High-Resolution Whispering Gallery Mode Force Micro-Sensor Based on Polymeric Spheres

T. Ioppolo1, U. K. Ayaz2 and M. V. Ötügen3

Southern Methodist University, Dallas TX, 75275

Previous experimental studies of whispering gallery mode (WGM) sensors have

indicated that microspheres with diameters ranging between 300-950 µm may have force resolutions reaching 10-5N [1]. In the present, we expand on the previous investigatons. Here, we carry out a systematic analysis and experiments to investigate the sensitivity, resolution and bandwidth limits of WGM-based force sensors. Expressions for WGM shifts due to applied force in the polar direction are obtained for microspheres of various dielectric materials, in the diameter range of 300-950 µm. The analyses are compared with experimental results for Polymethylmethacrylate (PMMA) Polydimethylsyloxane (PDMS) microsphere sensors. The present analysis shows that the strain effect on WGM shifts dominate over that of mechanical stress. It also indicates that force sensitivities of the order of a 1pN are possible using hollow PDMS spheres. The sensor bandwidths (based on the mechanical properties of the sensor material alone) range between 1 kHz and 1 MHz. These results have significance also from the point of view wall shear stress since the same force sensing concept can be used for the development of high-sensitivity skin friction sensors.

Nomenclature a = Sphere radius a0 = Contact area radius C1,C2,C = Elasto-optic constants c = Speed of sound dF = Incremental applied force dλ = WGM wavelength shift dn0 = Incremental refractive index change of the sphere F = Applied force G = Shear modulus l = Circumferential optical mode number n0 = Refractive index of the sphere Pn = Legendre polynomial Q = Optical quality factor r, ϑ ,φ = Radial, polar and azimuthal coordinates, respectively u = Vector displacement λ = Vacuum wavelength of laser light δλ = Optical resonance linewidth λ̂ , μ = Lame’ constants ωn = Microsphere resonance

1 Post Doctoral Associate, Mechanical Engineering Dept., AIAA Member 2 Doctoral student, Mechanical Engineering Dept.

3 Professor and Chair, Mechanical Engineering Dept., AIAA Associate Fellow

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition5 - 8 January 2009, Orlando, Florida

AIAA 2009-1450

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

I. Introduction

Whispering gallery mode (WGM) optical resonances 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 or 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 [1]. The modes are observed as sharp dips in the transmission spectrum at the output end of the fiber typically with very high optical quality factors, Q = λ/δλ (λ is the wavelength of the interrogating laser and δλ is the linewidth of the observed mode). The proposed WGM applications include those in spectroscopy [2], micro-cavity laser technology [3], and optical communications (switching [4] filtering [5] and wavelength division and multiplexing [6]). For example, mechanical strain [7] and thermooptical [4] 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 [8,9], trace gas detection [10], impurity detection in liquids [11] as well as mechanical sensing including force [1], pressure [12], temperature [13] and wall shear stress [14].

In the mechanical sensing applications, the dielectric microsphere is optically coupled to a single mode fiber which carries light from a tunable laser and serves as an input/output port for the microsphere [1]. When the microsphere comes into contact with an exposed section of the fiber core, its optical resonances (WGM) are observed as sharp dips in the transmission spectrum. These optical resonances, which are also known as the “morphology dependent resonances” (MDR), are extremely narrow due to the very high Q values and hence are highly sensitive to any change in the morphology of the sphere (shape, size or refractive index). A minute change in the morphology of the micro-sphere will cause a shift in the resonance positions allowing for the precise measurement of the mechanical effect causing the WGM shift.

Geometric optics provides the simplest interpretation of the WGM phenomenon. The laser light from the optical fiber is coupled to the microsphere nearly tangentially, and thus it approaches the interior surface of the sphere beyond the critical angle, thereby experiencing a total internal reflection along the interior surface of the microsphere. Trapped inside the microsphere, the light circumnavigates the interior surface of the sphere. A resonance (WGM) is realized when light returns to its starting location in phase. Thus, the approximate condition for resonance is

λπ lan =02 (1) where λ is the vacuum wavelength of laser, no and a are the refractive index and radius of sphere respectively, and l is an integer indicating the circumferential mode number. Eq. 1 is a first order approximation and holds for a >>λ. At resonance, light experiences constructive interference in the sphere which is observed as dips in the transmission spectrum through the optical fiber. A minute change in the size or the refractive index of the microsphere will lead to a shift in the resonance wavelength as

ada

ndnd

+=0

0

λλ

. (2)

In the WGM-based force sensor concept, uniaxial force is applied on the sphere along the polar direction (normal to the plane of light circulation) through two high-stiffness plates [1]. The applied force induces a change in both the radius (strain effect) and the refractive index (through mechanical stress) of the sphere. In Ref [1] the MDR of silica and Poly (methyl methacrylate) (PMMA) spheres have been force-tuned, experimentally demonstrating the feasibility of a WGM-based force sensor. The experiments also indicated that force resolutions as high as ~ 10-5 N were possible using hollow PMMA spheres.

