1

Whispering Gallery Mode Based-Micro-Optical Sensor for

Electromagnetic Field Detection

Tindaro Ioppolo , Ulas K. Ayaz§, M. Volkan Ötügen†

Southern Methodist University, Dallas, TX 75275

In this paper we investigate the electrostriction effect on the whispering gallery

modes (WGM) of polymeric microspheres and the feasibility of an WGM-based micro-

sensor for electric field measurements. 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 the strain and the 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 measurements of electric

fields as small as ~500V/m may be possible using water filled, hollow PDMS micro-spheres.

The electric field sensitivities for solid spheres, on the other hand, are significantly smaller.

The effect of dielectric constant perturbations in the ambient medium on sphere WGM has

also been investigated. A preliminary analysis indicates that changes of the order of ~10-3

in

dielectric constant of the medium surrounding the microsphere can be observed by using a

water-filled hollow PDMS sphere.

I. 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 whispering gallery modes or WGM) 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, Q = / ( is the wavelength of the interrogating laser and is the linewidth of the observed

mode). The proposed WGM applications include those in spectroscopy1, micro-cavity laser technology

2, and optical

communications (switching3 filtering

4 and wavelength division and multiplexing

5). For example, mechanical strain

6

and thermooptical4 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 applications7,8

, trace gas detection9, impurity detection in liquids

10 as well as mechanical sensing

including force 1, pressure

11, temperature

12 and wall shear stress

13.

In this paper we investigate the effect of an electrostatic field on the WGM of a polymeric microsphere.

Such electrostriction-induced shifts could be exploited for WGM-based gas composition and electric field sensors.

Potentially they could also be used for electrostatic-driven optical switches.

The simplest interpretation of the WGM phenomenon comes from geometric optics. When laser light is

coupled into the sphere nearly tangentially, 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

lan 02 (1)

Post Doctoral Associate, Mechanical Engineering Dept., AIAA Member

§ Graduate student, Mechanical Engineering Department

† Professor, Mechanical Engineering Dept., AIAA Associate Fellow

AIAA Infotech@Aerospace Conference <br>and <br>AIAA Unmanned...Unlimited Conference 6 - 9 April 2009, Seattle, Washington

AIAA 2009-1814

Copyright © 2009 by Tindaro Ioppolo. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2

where 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. Eq. (1) is a first order approximation and holds for a

>>. A minute change in the size or the refractive index of the microsphere will lead to a shift in the resonance

wavelength as

a

da

n

dnd

0

0

(2)

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 of the WGM shift of polymeric microspheres caused by the applied electrostatic field. The analysis

takes into account both the strain and stress effects.

II. Electrostatic Field-Induced Stress in a Solid Dielectric Sphere

We first consider an isotropic solid dielectric sphere of radius a and inductive capacity 1, embedded in an

inviscid dielectric fluid of inductive capacity 2. The sphere is subjected to a uniform electric field E0 in the direction

of negative z as shown in Fig.1. The force exerted by the electrostatic

field on the sphere will induce an elastic deformation (electrostriction)

that is governed by the Navier Equation14

:

021

12

G

fuu

(3)

where u is the displacement of a given point within the dielectric

sphere, , 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 15

:

2

211

2

4

1

2

1EaaEf

(4)

where E

is the electric field within the sphere, is the inductive

capacity, a1 and a2 are coefficients that describe the dielectric

properties. Physically, the parameter a1 represents the change of

inductive capacity due to an elongation parallel to the lines of the

field, while a2 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. The electric field inside the dielectric sphere is uniform and

parallel to the z axis, with its magnitude 15

:

0

21

2

2

3EE

(5)

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

021

12

uu

(6)

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

d

dPnrBrnnAu

PnrBrnnAu

nn

n

n

n

n

n

n

n

nr

cos451

cos421

11

11

(7)

Fig. 1: The sphere in the presence of

electric field

Figure 1. The sphere in the presence

of electric field

3

where ur and u are the components of displacement in the radial, r, and polar, directions. Pn’s 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:

cos12212 22

n

n

n

n

nrr PrnnBrnnnAG (8)

d

dPrBrnA

PrnBrnnnA

Gnn

n

n

n

n

n

n

n

n

coscot45

cos1224

22

222

(9)

d

dPrBrnA

PnrBrnnnA

Gnn

n

n

n

n

n

n

n

n

coscot45

cos4221

22

2

(10)

cos

12122 22 nn

n

n

nr

PrnBrnnAG (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 by15

12

2

2

12 nEnEnEEnEEP

(12)

where 𝑛 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 medium) The constants and are given as15

