Electric field distribution - SMU

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coupling light from a tunable laser into the sphere using an optical fiber. The modes are ... surface of the sphere through total internal reflection. A resonance ... uniform electric field E0 in the direction of negative z as shown in Fig. 1. Fig. 1.
T uning of whispering gallery modes of spherical resonators using an external electric field T indaro Ioppolo*1, Ulas A yaz 1, and M . Volkan Ö tügen 1 1

Department  of  Mechanical  Engineering,  Southern  Methodist  University,  Dallas,  TX  75275   *[email protected]

A bstract: In this paper we investigate the electrostriction effect on the whispering gallery modes (WGM) of polymeric microspheres and the feasibility of a WGM-based microsensor for electric field measurement. The electrostriction is the elastic deformation (strain) of a dielectric material under the force exerted by an electrostatic field. The deformation is accompanied by mechanical stress which perturbs the refractive index distribution in the sphere. Both strain and stress induce a shift in the WGM of the microsphere. In the present, we develop analytical expressions for the WGM shift due to electrostriction for solid and thin-walled hollow microspheres. Our analysis indicates that detection of electric fields as small as ~500V/m may be possible using water filled, hollow solid polydimethylsiloxane (PDMS) microspheres. The electric field sensitivities for solid spheres, on the other hand, are significantly smaller. Results of experiments carried out using solid PDMS spheres agree well with the analytical prediction. © 2009 Optical Society of America O C IS codes: (040.0040) Detectors; (080.0080) Geometric optics; (230.0230) Optical devices; (060.0060) Fiber optics and optical communications

References and links 1. W. von Klitzing³7XQDEOHwhispering modes for spectroscopy and CQED Experiments´New J. Phys. 3, 14.1± 14.14 (2001). 2. M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel³Fiber-coupled microsphere laser´Opt. Lett. 25(19), 1430± 1432 (2000). 3. H. C. Tapalian, J. P. Laine, and P. A. Lane³Thermooptical switches using coated microsphere resonators´ IEEE Photon. Technol. Lett. 14(8), 1118±1120 (2002). 4. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine³Microring resonator channel dropping filters´J. Lightwave Technol. 15(6), 998±1005 (1997). 5. B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona,Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks´IEEE J. Sel. Top. Quantum Electron. 5(5), 1400±1406 (1999). 6. V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche³Strain tunable high- Q optical microsphere resonator´Opt. Commun. 145(1-6), 86±90 (1998). 7. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold³Protein detection by optical shift of a resonant microcavity´Appl. Phys. Lett. 80(21), 4057±4059 (2002). 8. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer³Shift of whispering-gallery modes in microspheres by protein adsorption´Opt. Lett. 28(4), 272±274 (2003). 9. A. T. Rosenberger, and J. P. Rezac³Whispering-gallerymode evanescent-wave microsensor for trace-gas detection´Proc. SPIE 4265, 102±112 (2001). 10. N. Das, T. Ioppolo, and V. Ötügen, ³,QYHVWLJDWLRQRIDPLFUR-optical concentration sensor concept based on ZKLVSHULQJJDOOHU\PRGHUHVRQDWRUV´SUHVHQWHGDWWKHWK$,$$$HURVSDFH6FLHQFHV0HHWLQJDQG([KLELWLRQ Reno, Nev., January 8±11 2007. 11. T. Ioppolo, M. Kozhevnikov, V. Stepaniuk, M. V. Otügen, and V. Sheverev³Micro-optical force sensor concept based on whispering gallery mode resonators´Appl. Opt. 47(16), 3009±3014 (2008). 12. T. Ioppolo, U. K. Ayaz, and M. V. Ötügen³High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres´J. Appl. Phys. 105(1), 013535 (2009). 13. T. Ioppolo, and M. V. Ötügen³³3UHVVXUH7XQLQJRI:KLVSHULQJ*DOOHU\0RGH5HVRQDWRUV´J. Opt. Soc. Am. B 24(10), 2721±2726 (2007).

