nm for the silver film.
For independent verification of film thickness we turn to AFM. Using CAMCOR
facilities, thickness measurements for a sampling of deposited films were obtained by
scraping the silver films deposited on the smaller glass substrates and scanning across
the step left in its wake. This data obtained from this scan may then be analyzed to
measure the film thickness. A visualization of the silver film and substrate of one of
these samples, created using the NanoScope Analysis software package, is presented
in Figure 19. An alternate presentation of this data is the plot of Figure 20. Here
we plot the percentage of pixels at each height against the height as measured above
the lowest pixel height. When viewed in this way, the data exhibits two clear peaks,
36
the lower one representing the substrate surface and the higher one representing the
silver surface. The shoulder on the high end of the silver height distribution is a result
of pileup along the edge caused by the scraping procedure. By fitting each of these
peaks with a Gaussian distribution, we identify the film thickness as the difference
between the centers of these distributions and obtain a measure of the roughness of
the film via the full width at half maximum (FWHM). The resulting thickness, 50.5
nm, matches well with that from the optical data, as shown in Table 1. The roughness
of the silver film is estimated at 2.8 nm. This analysis acts as independent verification
of the film thickness value obtained via SPR and instills confidence in the thickness
measurements obtained by the fitting procedures to the optical data.
FIGURE 19. Visualization of silver film topography created from AFM data.
Atomic force microscopy was used to independently verify the thickness of the silver
film. The elevated surface on the left hand side of the figure is the top of the silver
film and the lower surface on the right is the exposed glass substrate where the
silver has been scraped off for a step height measurement. The image was generated
using the NanoScope Analysis software package.
Summary
The intent of this chapter was to highlight the physics behind propagating surface
plasmons and the accompanying resonance, as well as provide details regarding the
37
FIGURE 20. AFM step height analysis.
Pixel height data for the silver film was recorded via AFM. The peaks at the film
and substrate surfaces are fit with Gaussian distributions to determine the film
thickness as well as obtain a measure of the surface roughness. The shoulder on the
high side of the rightmost peak is due to pileup of silver caused by scraping.
experimental preparation and characterization of samples to be used in the surface
plasmon sensing experiments contained in the following chapters. We derived the
dispersion relation for SPPs and discussed the physical significance of the material
parameters which factor in to this equation. We discussed experimental configuration
and procedures necessary to excite SPPs and observe the corresponding resonance.
Finally, we demonstrated the ability to fabricate and characterize high quality silver
films for use in the SPR sensing experiments. Armed with both the mathematical
tools necessary to describe the system as well as the experimental techniques required
to produce high quality samples, we proceed to construct a theoretical model which
incorporates these features of the resonance within the architecture of the Sagnac
interferometer.
38
CHAPTER IV
RESONANTLY ENHANCED SAGNAC INTERFEROMETRY
Overview
A resonantly enhanced Sagnac interferometer is simply one that incorporates
a resonant system within the construct of the interferometer itself. Resonance is a
ubiquitous phenomenon encountered in a wide variety of systems, so a discussion of
resonantly enhanced Sagnac interferometers is broadly applicable. A few examples
of resonances that are already being pursued within this geometry include atomic
resonances [24–26], ring resonators [27–29], and coupled sequences of micro-resonators
[30–33]. While the particular resonance used within the system will heavily
influence the potential applications of said experiment, the mathematical formalism
remains quite general. This is because of the near-universal resonance features: an
approximately Lorentzian intensity profile accompanied by a steep change in phase
on resonance. Introducing these features within the structure of the interference
equations arrived upon in the second chapter produces interesting results and
motivates further research in resonantly enhanced Sagnac interferometry. In this
chapter we will describe the creation of a theoretical model to describe the system. We
begin by describing the system to be modeled and detail the methods for introducing
perturbations to the system. Then we display and analyze results for the normalized
intensity at the dark port and compute the accompanying sensitivity. Next, we
perform the same intensity and sensitivity analysis for the bright port. After that,
we touch on the impacts of an angular detuning from the resonance minimum before
finally summarizing the results obtained from this theoretical model.
39
Inclusion of a Resonance
In this work the interferometer is resonantly enhanced via surface plasmon
resonance (SPR). The results are more broadly applicable; however, a particular
resonance is chosen here to model a system that can be implemented and tested
experimentally. A schematic diagram of the interferometer configuration under
consideration is presented in Figure 21. Here the source beam is incident from the
left and splits evenly at a 50/50 beamsplitter. The two resulting beams traverse
the same overall path around the interferometer but in opposite directions. At the
prism-coupled SPR element both beams undergo the resonant interaction, and upon
returning to the beamsplitter the two beams interfere resulting in two interferometer
output signals.
The resonant interaction specific to this system involves the coupling of light
to SPPs. Upon reaching the SPR element, the counter-propagating beams excite
counter-propagating SPPs at the metal dielectric interface. For this to occur, the
conditions must be such that the momentum component of the incident light along
the interface is matched to that of the SPP mode for beams incident from both sides of
the prism. The metal film thickness used is tuned to match the intrinsic and radiative
damping and will be referred to as the critical film thickness for SPR. As the thickness
is detuned from this critical value, the magnitude of the differential phase decreases
as shown in Figure 22. A large slope in the phase is crucial to provide the desired
resonant enhancement effects. For this reason, we focus on metal film thicknesses at
or near the critical thickness for the duration of this work.
After the beams have undergone the resonant interaction and propagated
around the entirety of the interferometer loop, they recombine at the beamsplitter.
The normalized intensities at the two output ports of the resonantly enhanced
40
FIGURE 21. Resonantly enhanced Sagnac interferometer using SPPs.
In this schematic diagram, the Kretschmann prism configuration replaces one of the
mirrors and the counterpropagating beams excite counterpropagating SPPs at the
interface of the metal and dielectric.
interferometer are given by modifying the interference equations derived earlier to
include the resonant terms.
1 √
I± = [R1 +R2 ± 2 R1R2cos (φ2 − φ1)] (4.1)
4
Here R and φ represent the intensity and phase, respectively, due to reflection from
the SPR element. Subscripts 1 and 2 distinguish the two counterpropagating paths
around the interferometer while the plus and minus signs differentiate the two output
ports. Aside from the resonant interaction, the phase accumulation in the remainder
of the loop is assumed to be equal between the paths and the mirrors are assumed to
be perfectly reflecting. This interference equation is generalizable beyond just SPR,
41
FIGURE 22. Differential phase magnitude with variation in metal film thickness.
The magnitude of the differential phase is calculated on resonance. There is a
singularity at the critical film thickness and the magnitude falls off sharply on either
side.
and will serve as the basis of our investigation on the sensing capabilities of resonantly
enhanced Sagnac interferometers.
Perturbation Detection
To explore these sensing capabilities, we investigate the response of the system
to external perturbations. In this work, we choose to model perturbations as a
refractive index detuning applied to the lower-index dielectric opposite the prism
in the SPR configuration. This dielectric is chosen to exhibit perturbations as a
matter of practicality, since a common application of SPR is to detect ultrasmall
shifts in refractive index of this dielectric[46]. This refractive index value is one of
the parameters that goes into the calculation of the R and φ terms in Equation 4.1,
computed using the Fresnel equations. The reflectance and phase dependence on
the refractive index of this outer dielectric, for a fixed angle of incidence, reproduce
the expected general resonance features as shown in Figure 23. The reproduction of
42
a nearly-Lorentzian dip in the reflectance accompanied by a steep change in phase
on resonance justifies the choice to introduce perturbations to the refractive index
of the lower-index dielectric. However, the results are general to a detuning in any
parameter of a resonant system that similarly influences a resonance.
FIGURE 23. SPR reflectance and phase in the refractive index domain.
Computed for a configuration consisting of a glass prism, silver, and dielectric of
refractive index n. The film thickness and incidence angle are optimized for
maximum coupling to SPPs at the refractive index of water. The resonance features
in this domain are qualitatively similar to those in the angular domain.
Perturbations to the system fall into one of two classifications: reciprocal or
nonreciprocal. For reciprocal phenomena, the perturbation affects both the clockwise
and counter-clockwise paths of the Sagnac interferometer equivalently. In the
terms of Equation 4.1, this means that for purely reciprocal effects, R1 = R2 and
φ1 = φ2. Reciprocal perturbations are implemented in this work by adding ∆n to
the refractive index as seen by both the clockwise and counter-clockwise propagating
paths. Although no phase difference is introduced by a reciprocal effect, a change
in reflectance caused by the detuning has the potential to alter the interferometer
output according to Equation 4.1. One example of a reciprocal perturbation to the
lower-index dielectric in the Kretschmann configuration is a temperature drift. A
43
temperature drift shifts the refractive index according to the thermo-optic coefficient
of the material, but the shift is independent of the direction of beam propagation
around the loop [44].
On the other hand, a perturbation is nonreciprocal if it depends on the
direction of propagation. These phenomena break the symmetry between the counter-
propagating paths of a balanced interferometer such that φ1 =6 φ2. As seen from
Equation 4.1 this phase difference alters the nature of the interference, moving from
constructive toward destructive or vice versa. This, in combination with any intensity
changes introduced by the detuning, fully describes the change in signal measured at
the bright and dark ports. We choose to implement a nonreciprocal detuning by
adding ∆n to the refractive index of the lower-index dielectric as experienced by
one path and subtracting ∆n from the refractive index of the same dielectric as
seen by the counter-propagating path. For reference, the phase difference produced
by a nonreciprocal detuning of |∆n| = 10−7 is plotted in Figure 24 for metal film
thicknesses near the critical thickness. The phase difference peaks sharply at the
resonant angle, prompting operation of the sensor at this fixed angle of incidence. The
magnitude of the peak grows as the film thickness approaches the phase singularity
on resonance as seen in Figure 22, reflecting the benefit of a large differential
phase associated with the near-critical thickness films. An example nonreciprocal
perturbation of refractive index is uniform translational motion of the lower-index
dielectric adjacent to the metal film in the SPR configuration[78, 79]. For a moving
dielectric, there is a relativistic drag effect known as Fresnel-Fizeau drag [80, 81]
which imparts a phase shift dependent on the scalar product of the velocity vector of
the dielectric and the SPP propagation direction. This directional dependent phase
difference amounts to a nonreciprocal detuning in the refractive index of the dielectric.