:

2

12 aa ,

2

2a

(13)

For the case of a sphere embedded in a dielectric fluid, the constants a1 and a2 are defined by the Clausius-Mossotti

law 15

leading to:

= , 226

20 kk

(14)

for the fluid (medium 2). Here, 0 is the inductive capacity of vacuum, and k is the dielectric constant. Using Eq.(5)

and Eq.(12) the pressure acting at the dielectric interface is given by:

'2'' BCosBAP (15)

where A’ and B’ are defined as:

1122

2

2

1

2

0

21

2'

2

3

EA (16)

21

2

0

21

2'

2

3

EB (17)

Equation (14) 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

4

)2)(1(6

22

20 kkEP

(18)

For gas media, k1, thus P is negligible. In order to define the stress and strain distributions within the sphere, coefficient An and Bn have to be evaluated.

These coefficients are calculated by satisfying the following boundary conditions

0

a

Pa

r

rr

(19)

The coefficient An and Bn are determined by expanding the pressure P in terms of Legendre series as follows:

cosnn PZP (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:

'' 23

1BAZo

,

''

23

2BAZ (21)

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

112

2 ''

0G

BAA

756 2

''

2

Ga

BAA

756

72''

2

G

BAB

(22)

The radial deformation can be determined by using Eq.(7):

1cos32

1212122

2

2

3

20 rBrArAur (23)

III. WGM Shift in a Solid Sphere Due to Electrostriction

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 =) by plugging Eq. (22) into Eq. (23):

2122

2

2

1

21122

2

2

1

2

0

21

2

753

74

2316

21

2

3

G

GE

a

da (24)

As we can see from the above expression, the radial deformation, da/a, has a quadratic dependence on the electric

field intensity.

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 follows16

:

5

(25)

Here nnnr ,, are the refractive indices in the direction of the three principle stresses and 000 ,, nnn r are

those values for the unstressed material. Coefficients C1 and C2 are the elasto-optical constants of the material. For

PDMS this value are C1= C2 = C = -1.75x10-10

m2/N

17. Thus, for a spherical sensor, the fractional change in the

refractive index due to mechanical stress is reduced to:

n

C

n

nn

n

nn

n

nn

n

dn rr

o

or

o

o

or

orr

o

o

(26)

Thus, evaluating the

appropriate expressions for

stress in Eq.(8, 9, 10) at

=/2 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 a1 and a2 must be

evaluated. Very few reliable

measurements of these

constants for solids have been

reported in the literature.

Unfortunately, to our

knowledge there are not

experimental measurements of

a1 and a2 for PDMS. In our

analysis we take the values

developed for an ideal polar

rubber18

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. The calculations indicate that using WGMs, for a quality factor of Q~107

an electric field as small as

~ 20000 V/m can be resolved with a solid PDMS microsphere with the properties shown in the figure.

IV. Electrostatic Field-Induced Stress in a Hollow Dielectric Sphere

In this section we consider a dielectric spherical shell of inductive capacity 1 with inner radius a and outer radius b

that is placed in a uniform dielectric fluid of inductive capacity 2 as shown in Fig.3. The shell is filled with a fluid

of inductive capacity 3. 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 have to be determined. In general, both the pressure and the body

force are functions of the electric field distribution. The electric field distribution in a dielectric is governed by

Laplace's equation. The general solution of Laplace's equation in spherical coordinates (r,,) is given as:

21 CCnn rrorr

rro CCnn 21

rro CCnn 21

-0.00005

0

0.00005

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

0.0004

-70

-60

-50

-40

-30

-20

-10

0

0 200000 400000 600000

Δ

,pic

om

ete

r

Δ

, pic

om

eter

E, V/m

Strain effectStress effect

Figure 2. The WGM shifts of a solid PDMS (60:1) sphere due to

the applied electric field (for /a=2.6)

6

cos)(,, 1

0

i

i

i

i

i

i PrBrAr

(27)

where is the potential function. From the above equation, the

potential function in each medium can be written as:

cos

coscos

coscos

3

2

02

2

1

a

rA

r

bD

b

rbE

r

a

a

bC

a

r

b

aB

(28)

Constant A, B, C, D are determined by satisfying the boundary

condition at each interface, which are defined as:

bbaarrrr

bbaa

22

11

11

3

3

2113

(29)