In the present, an analytical framework is developed for the proposed force (and by extension, wall shear stress) sensor. Specifically, expressions are developed for WGM shifts in sphere sensors due to force applied in the polar direction. The analysis accounts for both the strain and stress effects in determining the WGM shifts. Using these expressions, a systemic analysis is carried out to determine the sensitivity, resolution and bandwidth limits of force and wall shear stress sensors based on WGM shifts of polymeric microspheres. The analytical expressions are compared to experimental results for Poly (methyl methacrylate) (PMMA) Polydimethylsyloxane (PDMS) microsphere sensors. The present results show that the strain effect (da/a) dominates over mechanical stress effect (dn0/n0). The results also indicate that force sensitivities of the order of a 1pN are possible using PDMS spheres.

II. Analysis

2.1. Stress and strain in a solid dielectric microsphere

Here we consider a solid dielectric sphere of radius a, that is compressed by two pads (plates) made of material with high stiffness compared to the sphere material (such as stainless steel) as shown in Fig. 1. The applied

force, F, will lead to deformation (strain) and a stress field distribution inside the microsphere. The distribution of stress field is obtained by solving the Navier equation:

021

12 =⋅∇∇−

+∇ uuν

(3)

where u is the displacement of a given point within the sphere and ν is the Poisson ratio. The solution of this equation for an azimuthally symmetric loading [15,16] is given by:

( )( )[ ] ( )ϑν cos42 11n

nn

n PnrBr1n

nr nnAu ∑ −+ ++−+=

(4) where r and ϑ are the radial and polar coordinates. respectively (Fig. 1); ur is the radial component of displacement; Pn are the Legendre polynomials; An and BBn are constants determined by the boundary condition at the sphere surface. Using Eq. (4) along with the stress-displacement relationship [22] the normal stress

distributions within the sphere are obtained as:

Figure 1: Microsphere with applied force

( )( ) ( )[ ] ( )ϑνσ cos12212 22

nn

nn

nrr PrnnBrnnnAG∑ −−+−−−+=

( )( )[ ] ( )

( )[ ] ( ) ( )∑⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+−+

++++++−=

−

−

ϑϑϑν

ϑνσϑϑ

ddPrBrnA

CosPrnBrnnnAG

nnn

nn

nn

nn

n

coscot45

12242

2

222

(5)

( )( )[ ] ( )

( )[ ] ( )∑⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+−+

++−−−+=

−

−

ϑϑ

ϑν

ϑννσ ϕϕ

ddP

rBrnA

PnrBrnnnAG

nnn

nn

nn

nn

n

coscot45

cos42212

2

2

( ) ( )[ ] ( )

ϑϑνσ ϑ d

dPrnBrnnAG nnn

nnr

cos12122 22∑ −−++−+=

Here G is the shear modulus of the sphere material. In defining the boundary conditions we neglect the friction at the contact point between the sphere and the plates. With that, the boundary conditions are:

( ) ( )

( ) πϑσϑπϑϑ

πϑϑπϑϑϑσ

ϑ ≤≤=⎩⎨⎧

−≤≤≤≤−≤≤−

=

000

0

00

00

a

andpa

r

rr (6)

h

( )

where angle ϑ0 defines the extent of the contact between the plate and the sphere as shown in Fig. 2. The pressure, p, ere is given by [15, 20]:

exerted by the plates of infinite stiffness on the sp

( )

( )⎪⎪

⎩

⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡ −=

−=

− 3/12

0

22203

0

413

sin23

EvFRa

aaaFp

πϑ

(7)

ere a0 is the radius of contact area as shown in Fig. 2. In order to

ϑ

Hobtain coefficients An and B Bn in Eqs. (4) and (5), the boundary condition (Eq.(6)) has to be expanded in terms of the Legendre polynomials in the following form [17]

( ) ( )ϑσ cosnn

nrr PHa ∑=

with coefficient H defined as:

(9)

Using the boundary condition, Eq. (6), for Eq.(5) and (8), coefficients An and Bn are determined as:

(10)

bstituting Eq.(10) into Eq.(4) and (5), respectively.