44
FIGURE 24. Phase difference produced by a nonreciprocal perturbation.
The perturbation is introduced as a shift in the refractive index with a magnitude of
|∆n| = 10−7. The phase difference peaks sharply at the resonant angle and the
magnitude is greater for film thicknesses near the critical value.
In many cases, both reciprocal and nonreciprocal phenomena will be present
simultaneously. This is often unwanted as it can contribute additional noise to the
system, but may be unavoidable under certain circumstances. For example, while
studying a nonreciprocal flow there will be a limit on how precisely the temperature
may be controlled. In these instances, one might wish to measure both nonreciprocal
and reciprocal effects simultaneously: such as the velocity of a moving dielectric and
any temperature change associated with it. In order to do so accurately, we should
utilize a device in which the response of the system to one type of perturbation is able
to be dependably distinguished from the response of the other. In this chapter we
will show that the resonantly enhanced Sagnac interferometer is capable of achieving
this level of discrimination. The dark port displays a high sensitivity to nonreciprocal
fluctuations and is robust against reciprocal phenomena. Meanwhile, the bright port
45
is highly sensitive to reciprocal drifts while maintaining a relative insensitivity to
nonreciprocal effects.
Dark Port Intensity
We begin by exploring the response of the dark port to both nonreciprocal and
reciprocal perturbations. For the nonreciprocal response, we plot I− from Equation
4.1 as a function of the nonreciprocal detuning. This nonreciprocal detuning alters the
refractive index of the dielectric by the same magnitude, but in opposite directions for
the two paths. It is performed for near-critical thickness metal films while maintaining
a fixed incidence angle in the prism, the resonant angle for the SPR system. Under
these initial conditions, the field reflectance from the SPR element is at its minimum.
As we detune from this resonant minimum, the reflectance values of the two counter-
propagating signals do increase but remain nearly equivalent, with differences owing
to a slight asymmetry in the resonance lineshape. Along with intensity changes, the
nonreciprocal detuning introduces a phase difference between the two paths. As this
phase difference accumulates it causes a switching from destructive to constructive
and then back to destructive interference at the dark port output. This results in the
nonmonotonic behavior of the normalized intensity output seen in Figure 25. This
general lineshape persists for film thickness on either side of the critical value. The
first shift from destructive to constructive interference happens rather abruptly due to
the π phase change on resonance. This resonant phase behavior provides a significant
phase difference with the introduction of only a slight detuning to the perturbed
parameter in the system, in this case the refractive index of the dielectric. The result
is a strong enhancement in the output intensity at the dark port. This resonantly
46
enhanced signal provides a sensitivity to nonreciprocal phenomena in the dark port
greater than that achievable in a nonresonant Sagnac interferometer.
FIGURE 25. Dark port response to nonreciprocal perturbations.
Normalized intensity output at the dark port, I−, exhibits nonmonotonic behavior
with the introduction of nonreciprocal phenomena. The film thickness is varied as
denoted in the legend.
In addition to the strong nonreciprocal response, a relative insensitivity of the
signal to reciprocal phenomena is needed to fully realize the potential of the dark port
as a sensor of nonreciprocal phenomena. We model reciprocal sensing in the dark port
by computing I− as a function of the reciprocal detuning. The reciprocal detuning
is implemented as described earlier, with ∆n equivalent for both interferometer
paths. Again, the incidence angle in the prism remains fixed at the resonant angle
of SPR, where the reflectance is at its minimum. As we detune the system, the
intensity upon reflection from the SPR element increases; identically in both paths
for reciprocal detuning. Unlike the nonreciprocal case, the phase remains equivalent
in both paths. Because of this, the output retains its destructive interference and
the intensity remains zero, despite the intensity increase in the counter-propagating
47
paths. The result is a signal completely insensitive to the introduction of purely
reciprocal phenomena.
FIGURE 26. Dark port perturbation response.
Normalized output intensity, indicated by the color bar, as a function of both
reciprocal and nonreciprocal detunings.
But what happens when reciprocal and nonreciprocal phenomena are present
simultaneously? To determine the robustness of the system, we consider the system
response in the presence of both types of phenomena. This response is plotted in
Figure 26. Here we see that the system maintains its nonmonotonic behavior as
a function of nonreciprocal detuning even in the presence of reciprocal phenomena.
However, unlike before, we observe that the reciprocal detuning can influence the
intensity output when occurring in conjunction with nonreciprocal effects. The
nonreciprocal effect introduces a phase difference at the output so there is no longer
complete destructive interference. Because of this, intensity variations of the counter-
propagating paths owing to reciprocal perturbations can begin to affect the output
signal. This result is unique to the resonantly enhanced interferometer and can
48
reintroduce noise that the nonresonant Sagnac interferometer is constructed to be
immune to. While this may be a subtle effect, it is important to consider when
utilizing a resonantly enhanced system.
FIGURE 27. Dark port response to small perturbations.
A zoomed-in view of Figure 26. The near-vertical contours suggest a strong response
to nonreciprocal perturbations and relative insensitivity to reciprocal phenomena.
Dark Port Sensitivity
Under certain circumstances, the sensitivity to reciprocal effects in the dark port
will be small enough that it may be safely neglected and the dark port signal used for
detection of nonreciprocal phenomena. As evident by Figure 27, a zoomed-in version
of Figure 26, in the region of small detunings the dark port remains largely insensitive
to the reciprocal perturbations. For larger detunings, the effect is large enough to
influence the behavior of the output intensity. In practice, this sets a limit on the
sensing capabilities of the dark port. Reciprocal detunings must be kept small enough
49
so as to be insignificant compared to nonreciprocal detunings in order to definitively
attribute the response of the dark port to a nonreciprocal perturbation.
To further quantify the performance of the resonantly enhanced Sagnac
interferometer we analyze its sensitivity. We define the sensitivity in the dark port
as S = ∂I−− , where ∆n is the magnitude of either the reciprocal or nonreciprocal∂∆n
detuning. The sensitivity is plotted against ∆n to illustrate how the sensitivity varies
with the strength of the perturbation. While one of the detunings is varied, the other
is held fixed at a constant detuning of |∆n| = 10−7 to exemplify the system behavior
in the presence of other perturbations. Figure 28 shows both the nonreciprocal and
reciprocal sensitivities in the dark output port as a function of the magnitude of the
respective detuning. The incidence angle in the prism and thickness of the metal film
are held fixed at the resonant angle and the critical SPR thickness, respectively. As
seen in the figure, the sensitivity to nonreciprocal effects is orders of magnitude larger
than the sensitivity to reciprocal effects in the dark port. This significant difference in
sensitivity persists over a wide range of detunings making the dark port an effective
detector of nonreciprocal phenomena.
As compared with other surface plasmon sensors, this eases detection of
nonreciprocal effects. While these effects are capable of detection in an intensity-
based, or even a Mach-Zehnder configuration phase-based scheme, these techniques
are more easily contaminated by reciprocal noise. For example, in detection of a
dielectric flow, the drag effect would induce a shift in the intensity and phase upon
reflection. However, so would a temperature drift, necessitating precise temperature
control in these experiments. In the resonantly enhanced Sagnac interferometer not
only is the sensitivity to nonreciprocal perturbation enhanced, but small reciprocal
perturbations are also, to a large extent, filtered out. This allows for a relaxation of
50
FIGURE 28. Nonreciprocal and reciprocal sensitivity in the dark port.
The reciprocal sensitivity is plotted for reciprocal ∆n varied and a constant
nonreciprocal detuning of |∆n| = 10−7. For the nonreciprocal sensitivity curve, ∆n
is the magnitude of the nonreciprocal detuning and the reciprocal detuning is held
fixed at 10−7.
constraints such as precise temperature control typically required in such experiments.
These same principles can be applied to any nonreciprocal perturbation that may also
be accompanied by reciprocal phenomena.
Persistence of Results
In experiments designed to detect both types of perturbations simultaneously,
the strength of both types of perturbations may exceed the |∆n| = 10−7 value that
was used to hold one type of perturbation constant in the previous calculation of
the sensitivity. The question then becomes, how much could these constraints be
relaxed while maintaining a large gap between the sensitivity to nonreciprocal and
reciprocal phenomena. To explore the extent of the nonreciprocal dominated regime,
we modify the calculation described above by increasing the magnitude of the ∆n
held constant. For example, Figure 29a displays reciprocal sensitivity with |∆n|
nonreciprocal held fixed at 10−6 and nonreciprocal sensitivity with ∆n reciprocal also
51
held fixed at 10−6. This is repeated in Figure 29b but with the parameter adjusted
down to 10−5. Comparison of these plots with that of Figure 28 shows that the trace of
the nonreciprocal sensitivity exhibits little change. However, the reciprocal sensitivity
grows as the magnitude of an also present nonreciprocal perturbation increases. This
significantly lessens the gap between the two on the left side of the plot, but the
discrepancy persists for higher detunings. As it turns out, the leftmost side of the
plot contains a bit of an incongruity; the red trace shows the reciprocal sensitivity
when a nonreciprocal perturbation of magnitude |∆n| = 10−5 is present. In the
presence of a nonreciprocal perturbation of this magnitude, it is no so instructive to
focus on the nonreciprocal sensitivity to a perturbation of magnitude |∆n| = 10−9.
Instead, it is more illuminating to look further along the x-axis at ∆n = 10−5 where
we see a discrepancy between the sensitivities of three orders of magnitude. This
illustrates the effectiveness of this device as a simultaneous detector when both effects
are present with larger strength in addition to the previously discussed potential of
enhanced nonreciprocal phenomena in the regime closer to pushing the lower limits
of perturbation detection.
