AN INVESTIGATION OF INTERMODAL COUPLING

EFFECTS IN OPTICAL MICRORESONATORS

By

ERIK KIRKLIND GONZALES

Bachelor of Science in Physics

East Central University

Ada, Oklahoma

2007

Submitted to the Faculty of the Graduate College of the

Oklahoma State University in partial fulfillment of the requirements for

the Degree of MASTER OF SCIENCE

December, 2011

ii

AN INVESTIGATION OF INTERMODAL COUPLING

EFFECTS IN OPTICAL MICRORESONATORS

Thesis Approved:

Dr. Albert T. Rosenberger Thesis Advisor

Dr. Gil Summy

Dr. Girish Agarwal

Dr. Sheryl A. Tucker Dean of the Graduate College

iii

TABLE OF CONTENTS

Chapter Page I. INTRODUCTION ......................................................................................................1

Probing the Resonance Structure of a Whispering-Gallery-Mode Microresonator ...................................................................................................2 Applications .............................................................................................................4 II. EXPERIMENTAL SETUP .......................................................................................5 III. CROSS POLARIZATION COUPLING (CPC) ......................................................7 3.1 Introduction .......................................................................................................7 3.2 Birefringence.....................................................................................................9 Experimental Results ........................................................................................10 3.3 Berry Phase ......................................................................................................10 Experimental Setup ...........................................................................................11 Results ...............................................................................................................12 Discussion .........................................................................................................12 3.4 Cylindrical Resonators ....................................................................................13 3.5 Modeling CPC ................................................................................................14 Data Fitting .......................................................................................................17 IV. CONCLUSIONS ...................................................................................................22 REFERENCES ............................................................................................................24 APPENDICES .............................................................................................................26

iv

LIST OF FIGURES

Figure Page 1 TE/TM pumped microsphere .....................................................................................3

2 Experimental Setup ....................................................................................................6

3 Microsphere CPC .......................................................................................................7

4 CPC turned on/off by Straining ...............................................................................10

5 Precessional Modes of a Prolate Sphere ..................................................................11

6 Berry Phase Experimental Setup .............................................................................12

7 Cylinder CPC ...........................................................................................................13

8 CPC Ring Cavity......................................................................................................15

9 Strained sphere data traces .......................................................................................19

10a CPC Model/Data Overlay ....................................................................................19

10b CPC Model/Data Overlay, strained sphere ..........................................................20

11 Close up image of a microsphere mounted in a strain tuner ..................................23

1

CHAPTER I

INTRODUCTION

Spherical symmetry is a classic and thoroughly covered topic in wave mechanics. In the

case of a spherical potential well or a dielectric sphere, it gives rise to cavity resonances known as

modes or eigenfrequencies, for example. It has produced collections of famous differential

equation structures like Bessel's and Legendre's equations that have seen generations of study. As

such, the thoroughness has consequently produced a broad field in optics particularly, and is

paired, of course, with countless applications. The ongoing research is a testimony to its intricacy

and importance. In the case of a sub-millimeter diameter optical cavity, a specialized level of

complexity is introduced as unusually higher mode orders are accessed.

A recently discovered process, known as cross polarization coupling (CPC), is a 'cross

talk' between modes that are orthogonally polarized. Due to the effect, some currently

understood phenomena in microresonator spectroscopy, for example, need to be updated.

Preliminary experiments motivated an extensive study of CPC [1]. In this report, an

attempt to clarify CPC is made by presenting a further investigation of its fundamental nature,

thereby providing a foundation which is crucial to the expansion of optical microresonator

applications.

2

Probing the Resonance Structure of a Whispering-Gallery-Mode Microresonator

Our resonators are typically made of fused silica and tunable laser light is coupled into

the cavity via evanescent coupling. The light is highly confined within the cavity where it faces

total internal reflection. When the effective optical path length is equal to an integer number of

wavelengths, resonance is achieved. These resonances are called whispering-gallery modes

(WGMs), the name coming from an analogous acoustic effect [2]. Toroids [3], cylinders [4], and

spheres [5] are examples of common geometrical shapes that can support WGMs. Spherical

cavity resonators can be both easily made and cheap so they are a common choice of study in the

lab. Tuning across a free spectral range reveals many narrow-width (typical linewidth of ~2

MHz) resonances which in turn leads to many uses such as microlasers [6], spectroscopy [7],

optical switching [8], and cavity quantum electrodynamics (QED) [9].

