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  1. 1. 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
  2. 2. 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
  3. 3. 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
  4. 4. 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
  5. 5. 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.
  6. 6. 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.
  7. 7. 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).
  8. 8. 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.
  9. 9. 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.
  10. 10. 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.
  11. 11. 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.
  12. 12. 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
  13. 13. 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
  14. 14. 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
  15. 15. 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.
  16. 16. 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 ο 1± . 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
  17. 17. 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).
  18. 18. 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 1fE and 2fE 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 1fE on and 2fE off will only excite the cavity's TE modes where TE corresponds to the intracavity field 1sE 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
  19. 19. 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 to 2sE . 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:
  20. 20. 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 1 1 sfc ErEiti L E ++−= δ α , (3) ]}[' 2 ' exp{ 22222 2 2 sfc ErEiti L E ++−= δ α , (4) are the fields just before the scattering center, and just before the output mirror we have: ])}['( 2 ' exp{ 211111 cscss EitEri LL E +−+ − −= δδα , (5) and ])}['( 2 ' exp{ 122222 cscss EitEri LL E +−+ − −= 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)
  21. 21. 17 2 ' 1 1 δ δ = , (8) and 2 ' 2 2 δ δ = . (9) After combining Eqs. (3) through (9) we end up with the intracavity fields 12 2 21 1 11 }) 2 exp{}( 2 exp{[ 1 fss Ei L rri L it D E δ α δ α +−−+−= ]} 24 exp{ 2 2121 2 fs EiLtt δδαα + + + −− , (10) and 21 1 12 2 22 }) 2 exp{}( 2 exp{[ 1 fss Ei L rri L it D E δ α δ α +−−+−= ]} 24 exp{ 1 1212 1 fs EiLtt δδαα + + + −− (11) Where } 2 exp{} 2 exp{1 2 2 21 1 1 δ α δ α i L rri L rrD ss +−−+−−= )}( 2 exp{ 21 21 21 δδ αα ++ + −+ 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.
  22. 22. 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 = ∆ = ∆ = k Ln Q kk k k k k 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 2 kkk tL +=∆ αδ (14) is the linewidth in δ ; (2) the loss ratio L t x k k k α 2 = ; (15) and (3) the round-trip scattering probability 2 st .
  23. 23. 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)
  24. 24. 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: 7 1 10)6.09.5( ×±=Q (TE), 7 2 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)
  25. 25. 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) φi f 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.
  26. 26. 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
  27. 27. 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.
  28. 28. 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. 617-620. [3] D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421(6926), 925-928 (2003). [4] R. Yang, W. H. Yu, Y. Bao, Y. X. Zhang, and X. Y. Pu, "Whispering-gallery modes based on evanescent field in cylindrical micro-cavity," Acta Physica Sinica 57(10), 6412-6418 (2008). [5] V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, "Quality-factor and nonlinear properties of optical whispering-gallery modes," Physics Letters A 137(7-8), 393-397 (1989). [6] S. I. Shopova, G. Farca, A. T. Rosenberger, W. M. S. Wickramanayake, and N. A. Kotov, “Microsphere whispering-gallery-mode laser using HgTe quantum dots,” Appl. Phys. Lett. 85, 6101-6103 (2004). [7] G. Farca, S. I. Shopova, and A. T. Rosenberger, “Cavity-enhanced laser absorption spectroscopy using microresonator whispering-gallery modes,” Opt. Express 15, 17443-17448 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25- 17443. [8] F. C. Blom, D. R. vanDijk, H. J. W. M. Hoekstra, A. Driessen, and T. J. A. Popma, "Experimental study of integrated-optics microcavity resonators: Toward an all- optical switching device," Appl Phys Lett 71, 747-749 (1997). [9] K. J. Vahala, "Optical microcavities," Nature 424(6950), 839-846 (2003). [10] V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, "Strain-tunable high-Q optical microsphere resonator," Optics Communications 145(1-6), 86-90 (1998).
  29. 29. 25 [11] J. Rezac and A. Rosenberger, "Locking a microsphere whispering-gallery mode to a laser," Opt. Express 8(11), 605-610 (2001). [12] D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, "Coupled-resonator-induced transparency," Physical Review A 69(6), 063804 (2004). [13] A. Naweed, G. Farca, S. I. Shopova, and A. T. Rosenberger, "Induced transparency and absorption in coupled whispering-gallery microresonators," Physical Review A 71(4), 043804 (2005). [15] M. V. Berry, "Quantal Phase Factors Accompanying Adiabatic Changes," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 392(1802), 45-57 (1984). [16] T. Lee, N. G. R. Broderick, and G. Brambilla, "Berry phase magnification in optical microcoil resonators," Opt. Lett. 36(15), 2839-2841 (2011). [17] M. L. Gorodetsky and V. S. Ilchenko, "High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers," Optics Communications 113(1-3), 133-143 (1994). [18] M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, "Ultrahigh-Q Tunable Whispering-Gallery-Mode Microresonator," Physical Review Letters 103(5), 053901 (2009). [19] J. Barnes, B. Carver, J. M. Fraser, G. Gagliardi, H. P. Loock, Z. Tian, M. W. B. Wilson, S. Yam, and O. Yastrubshak, "Loss determination in microsphere resonators by phase-shift cavity ring-down measurements," Opt. Express 16(17), 13158-13167 (2008). [14] D. S. Weiss, V. Sandoghdar, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, "Splitting of high-Q Mie modes induced by light backscattering in silica microspheres," Opt. Lett. 20(18), 1835-1837 (1995).
  30. 30. 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.
  31. 31. 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 1 tSinPP 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 2 037.2933.4)( pppppp VVVSR += GHz. (A.1) Now the tuning rate can easily be calculated as,
  32. 32. 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 ∆ = ∆ = c Q , (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.
  33. 33. 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.
  34. 34. 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.

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