(a) (b)
FIGURE 29. Dark port sensitivity for simultaneous detection.
(a) Same as Figure 28 except that the |∆n| not varied is held fixed at 10−6. (b)
Same again, but now with the |∆n| not varied is held fixed at 10−5.
52
We also consider the impact on sensitivity of variation in the silver film thickness.
Figure 30 shows the sensitivity to reciprocal and nonreciprocal phenomena for two
supercritical thicknesses. These films are 3 nm and 6 nm thicker than the critical
thickness film which reaches zero at the minimum of the resonance. As with Figure
28, the type of perturbation held fixed is done so at a constant detuning of |∆n| =
10−7. We observe very minimal deviation from the sensitivity at critical thickness
suggesting that the dark port sensitivity is quite forgiving with respect to deposited
film thickness.
(a) (b)
FIGURE 30. Dark port sensitivity for supercritical thickness.
(a) The reciprocal and nonreciprocal sensitivity are plotted for a silver film
thickness 3 nm greater than the critical thickness. (b) The reciprocal and
nonreciprocal sensitivity are plotted for a silver film thickness 6 nm greater than the
critical thickness. In both cases, very little deviation from the sensitivity at critical
thickness is observed.
Bright Port Intensity
In a balanced interferometer, the bright port exhibits perfectly constructive
interference. This remains the case for the resonantly enhanced system. Even still, the
bright port exhibits low signal intensity when the system is aligned at the resonant
angle of SPR. This is due to the reflectance from the SPR element being at its
minimum, specifically zero in the case of the critical thickness film. This serves as
53
our starting point in the bright port; we proceed to introduce perturbations to the
adjacent dielectric and investigate the system response.
We investigate the reciprocal sensitivity in the bright port by plotting I+ from
Equation 4.1 as a function of the reciprocal detuning. Because no phase difference is
introduced between the two counter-propagating paths in a reciprocal detuning, the
bright output port retains its constructive interference. However, the introduction of
a detuning causes a shift away from the reflectance minimum, increasing the intensity
in the counter-propagating paths and thus the intensity at the bright output port.
The intensity at the output as a function of the reciprocal detuning is shown in
Figure 31. Here we see that the introduction of a reciprocal perturbation results in
an interferometer output that is modulated by the resonance lineshape. Again, this
result is unique to the resonantly enhanced system. In the absence of the intensity
modulation provided by the resonance, for example at an off-resonant angle with flat
reflectivity response, the intensity at the bright port would be unaffected.
FIGURE 31. Bright port response to reciprocal perturbations.
Normalized intensity output at the bright port, I+, increases as perturbations shift
the system away from the resonance minimum. The film thickness is varied as
denoted in the legend.
54
Having looked at the bright port response to reciprocal detunings, we move
on to investigate the response to nonreciprocal detunings. We model nonreciprocal
sensing in the bright port by computing I+ as a function of nonreciprocal detuning.
Similarly to the dark port, nonreciprocal detuning increases the reflectance from the
SPR element as we detune from the minimum of the resonance. In addition, the
phase difference introduced by the nonreciprocal perturbation leads to a change in the
nature of the interference at the output. In the bright port, the interference switches
from constructive to destructive and then back to constructive. Because of this,
the bright port also exhibits nonmonotonic behavior as a function of nonreciprocal
detuning. Unlike the response of dark port, we do not see a resonant enhancement
of the interferometer output in the bright port. This is due to the fact that the π
change in phase across resonance shifts the output from constructive interference to
nearly complete destructive interference. The result is a near-perfect cancellation of
terms in the bright port, thus minimizing the effect of the intensity increase away
from resonance.
For a fuller picture, it is instructive to consider the output response in the
presence of both type of effects. Figure 32 shows the normalized intensity at the bright
port as a function of reciprocal and nonreciprocal detunings. The metal film thickness
and incident angle are held fixed at the critical and resonant values, respectively. The
overall trend of this density plot looks similar to that of the dark port but with a
valley running diagonally across the plot rather than a ridge. The key difference
comes when we focus in on smaller detuning and begin to analyze the sensitivity.
55
FIGURE 32. Bright port perturbation response.
Normalized output intensity, indicated by the color bar, as a function of both
reciprocal and nonreciprocal detunings.
Bright Port Sensitivity
For small nonreciprocal detunings the bright port signal is dominated by the
reciprocal response. Figure 33 displays the response for small detunings where the
reciprocal response dominates, as opposed to much of Fig 32 where the effects are
comparable in magnitude. This sets a constraint on the sensing capabilities of the
bright port. As long as the nonreciprocal signal is sufficiently small, as is often the
case particularly when pushing the lower limits of detection, the bright port can be
used as a monitor of the reciprocal drift. To illustrate this we perform a more rigorous
analysis of the sensitivity.
Using S = ∂I++ as our definition for sensitivity in the bright port, we perform∂∆n
similar calculations as was done for the dark port. Figure 34 shows the sensitivity to
reciprocal and nonreciprocal effects in the bright port. While one detuning is varied,
56
FIGURE 33. Bright port response to small perturbations.
A zoomed-in view of Figure 32. The near-horizontal contours suggest a strong
response to reciprocal perturbations and, by comparison, a weak response to
nonreciprocal phenomena.
the detuning of the alternate effect is held fixed at a constant detuning of |∆n| =
10−7. In this port we observe a reciprocal sensitivity orders of magnitude larger
than the nonreciprocal sensitivity. At a nonreciprocal detuning near 10−5 there is a
transition in the bright port nonreciprocal sensitivity from one slope to another and
the sensitivities begin to converge. This magnitude detuning is approximately where
the nonreciprocal detuning increases the phase difference at the interferometer output
sufficiently, such that the bright port moves from destructive back to constructive
interference. However, the phenomena that is of interest for sensing applications
will occur in the region of small ∆n where the sensitivity difference is roughly five
orders of magnitude. This enables clear detection of reciprocal effects using the bright
port. In combination with the results for the dark port, this opens new avenues of
57
simultaneous detection utilizing SPR. While current detection schemes are capable of
detection of one or the other type of perturbation, a resonantly enhanced Sagnac is
capable of concurrent monitoring of both. For example, one could monitor both the
concentration and magneto-optic properties of a magnetic fluid in the vicinity of the
metal film.
FIGURE 34. Reciprocal and nonreciprocal sensitivity in the bright port.
The reciprocal sensitivity is plotted for reciprocal ∆n varied and a constant
nonreciprocal detuning of |∆n| = 10−7. For the nonreciprocal sensitivity curve, ∆n
is the magnitude of the nonreciprocal detuning and the reciprocal detuning is held
fixed at 10−7.
Persistence of Results
In simultaneous detection experiments, the strength of both types of
perturbations may exceed the |∆n| = 10−7 value that was used to hold one type
of perturbation constant in the previous calculation of the sensitivity. To explore the
persistence of this large gap in sensitivity, we repeat the calculation described above,
but while increasing the magnitude of the ∆n held constant. Figure 35a shows the
bright port sensitivity with this parameter set to 10−6 and Figure 35b shows the bright
port sensitivity with this parameter set to 10−5. By comparison of these plots with
58
Figure 34, we observe the same qualitative features but note that the gap between
the reciprocal and nonreciprocal sensitivity does decrease slightly. For reciprocal
sensitivity with |∆n| nonreciprocal held fixed at 10−5 and nonreciprocal sensitivity
with ∆n reciprocal also held fixed at 10−5, the sensitivity difference has shrunk to
approximately four orders of magnitude. However, the fact that a sizable gap persists
between the two suggests that, for simultaneous detection, even while the strength
of a nonreciprocal perturbation increases, the reciprocal sensitivity dominates the
response of the bright port.
(a) (b)
FIGURE 35. Bright port sensitivity for simultaneous detection.
(a) Same as Figure 34 except that the |∆n| not varied is held fixed at 10−6. (b)
Same again, but now with the |∆n| not varied is held fixed at 10−5.
A second modification considered is that of the thickness of the silver film.
In the prior calculations the film thickness is assumed to be the critical thickness
which achieves identically zero reflectance at the resonance minimum. In practice,
experimental limitations will preclude the realization of a film at exactly this
thickness. To account for this, we explore the sensitivity of the bright port for
films other than the critical thickness. Figure 36 shows both the reciprocal and
nonreciprocal sensitivity for film thicknesses 3 nm greater than critical in panel (a)
and 6 nm greater in panel (b). As in the case of the critical film thickness, the
59
discrepancy starts out larger for smaller detunings and decreases as the strength of
the perturbation increases. For these supercritical thicknesses, the gap closes sooner,
starting at five orders of magnitude initially and dropping to one order of magnitude
difference at around ∆n = 10−5. There still exists a region over a range of parameter
values in which the response to reciprocal phenomena is multiple orders of magnitude
greater. However, if greater precision of film thickness is necessary in order to achieve
sufficient sensitivity contrast, other deposition techniques such as epitaxial growth of
single crystalline films may be considered [82, 83].
(a) (b)
FIGURE 36. Bright port sensitivity for supercritical thickness.
(a) The reciprocal and nonreciprocal sensitivity are plotted for a silver film
thickness 3 nm greater than the critical thickness. (b) The reciprocal and
nonreciprocal sensitivity are plotted for a silver film thickness 6 nm greater than the
critical thickness. In both cases, a similar disparity exists initially, but lessens more
quickly for larger perturbations.
