Optical Fiber Technology 17 (2011) 321–325

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Optical Fiber Technology

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Optical resonances in dielectric microspheres for transverse magnetic wave

Anisur RahmanMechanical Engineering Department, Polytechnic Institute of NYU, 6 MetroTech Center, Brooklyn, NY 11201, USA

a r t i c l e i n f o

Article history:Received 3 January 2009Available online 2 July 2011

Keywords:WGMOptical resonanceDielectric microsphereTM waveOptical sensors

1068-5200/$ - see front matter � 2010 Elsevier Inc. Adoi:10.1016/j.yofte.2010.11.002

E-mail address: mdanisur@gmail.com

a b s t r a c t

Optical resonances for Transverse Magnetic (TM) wave based on Whispering Gallery Modes (WGM) havebeen presented with comprehensive mathematical formulations. The present study is characterized boththeoretically and experimentally by considering the fact that the size parameter of the dielectric micro-spheres is very large at optical wavelengths. An asymptotic expression has been developed based onelectromagnetic theory for large size parameter. The developed expressions for optical resonance condi-tion of TM wave are very simple and can accurately characterize resonances in dielectric microspheres.The theoretical development is mathematically robust and significantly less complicated than existingapproaches based on quantum physics presented in the literatures. The theoretical result of size param-eter for consecutive morphology dependent resonance (MDR) peaks is validated by experimental data.The comparisons are shown to be very accurate for large size parameters.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

Over the decades optical dielectric microsphere resonators haveearned overwhelming research interests [1] due to its unique char-acteristics and potential applications such as acceleration sensors[2], optical biosensors [3,4], thermo-optical switches [5], atomicentanglement [6], evanescent spectroscopy [7], recurrent scatter-ing between dielectric microspheres [8], and wavelength-divisionmultiplexing [9]. In recent years study of MDR, also known aswhispering gallery modes (WGM), has been reported in theliterature both theoretically and experimentally [9–14]. However,theoretical approaches published in the literature are quite compli-cated. WGM and MDR presented in the literature [1,9–11] arebased on classical quantum mechanics that are difficult to follow.The reported theoretical approaches neither give simple usableresults nor can explain the MDR peaks sufficiently. The presentstudy is the first to our knowledge of a complete asymptoticsolution of TM wave in dielectric microspheres for large sizeparameter which is simpler and mathematically robust than exist-ing approaches presented in the literatures [1,9–11].

This paper describe resonances due to TM wave in the scatter-ing of radiation by microspheres that occur at certain discretewavelengths for a given sphere size and refractive index. Althoughresonant structures and their properties have been studied exten-sively in the literature via stringent electromagnetic analyses, thepresent study is the first to consider simpler approaches that takeadvantage of the fact that the size parameter of the microsphere isvery large.

ll rights reserved.

Depending on the wave polarization, electromagnetic wave canbe defined as three basic modes: Transverse Electric (TE) mode, TMmode and hybrid (combination of TE & TM) mode. In the presentstudy, radiation resonances due to TM wave in dielectric micro-spheres are characterized both theoretically and experimentallyfor large size parameter of microspheres. A new theoretical ap-proach has been developed to characterize MDR resonances inmicrospheres that are simpler than existing approaches in the lit-erature. The electromagnetic solutions from the literature are com-pared to the new expressions developed here that are significantlyencouraging. The theoretical result of size parameter for consecu-tive MDR peaks is validated by experimental data and the compar-isons are shown to be very accurate for large size parameters. Thequality factor of experimental resonance spectra observed in thelaboratory is calculated approximately in the order of 104 whichare sensitive enough to detect micro or nano level temperaturechanges in the surrounding medium.

2. Theory

It has been observed that when a dielectric microsphere andoptical fiber perfectly coupled, Electro-Magnetic (EM) wave entersinto microsphere from fiber and travels around inner surface of thesphere due to total internal reflection (TIR). While EM wave travelsinside the sphere, waves interfere each other in particular wave-length due to circular travel path of the sphere and produceMDR. Qualitatively, this is described as follows: after circumnavi-gating the microsphere, the light wave returns to the starting pointin phase to interfere constructively with itself. This constructiveinterference gives resonance peaks at certain discrete wavelengthsand these resonance peaks are known as MDR [11]. A robust

322 A. Rahman / Optical Fiber Technology 17 (2011) 321–325

asymptotic approach has been developed to explain MDR peaks forTM wave in dielectric microspheres.