To study these useful resonances, light must be coupled into and out of the resonator with

minimal perturbation to the system. Our most practiced method is using a single mode optical

fiber which is adiabatically tapered to a diameter on the order of a wavelength which is typically

1550 nm. The tapered region reveals an evanescent portion of light around the fiber. Bringing

this fiber into the proximity of a resonator allows a fraction of the light to tunnel out and into the

cavity; this familiar near field optical effect is known as evanescent coupling. The light inside of

the cavity is continuously reflected off the walls. At each reflection certain field components

remain continuous across the boundary in which the radial propagation constant simultaneously

goes imaginary, which in turn provides the resonator with its own evanescent field. This allows

the light to tunnel out of the cavity and back into the fiber, after which it falls on a detector.

Scanning the laser in frequency exposes the modes of the resonator as Lorentzian dips in the

detected throughput.

3

Our spheres are hand-made by melting the tip of an optical fiber in a hydrogen/oxygen

flame. The surface tension of the molten glass induces a nearly perfect spherical shape whose

diameter can be easily regulated to a sub-millimeter range, hence the name microresonators. In

terms of the wave equation, their symmetries provide many solutions. Our fabrication method

provides each sphere with a unique modal signature with free spectral ranges between 66 GHz

and 660 GHz for diameters of 1 mm and 0.1 mm, respectively. Because WGM's of many

transverse (radial and polar) orders may be excited, a typical observed WGM spectrum includes

several modes per GHz.

Figure 1 Oscilloscope image of throughput power versus time, demonstrating TE and TM modes. Yellow trace: TE; blue trace: TM.

An important feature of spherical cavities is the occurrence of well known transverse

electric (TE) and transverse magnetic (TM) mode families that fall from the boundary conditions

applied to the wave equation's various solutions. If a TE and a TM mode happen to share the

same resonant frequency, the coincidental overlap is called co-resonance. The two polarizations

are of course orthogonal to each other so we have two independent mode structures within the

cavity each with its unique signature (Fig. 1).

4

Applications

With such a rich selection of modes, microresonators have a range of functions.

Straining provides a means of easily tuning the resonator more than a free spectral range at a

MHz rate [10] and presents a method of locking a cavity mode to a tuning laser [11]. This is

readily advantageous for spectroscopy. For example, a microresonator can be tuned to a known

trace gas's absorption line. With an effective absorption path length of about 1 meter, our

resonators have shown to have high detection sensitivity [7]. A second resonator can be brought

into contact with another to introduce coupled mode effects such as mode splitting analogous to

electromagnetically induced transparency (EIT) [12, 13]. Coating a resonator with a gain

medium provides a low threshold microlaser [6]. The high quality factors found, combined with

a small mode volume, makes WGMs good candidates for cavity QED [9]. The main topic of this

report shows the potential for even broader applications: mode splitting in a single resonator that

can lead to an enhanced chemical sensor, as well as the possibility of a simple polarization

analyzer.

5

CHAPTER II

EXPERIMENTAL SETUP

Figure 2 shows the general setup. The light source is a tunable diode laser (linewidth <

300 kHz) operating in the IR (1550 nm) spectral range. Before the fiber coupling, the beam

passes through a set of wave plates which are used to control the polarization. The first

component in the fiber system is an optical isolator, which stops any back propagating light. The

fiber is also mounted in a compression based polarization controller for further regulation of the

input light. The cavity coupling component is an adiabatically bi-tapered single mode fiber

which is mounted on a 3D translation stage. In order to maintain good fiber-resonator alignment,

a microscope is in place above the fiber-resonator system. The throughput is sent through a fiber

splitter where one signal is sent to a fiber coupled fast detector, and the other is sent to a

polarization analyzer where the TE and TM modes can be simultaneously monitored (inset, Fig.

2). All photodiode voltages are sent to oscilloscopes.

The resonator is held by an apparatus for strain tuning (inset, Figure 2). A lock-in

stabilizer is connected through a voltage divider to the tuner which, depending on the resonator

shape, either cylinder or sphere, is a stretcher or compressor, respectively.

In most cases, the wave plates are adjusted to provide linearly polarized light. It is

common practice in the lab to adjust the polarization angle with respect to the resonator's basis.

