G. Adamovsky and S. Wrbanek

Glenn Research Center, Cleveland, Ohio

B. Floyd

Sierra Lobo Incorporated, Cleveland, Ohio

M. Crotty

Glenn Research Center, Cleveland, Ohio

Polarization Dependent Coupling of Whispering 

Gallery Modes in Microspheres

NASA/TM—2010-216902

November 2010
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G. Adamovsky and S. Wrbanek

Glenn Research Center, Cleveland, Ohio

B. Floyd

Sierra Lobo Incorporated, Cleveland, Ohio

M. Crotty

Glenn Research Center, Cleveland, Ohio

Polarization Dependent Coupling of Whispering 

Gallery Modes in Microspheres

NASA/TM—2010-216902

November 2010

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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Acknowledgments

The work has been supported by the Integrated Vehicle Health Management (IVHM) Project of the NASA Aviation Safety 

Program. M. Crotty also wishes to acknowledge the support of the NASA Undergraduate Student Research Program and 

NASA personnel involved in managing, organizing, and implementing the program at the Glenn Research Center.

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2010-216902 1 

Polarization Dependent Coupling of Whispering 

Gallery Modes in Microspheres 

G. Adamovsky and S. Wrbanek 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

B. Floyd 

Sierra Lobo Incorporated 

Cleveland, Ohio 44135 

M. Crotty1

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

Two sets of resonances in glass microspheres attached to a standard communication-grade single-mode 

optical fiber have been observed. It has been found that the strength of the resonances depends strongly on the 

polarization of the coupled light. Furthermore, the position of the resonances in the wavelength domain depends 

on the polarization of light in the optical fiber with maximum magnitudes shifted by approximately 45. 

1.0 Introduction 

The field of electromagnetic wave propagation is a well developed branch of physics and various physical 

models that have been supported by appropriate mathematical formulations(Refs. 1 to 3). That includes 

propagation of optical field in waveguides including optical fibers. 

In recent years, that area of physics has received further attention partially owing to the advances in 

micro- and nano-optics. Specifically, with the advent of a phenomenon called morphology-dependent 

resonances or whispering gallery modes in dielectric resonant cavities like spheres and cylinders, optical 

waveguides have been used to deliver the optical field to the cavities (Ref. 4). The resonances observed and 

reported have been, in most cases, explained only in terms of properties of the resonant cavities and the 

environment. 

The present paper documents our efforts to demonstrate and explain some effects of polarization of the 

incident light propagating in an optical fiber on the resonances triggered by coupling that light into 

microspheres. 

2.0 Theoretical 

In this work we deliver light to the microsphere using an optical fiber, which is a dielectric waveguide. 

The theory of wave propagation in cylindrical dielectric waveguides (Refs. 5 and 6) teaches that, from the 

formal solutions of Maxwell equations in cylindrical coordinates using matching boundary conditions on the 

core-cladding interface, two sets of solutions designated as the EH and HE modes could be extracted. Each of 

these two sets of modes by itself is a family of a sequence of modes defined by the parameter l. These sets, 

 

1

NASA Glenn Research Center, USRP Summer Intern. 
NASA/TM—2010-216902 2 

designated as EHlm and HElm modes, are called hybrid modes, and have contributions from both Ez and Hz

components. The relative contributions of Ez and Hz components to a transverse component of the field 

determines the designation of these hybrid modes. Furthermore, if l = 0, we arrive at a special case when the 

field components of the modes are radially symmetric and the modes have nonvanishing fields components 

Hr, Hz, E associated with TE for one set of modes and Er, Ez, H associated with the TM modes for the other 

set. These sets of modes are TE0m and TM0m modes, respectively. Analyses done by Yariv (Ref. 6) have also 

shown that HE11 does not have a cutoff frequency and the mode propagates in a fiber with any diameter. It 

also means that the mode HE11 propagates at any wavelength. 

Each propagating mode has its own propagation constant , which is usually expressed as a function of the 

normalized frequency V. Furthermore,  = nk0, where n the effective mode index or the ratio of the speed of 

light in vacuum to the mode phase velocity. The value of the normalized frequency determines the number of 

modes the optical fiber can propagate. If V2.405, then only one mode, HE11, could propagate and the fiber 

would be a single-mode fiber. 