In this study EM wave solution is considered when there is noincident plane wave inside the sphere. EM wave modes areassumed to be existed inside the sphere that is excited by theevanescent modes from the fiber optic waveguide. The solutionsinside the sphere are matched to the scattered field external tothe sphere at the boundaries, where it is expected that the exter-nally scattered field decays rapidly in the radial direction.

In case of fiber-sphere coupled condition with ~H tangential tosphere surface (TM mode), electromagnetic field inside the spherecan be expressed as [15]

~Ein ¼ �iX1n¼1

Endn~Nð1Þen ð1aÞ

~Hin ¼ �kin

xl1

X1n¼1

Endn~Mð1Þ

en ð1bÞ

Here,~E is the electric field vector and ~H is the magnetic field vector.The scattered field from the sphere can be expressed as [15]

~Eout ¼ iX1n¼1

Enan~Nð3Þen ð2aÞ

~Hout ¼kout

xl2

X1n¼1

Enan~Mð3Þ

en ð2bÞ

where k is wave number and defined as, kin ¼ 2pm1k0

, and kout ¼ 2pm2k0

,m1 and m2 are refractive indices of sphere and outside medium,respectively, m ¼ m1

m2is relative refractive index, k0 and k are wave-

lengths of light in free space and corresponding medium, respec-tively, x is angular frequency, l1 and l2 are permeability ofsphere and outside medium, respectively, a and d are constants,~M and ~N are vector spherical harmonics and subscript ‘e’ denoteeven.

The necessary boundary conditions for both electric andmagnetic fields can be expressed as

~Einjtan gential ¼~Eoutjtan gential at r ¼ R ð3aÞ~Hinjtan gential ¼ ~Houtjtan gential at r ¼ R ð3bÞ

Applying Eqs. (1a) and (2a) into Eq. (3a), the electric fieldboundary condition, and expanding the vector spherical harmonicsin component form [15] yields,

� iEndn cos /dPn

dhðcos hÞ 1

kinRd

dq½qjnðqÞ�jkinR

� �eh

�

� ðsin /ÞPncos hsin h

1kinR

ddq½qjnðqÞ�jkinR

� �e/

�

¼ iEnan cos /dPn

dhðcos hÞ 1

koutRd

dq½qhð1Þn ðqÞ�jkout R

� �eh

�

� ðsin /ÞPncos hsin h

1koutR

ddq

qhð1Þn ðqÞh i

jkout R

� �e/

�

which can be simplified to

�idn1

kinRd

dq½qjnðqÞ�jkinR¼ ian

1koutR

ddq

qhð1Þn ðqÞh i

jkout R . . . . . .ðeh termsÞ

�idn1

kinRd

dq½qjnðqÞ�jkinR¼ ian

1koutR

ddq

qhð1Þn ðqÞh i

jkout R . . . . . .ðe/ termsÞ

Since both eh and e/ terms yield same expression,

an1

kout

ddq

qhð1Þn ðqÞh i

jkout R þ dn1

kin

ddq½qjnðqÞ�jkinR ¼ 0 ð4Þ

Similarly, applying Eqs. (1b) and (2b) into Eq. (3b) for the mag-netic field boundary condition and expanding the vector sphericalharmonics in component form [15] yields,

� Kin

xl1Endn � sin /

sin hPnðcos hÞjnðkinRÞ

� �eh

�

� cos /dPnðcos hÞ

dhjnðkinRÞ

� �e/

�

¼ Kout

xl2Enan � sin /

sin hPnðcos hÞhð1Þn ðkoutRÞ

� �eh

�

� cos /dPn

dhðcos hÞhð1Þn ðkoutRÞ

� �e/

�

which could be simplified to

Kin

l1dnjnðkinRÞ ¼ �Kout

l2anhð1Þn ðkoutRÞ ðeh termsÞ

Kin

l1dnjnðkinRÞ ¼ �Kout

l2anhð1Þn ðkoutRÞ ðe/ termsÞ

Since both eh and e/ terms yield same expression,

Kout

l2anhð1Þn ðkoutRÞ þ

Kin

l1dnjnðkinRÞ ¼ 0 ð5Þ

So from Eqs. (4) and (5), there will be two equations for an anddn. Non trivial solution of an and dn can be found if