6

This freedom allows pure TE/TM excitement or the simultaneous pumping of the two. In other

words, power can be distributed among the two polarizations as needed. Directing all of the

power to the TE polarization reveals only the TE modes, for example. In all cases the resonator is

kept inside an airtight acrylic box to minimize temperature fluctuations and other effects of air

movement.

Figure 2 Laser light is provided by a New Focus Velocity tunable diode laser. The wave plate system consists of two quarter wave plates, and one half wave plate all of which can be rotated for polarization control of the input light. The tapered fiber is mounted on a 3D stage for accurate placement. All components in the polarization analyzer rotate together. Not shown is the function generator used to tune the laser.

7

CHAPTER III

CROSS POLARIZATION COUPLING

3.1 Introduction

It is rather common in the lab to see a leaking of power from one polarization to the

other. In the case of pure TE excitement, for example, a significant amount of power can be

found in its orthogonal counterpart (Figure 3). This phenomenon is known as cross polarization

coupling (CPC). Although CPC is frequently observed, certain criteria must be met in order to

see the effect. Figure 3 demonstrates two of these conditions: high quality factor and near-

critical input coupling for the driven mode, which ensure a high intracavity power.

Figure 3 Oscilloscope trace of throughput from a pure TE pumped microsphere. Significant power is found in the orthogonal polarization (blue trace) and appears as peaks. This is the case of internal polarization coupling (IPC). Note the pattern between the TM peaks and their corresponding TE dips: only narrow and deep modes produce CPC.

8

When CPC was first observed, many ideas were proposed to explain this non-intuitive

effect. Several experiments were carried out to confirm their either erroneous or legitimate

validity; specifically the polarization effects of the system were tested. The strength of cross

polarization input coupling at the fiber-resonator interface was calculated using coupled mode

theory and shown to be too weak to explain the observed CPC. Misalignment of the input field

with respect to the basis of the resonator can cause a significant amount of power in the "un-

pumped" mode; its main cause is experimental error. If the polarization analyzer is not aligned

to the resonator's basis, then undesired orthogonal power will be observed and produces a type of

CPC. This process is known as direct polarization conversion (DPC). However, a simple

protocol exists which guarantees cavity-analyzer alignment. When peaks are observed in this

case, there is most likely some internal mechanism of the cavity that transfers power between

polarizations; this is called intrinsic polarization coupling (IPC). In Fig. 3, IPC is demonstrated;

perfect alignment is rare to achieve, so DPC is usually observed which usually appears to mirror

the pumped trace, otherwise it can closely resemble IPC. It is then up to the experimentalist to

test the CPC for the two types.

The most likely explanation for IPC is scattering. Light incident on a scatterer can be

reemitted in an unknown direction. In the case of WGMs, back scattered light can lead to

identical but counter-propagating modes [10, 14]. It is also well known that the scattered light

can have a rotated polarization with respect to the incident field. The scattering process can

provide frequency shifted photons, for example, in the case of fluorescence. However, let's only

consider forward scattered light that has been rotated but with an unchanged frequency. If this

rotated light remains confined within the cavity it can very likely have a projected power in the

orthogonal polarization. If the frequency of the light is an eigenfrequency of the other

polarization, then a photon build up can occur which leads to CPC. So, our rotated light has been

9

transferred from one polarization to its orthogonal counterpart and the mechanism for the

coupling is scattering.

Referring to Fig. 3, note that the TM's (blue trace) vertical axis is scaled 5 times higher

than the pumped TE's axis. CPC is a relatively weak effect which is sensitive to three main

parameters: quality factor, co-resonance, and input coupling strength. The effects of both input

coupling strength and quality factor can be easily investigated in the lab; however, TE/TM co-

resonance is not so readily controlled and relies on coincidence.

It was hypothesized that co-resonance between the two polarizations is necessary to see

CPC. To test the co-resonance picture, it's necessary to shift the resonance structure of the two

mode families individually in order to control the overlap. A way to shift the eigenfrequencies of

a resonator is to adjust the index of refraction of the material, hence altering the optical path

length. One method is to take advantage of the index's temperature dependence. We used this

method previously to tune a second resonator with respect to another by mounting the second

sphere on a thermoelectric converter [1]. Coupled mode effects were readily seen in this

experiment. However, CPC occurs in a single resonator, so by changing the temperature of the

cavity both mode families shift together in frequency. In order to shift the modes with respect to

each other, we can take advantage of a well known physical characteristic of fused silica.