Angular Detuning
In the previous interferometer output results, the angle of incidence onto the
silver film from within the prism is taken to be precisely at the reflectance minimum of
SPR. However, in some instances a choice could be made to detune this angle slightly
above or below the resonance angle. Although the slope in the phase profile decreases
as we shift away from the resonance minimum, the slope of the reflected intensity
60
increases up until reaching the inflection point. Furthermore, the reflected intensity
itself will increase with this angular detuning leading to a larger normalized intensity
in the bright port. A particular angle of interest may be that which maximizes the
slope of the intensity. After computing this angle, we may use it in a similar fashion
as above to construct density plots which display the bright and dark port response
to both reciprocal and nonreciprocal perturbations. This computation confirms a
nonzero bright port signal even in the balanced interferometer configuration, as can
be seen by inspection of the scale bar in Figure 37. Beyond this distinction, the
qualitative behavior of this figure remains much the same as that of the system
without angular detuning. The nearly horizontal contours suggest a strong response
to the introduction of reciprocal perturbations, and by comparison very little response
to the presence of nonreciprocal perturbations continuing out to ∆n of nearly 10−3.
FIGURE 37. Angle detuned bright port intensity.
Normalized output intensity is plotted as a function of both reciprocal and
nonreciprocal perturbations. The incidence angle is fixed at the angle which
maximizes the slope in the surface plasmon reflectance.
The dark port, on the other hand, maintains perfectly destructive interference
under ideal conditions such that the intensity at this output remains zero for the
61
balanced interferometer, even with the angular detuning. Figure 38 displays the
normalized intensity as a function of both reciprocal and nonreciprocal perturbations.
The intensity contours in this plot are nearly vertical, representative of a response
dominated largely by the introduction of nonreciprocal perturbations. This holds for
reciprocal perturbations in ∆n on the order of roughly 10−4. As a point of reference,
this corresponds to a temperature change of approximately 1 ◦C in liquids pervasive
in scientific research such as water and ethanol.
FIGURE 38. Angle detuned dark port intensity.
Normalized output intensity is plotted as a function of both reciprocal and
nonreciprocal perturbations. The incidence angle is fixed at the angle which
maximizes the slope in the surface plasmon reflectance.
The foremost conclusion to be drawn from this discussion is that the results
obtained above are not exclusive to the resonance angle itself. The propensity toward
detection of reciprocal effects in the bright port and nonreciprocal perturbations in
the dark port holds true for additional angles of incidence away from the resonance
minimum. The nonzero bright port signal in the balanced interferometer represents
a departure from the earlier conclusions, nonetheless the residuum of the behavior
remains qualitatively the same.
62
Summary
The overall response of both the dark and bright ports of the interferometer
is succinctly summarized, with the response attributed to either intensity or phase
perturbations, in Table 2. The outputs displayed here were generated from the
interference equations, irrespective of the surface plasmon or any other particular
resonance. The equations were modified by introducing a small change, δ, directly to
each of the parameters that represent a round trip intensity transmission coefficient
or total accumulated phase for both the clockwise and counter-clockwise paths,
rather than indirectly through a refractive index shift, ∆n, as before. A reciprocal
intensity change, ∆I, was introduced by adding nonzero δ of the same sign and
magnitude to the round trip intensity transmission coefficient of the two paths; a
nonreciprocal intensity change by changing the sign of δ for one of the paths. Similarly,
a reciprocal phase change, ∆φ, was introduced by adding nonzero δ of the same sign
and magnitude to the total accumulated phase in both paths; a nonreciprocal phase
change by adding δ of the same magnitude but opposite sign. In reality, a perturbation
is likely to affect both the intensity and the phase of the system; however, here the
two were considered independently.
TABLE 2. Interferometer response to reciprocal and nonreciprocal effects.
Reciprocal Nonreciprocal
I−: constant I−: increase
∆I I+: increase I+: decrease
I− + I+: increase I− + I+: constant
I−: constant I−: increase
∆φ I+: constant I+: decrease
I− + I+: constant I− + I+: constant
63
In this scenario, the dark port signal remains constant in the presence of any
reciprocal perturbation and responds solely to nonreciprocal phenomena. Though
the dark port displays no reciprocal response, the bright port enables monitoring
of reciprocal intensity perturbations. And while it also responds to nonreciprocal
perturbations, the reciprocal response in the bright port overpowers the influence of
nonreciprocal phenomena.
For most any perturbation, Table 2 shows that the sum of the two interferometer
intensities, I−+ I+, remains constant. The exception is a reciprocal intensity change,
where movement along the resonance has altered the overall circulating intensity
within the interferometer. Whether this leads to an increase or decrease in the bright
port signal, and hence the sum, depends on the type of resonance and how it is
employed. For example, when using SPR aligned at the reflectance minimum, a
reciprocal ∆I will couple less strongly to the surface mode and lead to an increase
in signal, as is displayed in the table. On the other hand, if the system maximizes
transmission through a medium exhibiting electromagnetically induced transparency
(EIT), small perturbations could instead cause a decrease of signal.
In this chapter, we have developed a theoretical model which facilitates modeling
the introduction of perturbations, both reciprocal and nonreciprocal, to a resonantly
enhanced ring interferometer and computing the output response. The results of
this model point toward a dark port signal which is highly sensitive to nonreciprocal
phenomena and comparatively insensitive to reciprocal perturbations. Conversely,
the bright port signal is found to be dominated by sensitivity to reciprocal effects.
These results are robust to variation in the film thickness or resonance angle and to
the presence of competing effects. The end result is a device which appears well-suited
for the simultaneous detection of both nonreciprocal and reciprocal phenomena.
64
CHAPTER V
EXPERIMENTAL IMPLEMENTATION AND RESULTS
Overview
This chapter describes the construction of an experimental apparatus and
experimental results obtained from that device in an effort to support the results of
the theoretical modeling described in the prior chapter. We begin with a discussion
of the optical configuration, noting the components used and relaying the methods
for alignment of the interferometer. We then describe the experimental means by
which we introduce the perturbations to the system and the components needed
to carry out this procedure. We next present experimental results for detection of
reciprocal ohmic heating in the bright port. The data qualitatively matches the model
of the previous chapter but falls short quantitatively due to some logistical constraints.
After that, we present experimental results toward detection of nonreciprocal plasmon
drag in the dark port. A signal is observed with some characteristics matching
those our model but other features which appear in conflict with our expectation;
a full understanding of these observations is left unresolved. Finally, we discuss the
prospects of simultaneous detection and summarize the results of this chapter.
Configuration and Alignment
Incorporating the surface plasmon resonance system into the Sagnac
interferometer introduces a few additional experimental considerations to take
into account when building the resonantly enhanced interferometer. One such
consideration pertains to the angle of incidence of the light beam upon the prism. In a
65
typical Sagnac, the angles the beam makes with the reflecting elements are not crucial;
the important alignment issue is simply that the paths of the counterpropagating
beams overlap. However, due to the narrow angular width of the resonance and
the desire to probe a particular location along that resonance, the incident angle of
the laser beam upon the prism must be set precisely in this configuration. In the
initial alignment of the system, two mirrors and a right angle prism are configured
according to the schematic shown in Figure 39. The initial alignment is relatively
straightforward to implement and is aided by use of the rectangular grid of the optical
table and the back reflections from the lateral faces of the prism for alignment. The
motorized rotation stage holding the prism is zeroed while the back reflection from
the lateral face is aligned to pass back through the iris from which it came. The
second mirror is then positioned to close the loop, with final adjustments made by
projecting the dark port output onto a blank sheet. A micrometer dial is used to
precisely control the tilt of the mirror until maximum cancellation of the signal in
achieved in the dark port. This will be limited by the imbalance of the beamsplitter
split ratio, as discussed previously. The end result of this initial alignment procedure
is 45 degree incidence on the back of the prism from both directions; the configuration
must then be modified to achieve the desired angle dictated by the location of the
surface plasmon resonance.
In the first iteration of the experimental setup constructed in the lab, the general
configuration chosen was a symmetric one, as shown in the schematic of Figure
40, with the two mirrors mounted on rotation stages and the prism configuration
affixed to a translation stage oriented diagonally with respect to the grid of the optics
table. It is symmetric in the sense that the incidence angle upon both mirrors is
identical and the counter propagating beams travel the same distance before exciting
66
FIGURE 39. Initial alignment of the interferometer
The initial angles are set by zeroing the rotation stage below the prism while the
back reflection from the lateral face of the prism travels directly back along its
incident path. The second mirror is positioned to close the loop with a micrometer
used for fine tuning to achieve the lowest possible signal in the dark port.
propagating plasmons along the metal and dielectric interface. The alignment of this
configuration was executed first by rotating the two side mirrors to the same desired
angle followed by a translation of the prism configuration such that the hypotenuse
is aligned with the point where the two beams intersect. The intensity of the beams
at their intersection is large enough that the beam spots are visually discernable on
the back of the prism, facilitating the alignment procedure by allowing for initial
alignment by eye and fine tuning of the alignment by inspection of the interferometer
outputs. In practice, this procedure proved to be tedious in that it often required an
iterative process to achieve alignment at the desired angle and also did not allow for
easy characterization of samples to determine the thickness and dielectric properties
of the silver films as deposited.
67
FIGURE 40. A symmetric configuration for alignment at the desired angle.
This configuration is aligned first by equal rotation of the two mirrors followed by
translation of the SPR prism mount to close the beam paths. While it maintains an
inherent symmetry of the system, it does not facilitate seamless optical
characterization of silver films.
Instead, the optical configuration was modified to achieve the form illustrated in
Figure 41. In this conformation, one of the mirrors remains fixed, both in location
and orientation, with the beam reflected from this mirror directed toward the prism
configuration. The prism is mounted on a motorized rotation stage which can be
adjusted to achieve the desired angle of incidence within the prism without disturbing
the position of this first mirror. The motor angle is set to provide the desired incident
angle using the relation between the two described in an earlier chapter. Once
positioned appropriately, the beam reflected from the SPR configuration will cross
paths with the portion of the input beam which is initially transmitted through the
beamsplitter. The second mirror is then translated such that the plane of the surface
of the mirror contains this intersection point of the beams, and the mirror is rotated
such that the counterpropagating paths of the interferometer overlap entirely. This
68
gives rise to a certain asymmetry between the paths, as one beam travels a shorter
path to reach the interferometer; however, this difference in length is compensated on
the return trip to the beamsplitter, thus, maintaining equivalent phase accumulation
in the paths and not disrupting the symmetry of the interferometer as a whole.