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14 September 2009 / Vol. 17, No. 19 / OPTICS EXPRESS 16465

14. G. Guan, S. Arnold, and M. V. Ötügen³Temperature Measurements Using a Micro-Optical Sensor Based on Whispering Gallery Modes´AIAA J. 44(10), 2385±2389 (2006). 15. T. Ioppolo, U. K. Ayaz, M. V. Ötügen, and V. Sheverev³$0LFUR-Optical Wall Shear Stress Sensor Concept Based on WhisperinJ*DOOHU\0RGH5HVRQDWRUV´WK$,$$$HURVSDFH6FLHQFHV0HHWLQJDQG([KLELW±11 January 2008. 16. V. M. N. Passaro, and F. De Leonardis³Modeling and Design of a Novel High-Sensitivity Electric Field Silicon-on-Insulator Sensor Based on a Whispering-Gallery-Mode Resonator´IEEE J. Sel. Top. Quantum Electron. 12(1), 124±133 (2006). 17. R. W. Soutas-Little, Elasticity, (Dover Publications Inc., Mineola, NY, 1999). 18. J. A. Stratton, Electromagnetic Theory (Mcgraw-Hill Book Company, Inc., New York and London, 1941). 19. F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan³Prism coupling technique investigation of elastooptical properties of thin polymer films´J. Appl. Phys. 96(12), 341±345 (2004). 20. J. E. Mark, Polymer Data Handbook (Oxford University Press, 1999). 21. T. Yamwong, A. M. Voice, and G. R. Davies³Electrostrictive response of an ideal polar rubber´J. Appl. Phys. 91(3), 1472±1476 (2002). 22. A. E. H. Love, The Mathematical Theory of Elasticity (Dover, 1926). 23. K. C. Kao, Dielectric Phenomena in Solids (Elsevier Academic Press, 2004).

1. Introduction Whispering gallery modes (WGM) of dielectric microspheres have attracted interest with proposed applications in a wide range of areas due to the high optical quality factors that they can exhibit. The WGM (also called the morphology dependent resonances MDR) are optical modes of dielectric cavities such as spheres. These modes can be excited, for example, by coupling light from a tunable laser into the sphere using an optical fiber. The modes are observed as sharp dips in the transmission spectrum at the output end of the fiber typically with very high quality factors, 4 ȜįȜ (Ȝ is the wavelength of the interrogating laser and įȜ is the linewidth of the observed mode). The proposed WGM applications include those in spectroscopy [1], micro-cavity laser technology [2], and optical communications (switching [3] filtering [4] and wavelength division and multiplexing [5]). For example, mechanical strain [6] and thermooptical [3] tuning of microsphere WGM have been demonstrated for potential applications in optical switching. Several sensor concepts have also been proposed exploiting the WGM shifts of microspheres for biological applications [7,8] trace gas detection [9], impurity detection in liquids [10] as well as mechanical sensing including force [11,12], pressure [13], temperature [14] and wall shear stress [15]. In this paper we investigate the effect of an electrostatic field on the WGM shifts of a polymeric microsphere. Such electrostriction-induced shifts could be exploited for WGM-based gas composition and electric field sensors. The concept of an electric field detector based on the WGM of a microdisk was discussed recently [16]. Potentially, the electrostatic field-driven micro-cavities could also be used as fast, narrowband optical switches and filters. The simplest interpretation of the WGM phenomenon comes from geometric optics. When laser light is coupled into the sphere nearly tangentially, it circumnavigates along the interior surface of the sphere through total internal reflection. A resonance (WGM) is realized when light returns to its starting location in phase. A common method to excite WGMs of spheres is by coupling tunable laser light into the sphere via an optical fiber [5,10]. The approximate condition for resonance is 2S n0 a

lO

(1)

where Ȝ is the vacuum wavelength of laser, no and a are the refractive index and radius of sphere respectively, and l is an integer representing the circumferential mode number. Equation (1) is a first order approximation and holds for a >>Ȝ. A minute change in the size or the refractive index of the microsphere will lead to a shift in the resonance wavelength as

dO O

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dn0 da  n0 a

(2)

Received 1 Jun 2009; revised 17 Jul 2009; accepted 31 Jul 2009; published 1 Sep 2009