1kout

ddq qhð1Þn ðqÞh i

jkout R1

kin

ddq ½qjnðqÞ�jkinR

koutl2

hð1Þn ðkoutRÞ kinl1

jnðkinRÞ

������������ ¼ 0

which yields

kin

l1koutjnðkinRÞ d

dqqhð1Þn ðqÞh i

jkout R ¼kout

l2kin

ddq½qjnðqÞ�jkinRhð1Þn ðkoutRÞ

ð6Þ

Introducing size parameter, x ¼ 2pRk ¼

2pm2Rk0¼ koutR, and

kinR ¼ 2pm1Rk0¼ 2pm2R

k0

m1m2¼ mx, in Eq. (6)

ddq qhð1Þn ðqÞh i

jxhð1Þn ðxÞ

¼ l1

l2

1m2

ddq ½qjnðqÞ�jmx

jnðmxÞ ð7aÞ

as l1 = l2 for continuity,

xhð1Þn ðxÞh i0

hð1Þn ðxÞ¼ 1

m2

½mxjnðmxÞ�0

jnðmxÞ ð7bÞ

Eq. (7) is the resonance condition of TM modes for the dielectricmicrosphere with large size parameter. For a given n (integer) itwill give a value of x (or more than one value of x). jn is sphericalBessel function of first kind and h1

k is spherical Bessel function ofthird kind. Since h1

k is complex (ffi jn + iyn), the solution of Eq. (7)will be a complex x. Eq. (7) can be further simplified as follows[16]:

x ddq hð1Þn ðqÞh i���

xþ hð1Þn ðxÞ

hð1Þn ðxÞ¼ 1

m2

mx ddq ½jnðqÞ�jmx þ jnðmxÞ

jnðmxÞhð1Þnþ1ðxÞhð1Þn ðxÞ

¼ 1m

jnþ1ðmxÞjnðmxÞ þ

1xðnþ 1Þ 1� 1

m2

� �ð8Þ

After expanding spherical Bessel function of first kind (jn) andthird kind h1

n

for large x, Eq. (8) will be simplified as

1i¼ � 1

mcos zsin z

þ 1xðnþ 1Þ 1� 1

m2

� �

as mx� 12 np

� �¼ z (complex)

Table 1Comparison between experimental results and calculated values based on Eq. (11).

MDRresonancepeak

MDR resonancewavelength (nm), kexp

Size parameter at resonancewavelength xexp ¼ 2pR

kexpðR ¼ 500 lmÞ

Acceptableinteger, n

Calculated size parameter with acceptable

integer xcal ¼ p2m nþ 2ðnþ1Þ

mnp ðm ¼ 1:517ÞError (%)

e ¼ jxcal�xexp jxcal

� 100%

1 785.07 4001.67 3865 4002.48 0.022 785.23 4000.86 3864 4001.45 0.0153 785.41 3999.94 3863 4000.41 0.0124 785.60 3998.97 3862 3999.38 0.015 785.81 3997.90 3861 3998.34 0.01

Table 2Comparison between experimental results and calculated values based on Eq. (13).

MDRresonancepeak

MDR resonancewavelength (nm), kexp

Size parameter at resonancewavelength xexp ¼ 2pR

kexpðR ¼ 500 lmÞ

Acceptableinteger, n

Calculated Size parameter with acceptableinteger xcal ¼ p

2m nðm ¼ 1:517ÞError (%)

e ¼ jxcal�xexp jxcal

� 100%

1 785.07 4001.67 3865 4002.06 0.00972 785.23 4000.86 3864 4001.03 0.00423 785.41 3999.94 3863 3999.99 0.00134 785.60 3998.97 3862 3998.96 0.00035 785.81 3997.90 3861 3997.92 0.0005