Additionally, the difference in travel time from the beamsplitter to the prism for one
path compared to the other is less than one nanosecond, a timescale much smaller
than that of any of the perturbations to be studied in this dissertation.
FIGURE 41. An asymmetric configuration for alignment at the desired angle.
This configuration fixes entirely the position of one of the mirrors and includes a
rotation stage for the prism to achieve the desired angle of incidence. The second
mirror is then positioned to close the interferometer loop. While not symmetric in
the same manner as the prior system, it does maintain equivalent optical path
lengths while also facilitating seamless optical characterization of silver films.
One potential drawback to the alignment of this configuration is that the beam
reflected from the prism, if aligned to the minimum of the resonance, can be difficult to
observe visually due to the largely diminished intensity. This can affect the alignment
of the second mirror, though the problem can be mitigated through exploration
of other avenues, such as the interference pattern at the output, to perform the
fine tuning of the alignment. On the other hand, an overwhelming success of this
69
configuration is the simplicity of achieving the desired angle with one simple rotation
of the prism and, subsequently, the ease with which it allows for characterization of
samples.
When characterizing a sample, the beam transmitted by the 50/50 beamsplitter
is blocked so that only a single beam is incident upon the prism configuration.
The motorized rotation stage on which it is mounted can be programmed through
LabVIEW to methodically vary the incidence angle through a specified range. The
reflected beam is diverted toward a silicon photodetector by placing an additional
mirror within the interferometer beam path. The diverted beam is further redirected
by a combination of two lenses positioned precisely to enable an angular scan without
need for repositioning the detector. The first of two lenses is placed a cumulative
distance of a focal length away from the spot on the back of the prism from which
the reflected beam originates. This positioning generates a beam normal to the plane
of the lens upon exit from the lens. A second lens is then positioned parallel to the
first and a focal length away from the detector so that the beam incident on this lens
is refracted toward the detector. The distance between the lenses is inconsequential
because the beam traverses this segment normal to the plane of both lenses. This
setup enables an angular scan over a range of nearly ten degrees, limited primarily
by the size of the optical elements used.
This interferometer configuration is the pivotal part of a larger optical setup
shown schematically in Figure 42, and as configured on the optical table in Figure 43.
The optical source used is a 5 mW HeNe laser (JDS Uniphase model 1125/P). The
output beam from this source first passes through an optical isolator (OFR IO-3D-
633-VLP), which eliminates optical feedback and maintains the stability of the source.
After redirection of the beam with 1” mirrors, a spatial filter (Newport Model 900)
70
is used to eliminate high k spatial frequency components. The Airy pattern output
produced by the pinhole of the spatial filter is filtered by an iris with aperture set at
approximately 1 cm, large enough to select the central bright spot. This cleaned-up
beam is then telescoped down to roughly 2 mm beam diameter using a configuration
of Newport AR.14 anti-reflection coated lenses. Just prior to the interferometer is a
linear polarizer (Newport 10GT04AR.14) to reinforce the p-polarization of the beam
and a beamsplitter with roughly 15:85 (R:T) split ratio for HeNe wavelength (Newport
10RQ00UB.2). This beamsplitter is used both as a reference to monitor the input
beam for intensity fluctuations and to divert the bright port signal for detection upon
its exit from the interferometer. After this beamsplitter is a second, non-polarizing,
laser line, 50/50 beamsplitter (Newport 10B20NP.25) used to split the beam into
two components which then begin their trips around the interferometer. These two
beams reflect off silvered 2” mirrors and cross at the hypotenuse of a BK7 right
angle prism (Thorlabs PS911), incident at an angle beyond the critical angle for total
internal reflection of a glass and air interface. The prism is mounted on a motorized
rotation stage (Newport URS75CC) driven by motor controller (Newport ESP300).
The beam is modulated at 911 Hz by optical chopper (Thorlabs MC1000). The output
and reference signals are collected with a silicon photodiode (Thorlabs DET36A) or
photomultiplier tube (Hamamatsu R955), depending on the level of intensity, each of
which is connected to a Lock-In Amplifier (SR810) with a 50 Ω terminating resistor
for impedance matching. All motor control and data collection are integrated with
the LabVIEW software so the experiments may be run easily through the laboratory
computer.
71
FIGURE 42. Schematic of full experimental configuration.
Perturbation Implementation
With the optical configuration assembled, the next step toward experimentally
investigating the findings of the previous chapter is to choose a method through
which the reciprocal and nonreciprocal perturbations may be enacted upon the
physical system. For this work, a choice was made to implement these perturbations
by application of an electrical current across the silver film which supports the
propagation of SPPs. This decision was made due to the ability of the current to
simultaneously produce both types of perturbations, and also because of the minimal
72
FIGURE 43. Experimental configuration as constructed in the laboratory.
extra equipment and cost associated with pursuit of this route. The presence of
a current through the silver film leads to ohmic heating, which will then drive a
temperature change in the layered film system. This, in turn, will generate an optical
response at the interferometer output owing to the thermo-optic coefficient of the
relevant material, which describes the refractive index response of the material to
a change in temperature. As this will impart equivalent phase accumulation on
the interferometer paths, independent of the direction of propagation, this thermal
induced heating will manifest as a reciprocal perturbation within the system.
In addition to generating heat, the electric current in the film is also expected
to give rise to a nonreciprocal dragging effect. The presence of the current breaks a
symmetry of the system and sets up a preferred direction of electron motion within
73
the silver film. The SPP propagation direction will then be parallel to the direction
of the electric current for one path around the interferometer and antiparallel for the
counterpropagating path. This is expected to produce a relativistic dragging effect
referred to as plasmon drag, which acts somewhat analogous to that of the Fresnel-
Fizeau drag effect experienced by a beam of light traveling through a moving dielectric
[84]. This perturbation is presumed to manifest as a difference in phase accumulation
of the two beams around the path and, therefore, result in an interferometer response
as predicted in the previous chapter for a nonreciprocal perturbation.
In practice, this electrical current is implemented by affixing short wire segments
to both ends of a deposited metal film with a two-part, high conductivity, silver
epoxy. The newly attached leads add only a small bit of additional resistance; the
film with leads is measured to have resistance on the order of 10 ohms, with some
variability owing to slight differences in the actual dimensions of the deposited silver
film. The electric current is supplied to the film by connecting the leads to either a
power supply or a function generator, with the latter capable of supplying a variety
of pulse shapes. If desired, a current limiting resistor can also be added to the system
to allow for further control of the current. The magnitude of the current passing
through the system can be measured by insertion of an ammeter into the electronic
circuit.
After a two hour cure time for the silver epoxy, the sample may be put to use
within the optical system. The glass substrate, with vapor deposited silver film and
cold soldered wire leads, is affixed to the back of a prism with the metal film facing
outward, away from the prism. A refractive index matching liquid is used to eliminate
any refractive air gaps between the prism and glass slide. This prism, now with metal
film attached is placed within a prism mount fabricated by a prior graduate student,
74
Lawrence Davis, schematically shown in Figure 44 [44]. The micro-volume enclosed
dielectric cavity and close proximity of the thermometer to the dielectric are the
desired features motivating the use of this apparatus; however, a few modifications
are made for this experiment. In order to prevent electrical contact between the
silver film and brass mass, a layer of plastic cling wrap is situated so as to provide
physical separation between the two. Also, rather than filling the cavity with a liquid,
it is comprised of ambient, room temperature air. The enclosed system provides a
stable environment for the air and limits the formation of an oxide layer on the silver
film. In addition, the heater resistor is unused as the heat for this experiment will be
generated by a current in the silver film itself.
FIGURE 44. Diagram of the SPR prism mount used by L. J. Davis.
This image was taken from reference [44]. The setup was modified for this
experiment by filling the reservoir with air rather than liquid, removal of the heater
resistor, and use of plastic cling wrap in place of metal foil.
The prism, now in the prism holder, can then be mounted on the rotation stage
and aligned within the optical system as described above. Using the configuration for
sample characterization, an angular reflectance measurement is performed to observe
the surface plasmon resonance and a least squares fit to the optical data is executed
to obtain the sample specific material parameters. From there, the motor angle can
be set to probe the desired location along the resonance, whether that be at the
75
resonance minimum or some other off-resonant angle. The mirror redirecting the
beam for characterization is then removed and the remaining interferometer mirror
positioned to align the interferometer, first but translating it so that the counter
propagating beams meet at the mirror surface and then by adjusting the angle so
that the paths fully overlap. At this point the system is set up and ready to perform
experiments.
Reciprocal Ohmic Heating in the Bright Port
We begin our discussion of experimental results obtained from the resonantly
enhanced interferometer with optical data collected from the bright port. While
applying a current in the silver film, the temperature of the film and its surroundings
increases in response to ohmic heating. Then, according to the thermo-optic
coefficient which describes the material, the refractive index shifts as a result of the
increased temperature. As discussed in the previous chapter, this will manifest most
strongly as a response in the bright port signal. In order to verify that a response
in the output signal can be traced to temperature increase of ohmic heating, an
independent measure of temperature is collected by placing a thermometer in the
location as shown in Figure 44, 250 µm from the back of the air-filled cavity. In order
to more strongly observe the response of the interferometer to temperature change,
the angle of incidence within the prism is set to the angle of maximum slope on either
side of the resonance. For the particular sample under investigation those angles are
found to be 42.92◦ and 43.12◦. The locations of these angles with respect to the
resonance are indicated in Figure 45. In order to heat the sample quickly and reduce
time of the experiment, a potential difference of five volts is applied across the metal
film resulting in a direct current of approximately 175 mA. In future applications of
76
this device, temperature fluctuations may be a side effect of some other phenomena
under investigation and vary more slowly than the time scales resulting from this
particular procedure. However, here we aim for a proof of principle result to show
that the bright port responds to change in temperature and, thus, may be used
to monitor temperature change, or other effects similar in nature, while performing
interferometric measurements.