.2. Force-induced WGM shift in a solid dielectric microsphere For the solid microsphere, the last term in Eq. (2) can be calculated by evaluating Eq. (4) at r = a and then

dividing (11)

urbation, dn0/n0, in Eq. (2). Neumann-Ma ws [18]:

(12)

(8)

Figure 2: Contact geometry

Once coefficients An and Bn are obtained, strain (displacement) and the stress fields can be calculated by su

2

it by the sphere radius, a:

Next we determine the effect of stress on refractive index pert

xwell equations provide the relationship between stress and refractive index as follo

( )φφϑϑ σσσ +++= 21 CCnn rrorr

( )φφϑϑϑϑ σσσ +++= rro CCnn 21

( )rro CCnn σσσ ϑϑφφφφ +++= 21

( ) ( ) ϑϑϑϑπ

π

dPaaaF

n sincossin23 223

0

2/

030

−∫

)

nH n 212 +

=

( )( ) ([ ]νν 2122212 22 +−+−−−−+=

nnnnnnGaA n

nn

H

( )( ) )0(424 2

nn Pnn

⎥⎤−−ν

( )( ) ( )[ ]( )

( )

1124 2 nnnnGa ∑ ⎢ −++++ νν42 2nHda ⎡ +−

n ⎦⎣=

12

2122212

2

22 −−+

+−+−−−−+−=

nnn

nnnnnnGHB n

n νν21 2+ −a nν

ere are the refractive indices in the direction of the three principle stresses and are

ues for the unstressed material. Coefficients C1 and C2 are the elasto-optical constants of the material, and r both PMMA and PDMS, these two constants are equal. For PMMA, C1= C2 = C = -10-10 m2/N [18], and for

this valu -10 2 change in

φϑ nnnr ,, φϑ 000 ,, nnn rH

those valfoPDMS e is C1= C2 = C = -1.75x10 m /N [19]. Thus, for a spherical sensor, the fractional the refractive index due to mechanical stress is reduced to:

( )n

Cn

nnn

nnn

nnndn rror

o

o

or

orr

o

o φφϑϑφ

ϑ

ϑϑ

oφ

σσσ ++=

−=

−=

−= (13)

In the present WGM optical sensor, light is traveling in a plane that is normal to the appthe appropriate expressions for stress in Eq.(5) at ϑ =π/2 and r = a, and introducing thchange in the refractive index due to force F is obtained as:

lied force. Thus, evaluating em into Eq. (13) the relative

( )( )

( ) )0(2

444312 2

2

00

0n

n

n Pnnn

nnnnHnC

ndn ∑ ⎥

⎤⎢⎣

⎡+++

+++++=

ννν

(14) 1 ⎦+ν

Equations (11) and (14) represent the effect of strain and stress, respectively, on thesphere. Plugging these equations in Eq.(2) we obtain the total WGM shift as:

WGM shift of the solid dielectric

( )( )

( )( )( ) )0(44431

2142442

122 2 nn

n PnnnnCGn

nnnnnn

Hd ∑ ⎥⎤

⎢⎣

⎡ ++++++

−−−+−

++++=

νννννλ

λ(15)

0

222

n ⎦

2.3. Stress and strain in a hollow dielectric microsphere ere, we consider a hollow dielectric microsphere with outer and inner radii of a and b, respectively. Again,

in order to obtain the WGM shift induced by the applied force, the strain and stress at the sphere's outer surface must be known. To obtain the strain and stress distributions for a hollow microsphere, we superimpose the solution of a

a hollow cavity in an infinite medium [15,16]. The displace

H

solid microsphere (Eqs.(4) and (5)) and the solution of ment for a hollow cavity in any given medium is given by:

( ) ( ) ( )ϑν cos143 2 nnn

nn

and the corresponding stress field is given by [22]:

r PrnDnn

rCu ∑ ⎥⎦

⎤⎢⎣⎡ +

−−+= + (16)

( ) ( )( ) ( )ϑνσ cos

21232 3

21nrr r⎢⎣ + nn

nn Pr

nnDnn

nCG∑ ⎥⎦

⎤⎡ +++−+−= +

( ) ( ) ( )