The mathematical treatment of mode propagation in a cylindrical fiber is significantly simplified with the 

introduction of the weakly guiding modes approximation (Ref. 7). It follows from the approximation that the 

longitudinal components of the propagating fields are small compared with the transverse ones. Furthermore, 

the modes are considered to be linearly polarized in the transverse plane (Ref. 8). Such modes are called 

linearly polarized and designated as LPlm modes. In the LP modes the transverse fields inside the core are 

presented as products of Bessel functions in the radial direction and trigonometric functions sines and cosines 

in the angular direction (Refs. 9 and 10). The subscript l represents the lth-order Bessel function 

corresponding to the cutoff condition of the mode and m is the number of successive zeros in that Bessel 

function. In other words, the subscript l gives the number of angular nodes or azimuthal periodicity in the 

field distribution and the subscript m gives the number of radial nodes. 

Each of the LPlm modes has also two mutually orthogonal components associated with E and H

components of the electromagnetic field. Thus, modes with l = 0 have two orthogonal states of polarization 

and “donut shaped” distributions of field intensities. Furthermore, the lowest order linearly polarized mode 

LP01 includes the conventional lowest order mode He11. 

The next linearly polarized mode, LP11, is a combination of HE21, TM01, and TE01 conventional modes. It 

can be constructed choosing sin(l) or cos(l) in the transverse angular distribution ( is the angular 

coordinate) of the field intensity and two orthogonal states of polarization. As a result, the LP11 mode has four 

possible distributions with 90° symmetry in the circumferential (angular)  direction. The LP11 mode occurs 

when 2.405V3.832. As the value of V increases even higher order modes become guided. Eventually we 

will have a multimode fiber with a large number of modes propagating through it. 

When the propagating LP modes reach a microsphere, they are trapped in the sphere and form whispering 

gallery modes—also known as morphology dependent resonances. Those resonances also occur in resonant 

cavities with shapes formed by bodies of revolution such as cylinders, ellipsoids, etc. Resonant cavities 

discriminate certain wavelengths by selectively insolating them. Geometry and dimensions of the cavity, 

material properties, and the environmental conditions are the main factors in determining the wavelengths. 

Descriptions of optical processes in microspheres could be found elsewhere (Ref. 11). Owing to the 

dependency of resonances on geometric and material properties of microspheres, as well as the environment, 

such microspheres have been proposed as sensing structures for various applications including medical, 

chemical, and aerospace (Ref. 12). 

3.0 Experimental 

The basic apparatus to generate whispering gallery modes in microspheres has been already published 

(Ref. 11). It comprises a light source, usually a tunable semiconductor laser diode, a microsphere about 

450 μm in diameter and usually mounted on a stem for easy handling and manipulation, a waveguide to 

deliver light from the source to the microsphere, and a photodetector. For instance, Figure 1, adopted from 
NASA/TM—2010-216902 3 

Reference 12, shows a schematics setup to generate optical resonances in a microsphere via a side-coupled 

optical fiber. 

The operating wavelength of the laser is between 1290 and 1330 nm, and the output power is about 

20 mW. The laser operates in a tunable mode and emits light whose wavelength varies in time. This is 

achieved by sending a monotonically changing current through the laser diode junction. In response, the 

wavelength changes accordingly. The light generated by the laser is coupled into an optical fiber. The 

microsphere is placed close to the fiber in such a manner that light at certain wavelengths is coupled into the 

microsphere. The photodetector detects the light that passes through the fiber. When the microsphere is 

absent, the photodetector displays a continuing spectrum across all wavelengths emitted by the laser over the 

given period of time. However, the microsphere extracts certain wavelengths, and signals at these 

wavelengths are not being carried toward the photodetector. The result appears as sharp decreases in light 

intensity at wavelengths that couple into the sphere. 

To obtain stable operation of the laser, the process of changing current through a laser junction is 

accompanied by keeping the junction temperature constant. However, the resultant tuning range of the laser 

diode does not exceed 0.6 nm. This range has been often found insufficient to observe the resonances. To 

increase the tuning range, the process of changing current is combined with changing the constant 

temperature ranges (Ref. 12). 

Various techniques to deliver light to the microsphere via an optical fiber have been proposed (Ref. 13). 