A. Rahman / Optical Fiber Technology 17 (2011) 321–325 323

e2iz ¼ ðmþ 1Þðm� 1Þ 1þ i

ðnþ 1Þmx

ðmþ 1�mþ 1Þ� �

as x is large, ignoring 1x2 terms, which yields

mx ¼ 12

npþ ðnþ 1Þmx

� i ln

ffiffiffiffiffiffiffiffiffiffiffiffiffimþ 1m� 1

rð9Þ

Since imaginary part of Eq. (9) will yield significantly smallernumber compare to real parts, only real parts are considered inthe following formulation:

x ¼ np4m� np

4m1þ 8ðnþ 1Þ

ðnpÞ2

" #ð10Þ

Eq. (10) will give two solutions for x:

x ¼ np2mþ 2ðnþ 1Þ

mnp

and

x ¼ �2ðnþ 1Þmnp

However, size parameter, x cannot be negative value, hence asimple expression of size parameter x, for TM mode can be writtenas,

x ¼ np2mþ 2ðnþ 1Þ

mnpð11Þ

where, n is an integer. Further simplification of Eq. (11) yields,

x ¼ np

2mþ 2

mp1nþ 1

n2

� �� �ð12Þ

When n is very large, i.e.; n >> 1, & hence 1n << 1, and Eq. (12) can be

reduced to

x ¼ p2m

n ð13Þ

Eq. (13) reveals that size parameter for any MDR wavelengthwill be the product of an integer and a constant p

2m

� �. Therefore,

the size parameters for consecutive MDR peak wavelengths willbe multiple of consecutive integer numbers.

3. Validation of the theory

When laser current varies, the wavelength also varied due tochange in temperature inside the cavity of semiconductor laserdiode. As laser wavelength depends on laser current, the currentversus wavelength calibration gives the corresponding wavelengthwith respect to current. Experimental results show that the MDRpeaks are almost equally spaced with the change of wavelength[14]. However, some random small peaks are captured betweenthe larger peaks due to surface roughness of the dielectric micro-sphere which are ignored for their random appearances.

The size parameter for each MDR peak wavelength will be a prod-uct of a constant (p/2m) and an integer n. The integers will be con-secutive numbers for consecutive MDR peaks. Table 1 contains thecomparison between developed theoretical model and experimen-tal results found in the laboratory. The size parameter calculatedwith experimental MDR wavelength is very close to the size param-eter calculated by Eq. (11). The error between experimental resultand calculated results are in the negligible range (�10�2%) whichsupports the validation of theoretical development.

Using simplified Eq. (13), the calculated size parameter will bevery close to experimental MDR wavelength which is exactlysimilar to result presented in the literature [14] for TE wave. Theseresults show that MDR peaks behave same for both TE and TMwaves. Table 2 contains the comparison based on simplified Eq.(13). Using Eq. (13), error is reduced to negligible range (�10�3%)which supports the validation of developed theory. The acceptableintegers n are obtained by finding the nearest integer to (2mxexp/p).

Fig. 1 shows the comparison between experimental results andcalculated values for TM modes by using developed theory (Eq.(13)) for large size parameter at five different MDR peaks. Calcu-lated plots fit with experimental plots which strongly supportthe validation of the theoretical development for TM modespresented here.

Fig. 2 shows the comparison plot between the published exper-imental results [9] and calculated values based on Eq. (13) for fivedifferent MDR peaks with TM waves. Even though some of thepoints on Fig. 2 are off by few digits, the comparison plot reason-ably support the developed theory. It is important to note that dataextracted from the literature [9] for MDR peak wavelengths aredone by interpolation method. Hence, resonance wavelengths usedfor calculating the size parameter (x) have an increased source oferror, which is reflected in Fig. 2.

1 2 3 4 5 3997.5

3998

3998.5

3999

3999.5

4000

4000.5

4001

4001.5

4002

4002.5

MDR Peaks

Calculated with TM Eq. (13)

Experimental

Size

Par

amet

er, x

Fig. 1. Comparison between experimental results and calculated values for TMmodes.

1 2 3 4 5 3866

3866.5

3867

3867.5

3868

3868.5

3869

3869.5

3870

3870.5

MDR Peaks

Size

Par

amet

er, x

TM Calculated Eq. (13) TM Experimental [9]

Fig. 2. Comparison between experimental results for TM mode [9] and calculatedvalues by using developed theory for size parameter (x) at five different MDR peaks.