14 September 2009 / Vol. 17, No. 19 / OPTICS EXPRESS 16466

Variation of the electrostatic field will cause changes both in the sphere radius (strain effect) and index of refraction (stress effect) leading to a WGM shift, as indicated in Eq. (2). In the following, we develop analytical expressions to describe the WGM shift of polymeric microspheres caused by an external electrostatic field. The analysis takes into account both the strain and stress effects. 2. E lectrostatic F ield-Induced Stress in a Solid Dielectric Sphere We first consider an isotropic solid dielectric sphere of radius a and inductive capacity İ1, embedded in an inviscid dielectric fluid of inductive capacity İ2. The sphere is subjected to a uniform electric field E 0 in the direction of negative z as shown in Fig. 1.

Fig. 1. The sphere in the presence of electric field.

The force exerted by the electrostatic field on the sphere will induce an elastic deformation (electrostriction) that is governed by the Navier Equation [17]:

’2 u 

1 f ’’ ˜ u  1  2Q G

0

(3)

where u is the displacement of a given point within the dielectric sphere, Ȟ is the Poisson ratio, G is the shear modulus, and f is the body force. Neglecting gravitational effect, the body force is due to the electric field, and is given by [18]:

f



1 2 1 E ’H1  a1  a 2 ’ E 2 2 4

(4)

where E is the electric field within the sphere, İ is the inductive capacity, a 1 and a 2 are coefficients that describe the dielectric properties. Physically, the parameter a 1 represents the change of inductive capacity İ due to an elongation parallel to the lines of the field, while a 2 determines this change for elongation in normal direction to the field. In this analysis, we assume that the electric and elastic properties of the microsphere in the unstrained configuration are isotropic. Therefore the first term on the right hand side of Eq. (4) is zero. #112051 - $15.00 USD

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14 September 2009 / Vol. 17, No. 19 / OPTICS EXPRESS 16467

The electric field inside the dielectric sphere is uniform and parallel to the z axis, with its magnitude [18]:

E

3H 2 E0 H1  2H 2

(5)

Therefore, the second term on the right hand side of Eq. (4) is also zero. Thus, Eq. (3) becomes:

’2u 

1 ’’ ˜ u 1  2Q

(6)

0

The solution of this equation in spherical coordinates is given by [14]:

ur u-

¦ ª¬ An n  1 n  2  4Q r n

 Bn nr n 1 º¼Pn cos - ½ °° dP cos - ¾ ¦ ª¬ An n  1 n  5  4Q r n 1  Bn nr n 1 º¼ n d- °°¿ 1

(7)

where ur and u- are the components of displacement in the radial, r, and polar, - directions. Pn¶V represent the Legendre polynomials, and An and Bn are constants that are determined by satisfying the boundary conditions. Using the stress displacement equations, the components of stress can be expressed as: 2G ¦ ª¬ An n  1 n 2  n  2  2Q r n  Bn n n  1 r n  2 º¼Pn cos -

(8)

­ ª An n 2  4n  2  2Q n  1 r n  Bn n 2 r n  2 º Pn cos -  ½ ¼ °¬ ° 2 G ¦ ® ¾ dP cos ° ª¬ An n  5  4Q r n  Bn r n  2 º¼ cot - n ° d¯ ¿

(9)

­ ª An n  1 n  2  2Q  4nQ r n  Bn nr n  2 º Pn cos -  ½ ¼ °¬ ° 2G ¦ ® ¾ dP cos ° ª¬ An n  5  4Q r n  Bn r n  2 º¼ cot - n ° d¯ ¿

(10)

V rr

V --

V II

2G ¦ ª¬ An n2  2n  1  2Q r n  Bn n  1 r n  2 º¼

V r-

wPn cos - w-

(11)

In an inviscid fluid, only normal (pressure) forces are acting on the sphere. The normal force per unit area acting on the interface of the two dielectrics (the sphere and its surrounding) is given by [18]

P









ªD E E ˜ n º  ªD E E ˜ n º  ª E E 2 n º  ª E E 2 n º ¼2 ¬ ¼1 ¬ ¼2 ¬ ¼1 ¬

(12)

where is the unit surface normal vector. The subscripts indicate that the values are to be taken on either side of the interface (1 represents the sphere and 2 represents the surrounding PHGLXP 7KHFRQVWDQWVĮDQGȕDUHJLYHQDV>18]:

D

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H

a 2  a1 2

,

E

H  a2 2

(13)

Received 1 Jun 2009; revised 17 Jul 2009; accepted 31 Jul 2009; published 1 Sep 2009

14 September 2009 / Vol. 17, No. 19 / OPTICS EXPRESS 16468

For the case of a sphere embedded in a dielectric fluid, the constants a 1 and a 2 are defined by the Clausius-Mossotti law [15] leading to:

D

H

E

,



H0 6

k

2

 2k  2

(14)

for the fluid (medium 2). Here, İ0 is the inductive capacity of vacuum, and k (N İİ0) is the dielectric constant. Using Eq. (5) and Eq. (12) the pressure acting at the dielectric interface is given by:

A  B Cos -

P

'

'

2

 B'

(15)

where $¶ and %¶ are defined as: 2 2 º § 3H 2 · ª§ H 1 · « E ¨ ¨ ¸ D 2  E 2  D1  E1 » 0¸ »¼ © H1  2H 2 ¹ «¬© H 2 ¹

A'

(16)

2

§ 3H 2 · E 0 ¸ E1  E 2 ¨ © H1  2H 2 ¹

B'

(17)

Equation (15) represents the pressure acting on the sphere surface due to the inductive capacity discontinuity at the sphere-fluid interface. Apart from this, the electric field induces a pressure perturbation in the fluid as well. This is given by

H0

P

6

E 2 ( k2  1)( k2  2)

(18)

For gas media, k |1, thus į3 is negligible. In order to define the stress and strain distributions within the sphere, coefficient A n and B n have to be evaluated. These coefficients are calculated by satisfying the following boundary conditions

V rr a

P

(19)

V r- a 0

The coefficient A n and B n are determined by expanding the pressure P in terms of Legendre series as follows:

¦ Z n Pn cos-

P

(20)

From Eq. (15), it can be noted that only two terms of the series in Eq. (20) are needed to describe the pressure distribution, from which the coefficients Zn are defined as:

Zo

1 ' A  2B' 3

Z2

,

2 ' A  B' 3

(21)

Plugging Eq. (8) and (11) and Eq. (20) and (21), into Eq. (19), the coefficients A n and B n are determinate as follows:

A  2B , A  '

A0

'

12G 1 Q

2

A B '



'

6Ga 2 5Q  7

A  B 2Q  7 '

, B2

'

6 G 5Q  7

(22)

The radial deformation can be determined by using Eq. (7): #112051 - $15.00 USD

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ur

2 A0 2Q  1 r  12 A2Q r 3  2 B2 r





1 2 3cos -  1 2

(23)

3. W G M Shift in a Solid Sphere Due to E lectrostriction We can evaluate the last term in Eq. (2) (the relative change in the optical path length in the equatorial belt of the microsphere at r = a and - ʌ) by plugging Eq. (22) into Eq. (23):

da a

­ 1  2Q ª§ § H ·2 ·º ½ ° «¨ ¨ 1 ¸ D 2  E 2  D1  3E1  2E 2 ¸ »  ° 2 ¨ ¸ ° § 3H 2 · °° 6 G 1  Q «¬© © H 2 ¹ ¹ »¼ ° E ® ¾ (24) ¨ 0¸ º © H1  2H 2 ¹ ° 4Q  7 ª § H ·2 ° «  ¨ 1 ¸ D 2  E 2  D1  E 2 » ° ° 3G 5Q  7 « © H 2 ¹ »¼ ¬ ¯° ¿°

As we can see from the above expression, the radial deformation, da/a , has a quadratic dependence on the electric field strength. Next we determine the effect of stress on refractive index perturbation, dn0/n0, in Eq. (2). Here we neglect the effect of the electric field on the index of refraction of the microsphere. The Neumann-Maxwell equations provide a relationship between stress and refractive index as follows [19]:

nr

nor  C 1V rr  C 2 V --  V MM

n-

no-  C 1V --  C 2 V rr  V MM

nM

noM  C 1V MM  C 2 V --  V rr

(25)