FIGURE 45. Angles of incidence used in bright port data collection.
The black trace represents the normalized angular reflectance data collected from
the sample. The blue dot indicates 42.92◦ while the green dot corresponds to 43.12◦.
These correspond to approximately the angle of maximum slope on either side of
the resonance curve.
The data collection for this experiment is automated and run by LabVIEW
software. When the potential difference is applied across the silver film, the time,
temperature, laser reference signal, and bright port signal are recorded on the
computer at regular time intervals as the system heats up. The recorded time is,
for the most part, inconsequential for this experiment as we are primarily concerned
with the temperature at that particular moment and not how long it took to reach
that temperature. The recorded time stamps will serve a greater purpose in the
experiments to come, particularly when probing the nonreciprocal perturbations.
The laser reference signal is collected to be used in the normalization of the bright
77
port signal, accounting for any fluctuations in the intensity of the input beam. The
normalized bright port data is sorted by temperature with bin sizes of 0.1◦C, the
nominal resolution of the thermometer used. Data are then averaged to find the
mean bright port signal at a given temperature and the standard error is computed
for that set of data points. This data is plotted for the incidence angle of 42.92◦ in
Figure 46a. A linear fit to this data is performed with the resulting equation displayed
in the plot legend. This data can then be compared with the theoretical model using
the results from the previous chapter.
(a) (b)
FIGURE 46. Bright port response to reciprocal ohmic heating for θ =42.92◦.
(a) Normalized experimental optical data with error bars to indicate the standard
error. The dotted line represents a linear fit to the data with the equation shown in
the legend. (b) Theoretical model of the same scenario using a temperature
dependent refractive index of air.
This experimental process is modeled theoretically using the normalized output
intensity equations presented earlier. The surface plasmon reflectance term is
computed using the sample-specific material parameters for that particular silver
film obtained from a least squares fit to the angular reflectance data of Figure 45. In
this model, we consider only the thermo-optic behavior of the dielectric, air in this
case. The temperature dependent refractive index of the air is calculated based on an
78
updated Edlén equation from Birch and Downs[85]. The result of this model, plotted
across the range of temperatures observed experimentally, is displayed in Figure 46b.
By comparison of experiment with the theory, we first observe that the
normalized values of the theoretical model differ from the experimental data by
approximately 10%. To account for this discrepancy, we consider a few potential
contributing factors. First, while our best efforts were made toward alignment of the
beams it is feasible that there exists some residual misalignment. This would result in
less than perfectly constructive interference, even beyond the limit set by beamsplitter
imbalance, and lower the values of the experimental data, as is observed. Secondly,
this discrepancy may also be in part due to the methods used for normalization
of the data. The signal was normalized to the bright port measurement with the
interferometer aligned to an angle just beyond the angle of total internal reflection.
This accounts for losses in the system but assumes they are constant with angle;
angular dependencies of loss mechanisms such as reflections from optical components
will introduce a small error to the normalized bright port signal when aligned at
other angles. In addition, the refractive index values used for the dielectric material
may not accurately describe the air used in this experiment owing to reasons such
as differences in composition, for one. Despite the discrepancy in the value of the
normalized output, it is instructive to compare the slopes, as this gives an indication
of the output response to temperature change.
Visually, we observe that the sign of the computed slope matches that of the
experimental data. In comparing the magnitudes of the slopes we find that the
theoretical slopes is smaller than the experimental by just more than a factor of ten:
0.00018 for the theoretical and 0.0026 for the experimental. Another way of stating
this is that, based on these values for 42.92◦ incidence, a change in the normalized
79
reflectance of 0.001 corresponds to an independently measured temperature change
of about 0.4◦C. Conversely, the theoretical model predicts that, for the material
parameters used, the same change in reflectance requires a temperature change of
5.5◦C.
This same discrepancy exists in when the interferometer is aligned on the higher
angle side of the resonance, at an angle of 43.12◦. The data collected for this angle
is shown in Figure 47a along with a linear fit to the data. The sign of the slope in
the temperature data is opposite that of the previous data set due to the change in
sign of the slope across the resonance itself. The theoretically modeled values for this
angle are shown in Figure 47b. Again we observe that normalized intensity values are
greater in the theoretical depiction. Comparing the slopes, we again observe an order
of magnitude difference, 0.00014 for the theoretical and 0.0021 for the experimental,
predicting a larger temperature change than is experimentally measured.
(a) (b)
FIGURE 47. Bright port response to reciprocal ohmic heating for θ =43.12◦.
(a) Normalized experimental optical data with error bars to indicate the standard
error. The dotted line represents a linear fit to the data with the equation shown in
the legend. (b) Theoretical model of the same scenario using a temperature
dependent refractive index of air.
However, recall that the SPR sensing techniques probe the immediate vicinity
of the interface, with SPP field penetration depths on the order of hundreds of
80
nanometers. It is possible that the actual temperature change experienced by
the few hundred nanometers of material in the immediate vicinity of the interface
is closer to the higher, theoretically predicted temperature than the lower value
measured experimentally. In any case, we expect the measured temperature to be an
underestimate of the actual temperature change experienced at the interface due to
the difference in location of the temperature measurement as performed by the optical
technique and independently by the thermometer. The thermometer is positioned
.25 mm from the base of a 0.4 mm dielectric reservoir as indicated in Fig. 44. This
amounts to a 0.65 mm total displacement from the source of the heat, the silver film.
At a minimum, we can use this measured change in temperature to set a lower bound.
Toward further resolving this discrepancy, one avenue of investigation would be to
perform a transient heat transfer analysis and solve the temperature distribution in
the system providing a relation in temperature between the two locations of relevance.
A second consideration is that the model accounts only for a change in the refractive
index of the dielectric. A thermo-optic analysis of the remaining constituents, such
as the silver film, could be added to the model. That being said, this result does act
in support of the general conclusion that the bright port of the resonantly enhanced
Sagnac exhibits sensitivity to reciprocal perturbations. It is feasible that this output
could be calibrated and optimized for use as a monitor of reciprocal phenomena,
which could be done in conjunction with detection at the dark port, the next area of
discussion.
Nonreciprocal Plasmon Drag in the Dark Port
We proceed to discuss our experimental probing of the dark port in search of
response to nonreciprocal phenomena. In this experiment, the perturbation of interest
81
is introduced to the system in the same manner as that of the reciprocal ohmic heating,
by applying a voltage across the silver film of the SPR configuration. This establishes
an electric field within the silver giving rise to a preferred direction of motion for
the electrons in the silver film. The beams of the interferometer then will excite
SPPs, one of which has a propagation direction matching the direction of current
flow in the silver and the other of which is counter to this flow. We expect to observe
an electron-plasmon interaction similar to that of the plasmon drag effect in which
there is a momentum transfer between the SPP mode and the electrons. Much of
the previous work on this interaction pertains to the characterization of the electrical
current induced by the optical interaction [84, 86–88]. The magnitude of the resulting
current peaks sharply on resonance, where there is maximal coupling to SPPs, and
the direction matches that of the propagating plasmons.
On the other hand, there is some precedent, but much less understanding, in
the literature for observation of coupling in the opposite direction; where an applied
electrical signal modifies an optical signal through interaction of the electrons with
SPPs. In this intensity based measurement, application of a square wave pulse
resulted in a SPR reflectance shift with timing to match that of the pulse [89].
After modeling the expected reflectance profile due to variation in the film thickness
and permittivity, the authors concluded that, “the explanation of the observed
phenomenon is beyond simple current-induced changes of the basic film parameters”
[89]. Use of an interferometric scheme, by employing the SPR configuration within a
Sagnac, for further exploration of this phenomenon has the potential to provide new
details, including phase information, in order to better understand and describe this
interaction between SPPs and electrons. Further insight into this interaction may
82
have broader impacts as well, as plasmonics are viewed as a potential intermediary
between photonics and electronics in the area of information processing [90, 91].
To probe for nonreciprocal response in the bright port we utilize a function
generator to drive an electrical current in the silver film while collecting the intensity
output at the dark port with a photomultiplier tube and through use of a lock-in
amplifier. Neutral density filters are used in front of the detector as necessary to avoid
saturation. We choose to apply a square wave voltage and look for a modulation in
the output at the same frequency. It is important that the square wave be vertically
offset so that it is not centered around zero because alternation between positive and
negative values of the same magnitude is not expected to elicit a response. This will
flip the direction of the current, but in both cases SPPs will be propagating both
with and against the current, producing the same phase difference in both scenarios.
Here we use a square wave with a 5 volt peak-to-peak amplitude and 2.5 volt offset,
as this +5 volts is the largest voltage capable of being produced by the function
generator used. We expect the strength of the interaction to scale positively with
the current in the film, in which case increasing the voltage will produce a larger
response. A range of frequencies was examined, spanning from tens of kHz down to
roughly tenths of mHz. The higher frequencies were analyzed via digital oscilloscope
because the timing of the LabVIEW program subroutine, as written, was too slow
to process this data in real time. Much of this search came up empty; a null result
was observed in which the dark port signal did not vary on the timescale to match
the input frequency, exhibiting only noise fluctuations. However, in some instances,
a modulation of the dark port signal was observed. A more or less representative
sampling of this data will be discussed and analyzed below.
83
Angle Greater Than Resonance Minimum
The results of one such data run are shown in Figure 48. For this data set, the
incidence angle within the prism was fixed at an angle greater than the resonance
angle, specifically 43.25◦. An input voltage with frequency of 5 mHz was used as is
shown in Figure 48a. We concurrently collect temperature data with the expectation
that, as before, the current induced heat will raise the temperature of the system over
the course of the experiment. This is indeed the case as is seen in Figure 48b.