3.2 Birefringence

Under stress, fused silica loses its isotropic structure and experiences birefringence, an

effect that causes the TE and TM modes to see different indices of refraction. It is this exact

mechanism that compression based polarization controllers exploit in order to rotate the

polarization state of the fiber field.

Compression tuners were built to easily shift a spherical resonator's frequency pattern

[10, 11]. The device is constructed of aluminum where the sphere can be mounted between two

10

flexible stands. Below one is a piezoelectric transducer (PZT) which expands when a voltage is

applied (inset, Figure 2). The straining tunes the TE/TM modes independently of each other and

is enough to bring many orthogonal modes into co-resonance. Further, we also know, a priori,

how much a sphere can be tuned by straining [10, 11] which is more than a FSR.

Experimental Results

As voltage was applied to the tuner, CPC peaks would disappear, and new peaks would

appear as the co-resonance condition was altered throughout the scan range (Fig. 4). As

predicted, co-resonance can be controlled by strain tuning a sphere and provides a means of

switching the CPC on or off. This result also provides a test of our basis alignment protocol; if

CPC is an alignment error then the orthogonal peak would exist even if the TE/TM modes are not

co-resonant. We have now shown that this intracavity coupling effect requires co-resonance.

Figure 4 CPC (peak in yellow trace) is turned on/off by differential strain tuning of the two mode families.

3.3 Berry Phase

Another possible explanation for CPC is a geometric effect known as Berry phase [15].

CPC was found in a different type of resonator: a coil of tightly wound, small radius optical fiber

11

[16]. The researchers explained the observed CPC with Berry phase. A WGM excited in a

perfect sphere can accumulate no Berry phase; however, our spheres are slightly prolate which

allows access to the so-called precessional modes [17] if the input fiber is not parallel to the

equatorial plane (Fig. 5). A precessing mode's polarization vector might be rotated via Berry

phase, evidently with a much larger effect acquired from stronger precession.

Figure 5 A perfect sphere (left) does not support precessional modes. An angularly offset input field will always generate modes in the same plane as the fiber (inset left). A more realistic sphere is a prolate spheroid (right). The precessing modes have an effective build up nearer the poles as in a bottle resonator [18].

An easy way to excite precessional modes is to tilt the input fiber with respect to the

sphere's equatorial plane as in the inset of Fig. 5. The microscope was only recently added to the

experimental setup, so in previous CPC experiments the input angle had a significant degree of

uncertainty. Also, the fiber's displacement from the sphere's equatorial plane had a degree of

uncertainty, which is another way to excite precessional modes. Although there are tricks of the

trade available to ensure 'good' placement, the possible CPC ambiguity had to be addressed.

12

Experimental Setup

A rotating stage was added to the system in order to control the angle between the fiber

and the equatorial plane (Fig. 6). A small servo was attached to the rotatable base for

repeatability.

Figure 6 Using a simple hobby servo to manipulate a rotating base. The servo can be positioned

within ο1± .

Results

There was no notable increase/decrease in orthogonal power based on the input angle. A

similar experiment with a bottle shaped resonator provided comparable results [1]. However,

even with a microscope, it is difficult to determine the equatorial plane of the spheres because

they are only slightly prolate (cross section eccentricity of a few percent). In spite of this,

empirical evidence showed that the average CPC power was fairly constant across a range of

about °±10 in input angle. Furthermore, CPC has been observed almost routinely since the

polarization analyzer was added and in all cases it has a very typical behavior that the

experimentalist gets accustomed to quickly. With such a potential range of uncertainty in the

Sphere

Rotating Stage

Servo

13

fiber placement and angle, average CPC power would be unlikely to have the consistency and

repeatability that is observed in the lab if Berry phase is the cause.

Discussion

With image processing software one could possibly take a high resolution image of a

magnified sphere and determine physical quantities such as prolateness and equatorial plane

location. By adding a drop fiber on the opposite side of the resonator the angular displacement of

the precessing WGMs can be probed to confirm these geometric quantities. Then the field

rotation can be accurately calculated with reasonable accuracy. Such quantities would be

relatively simple to measure and would be necessary to give a quantitative argument against/for

Berry phase. However, since mode coupling has been found from backscattering [14], this

finding greatly supports the case for forward scattering as the source of IPC.