( ) ( )ϑ

ϑϑν

ϑνσ ϑϑ

ddP

rD

nr

nC

PrnD

nnrC

G

nn

nnn

nnn

nn

coscot44

cos1

2122

31

3

22

1

⎥⎦⎤

⎢⎣⎡ +−+−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡ +−+−−=

++

++∑

(17)

( ) ( ) ( )

( ) ( )ϑ

ϑϑν

ϑννσ φφ

ddP

rD

nrC

Pr

nDnn

rnC

G

nn

nn

n

nnn

nn

coscot44

cos1

2432

31

31

⎥⎦⎤

⎢⎣⎡ +−+−+

⎥⎦⎤

⎢⎣⎡ +

−−−+=

++

++∑

( ) ( ) ( )

ϑϑ

νσ ϑ ddP

rnD

nrC

G nn

nn

nr

cos2222 3

21∑ ⎥⎦

⎤⎢⎣⎡ +

−+−= ++

The strain distribution in the hollow sphere is obtained by adding the Eqs. (4) and (16). Similarly, the stress

r the hollow sphere is obtained by adding Eqs. (5) and (17). coefficients An, Bn, Cn, Dn are determined by satisfying the boundary condition at the inner and outer

icrosphere. Again at the outer surface we assume that the friction at the contact surface between the sphere and the plates is negligible. We also assume that the sphere wall is thick enough such that when force, F, is applied to the sphere, the pressure inside the sphere cavity remains c stant. With these assumptions, the

itions are

These boundary conditions lead to a linear system of four equations with unknowns be presented in the following matrix form:

⎞⎜⎛

⎟⎞

⎜⎛

⎟⎞

⎜⎛

0

0

414342

14131211

n

n

C

A

αααα

αααα

where the coefficients αij are determined by the imposed boundary conditions. Once the coefficients An, Bn, Cn, Dn are known the strain and stress components can be evaluated. 2.4. Force-Induced WGM shift in a hollow dielectric microsphere The resonance shift for the hollow microsphere is obtained using the same procedure followed for the solid microsphere. Doing so, WGM shifts in a hollow polymeric sphere is obtained as follows:

distribution foThe

surfaces of the hollow m

onoundary condb

( )

( ) πϑσ

πϑσ ϑ

≤≤=

≤≤=

00

00

b

a

rr

r

( )( )

⎪⎩

⎪⎨

⎧

−≤≤

≤≤−≤≤−=

00

0

0

ϑπϑϑ

πϑϑπϑϑϑσ

andparr

( ϑσ ) πϑ ≤≤= 0r 0b

00

(18)

An, Bn, Cn and Dn which can

⎟⎟⎠

⎜⎜⎝

⎟⎟⎠

⎜⎜⎝

⎟⎟⎠

⎜⎜⎝ 041

34333231 n

Dαααα

⎟⎟

⎜⎜=⎟

⎟⎜⎜⋅⎟

⎟⎜⎜ 24232221 nn HBαααα

(19)

⎟

( )( ) ( ) ( )

( )( ) ( ) ( ))0(

2324

1

3

2

1

2

10

nnn

nnnn

PnnDnnnnC

nDCdnda

⎥⎤⎡ ++

−+−−

+⎤⎡ +−−++++−+=+=

+

−

ν

ννλ

. (20)

)0(43421 310

nn

nnnn Pa

nna

naBannAna ∑ ⎥⎦⎢⎣ ++λ

d

44431 2n

nnn annnnA

n ∑⎢ ++++++− +ννC

0 n aa ⎦⎣

Here the first term on the right hand side of Eq. (20) represents the strain effect the second term represents that of stress. Figure 3 shows the WGM shifts of solidhollow PMMA spheres

sufor

volume ratio of 50:1 (E=10 kPa). The figure indicates a strong agreement between the

II

of the other matre een force and MDR shift is essentially linear in the force range stushift (sensitivity),

and

and

(D=400 µm) under compressive force. The effect of force on da/a (strain) and dn0/n0 (stress) are shown separately to compare the contribution of each of these terms to the WGM shift, dλ/λ. The figure shows that strain effect is dominant and the contribution of dn0/n0 to WGM shift for both solid and hollow spheres is negligible. The lts of Fig.3 also indicate that force range considered, the

dependence of dλ/λ, on force is essentially linear. In Fig. 4, the experimental results for solid and hollow PMMA spheres are compared to the analytical results. For this calculation the Young's Modulus for PMMA was taken as E= 3.3 GPa. The diameter of the solid PMMA sphere is ~470 µm and the diameter of the hollow sphere is ~ 960 µm with a wall thickness of about 20 μm. The figure shows a good agreement between experimental results of Ref [1] and the presented analytical predictions for both solid and hollow spheres. Figure 5 shows the WGM shift dependence on the applied force for a solid PDMS microsphere of 910 µm with base-to-curing agent

experiments and Eq (15).