Here nr , n- , nM are the refractive indices in the direction of the three principle stresses and

n0 r , n0- , n0M are those values for the unstressed material. Coefficients C 1 and C 2 are the elastooptical constants of the material. In our analysis we consider PDMS microspheres that are manufactured as described in Ref [15]. For PDMS these values are C 1 = C 2 = C = 1.75x1010 m2/N [20]. Thus, for a spherical sensor, the fractional change in the refractive index due to mechanical stress is reduced to:

dno no

nr  nor nor

n-  nono-

nr  noM noM

C V rr  V --  V MM n

(26)

Thus, evaluating the appropriate expressions for stress in Eq. (8), 9, 10) at - = ʌ and r = a , and introducing them into Eq. (26) the relative change in the refractive index can be obtained. In order to evaluate the WGM shift due to the applied electric field, the constants a 1 and a 2 must be evaluated. Very few reliable measurements of these constants for solids have been reported in the literature. Unfortunately, to our knowledge there are no experimental measurements of a 1 and a 2 for polymeric material including PDMS. In our analysis we take the values developed for an ideal polar rubber [21]. In Fig. 2, the strain (da/a ) and stress (dn0/n0) effects on the WGM shifts due to an electric field are shown. The stress and strain have opposite effects on WGM shifts, but as seen in the figure, the strain effect dominates over that of stress and thus, the latter effect can be ignored in calculations. If we assume that the minimum measurable WGM shift is ¨Ȝ Ȝ4, the measurement resolution is defined as

G E O Q d O dE . The results of Fig. 2 indicate that for a quality factor of Q~10 7 an electric field as small as ~20 kV/m can be resolved with a solid PDMS microsphere (polymeric base to curing agent ratio of 60:1 by volume). 1

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Fig. 2. The WGM shift of a solid 1 mm diameter PDMS sphere due to applied electric field (base-to-curing-DJHQWUDWLRRIDȜ  

4. E lectrostatic F ield-Induced Stress in a Hollow Dielectric Sphere In this section we consider a dielectric spherical shell of inductive capacity İ1 with inner radius a and outer radius b that is placed in a uniform dielectric fluid of inductive capacity İ2 as shown in Fig. 3. The shell is filled with a fluid of inductive capacity İ3. As in the solid microsphere case, in order to determine the WGM shift, the strain distribution at the sphere outer surface must be known. In order to find this distribution the pressure acting at the surfaces, as well as the body force inside the shell has to be determined. In general, both the pressure and the body force are functions of the electric field distribution.

Fig. 3. Notation for a hollow dielectric sphere.

The electric field distribution in a dielectric is governed by Laplace's equation. The general solution of Laplace's equation in spherical coordinates ( r, -ij) is given as:

) r ,- , I

( Ai r i  Bi r ¦ i

 i 1

) Pi cos -

(27)

0

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14 September 2009 / Vol. 17, No. 19 / OPTICS EXPRESS 16471

where Ɏ is the potential function. From the above equation, the potential function in each medium can be written as: 2

)1

a§r· § b ·§ a · ¨ ¸ cos -  C ¨ ¸¨ ¸ cos b©a¹ © a ¹© r ¹

B

2

)2 )3

§r· §b·  E 0 b ¨ ¸ cos -  D ¨ ¸ cos b © ¹ ©r¹ §r· A ¨ ¸ cos ©a¹

(28)

Constant A , B , C , D are determined by satisfying the boundary condition at each interface, which are defined as:

)3 a

H3

w)3 wr

)1 a

H1 a

)1 b

w)1 wr

H1 a

w)1 wr

)2 b

H2 b

w) 2 wr

(29) b

The coefficients are obtained by solving the following linear system

§ D12 ¨ ¨ D 21 ¨ D 31 ¨ © D 41

D12 D 22 D 32 D 42

D13 D 23 D 33 D 43

D14 · § A · D 24 ¸¸ ¨¨ B ¸¸ ˜ D 34 ¸ ¨ C ¸ ¸ ¨ ¸ D 44 ¹ © D ¹

§ J1 · ¨ ¸ ¨J 2 ¸ ¨J3 ¸ ¨ ¸ ©J 4 ¹

(30)