3.4 Cylindrical Resonators

CPC also appears in cylindrical resonators (Fig. 7). Similar to the spheres, strain induced

birefringence provides a means of controlling co-resonance between the two polarizations. This

gives results very similar to the CPC effects found in the compressed spheres.

Figure 7 CPC in a cylinder (far right yellow peak). Note the mode splitting in the co-resonant TE mode (blue trace).

14

A cylindrically shaped resonator does not support precessional modes. However a slight

tilt of the input fiber can produce modes that spiral away from the input fiber, forming a helical

shape. In this case, considerable field overlap between adjacent turns in the helix occurs, leaving

the cylinder with an analogous structure to that of the aforementioned microcoil resonator which

can provide CPC via Berry Phase. To test this, a similar experiment to the sphere with tilted

input fiber was carried out with the cylinder/fiber alignment and no notable alignment

dependence was found. However, the rarity of cylinder CPC peaks provided little statistical

support for this experiment and the findings are left inconclusive.

3.5 Modeling CPC

CPC is a coupling between two orthogonal modes, and as such modal coupling effects

can be observed. Induced transparency features as a type of mode splitting are consistently

observed in the lab. An intriguing outcome of the intermodal coupling is the dependence it has

on the input field polarization. This effect can be used as a means of constructing a polarization

analyzer. Such a tool would be readily applicable in the optical fiber industry. In order to realize

these possibilities, it is necessary to construct a theoretical model of the system. A successful

model allows one to fit the data to determine system parameters such as coupling coefficients that

are not easily measured in the lab.

A promising model was recently developed and is based on the ring cavity (Fig. 8). The

input field, of any polarization, is represented by two orthogonal components 1f

E and2f

E with

arbitrary relative phase. The input field polarization basis is assumed to be lined up perfectly

with the cavity's natural polarization's TE and TM. In other words, turning 1f

E on and 2f

E off

will only excite the cavity's TE modes where TE corresponds to the intracavity field 1s

E in Fig. 8.

Since the fields have different spatial profiles, they have their unique reflection and transmission

coefficients at the input/output mirror. Energy conservation is imposed by making the reflection

15

coefficients real and the transmission coefficients imaginary. The output field polarization basis

also corresponds to the cavity's TE and TM. At the scattering center, the probability amplitude

for polarization rotation by 2

π (the CPC mechanism) is sit . The

2

π phase shift (imaginary

coefficient) ensures energy conservation. So we have 122 =+ kk tr for =k 1, 2, s.

Figure 8 Ring cavity simulation of CPC. Input fields 21, ff EE are the two polarization

components and are coupled to the cavity through a partially transmitting mirror. The fields face total internal reflection at the lossless mirrors and field build-up occurs at resonance. Similar to how the mirror couples the input fields to the cavity, the scattering

center rotates an intracavity field 1sE , for example ,by 90 degrees and couples to2sE .

Let's illuminate the ring cavity with this incident field. The output fields will be a

combination of directly reflected input fields and whatever is transmitted out of the cavity:

16

11111 sfr EitErE += , (1)

22222 sfr EitErE += , (2)

where the intracavity fields have picked up an i through transmittance. Next we define the

intracavity fields. Here we treat, at first, two sets of intracavity fields:

]['2

'exp 11111

11 sfc ErEiti

LE ++−= δ

α, (3)

]['2

'exp 22222

22 sfc ErEiti

LE ++−= δ

α, (4)

are the fields just before the scattering center, and just before the output mirror we have:

])['(2

'exp 211111 cscss EitEri

LLE +−+

−−= δδα , (5)

and ])['(2

'exp 122222 cscss EitEri

LLE +−+

−−= δδα . (6)

Where the α 's are the effective intrinsic loss coefficients. L is the round trip cavity length and L'

is the length from the input mirror to the scattering center. δ defines the field detuning from the

cavity's natural resonant frequency normalized to a phase modulo π2 . After each round trip

each field picks up more loss, input field, detuning, etc. Summing over the round trips reveals a

geometric series which converges to a closed form. Such details have been thoroughly covered in

the literature so here we just give the end result.