The coefficients are obtained by solving the following linear system

4

3

2

1

44434241

34333231

24232221

14131212

D

C

B

A

(30)

The matrix coefficient αij and γi 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:

sincos

sincos2

sin1

cos21

,3,3

03

2

,203

2

,2

3,13,1

a

AE

a

AE

Er

bDEE

r

bDE

r

abC

abBE

r

abC

abBE

r

r

r

(32)

Where Er 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

Fig.3: Schematics of a hollow

dielectric sphere

2

z

x a

b 3

1

Figure 3. Schematics of a hollow

dielectric sphere

7

baba EEnEEnEEP 22

(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:

YYZP 2

3,1 cos (34)

WWKP 2

2,1 cos

Where P1,3 is the pressure at the inner surface of the shell, while P1,2 is the pressure on the outer surface. The

constant Z, Y, K and W are defined as:

13

2

3311

2

1

3

2

,

a

AY

a

AZ (35)

21

2

21122

2

2

1

2

2

1,2

1

b

aC

ab

aBW

b

aC

abBK (36)

Note that these pressures are due to surface discontinuity in the dielectric properties of the medium. If the hollow

cavity is filled with a liquid (k>1), there will be an increment of the fluid pressure due to electrostriction 15

. This

change in pressure due to applied electric field is given by:

)2)(1(6

332

20

3 kka

AP

(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:

47

222

47

2222

47

222

21

632

661818

)(4

1

r

abBC

r

baCSin

rr

abBC

r

baCCos

r

abBC

r

baC

aaf

(38)

where the constants B,

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), 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. The

figure shows that the

effect of body force on

Bt,

N/m

2 x

10

15

Fig. 4: Pressure and body force distributions for a spherical PDMS shell

(60:1, 300 m radius) due to the applied electric field

Fig. 4: Pressure and body force distributions for a spherical PDMS shell

(60:1, 300 m radius) due to the applied electric field

-30

20

70

120

170

220

270

0 20 40 60 80 100 120 140 160 180

, deg rees

Pre

ss

ure

, N

/m2

x1

09

-3.37

-2.87

-2.37

-1.87

-1.37

-0.87

-0.37

0.13

0.63

1.13

1.63

Bo

dy

fo

rce

, N

/m2

x1

01

5

P res s ure

R adial body force

Tangential body force

Fig. 4: The pressure and body force distributions for a spherical PDMS shell (60:1,

diameter 600 µm) due to the electric fieldFigure 4. Pressure and body force distributions for a

spherical PDMS shell (60:1, a/b= 0.93 ) due to the applied

electric field

8

hollow microspheres is several orders of magnitude smaller than the pressure force exerted on the sphere due to the

external electric field. Thus, we neglect the body force in the analysis.

The components of the displacement in the radial direction is given by 19

:

cos21

23

cos421

2

2

11

nn

n

n

n

n

n

n

n

nr

PR

nnDnn

R

C

PnRBRnnAu

(39)

whereas the corresponding stress components are:

cos21

23

cos12212

3

2

1

22

nn

n

n

n

n

n

n

n

nrr

PR

nnDnn

R

nC

PRnnBRnnnAG

(40)

d

dP

r

Dn

r

CrBrnA

Pr

nDnn

r

nCrnBrnnnAG

n

n

n

n

nn

n

n

n

nn

n

n

nn

n

n

n

coscot4445

cos1

21212242

31

2

3

2

2

1

222

(41)

d

dP

r

Dn

r

CrBrnA

Pr

nDnn

r

nCnrBrnnnAG

n

n

n

n

nn

n

n

n

nn

n

n

nn

n

n

n

coscot4445

cos1

24342212

31

2

31

2

(42)

cos222

cos12122

3

2

1

22

n

n

n

n

n

nn

n

n

nr

P

R

nDn

R

C

PRnBRnnAG

(43)

The constants An, Bn, Cn and Dn are determined by satisfying the boundary conditions. The boundary conditions are

defined as follow:

00

2,13,13

ba

PbPPa

rr

rrrr

(44)

The pressure acting at the boundaries of the hollow sphere can be expanded into Fourier-Legendre series as

9

WWKPFP

YYZPEP

nn

nn

2

2,1

2

3,1

coscos

coscos

(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:

WKFYZE

WKFYZE

3

2

3

2

23

12

3

1

22

00

(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

(47)

The matrix coefficients βij, φi, δij, ρij are

presented in Appendix A. Once the

constants An, Bn, Cn and Dn are known,

the change in WGM due to strain (da/a)

can be calculated by using Eq. (38).