The matrix coefficient Įij and Ȗi are presented in Appendix A. The electric field distribution in each medium is obtained by

E

’)

(31)

From the above equation each component of the electric field can be obtained, and are listed as follows:

E1, r E 2, r E3, r

ª 1 § ab · º  2 C ¨ 3 ¸ » cos « B ab © r ¹¼ ¬

E1,-

ª º b2 2 D  E 0 » cos « 3 r ¬ ¼ A  cos a

ª 1 ab º  C 3 » sin «B ab r ¼ ¬

ª b2 º « D 3  E 0 » sin ¬ r ¼ A E3,sin a

E 2,-

(32)

Where E r and E - are the radial and polar component of the electric field in each medium. As done for the solid sphere the surface force acting at each interface can be written as

P









ªD E E ˜ n º  ªD E E ˜ n º  ª E E 2 º  ª E E 2 º ¼a ¬ ¼b ¬ ¼a ¬ ¼b ¬

(33)

where a and b represent the media on the two sides of the interface. Using Eq. (7) and Eq. (12) the pressure distributions at the inner and outer interface are given as follows:

P1,3 P1,2 #112051 - $15.00 USD

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Z  Y cos -  Y K  W cos - 2  W 2

(34)

Received 1 Jun 2009; revised 17 Jul 2009; accepted 31 Jul 2009; published 1 Sep 2009

14 September 2009 / Vol. 17, No. 19 / OPTICS EXPRESS 16472

Where P 1,3 is the pressure at the inner surface of the shell, while P 1,2 is the pressure on the outer surface. The constant Z, Y, K and W are defined as: 2 2 º § A · ª§ H 3 · « D1  E1  D3  E3 » ¨ ¸ ¨ ¸ © a ¹ «¬© H1 ¹ »¼

Z

· ¸¸ ¹

2

2

, Y

§ A· ¨ ¸ E3  E1 ©a¹

ª§ H · 2 º «¨ 1 ¸ D 2  E 2  D1  E1 » , «¬© H 2 ¹ »¼

K

§ 1 a  2C 2 ¨¨ B ab b ©

W

§ a 1 a · ¨¨ B b a  C b 2 ¸¸ E1  E 2 © ¹

2

(35)

(36)

Note that these pressures are due to the inductive capacity discontinuity at the interface separating the media. If the hollow cavity is filled with a liquid ( k >1), there will be an increment of the fluid pressure due to electrostriction. This change in pressure due to applied electric field is given by [18]:

P3

H 0 A2 ( k3  1)( k3  2) 6 a2

(37)

The effect of the body force inside the shell due to the applied electrostatic field can be calculated using Eq. (4). Considering an isotropic dielectric, the first term on the right hand side of Eq. (4) becomes zero. However, the electric field within the shell is not constant, hence, the second term on the right hand side of Eq. (4) is finite. Using the expression given by Eq. (33), we can find the body force (per unit volume) as:

f

­ ª§ 18C 2 a 2 b 2 18 BC ab · º ½ 2  ° «¨¨  ¸ Cos - » ° 7 4 ¸ r r ° «© »r° ¹ °« » ° 2 2 2 1 ° » °¾  ( a1  a2 ) ® «  6 C a b  6 BC ab 7 4 « »¼ ° 4 r r °¬ ° ° § 2 2 2 · ° Sin 2- ¨ 3C a b  6 BC ab ¸ ° ¨ r7 ¸ °¯ °¿ r4 © ¹

(38)

where the constants B, and C are constants determined from Eq. (28), For a thin walled shell, the body force along the radial direction is nearly constant. In Fig. 4, the net surface pressure distribution along the polar direction (-) is compared to the distribution of radial and polar body force per unit volume times the shell thickness, Bt.

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Fig. 4. Pressure and body force distributions for a spherical PDMS shell (base-to-curing-agent UDWLRD ȝPDȜ  GXHWRDSSOied electric field.