In reality, scattering is a random process and can, in general, happen anywhere in the

cavity. We therefore average the position L' of the scattering center and phase '1δ or '2δ

accumulated from the input mirror to the scattering center, over the cavity length, L, and we get

2

'L

L = , (7)

17

2

' 11

δδ = , (8)

and 2

' 22

δδ = . (9)

After combining Eqs. (3) through (9) we end up with the intracavity fields

122

211

11 )2

exp(2

exp[1

fss EiL

rriL

itD

E δα

δα

+−−+−=

]24

exp 22121

2 fs EiLttδδαα +

++

−− ,

(10)

and 211

122

22 )2

exp(2

exp[1

fss EiL

rriL

itD

E δα

δα

+−−+−=

]24

exp 11212

1 fs EiLttδδαα +

++

−−

(11)

Where

2

exp2

exp1 22

211

1 δα

δα

iL

rriL

rrD ss +−−+−−=

)(2

exp 2121

21 δδαα

+++

−+ iLrr

(12)

Equations (10) and (11) are substituted into Eqs. (1) and (2) to give the output fields of

the two polarizations. Their square moduli are then proportional to the throughput powers and

may be compared to experimental results. The model throughput powers are plotted as a function

of detuning in a dynamic environment so that the physical parameters of the system, such as

scattering amplitude, may be adjusted to fit experimental data. Realistically, these parameters are

physical representations of the system and are therefore constants. Now let's test the model.

18

Data Fitting

Data was collected from a strain-tuned microsphere exhibiting CPC. The sphere was

tuned into perfect co-resonance and then slightly detuned in both directions to display the CPC

switching (Fig. 9) on both sides of the directly pumped mode. This data was then overlaid with

the model for fitting (Fig. 10a-b). After adjusting the dynamic parameters to fit the co-resonant

data trace, only the detuning was adjusted to simulate the conditions of the tuned system. The

experimentally relevant parameters are: (1) the WGM quality factor

)2,1(2

=∆

=∆

= kLn

Qkk

k

k

kk

δλ

π

ν

ν (13)

where kν , kλ , and kn are the WGM's resonant frequency, wavelength and effective refractive

index, respectively. kν∆ is the WGM linewidth, and

2kkk tL +=∆ αδ (14)

is the linewidth in δ ; (2) the loss ratio

L

tx

k

kk

α

2

= ; (15)

and (3) the round-trip scattering probability 2st .

19

Figure 9 Overlay of five different data traces. Here, the TM traces are detuned relative to the TE

resonance at 1ν , each with differing magnitudes of strain. The sphere is pumped with

pure TE light; the TM traces are magnified 10 times for clarity. Note the asymmetry in each of the TE modes. This is a transparency feature due to the strong intermodal coupling.

Figure 10a TE/TM co-resonance. The model's trace is represented by the dashed and dotted lines. In this trace, the model's parameters are adjusted to fit the data.

2

1f

rk

E

E

1νν − (Hz)

2

1f

rk

E

E

1νν − (Hz)

20

Figure 10b Now, both the model and system have been detuned to test the model/data agreement. Similar results are found with the other traces from Fig. 9

For the fit shown in Fig. 10a-b, the values are:

71 10)6.09.5( ×±=Q (TE),

72 10)6.06.5( ×±=Q (TM),

03.036.01 ±=x ,

03.012.02 ±=x ,

92 10)8.02.9( −×±=st .

(16)

The scattering probability is particularly interesting because it is much greater than the value

typical for backscattering [14], and so produces noticeable splitting even at these relatively low Q

values.

2

1f

rk

E

E

1νν − (Hz)

21

Discussion

With a working model at our disposal, physical insight into CPC can be readily gathered

and offers a way to investigate CPC's possible applications. CPC is sensitive to the polarization

state of the input light, with wide implications. By modifying the model to include an arbitrary

phase on the input field, this property can be readily investigated. Recall the input fields

1fE and 2fE . A phase term can be multiplied to the TM input, for example. We have,

][1 Ω= CosE f (17)

φif eSinE ][2 Ω= (18)

This is the total input. Here, I added two parameters Ω and φ . Ω is the angle between the

input polarization and the cavity's polarization basis, so this is how one can set any mixture of the

two fields. φ is of course the phase between the two fields.