(a) (b)
(c)
FIGURE 48. Dark port response to nonreciprocal plasmon drag.
(a) Square wave voltage applied by the generator as a function of time. (b)
Temperature recorded by the thermometer used for independent measure of
temperature. A steady temperature increase is observed. (c) The dark port signal
shows a modulated response with a frequency matching that of the applied voltage
on top of a slight overall trend upward with time. Color is consistent throughout
and used purely as a visual aid to distinguish low and high voltage regions.
84
In the dark port signal we observe a modulated response with a frequency
matching that of the input square wave. The plot in Figure 48c is color coded to
clearly demarcate the time intervals in which a voltage is applied across the silver
film from those in which no voltage is applied. The signal in regions with no current
is distinctly higher than it is in regions with voltage applied. In addition to this
modulation, there is an overall upward trend in the data. This is attributed to
the temperature change, which also increases steadily over the course of the data
collection. While the dark port is generally impervious to the effects of temperature
change, the presence of a nonreciprocal perturbation to break the symmetry of the
paths does introduce a small sensitivity to reciprocal effects.
Qualitatively similar behavior can be reproduced using the theoretical model
developed in the previous chapter. To account for the fact that the dark port intensity
does not reach identically zero in experiment, we first introduce a disruption to the
system which raises the baseline of the dark port signal to some nonzero value.
Then, on top of that signal, we introduce two perturbations: one to model the
reciprocal ohmic heating and a second to model the nonreciprocal plasmon drag.
We observed in Figure 48b that the temperature of the system increased more or
less linearly with time. For this reason, we introduce the reciprocal perturbation into
the model simply as a linear increase in the refractive index of the dielectric. For the
nonreciprocal perturbation, we assume, as a first pass, a current induced change to the
real component of the relative permittivity of silver. We enter this into the model with
a frequency matching that of the applied voltage in the experiment. When no voltage
is applied across the film, no adjustments are made to the permittivity of silver. But
when a voltage is applied, we model this as a nonreciprocal change in the permittivity:
the permittivity increases as seen by one path and decreases as experienced by the
85
counterpropagating path. With some adjustment to the magnitude of the two effects
entered into the model, we obtain the plot shown in Figure 49, which exhibits behavior
qualitatively similar to the data of Figure 48c.
FIGURE 49. Dark port model of plasmon drag.
We use the theoretical model to introduce a nonreciprocal perturbation to the real
component of the silver permittivity at a frequency of 5 mHz and a reciprocal
perturbation to the dielectric that grows linearly with time. This produces
qualitatively the same behavior as observed in the experimental data.
For just one trace, there are likely multiple sets of fit parameters which could
used to match the data fairly well with experimental observation. However, if we are
able to build up a repository of data collections, spanning the parameter space of
incidence angle and magnitude of the applied voltage, we may be better able to pin
down a set of fit parameters which describe the experimental data. This will likely
involve moving beyond our assumption of nonreciprocal perturbation manifest only in
the real component of the silver permittivity, and expanding it to also involve other
parameters such as the imaginary component or the thickness. While this output
appears suggestive of a nonreciprocal dragging effect, and the theoretical model can
86
be employed to reproduce similar qualitative behavior, further measurements are
necessary to fully characterize the interaction.
Angle at Resonance Minimum
We next move to a data set collected with the angle aligned at the resonance
minimum. A square wave signal output from the function generator is again used to
apply and remove a 5 volt potential difference across the silver film with a frequency
of 5mHz. The dark port data associated with this measurement, shown in Figure 50,
again displays a modulated response with frequency matching that of the input signal.
However, a striking difference is observed between this signal and the previous one.
In this data we observe a curvature in the response indicative of a nonzero optical
response time to the changes in the electrical current. This noticeable response time
is unexpected for the plasmon drag effect; previous results in the literature exhibit a
lack of noticeable response time, even with pulse durations as short as a millisecond
[89]. This response time is characterized by a time constant that may be found by
fitting the data with equations of the form
Ae−α(t−t( 0
) +)B (5.1)
A 1− e−β(t−t0) +B (5.2)
where A and B are constants representing the amplitude and the vertical shift of the
data, and t0 represents the time at which the voltage changes. The results of fitting
this equation to the data are shown as the solid lines in Figure 50. The parameters α
and β are the inverse of the time constant for the fall and rise of the dark port signal,
respectively. The best fit parameters to the data using these equations are displayed
87
(a)
(b)
FIGURE 50. Dark port nonreciprocal response at the resonance minimum.
(a) Square wave voltage applied by the generator as a function of time. (b) Dark
port signal as a function of time. The experimental data points show a modulated
response with a frequency matching that of the applied voltage, but with an
extended response time. The solid curves represent the exponential fit functions
which characterize the response time.
in the Table 3. From this, we observe that the rise in signal happens on a time scale
approximately 3–4 times faster than the drop; roughly 22 seconds for the former and
6 seconds for the latter. In revisiting the data of Figure 48c, it appears as though
there could be a similar effect present in that data, albeit on a much shorter time
scale. In any case, the effect is not nearly as pronounced as seen here with the angle
of incidence aligned to the minimum of the resonance. To test whether this behavior
is unique to this angle, we move to a second off-resonant angle.
88
TABLE 3. Decay constants of the dark port signal at the resonance minimum.
Based on the data shown in Figure 50b, α and β display the parameters resulting
from a fit to the data of the form of Equations 5.1 and 5.2.
Time Frame α β
0-199 s 0.1847 0.0423
200-399 s 0.1606 0.0433
400-599 s 0.1438 0.0479
600-799 s 0.1871 0.0492
800-999 s 0.1657 0.0488
Angle Less Than Resonance Minimum
Figure 51 displays the dark port signal with the interferometer aligned for an
incidence on the lower angle side of the resonance, at 43.02◦. The voltage signal
applied across the silver film remains the same. Here we again see a modulation
of the signal at the expected frequency; furthermore, we observe that the extended
response time of the signal is persistent, even away from resonance.
Equations 5.1 and 5.2 are applied as a fit to this data, resulting in the parameters
shown in Table 4. The magnitude of the values obtained are consistent with those
shown above; however, in this scenario the time constant is longer for the drop in the
signal, approximately 17 seconds, and shorter when the signal rises again, roughly 7
seconds. What does remain consistent from the previous angle is that the shorter time
constant is associated with the disappearance of a current from the silver film while
the longer time constant is associated with the appearance of a current. We explored
the possibility of this effect arising from an electronics issue, such as an impedance
mismatch somewhere in the system, but efforts to track down a potential source
were ultimately unsuccessful. The magnitude of the time constant could provide a
clue about the source, for example if it were related to an RC time constant located
within the system; though here the resistances of interest are of the order of tens of
89
(a)
(b)
FIGURE 51. Dark port nonreciprocal response at an angle less than resonance.
(a) Square wave voltage applied by the function generator as a function of time. (b)
Dark port signal as a function of time. The experimental data points show a
modulated response with a frequency matching that of the applied voltage, but with
an extended response time. The solid curves represent the exponential fit functions
which characterize the response time.
ohms corresponding to a capacitance on the order of a farad, which seems largely
implausible.
TABLE 4. Decay constants for an angle less than the resonance minimum.
Based on the data shown in Figure 50b, α and β display the parameters resulting
from a fit to the data of the form of Equations 5.1 and 5.2.
Time Frame α β
0-199 s 0.0475 0.1698
200-399 s 0.0545 0.1557
400-599 s 0.0591 0.1255
600-799 s 0.0792 0.1012
800-999 s 0.0796 0.0856
90
Rotation of Film Orientation
In an attempt to determine whether the modulation of the output seen occurs as
a result of the plasmon drag effect, we devised an experiment involving a rotation of
the silver film orientation. If we posit that the magnitude of the effect is proportional
to the scalar product of the drift velocity of the electrons with the momentum vector
of the SPPs, then rotating the film such that the direction of current flow in the
film is perpendicular to the propagation of SPPs should result in a disappearance
of the modulation in the signal. The construction of the brass mass from the prism
mount shown in Figure 44 is not compatible with the epoxied leads for this alternate
orientation of the sample. Instead, we remove the prism from the mount and position
the film as shown in Figure 52. This configuration should give rise to an electron drift
velocity with a near vertical orientation while the SPP propagation remains primarily
within the horizontal beam plane.
FIGURE 52. Photo of apparatus with film oriented vertically.
In order to generate an electrical current perpendicular to the direction of plasmon
propagation, the glass substrate onto which the silver was deposited was rotated on
the back of the prism by an angle of 90 degrees.
In this experiment, the interferometer is aligned at the same angle of incidence
on the lower angle side of the resonance as before, that being 43.02◦. A 10 mHz
square wave with 5 volt peak-to-peak amplitude was applied across the silver film by
91
a function generator. The interferometer output was collected at the dark port and
the results are shown in Figure 53a. Despite the orientation of the film such that
the electron drift velocity is perpendicular to the momentum vector of the SPPs, the
output signal is modulated at the frequency of the voltage input. The amplitude of
this modulation is on the same order as similar experiments performed with silver
film in the original orientation. This goes against our intuition and calls into question
whether or not the observed effects are as a result of plasmon drag. In addition, we
continue to observe an exponential growth and decay in the signal response with a
timescale on the order of seconds.
(a) (b)
(c) (d)
FIGURE 53. Dark port response for varied current with vertically oriented film.
(a) Dark port response for a 5 volt peak-to-peak applied voltage. The solid lines
represent the voltage applied while the dots display the bright port signal. (b) Dark
port intensity and applied voltage for a 4 volt peak-to-peak signal. (c) Dark port
intensity and applied voltage for a 3 volt peak-to-peak signal. (d) Dark port
intensity and applied voltage for a 2 volt peak-to-peak signal.