The figure shows that the effect of body force on hollow microspheres is several orders of magnitude smaller than the pressure force exerted on the sphere. Thus, we neglect the body force in the analysis. The components of the displacement in the radial direction is given by [22]:

ur

¦ ª¬ An n  1 n  2  4Q R n

1

 Bn nR n 1 º¼Pn cos - 

(39)

Dn n  1 n  2 º ª Cn 2 « n n  3n  2Q  » Pn cos - Rn2 ¬R ¼ whereas the corresponding stress components are:

V rr

2 G ¦ ª¬ An n  1 n 2  n  2  2Q R n  Bn n n  1 R n  2 º¼Pn cos -  (40)

Dn n  1 n  2 º ª Cn n 2 «  n 1 n  3n  2Q  » Pn cos - Rn 3 ¬ R ¼ V --

V MM

2 ª D n  1 º Cn 2 G ¦ « An n 2  4n  2  2Q n  1 r n  Bn n 2 r n  2  nn1 n 2  2n  1  2Q  n n  3 »Pn cos - r r «¬ »¼ (41) dPn cos - Cn Dn º ª n n2 « An n  5  4Q r  Bn r  r n 1 n  4  4Q  r n  3 » cot d¬ ¼

D n  1 º ª Cn 2 G ¦ « An n  1 n  2  2Q  4nQ r n  Bn nr n  2  nn1 n  3  4nQ  2Q  n n  3 »Pn cos - r r ¬ ¼ (42) dPn cos - Cn Dn º ª n n2  « An n  5  4Q r  Bn r  n 1 n  4  4Q  n  3 » cot dr r ¼ ¬

V r-

2 G ¦ ª¬ An n 2  2n  1  2Q R n  Bn n  1 R n  2 º¼

Dn n  2 º wPn cos - ª Cn 2 « n 1 n  2  2Q  » wR R n3 ¼ ¬ #112051 - $15.00 USD

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wPn cos - w-



(43)

Received 1 Jun 2009; revised 17 Jul 2009; accepted 31 Jul 2009; published 1 Sep 2009

14 September 2009 / Vol. 17, No. 19 / OPTICS EXPRESS 16474

The constants A n, B n, C n and D n are determined by satisfying the boundary conditions. The boundary conditions are defined as follow:

Fig. 5. The WGM shifts of a hollow PDMS (60:1) sphere with the applied electric field due to strain effects (a/b = EȜ  

V rr a  P3  P1,3

V rr b

V r- a 0

V r- b 0

P1,2

(44)

The pressure acting at the boundaries of the hollow sphere can be expanded into FourierLegendre series as

P1,3

¦ E n Pn cos- Z  Y cos -

P1,2

¦ Fn Pn cos- K  W cos -

2

Y 2

W

(45)

Again, only two terms of the series are needed to represent the pressure on the inner and outer surfaces of the hollow sphere. These are:

E0 E2

1 Z  2Y 3 2 Z Y 3

1 K  2W 3 2 F2 K W 3

F0

(46)

Substituting Eq. (46) into Eq. (45) and then into Eq. (44) we obtained the constants of Eq. (39). They are determined by solving the following two linear systems

§ E11 ¨ ¨ E 21 ¨ E31 ¨ © E 41

E12 E 22 E32 E 42

E13 E 23 E33 E 43

E14 · § A0 · E 24 ¸¸ ¨¨ B0 ¸¸ ˜ E 34 ¸ ¨ C 0 ¸ ¸ ¨ ¸ E 44 ¹ © D0 ¹

§ I1 · ¨ ¸ ¨ I2 ¸ ¨ I3 ¸ ¨ ¸ © I4 ¹

§ G11 ¨ ¨ G 21 ¨ G 31 ¨ © G 41

G12 G 22 G 32 G 42

G13 G 23 G 33 G 43

G14 · § A2 · G 24 ¸¸ ¨¨ B2 ¸¸ ˜ G 34 ¸ ¨ C 2 ¸ ¸ ¨ ¸ G 44 ¹ © D2 ¹

§ U1 · ¨ ¸ ¨ U 2 ¸ (47) ¨ U3 ¸ ¨ ¸ © U4 ¹

The matrix coefficients ȕij, iji, įij, ȡij are presented in Appendix A. Once the constants A n, B n, C n and D n are known, the change in WGM due to strain ( da/a ) can be calculated by using Eq. (39). However, as discussed earlier, dn0/n0 <