Preliminary results are promising. For an initial check, I turned off the intermodal

coupling and found that the output power is independent of any phase shift φ which is what we

expect. Although CPC is a relatively weak process, even a small amount of scattering shows

strong sensitivity to the input polarization which is helpful for the experimentalist. However, in

order to realize a polarization analyzer, a careful setup would be needed to minimize unwanted

polarization effects. As an example, the fiber in between the collimator and taper (Fig. 2) is

prone to birefringence at the slightest disturbance. This loose fiber must be eliminated from the

system in order to guarantee polarization maintenance.

22

CHAPTER IV

CONCLUSIONS

Cross polarization coupling is a newly discovered process that deserves stringent care and

experimentation. As previously mentioned, some microresonator spectroscopy methods need to

be updated. Notice the effect CPC has on the mode dip depth and linewidth (Fig. 9). If an

absorption line profile is being measured using a tuned microresonator, changes in dip depth

would occur via CPC which would be erroneously calculated as absorption into the medium of

interest. Common chemical absorption linewidths are wide enough that there would be a high

probability of TE/TM co-resonance. As such, CPC would be effectively turning on and off

throughout the scan range, impacting the data. However, in the case of strong intermodal

coupling, mode splitting occurs (Fig. 10a). The separation in the splitting is dependent on

absorption and coupling strength, so this could be utilized as an alternative sensor. It has been

shown that the coupled resonator mode splitting is sensitive to analyte absorption [13]; repeating

this experiment with CPC induced mode splitting could lead to a new method of spectroscopy.

A preliminary experiment could be easily set up to test the resonator's ability to

interrogate the polarization status of the input field. The success of the model has shown its

potential to predict the throughput response to a known input polarization. The input polarization

state is easily regulated in the lab using wave plates. However, there are several aspects of the

23

system that need to be addressed, as previously mentioned. DPC is extremely common and

provides the greatest source of error.

A notable characteristic was observed in the strain tuning experiments. Compressing a

prolate spheroid reveals modes that do not tune at the same rate: each individual mode's

frequency shift is a function of induced strain, so in some cases there are modes that shift up in

frequency at a particular tuning rate while others tune faster. There are even modes that shift

down in frequency while others shift up, with the same applied strain. There are also modes

whose resonances go unchanged. This effect is due to the off-axis pressure that is applied by the

tuner (Fig. 11) and causes an asymmetrical strain distribution (which does not occur in cylinders).

Since each mode propagates in a unique section of the sphere, it will see its particular respective

change in index as pressure is applied to the cavity. This phenomenon could be utilized to

identify what family a mode belongs to, which could be extended to find where CPC occurs

within the cavity, or outside for that matter. As a result, this would provide an even broader

understanding of CPC and its consequences.

Figure 11 Close up image of a microsphere mounted in a compression tuner. The red line represents the sphere's polar axis. The yellow arrows indicate the direction of the induced strain.

24

REFERENCES

[1] E. B. Dale, Coupling Effects in Dielectric Microcavities, PhD dissertation,

Oklahoma State University, 2010; http://physics.okstate.edu/rosenber/Material/Dale%20Dissertation.pdf.

[2] L. Rayleigh, Scientific Papers (Cambridge University Press, London, 1912), pp.

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25

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26

APPPENDIX

A.1 Preparation of Cylindrical Resonators

The cylindrical resonators used in this report were constructed from fused silica capillary

tubing, available from Polymicro Technologies. The first step is to strip the coating from the

fiber. The method used will depend on the coating's physical composition (i.e. polyimide,

acrylic, etc.). It is important that the coating is removed as gently as possible, usually by a

chemical means, in order to achieve an acceptable quality factor. In the case of polyimide, for

example, dip the tubing in a hot (180 - 200 C) sulfuric acid bath. Caution is advised, as the acid

is effectively being removed from the container and into the working room via capillary action.

This can be regulated by plugging the top end of the tubing with epoxy prior to the acid dip (and

watching for any chemical reaction between the acid and epoxy). Within a few minutes, the

coating is stripped away and the tubing can be rinsed in water, then acetone. Break off the

plugged end and force out any remaining liquid with the following: water, acetone, compressed

nitrogen. After drying, the fiber must be flame polished, which surprisingly is worth an order of

magnitude in quality factor. Best results are achieved by rotating the tubing at a slow rate while

simultaneously brushing with a hydrogen + oxygen flame. I have achieved a quality factor of low

to mid 108 using this method.