92
The continued observance of a modulated signal matching the frequency of the
applied voltage suggests, if not plasmon drag, some other link between the two. To
further investigate this link, we vary the amplitude of the square wave potential
applied across the silver film at a fixed frequency. Figure 53 shows the results of
stepping down the voltage from 5 volts, to 4 volts, then 3 volts, and finally 2 volts.
The signal in the regions with no current present in the film remains fairly consistent
across all the plots shown. However, by looking at the regions of the plot where the
current is present we observe a dark port signal that increases with the voltage applied
across the film. An amplitude of this modulation is estimated by averaging the data
values in the presence of a voltage and in the absence of a voltage and computing the
difference; standard error is propagated through this calculation. These amplitudes
are plotted with respect to the amplitude of the applied voltage in Figure 54. The
data appears to exhibit an exponential scaling of modulation amplitude with respect
to the peak-to-peak voltage supplied by the function generator.
FIGURE 54. Scaling of modulation amplitude.
Calculated from the data displayed in Figure 53, the amplitude of modulation in the
dark port signal appears to scale exponentially with the amplitude of the applied
voltage. Errors bars represent the standard error, propagated through the
calculation of the modulation amplitude.
93
Analysis
Unfortunately, a modulated response of the dark port output was not measured
with sufficient repeatability and predictably to fully characterize the SPP interaction
with electrons by use of our model and in terms of a current induced shift in the
material parameters of the silver film. We anticipate the amplitude of this modulation
to vary with position along the resonance based on the change in the amount of
energy that is coupled into the SPP mode for each angle. With measured amplitudes
of modulation across a range of angles along the resonance, we could search for some
singular combination of parameters, involving the relative permittivity and thickness
of the silver film, capable of reproducing this behavior when applied in our model.
For any particular angle, we would also expect the amplitude of the modulation to
vary with the amplitude of the applied voltage. Collection of a similar set of data for
a different applied voltage could provide insight into how any current induced changes
to material properties of the film scale with the strength of the current. Initial findings
suggest an exponential dependence of the intensity modulation on the amplitude of
the applied voltage.
Simultaneous Detection
While the data presented here was collected for only one interferometer output
at a time, there were no hindrances encountered in performing these experiments
that would preclude the simultaneous detection of both ports. The aim of this work
was to provide proof of principle that the bright port could be used for detection
of reciprocal perturbations in a resonantly enhanced interferometer and that the
dark port response is dominated by sensitivity to nonreciprocal phenomena. Future
94
experiments could utilize both, in tandem, to further characterize the response of the
resonantly enhanced system or in applications as a device for simultaneous detection.
Summary
In this chapter we have presented the assembly, alignment procedures, and
experimental results of a resonantly enhanced Sagnac interferometer using surface
plasmon polaritons. We first discussed the choice of configuration and the components
used in building a balanced interferometer. We then presented the methods used
to introduce both reciprocal and nonreciprocal perturbations to the system in the
form of ohmic heating and plasmon drag. The first set of experimental results
presented pertain to the detection of reciprocal ohmic heating in the bright port
supplied via an electrical current in the silver film. A response in the interferometer
output was observed which qualitatively matches our model. We examined various
sources of quantitative disagreement and suggested paths for improvement. The
following section displayed and analyzed data collected from the dark port of the
interferometer. A modulation of the dark port signal was observed exhibiting some
behavior consistent with plasmon drag and was qualitatively reproducible by use of
the theoretical model. An unexpected response time of the signal appear in some of
the data and was characterized via exponential fitting procedures, but the ultimate
source of this behavior remains unknown. In addition, a dependencies on the direction
and magnitude of the electrical current were explored by rotating the orientation of
the silver film and varying the voltage applied across it. We observed a modulation
of similar amplitude with the rotated orientation and an exponential scaling with
voltage. Finally, we reviewed the prospects of the device for future simultaneous
detection experiments.
95
CHAPTER VI
CONCLUSION
This dissertation has sought to convey our exploration into the field of resonantly
enhanced Sagnac interferometry using surface plasmon polaritons. We began with a
brief overview of both Sagnac interferometry and surface plasmon resonance sensors,
covering topics ranging from historical development to modern applications. We
calculated the interference equations which give rise to the normalized intensity at the
two interferometer outputs. We dealt with the physics of SPPs, from the dispersion
relation to coupling schemes for excitation, as well as the resulting surface plasmon
resonance. We learned how to engineer the resonance, adjusting the appropriate
material parameters to give rise to the desired lineshape. We fabricated samples which
displayed this resonance. We then theoretically modeled the outputs and sensitivity
of a resonantly enhanced Sagnac interferometer and finally assembled an experimental
apparatus and performed initial experiments described by this model.
Through our investigation we found that the inclusion of a resonance within a
Sagnac interferometer gives rise to some intriguing results. The dark port behaves
mostly as expected for a Sagnac interferometer: rejecting low frequency reciprocal
noise and exhibiting sensitivity to nonreciprocal, symmetry breaking perturbations
to the system. However, the rapidly changing phase on resonance acts to enhance
this signal, improving nonreciprocal sensitivity as compared to the system without
resonance. The bright port was also found to respond to these nonreciprocal
perturbations; however, the more predominant response is to perturbations that
are reciprocal in nature. The intensity modulation imparted on the system by the
resonance gives this interferometer, unlike a nonresonant Sagnac, the capacity to
96
monitor reciprocal drifts within the system. The preferential response to a particular
type of perturbation in each of the interferometer outputs gives this device the ability
to distinguish between reciprocal and nonreciprocal phenomena in cases where both
perturbations may be present, and enables the possibility of simultaneous detection
all contained in one device with a single optical source.
This scheme was realized experimentally by applying an electrical current to the
silver film supporting SPPs. Optical data collected from either side of the resonance
exhibited a response consistent with the reciprocal effect of ohmic heating. The
magnitude of the effect suggests a larger temperature change than that which was
measured; however, this is not surprising given that the independent measurement
of temperature was spatially separated from the optically interrogated region, in a
location further away from the source of heat generation. The optical data collected
from the dark port exhibited a periodic response with a frequency matching that of
the applied electrical signal. However, the nature of the response was not entirely
consistent with that expected for the nonreciprocal plasmon drag, and therefore we
concede that the origin of the signal may be due, at least in part, to other sources. In
addition, the strength of the applied voltage was varied and the modulation amplitude
appeared to scale exponentially with this voltage.
Future Directions
There are a number of areas that exist for continued study on this topic. As far as
the experimental detection of reciprocal effects in the bright port, the largest obstacle
in this work was the physical displacement between the location of the independent
measure of temperature and the region probed optically giving rise to a temperature
discrepancy. One way to address this could be through a transient heat transfer
97
analysis to compute a temperature distribution relating the temperature in these two
locations. Another possibility could be to prolong the temperature measurements
allowing for greater equilibration in hopes to minimize the temperature gradient. A
further possibility would be to devise a new configuration or alternate method for
independent measure of temperature that is capable of providing a more accurate
temperature measurement.
As for the nonreciprocal perturbation detection in the dark port, the results
of this dissertation contained some features suggestive of the plasmon drag effect,
but other features were not consistent with being produced by this effect. If the
interferometer response could be reliably linked to the plasmon drag effect there are
some additional experiments to be performed that could further the understanding
of the electron-SPP interaction. Two important parameters to vary in these
experiments, which were explored in some detail in this work, would be the voltage
applied across the silver film and the incidence angle within the prism. The voltage
acts as a knob for the current flow, and hence the electron drift velocity, in the metal
while the incidence angle controls the amount of energy transferred into the surface
mode through the degree of coupling to SPPs. A more complete data set within
this parameter space, in combination with the theoretical model to explore these
dependencies, could relate the change in optical signal resulting from these affects to
modifications in the parameter values that go into the calculations.
Other avenues exist for exploration of the Sagnac interferometer resonantly
enhanced by SPR that do not involve introduction of an electrical current within
the silver film. For instance, the perturbations could instead be introduced by use of
a microfluidic flow channel as the outer dielectric of the SPR configuration. In this
scheme, Fresnel-Fizeau drag associated with the moving dielectric would provide the
98
nonreciprocal perturbation, while the reciprocal phenomena could again be manifest
as a temperature or instead, for example, by concentration changes in the fluid.
In addition to the many experimental avenues, another future direction involves
modifications to the theoretical model. One specific path of interest would be
the inclusion of a metal/insulator stack within the SPR configuration for greater
tunability of the resonance lineshape. The use of a Fano resonance to further engineer
the lineshape could be optimized for greater sensitivity or improvement of signal to
noise. The implementation of perturbations could also be adjusted to include more
than just a change in refractive index of the outer dielectric. While this general
behavior is able to qualitatively portray much of the system behavior, quantitative
description of specific phenomena may involve manipulation of multiple parameters
in the system.
Closing Remarks
Throughout my work toward this dissertation, and more broadly during my
time at the University of Oregon, I have experienced significant growth in both the
academic and personal aspects of my life. Through graduate level coursework, I
have accumulated a wealth of knowledge describing the mechanisms through which
our physical world operates while simultaneously exploring and expanding the limits
of my resolve and understanding. In my experiences teaching and interacting with
students, I have cultivated a quiet confidence in clear and effective communication of
concepts and affirmed the deeply rooted joy found in positively impacting others
toward growth and success. By conducting advanced scientific research, I have
exercised and augmented my problem solving capabilities and accessed a formerly
untapped level of persistence to move forward, even in the face of adversity. I have
99
delighted in the thrill of discovery and taken pride in that which I have accomplished. I
have also been humbled, time and time again, and faced with the stark realization that
I am wholly incapable of fully actualizing the totality of my own lofty expectations.
Through it all, I have become a more amicable colleague, a more compassionate
friend, a more curious student, a more forgiving husband, a more diligent teacher, a
more patient father, a more purposeful scientist, and a more genuine individual. For
this I am especially grateful.
100
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