erials used previously [1], the died. The slope of the MDR

re

diameter

I. Force sensitivity and resolution Figure 5 indicates that for PDMS spheres, as in the case

lationship betwdFd /λ , is a function of the microsphere material and geometry (size and whether it is a solid or

Fig. 4: Comparison between experimental and analytical results for solid (D ~470 µm) and hollow (D ~960 µm) PMMA microspheres

Figure 3: WGM shift as a function of applied force on PMMA sphere(D=400μm diameter)

Figure 4: Comparison between experimental and analytical results for solid (D=470 μm) and hollow (D=960μm) PMMA microsphere

choosing different materials and varying sphere geometry, a wide range of sensitivities can be obtained. Using the previous analyses, the sensitivity of PMMA and PDMS microspheres are investiga

membr

hollow microsphere). Thus, by

ted. The sensitivity of hollow PMMA (E=3.3GPa) sensors with varying sphere diameters, D, and inner-to-outer radius ratios, b/a, is shown in Fig. 6 (a). The figure indicates that for very thin walled spheres (b/a> 0.9), the sphere behaves like a ane and for a given b/a value the sensitivity is essentially independent of the sphere diameter. The data of Fig. 6 (a) can be represented by the following approximate equation:

Figure 5: Comparison between experimental and analytical results for a solid PDMS (50:1) microsphere with D=910 µm

abR

eKDdFd 2−=

λ (25)

=9.687x10 where K

K=0.olid m

b) sho(for a ra

equation:

4 and R=3.54 for b/a<0.91 0362 and R=20 for b/a>0.9. In the case

icrosphere K=2.217x10andof s6 (

5, R=0. Figure ws the effect of Young’s modulus, E

nge corresponding to that for PDMS) and b/a on sensor sensitivity, dλ/dF. The data of Fig. 6 (b) can be expressed by the approximate

abR

eEKDdF

12 −−= (26)

her sensitivities than their PMMA counterparts.

Figure 7 shows the influence of sphere

diameter, D, on sensitivity d PDsensors as a function of Young’s modulus, E.

dλ

where K=6.884x104 and R=3.3 for b/a<0.91 and K=1.744x10-2 and R=22 for b/a>0.9. A comparison of Fig. 6 (a) and (b) shows that PDMS sensors can provide significantly hig

Figure 6: Sensitivity curves for microspheres

of soli MS

The results of Fig. 7 indicate that the sensitivity for the solid sphere can be

expressed as:

types; a) Hollow PMMA; b) Hollow PDMS (400μm diameter)

of different dλ 12810388.8 −−= EDxdF

(27)

an increase in sensitivity.

Together with the sensitivity, dλ/dF, the optical quality factor, Q=λ/ λ, determines the minimum measurement resolution [1]. If we assume that the minimum measurable WGM shift

As expected, a decrease in both the sphere diameter and Young’s modulus results inδ

is ∆λ= λ/Q, the measurement resolution is: 1−

⎟⎠

⎜⎝

=dFQ

Fδ (28)

Figure 7 indicates that for a solid PDMS sensor of ~900 µm diameter and a mixture ratio of 50:1, a force resolution

⎞⎛ dλλ

of ~10 nN is possible with a Q-factor of 107. Sensitivities and force resohave been listed in Table I. The data listed for PMMA are results fro

thus, are estimates. As it is indicated in

Material dλ/dF, Resolution, N

lutions of several PMMA and PDMS spheres m previous experimental studies [1], whereas

the data for PDMS are calculated analytically and, Table I: Sensitivity and estimated force resolutions of polymeric spheres (D=400µm), with Q=107

Solid PMMA 2 nm/N 5x10-4

Hollow PMMA -6 50 nm/N 2x10

Sol 1) 0.5 pm -10id PDMS (60: /nN 2x10

Hollow PDMS (60:1) -12 50 pm/nN 2x10

the table, choosing h geometry for the polymeric microspheres may improve the sensitivity around two orders of magnitude. We note that these

ts also indicate that with hollow, 60:1 mixture

m asurements. We take the highest frequency he first natural freq en approach, the other movi

Ws

ollow

resulratio PDMS sensors, force resolutions of the order of a pN may be possible.