27

A.2 Measuring Q in the Lab

For many reasons, it is necessary to measure the quality factor of a mode. It is the most

consistently measured quantity in the lab, therefore our methods are outlined here. There are

many ways to measure this quantity; analyzing a mode's transient response to a modulated field

and the direct measurement of the FWHM of the mode are two common techniques. In cavity

ringdown, the input field is regulated with an electrooptic modulator, for example. By

introducing a square wave pulse to the cavity, the field decays exponentially with time after the

field is turned off, analogous to a discharging capacitor. An exponential regression can then be

fitted to the data and its decay constant readily analyzed. In the case of phase sensitive cavity

ringdown, the input laser power can be modulated with a sine wave )1(2

1tSinPP oin ω+= at a

frequency comparable to the cavity linewidth which provides even greater certainty [1, 19].

Measuring the FWHM of the mode is the quickest method of acquiring the quality factor.

The oscilloscope provides voltage vs. time data so measuring the FWHM of the dip provides a

width in seconds t∆ , ν

ν

∆=Q so we need the width in terms of frequency ν∆ . The laser is

tuned by applying a voltage to a piezoelectric transducer mounted on the output coupler of the

laser cavity. This voltage is provided by a function generator that supplies a triangle wave signal

to the transducer. So, the tuning rate depends on the frequency of the signal sν and the tuning

range depends on the peak-to-peak voltage amplitude ppV . The transducer will displace the

mirror linearly as long as the wave amplitude is in a certain voltage range, however, there is a

slight nonlinear response with a typical voltage. With respect to ppV measured in volts, the scan

range (SR) from the laser is

2 037.2 933.4)( pppppp VVVSR += GHz. (A.1)

Now the tuning rate can easily be calculated as,

28

sSRrate ν⋅= 2 . (A.2)

We need to include only the voltage rise of a single pulse which only occurs for half of a wave,

hence the factor of 2. Now the time width of the mode can be converted to a width in frequency,

ratet ⋅∆=∆ν (A.3)

Finally,

νλν

ν

∆=

∆=

cQ , (A.4)

where c is the speed of light and the final expression is in terms of loaded Q which implies that

both intrinsic and coupling loss are contributing to the cavity linewidth.

VITA

Erik Kirklind Gonzales

Candidate for the Degree of

Master of Science

Thesis: AN INVESTIGATION OF INTERMODAL COUPLING EFFECTS IN

OPTICAL MICRORESONATORS Major Field: Physics Biographical:

Education: Received a Bachelor of Science degree in Physics from East Central University, Ada, OK in May 2007; completed requirements for the Master of Science degree in Physics from Oklahoma State University, Stillwater, OK in December 2011.

Experience: Machinist for Gonzales Machine, Inc. 1999-2003, Environmental

Scientist for the Environmental Protection Agency 2007, Bridge to the Doctorate Fellow, Oklahoma State University, 2009-2011, Graduate Teaching Associate, Oklahoma State University, Physics Department, 2010-2011.

Professional Memberships: American Physical Society, Optical Society of

America.

ADVISER’S APPROVAL: Dr. Albert T. Rosenberger

Name: Erik Kirklind Gonzales Date of Degree: December, 2011 Institution: Oklahoma State University Location: Stillwater, Oklahoma Title of Study: AN INVESTIGATION OF INTERMODAL COUPLING EFFECTS IN

OPTICAL MICRORESONATORS Pages in Study: 28 Candidate for the Degree of Master of Science

Major Field: Physics Scope and Method of Study: The goal in this report was to examine cross polarization

coupling in order to broaden the understanding of its cause and extend the promise of its applications. Experimental methods were used to directly test the co-resonance hypothesis. A theoretical model was developed to provide insight for future experimentation.

Findings and Conclusions: It has been concluded that co-resonance is required between

orthogonally polarized modes for CPC to occur. This research has also supported the belief that forward scattering is the mechanism for intrinsic polarization coupling. The possibility of a polarization analyzer utilizing CPC has been hypothetically demonstrated and should therefore be investigated more thoroughly to realize this potentially practical tool. The success of the theoretical model has increased the understanding of CPC's fundamental nature. The excellent replication of experimental data has further demonstrated the power of the simple ring cavity model and allows one to safely skip, for example, the direct approach of solving Maxwell's equations.