1 2 3 4 5 123

123.5

124

124.5

125

125.5

126

126.5

127

127.5

MDR Peaks

Size

Par

amet

er, x

Calculated-TM with Eq. (13) Experimental-TM [10]

Fig. 3. Comparison between experimental results for TM mode [10] and calculatedvalues by using developed theory (Eq. (13)) for size parameter (x) at five differentMDR peaks.

324 A. Rahman / Optical Fiber Technology 17 (2011) 321–325

Comparison between experimental results found in the labora-tory and experimental results found from the literature [9]strongly supports the validation of the developed theoreticalmodel. The size parameter calculated with experimental MDRwavelength [9] is almost similar to the size parameter calculatedby Eq. (13). The error again is small (�10�2%), although larger thanthat in Table 2.

Comparison between published results for the TM mode [10]and calculated values based on Eq. (13) are presented in Table 3.The error levels are small but significantly larger (�10�1%) than be-fore. The error level increases as size parameter decreases. As dis-cussed before, it could be argued that error level shown in Table 3increases due to mismatch of the size parameter between experi-mental [10] and theoretical assumptions, i.e., theory developedhere considering the fact that the size parameter is very large.Fig. 3 shows the comparison between experimental results for

Table 3Comparison between published experimental results for TM mode [10] and calculated va

MDRresonancepeak

MDR resonancewavelength (nm),kexp

Size parameter at resonancewavelength xexp ¼ 2pR

kexpðR ¼ 12 lmÞ

Accepta

I � 4m1 Rkexp

1 595.06 126.71 1292 598.05 126.07 1283 601.05 125.44 1274 604.05 124.82 1265 607.05 124.20 125

the TM mode [10] and calculated values by using the developedtheory (Eq. (13)) for size parameter (x) at five different MDR peaks.

4. Discussion

In this study, after solving the Maxwell’s equations for TMwaves in dielectric microsphere, a simple and usable new expres-sion for the size parameter has been developed (Eq. (13)). It isfound that the simplified mathematical formulation of the sizeparameter gives similar result to those obtained by experiment,as presented in Table 1. The error level is very negligible(�10�3%). EM solutions of TM wave based on quantum physicsfrom the literatures [1,9–11] are compared with the new approachdeveloped here. It is found that the present approach is producedvery accurate result for large size parameters. The new developedexpressions of size parameter for TM waves are found very similarwith TE waves [14], which demonstrate that MDR peaks behavesame for both TE and TM waves. This theoretical fact reveal thatoptical resonances in dielectric microsphere will exhibit similarshifts in MDR peaks for both TE and TM waves, i.e., wavelengthshifts in MDR peaks will be same regardless of polarization of inci-dent EM waves. Therefore, polarization of incident beam has notbeen considered and only hybrid mode is used in the presentexperimental setup. However, effect of polarization will be verifiedin future study. The wavelength of the excited EM wave has beenmeasured by using spectrometer with the resolution of 0.05 nm.

The expression for the size parameter found by the new asymp-totic approach for TM wave is compared with published results inthe literature [9,10]. Comparison between reported results [9,10]and calculated values based on the new approach show that thesize parameter calculated with the experimental MDR wavelengthsis very similar to the size parameters calculated by Eq. (13). Theerror level is on the order of hundredths of a percent (�10�2%).

lues based on newly developed theoretical expression (Eq. (13)).

ble integer, n

ðm1 ¼ 1:59ÞCalculated Size parameter withacceptable integer xcal ¼ p

2m1Iðm1 ¼ 1:59Þ

Error (%)

e ¼ jxcal�xexp jxcal

� 100%

127.4420 0.5768126.4540 0.3010125.4661 0.0175124.4782 0.2755123.4903 0.5782

A. Rahman / Optical Fiber Technology 17 (2011) 321–325 325

5. Conclusion

Comparison between theoretical and experimental results forTM waves reveals that theoretical development for the size param-eter in a dielectric microcavity could successfully explain the MDRpeaks. The theoretical result of the size parameter for consecutiveTM-MDR peaks are validated by experimental data reported in theliteratures [9,10], and the comparisons are shown to be very accu-rate for large size parameters. The present theoretical and experi-mental approaches could potentially be used in applications ofnano science, proteomics, Surface Plasmon Resonance (SPR), Mi-cro-Electro-Mechanical Systems (MEMS), biomedical, etc.

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