One of the most efficient arrangements to couple light into a microsphere using optical fibers involves a 

device called either fiber half-block or half-coupler (Ref. 14). The half-coupler is constructed by cementing a 

piece of fiber in a glass block and polishing a flat section on the fiber cylindrical surface. The polishing 

removes part of the cladding, therefore a microsphere placed on the flat section is located closer to the core. 

Figure 2 shows our experimental setup used to demonstrate the effects of using a fiber to deliver light to a 

microsphere. It has a tunable laser, a laser controller, a half-coupler with a microsphere positioned on the top 

of it, and an optical fiber. Electrical current from the laser controller is applied to a tunable laser. The current 

is time dependent and, as a result, the wavelength of light emitted by the laser has an identical time 

dependency. The light is coupled into a commercial single-mode fiber with parameters described in the 

following paragraph. Also, a linear polarizer is incorporated in the path of the light. This is done by putting 

the polarizer between two fibers as shown in Figure 2 and aligning the fibers to achieve the maximum 

coupling of light from one fiber to another through the polarizer. 

The fiber used in the setup is a commercial-grade single-mode fiber specified for 1300/1550 nm 

wavelengths. The fiber is a standard in the communication industry and has the following parameters: the core 

diameter 8.2 μm, refractive index difference  = 0.36 percent, and the effective index of refraction n is 1.4677 

and 1.4682 for 1310 and 1550 nm wavelength, respectively (Ref. 15). Calculations show that the 

corresponding normalized frequency V, a parameter defined earlier, is 2.449 and 2.071 for those respective 

wavelengths. Thus, the parameter V for the wavelength of 1310 nm and below is above the normalized cutoff 

frequency of 2.405. Technically, we do not have a single-mode operation anymore and rather are dealing with 

a two-mode case. The presence of the second mode does not practically change the propagating properties of 

the fiber because that mode propagates very close to the core-cladding interface and carries a very 

insignificant amount of power. 

Figure 1.—Schematic setup to generate optical resonances in a microsphere by side-coupled optical fiber. 

[Adopted from Ref. 12] 

Tunable laser 

Microsphere 

Photodetector 

Optical fiber 
NASA/TM—2010-216902 4 

Figure 2.—Diagram of the experimental setup. 

TABLE 1.—TABULATED CALIBRATION DATA FOR THE LASER DIODE 

Current, 

mA 

Temperature, 

5 °C 

Temperature, 

10 °C 

Temperature, 

15 °C 

Temperature, 

20 °C 

Temperature, 

25 °C 

 Wavelength, 

nm 

Wavelength, 

nm 

Wavelength, 

nm 

Wavelength, 

nm 

Wavelength, 

nm 

20 1303.548 1304.023 1304.5 1304.978 1305.458 

30 1303.576 1304.051 1304.515 1304.993 1305.49 

40 1303.607 1304.084 1304.53 1305.01 1305.509 

50 1303.641 1304.101 1304.563 1305.044 1305.527 

60 1303.678 1304.119 1304.6 1305.082 1305.567 

70 1303.717 1304.157 1304.639 1305.124 1305.609 

80 1303.76 1304.198 1304.682 1305.168 1305.656 

90 1303.806 1304.242 1304.728 1305.216 1305.705 

100 1303.855 1304.29 1304.778 1305.267 1305.758 

110 1303.908 1304.341 1304.831 1305.321 1305.814 

120 1303.963 1304.395 1304.886 1305.379 1305.874 

125 1303.992 1304.452 1304.945 1305.44 1305.936 

130 1304.021 1304.512 1305.007 1305.503 1306.002 

140 1304.099 1304.575 1305.073 1305.571 1306.071 

Prior to starting the experiment, a calibration procedure is performed to assure the continuity of the laser 

performance at different temperatures and different currents. We used five fixed temperatures regimes, 5, 10, 

15, 20, and 25 C, and changed the current at each temperature. The calibration involved finding the 

wavelength-current relationship at every fixed laser diode temperature and doing a continuous mapping. 

Results of these measurements are presented in Table 1. 