The contribution of the stress effect

(dn/n) on the total WGM can be

calculated as before. The stress and

strain effects at the equatorial belt

(=/2, r=b) of a hollow PDMS

microsphere of 600µm diameter and

b/a=0.95 are shown in Fig (5). In this

configuration, the PDMS shell is filled

with and also surrounded by air (Note

here that the stress effect is several

orders of magnitude smaller than that

of strain and hence, does not play a role

in WGM shift). Comparing Fig. (5) to

Fig. (2), we see that the effects of

electric field on shape distortion of the

spheres are opposite: The solid sphere

becomes elongated in the direction of

the static field, on the other hand, the

hollow sphere shortens in the direction

of the applied field.

Next we look at the case

where the fluid inside the sphere has a

higher relative dielectric constant than

the surrounding medium (3>1). For

4

3

2

1

2

2

2

2

44434241

34333231

24232221

14131211

D

C

B

A

4

3

2

1

0

0

0

0

44434241

34333231

24232221

14131211

D

C

B

A

0

2

4

6

8

10

12

0 100000 200000 300000 400000 500000 600000

D

, p

ico

met

er

E, V/m

Figure 5. The WGM shifts of a hollow PDMS (60:1) sphere with the

applied electric field due to strain effects (for /b=2.6, a/b =0.95)

-2.5

-2

-1.5

-1

-0.5

0

0 500 1000 1500 2000 2500 3000

D

, pic

om

ete

r

E, V/m

Figure 6. The WGM shifts of a thin shell PDMS (60:1)

sphere filled with water (for /b=2.6, a/b =0.95)

10

this, we consider the case of a

thin spherical shell of PDMS

that is filled with water

(k=80.1) and surrounded by air

on the outside. Figure (6)

illustrates the solution for this

particular configuration.

Another interesting observation

is made from Fig. (6). Filling

the sphere with water increases

the sensitivity significantly.

With a Q-factor of 107, the

resolution of the sensor is

estimated to be ~500 V/m. The

next question we address is:

Can such a sensor be used to

detect the contamination of the

surrounding medium due to, for example, an additional gas mixture being present? Figure (7) illustrates this. Using

the same configuration as before (spherical PDMS shell filled with water inside and surrounded by air), the electric

field applied on the sphere is kept at 10000 V/m and the dielectric constant of the outside medium is changed. The

resulting WGM shift is given in Fig. (7). Again, with Q-factor of ~107, the sensor can detect changes in relative

inductive capacitances of ~10-3

. This result shows that a sensor could be developed for the detection of contaminants

in air or liquids.

V. Conclusion Electrostriction effect on WGM microsphere sensors has been investigated analytically. The analysis shows

that the magnitude of external electric field can be monitored by monitoring the WGM shifts. The results indicate

that WGM-based electric field sensor are possible. Hollow PDMS spheres that are filled with air showed less

sensitivity than their solid counterparts. However, when a hollow PDMS sphere is filled with a dielectric liquid, the

sensitivity of the WGM sensor increases drastically. An analysis is also carried out to determine the WGM shift

dependence on dielectric constant perturbations of the surrounding medium (with the dielectric shell subjected to

constant electric field). The results indicate that sensor system may be feasible for impurity detection in gases or

liquids.

Appendix

0,

6

2),2)(1(

66

2

2,1

2,

1,

2,1

2,12

0,1

,2

,2

,12

432

2

01

34443242334334131

232213122323243142111

G

WKkk

a

A

G

YZ

bbbaav

aba

0

2

340231

2

3444243233

32

2

131232234211413122411

,,0,2

,1

,2

,2

1,,,,0,,,1

EbEbabb

a

a

b

abab

a

a

b

a

b

b

a

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

1 1.001 1.002 1.003 1.004

D

, p

ico

met

er

k

Figure 7. WGM shifts of a thin spherical shell of PDMS (60:1, for /b=2.6,

a/b =0.95) filled with water. The WGM shifts obtained here are due to the

change of dielectric constant of the surrounding medium.

11

Acknowledgments

This research was support by the National Science Foundation (through grant CBET-0809240) and Department of

Energy (through grant DE-FG02-08ER85099). We also acknowledge Ms. Kaley Marcis’ contribution in carrying

out some of the numerical calculations.

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