4.0 Results and Discussion 

The experiment involved scanning the laser wavelength through all five temperature segments and 

observing and recording resonances at each temperature segment separately. The experiment consisted of two 

parts. The first part was conducted without the polarization filter to observe the resonances that occurred over 

the entire tuning range of the laser diode at all temperature regimes. Figures 3(a) to (e) show resonances 

obtained during the first part of the experiment over five individual temperature segments. 

Laser 

controller 

Tunable laser 

Time Time 

I 

Photodetector 

Linear 

polarizer 

Halfblock 

Microsphere 

λ

Fiber Fiber
NASA/TM—2010-216902 5 

a) 

Figure 3.—Resonances obtained without the polarizer for the laser diode temperatures: a) 5 C; b) 10 C; 

c) 15 C; d) 20 C; e) 25 C. 

b) 

Figure 3.—Continued.
NASA/TM—2010-216902 6 

c) 

Figure 3.—Continued. 

d) 

Figure 3.—Continued. 
NASA/TM—2010-216902 7 

e) 

Figure 3.—Concluded.

The second part of the experiment was conducted with the polarizer inserted in the gap between the fiber 

from the laser diode and the fiber to the microsphere as is shown in Figure 2. The polarizer was placed in a 

special mount that permitted rotation of the polarizer around its optical axis. The mount has a dial with 360 

angular gradations separated by 1 angular degree. The dial enables an accurate rotation of the polarizer. 

In the process of conducting the second part of experiment we set the temperature of the laser at a certain 

fixed temperature and rotated the polarizer with a 10 angular increment. At each angular position of the 

polarizer we conducted the wavelength scan over the appropriate wavelength range and recorded the 

resonances. That process was repeated at all five temperature segments. The recorded resonances for the first 

temperature segment, 5 C, are shown in Figure 4. The resonances are grouped in columns with four angular 

positions per row. The resonances in each row are obtained by rotating the polarization filter by 90° and they 

are almost identical. The columns show resonances obtained under incremental change in the angular position 

of the filter. 

Owing to the clear 90° symmetry in resonances displayed in Figure 4, the next figure (Fig. 5) shows those 

resonances selected from the complete set in Figure 4 that occurred within the first polarization quadrant. 

Similar sets of resonances were also obtained with the laser diode thermally stabilized at 15, 20, and 

25 C. In each position of the polarizer we conducted the wavelength scan over the entire tuning range over 

all five temperature segments. Resonances over the entire tuning range for four angular positions of the 

polarizer, 0°, 60°, 120°, and 180°, are shown in Figure 6. 

The columns in Figure 6 represent resonances observed over the entire tuning range of the laser for the 

selected angular positions of the polarizer. The rows in the figure represent resonances at the selected angular 

positions obtained at a given stabilized temperature of the laser. 

 
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 40° 130° 220° 310° 

Figure 4.—Resonances obtained with the laser diode thermally stabilized at 5 C and the polarizer rotated 

with a 10 increment. 

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 80° 170° 260° 350°

Figure 4.—Concluded. 

 
NASA/TM—2010-216902 10 

0° 10° 20° 

 30° 40° 50° 

 60° 70° 80° 

Figure 5.—Resonances from the set shown in Figure 4 that occurred within the first quadrant. 

 
NASA/TM—2010-216902 11 

 0° 60° 120° 180° 

5 °C 

10 °C 

15 °C 

20 °C

25 °C 

 

Figure 6.—Resonances over the entire tuning range for four angular positions of the polarizer. 

5.0 Conclusions 

The patterns presented in Figures 4 and 5 clearly show a presence of two sets of resonances. The 

strength of the resonances varies with the angular polarization and each of the sets has a 90° angular 

periodicity. Moreover, the sets appear to have angular positions of the polarizer that correspond the 

maximum strengths of the resonances shifted by, what could be estimated as, about 45°. 
NASA/TM—2010-216902 12 

The behavior of the resonances is characteristic of the behavior of two linearly polarized modes, LP01 

and LP11. The 90° symmetry in the resonance patterns could be explained by the existence of two 

mutually orthogonal components associated with E and H components of the electromagnetic field. The 

fact that the maxima in the intensities for resultant resonances in the microsphere do not occur at the same 

angle could be explained by the differences in the electric (and magnetic) field amplitude profiles for LP01

and LP11 modes. 

We have demonstrated experimentally that the strength and wavelength of whispering gallery 

resonances in microspheres depends on the polarization of the incident light. We also demonstrated 

appearance of two sets of resonances that could be attributed to the fact that at our operating wavelengths 

the fiber used was not single-mode and two linearly polarized modes, LP01 and LP11, were present. The 

LP11 mode propagates close to the core—cladding interface and is also coupled into the microsphere. 

The setup that we presented and arrangement of the experiment demonstrated polarization dependent 

coupling of whispering gallery modes in microspheres. The strength and location of these modes in 

wavelength domain appeared to be in agreement with the two lowest linearly polarized modes 

propagating in the fiber, LP01 and LP11. 

References 

1. Collin, R.E.: Field Theory of Guided Waves. Second ed., IEEE Press, New York, NY, 1990. 

2. Balanis, C.A.: Advanced Engineering Electromagnetics. John Wiley & Sons, New York, NY, 1989. 

3. Ishimary, A.: Electromagnetic Wave Propagation, Radiation, and Scattering. Prentice Hall, 

Englewood Cliffs, NJ, 1991. 

4. Arnold, S.: Microspheres, Photonic Atoms and the Physics of Nothing. American Scientist, vol. 89, 

no. 5, 2001, pp. 414–421. 

5. Marcuse, D.: Light Transmission Optics. Second ed., ch. 8, Van Nostrand Reinhold Company, 

New York, NY, 1982. 

6. Yariv, A.: Optical Electronics. Propagation of Optical Beams in Fibers, Third ed., ch. 3, CBS College 

Publishing, New York, NY, 1985, pp. 54–86. 

7. Gloge, D.: Weakly Guiding Fibers. Appl. Opt., vol. 10, no. 10, 1971, pp. 2252–2258. 

8. Jeunhomme, L.B.: Single-Mode Fiber Optics: Principles and Applications. Second ed., ch. 1, Optical 

Engineering Series, B.J. Thompson, ed., vol. 23, Marcel Dekker, New York, NY, 1990, pp. 1–59. 

9. Gloge, D.: Propagation Effects in Optical Fibers. IEEE Trans. Microwave Theory Tech., MTT–23, 

no. 1, 1975, pp. 106–120. 

10. Kumar D.; and Choudhury, P.K.: Introduction to Modes and Their Designation in Circular and 

Elliptical Fibers. Am. J. Phys., vol. 75, no. 6, 2007, pp. 546–551. 

11. Chang, R.K.; and Campilla, A.J., eds.: Optical Processes in Microcavities, Advanced Series in 

Applied Physics, vol. 3, World Scientific Publishing Co., Singapore, 1996. 

12. Adamovsky, G.; and Ötügen, V.: Morphology-Dependent Resonances and Their Applications to 

Sensing in Aerospace Environments. AIAA J. Aerospace Computing, Information, and 

Communication, JACIC, vol. 5, no. 10, 2008, pp. 409–424. 

13. Little, B.E., et al.: Pedestal Antiresonant Reflecting Waveguides for Robust Coupling to Microsphere 

Resonators and for Microphotonic Circuits. Opt. Lett., vol. 25, no. 1, 2000, pp. 73–75. 

14. Serpengüzel, A.; Arnold, S.; and Griffel, G.: Excitation of Resonances of Microspheres on an Optical 

Fiber. Opt. Let., vol. 20, no. 7, 1995, pp. 654–656. 

15. Corning® Single-Mode Optical Fiber, Corning Inc., 2000. 

http://cc.ee.ntu.edu.tw/~ypchiou/Photonic_Crystals/SMF28Fiber.pdf 
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Polarization Dependent Coupling of Whispering Gallery Modes in Microspheres 

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14. ABSTRACT

Two sets of resonances in glass microspheres attached to a standard communication-grade single-mode optical fiber have been observed. It 

has been found that the strength of the resonances depends strongly on the polarization of the coupled light. Furthermore, the position of the 

resonances in the wavelength domain depends on the polarization of light in the optical fiber with maximum magnitudes shifted by 

approximately 45°. 

15. SUBJECT TERMS

Optics; Fiber optics; Polarization (Waves); Resonance; Spheres; Optical coupling; Waveguide; Microspheres; Optical resonances 

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