V. Sensor bandwidth

Sensor bandwidth is an important

parameter in most force e response of the sensor as t

cy of the sphere. In this ung parts of the sensor are not taken into

account and the sphere is considered undamped. It should be noted that the additional mass of the other moving parts will reduce the bandwidth estimate.

e calculate the resonant frequency by numerically olving the characteristic equation for the sphere [16]

Figure 7: Force sensitivity curves for solid PDMS microsphere

0*23 =⎟

⎠⎞

⎜⎝⎛

CaJ nω

(29) 22 **2

1* ⎟⎠⎞

⎜⎝

−⎟⎠

⎜⎝

⎟⎠

⎜⎝− C

aC

aJC

a n

ν

11 2

⎛⎞⎛⎞⎛− nn ωωων

where ωn is the angular frequency, Jn's are the Bessel functions of first kind and C* is the compressive wave velocity defined as

ρ2* =C (30)

μλ̂ +

where and μ é constants and the mi osphere material density The ressolid PDMS sphere sensors are shown in Fig.8. Again, the values in this figure represent the

p

shell filled with air is considered. The resonant (frequency) modesthe equation [18]:

λ̂ are the Lam ρ is cr . ults for

Figure 8: Estimated sensor bandwidths for solid PDMS spheres

highest ossible frequency responses that are associated with the force sensor. They do not take into account any other moving parts of these sensors other than the sphere itself. In the previous analysis, it was shown that the lower the Young's modulus the better the force resolution. However, there is a trade off. As it is seen in the figure, the sensor bandwidth is significantly smaller for sphere sensors with low Young's moduli than those with the higher values. We also note that for a given E, the smaller the sensor, the larger the bandwidth due to the effect of sensor mass.

In the case of hollow microsphere, a thin for the case (b/a > 0.91) are obtained by solving

( ) 01

1111

11)1(12

11 3

2

22 ⎥⎤

⎢⎡

⎟⎞

⎜⎛

⎪⎫

⎪⎪

⎨

⎥⎤

⎢ ⎟⎞

⎜⎛

⎤⎡⎤⎢⎡

⎟⎞

⎜⎛ − + c

RJRv n

n ωωβ

ωω

22

21

2

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

+−

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎥

⎦⎢⎢⎢

⎣⎟⎠⎞

⎜⎝⎛

⎠⎝−

⎪⎪

⎭

⎪⎬

⎪⎪

⎩

⎧

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

⎡

−⎠⎝−++

⎥⎥⎦⎢

⎢⎣

⎟⎠

⎞⎜⎝

⎛+⎥⎥⎦⎢⎣

−⎟⎠

⎞⎜⎝

⎛⎠⎝ +

+

vnn

cR

vv

cR

hR

cRJ

znv

cnn

cR

cR

v

s

n

nn

ωβωβρρ

ωββ

(31)

where Jn's are the Bessel functions of first kind; c is the speed of the sound of air in sphere shell; R is the average radius (average of inner and outer radius) and h=a-b. ρs and ρ are the density of the shell material and the fluid inside the microsphere (air for the present case) respectively; and β is defined as:

( )vEcs += 1

2ρβ (32)

The sensor upper bandwidth limits for a solid and hollow PMMA as well as hollow PDMS spheres are shown in Fig. 9. As expected, the sensor bandwidth is narrower for sensors of higherb/a>0.91, the system is essentially a shell and the resonant frequ

ominated by the compressibility of the gas inside the shell. Thus, for shell sensors b/a and E do not have any

sensitivity. Note that for hollow spheres with encies of the system (sensor bandwidth) is

dsignificant influence on bandwidth. Also, while hollow PMMA spheres have been shown to have better force resolutions than those of the solid PMMA [1], they have narrower sensor bandwidth when compared with their solid counterpart. For PDMS microsphere sensors, however, the solution of Eq. (31) for sphere shells indicate better

sensor bandwidths than those of the solid ones predicted by Eq. (29). This is due to the small mass of the thin walled sphere shell as compared to the solid sphere. Thus, hollow PDMS microsphere sensors provide both high force resolutions as well as higher bandwidths.

VI. Conclusion Analytical studies of polymeric

spheres under the effect of uniaxial compressive force have been carried out and these findings have been validated through xperimental results. Overall, the results of e present study show that the force

sensitivity for a function of the sphe

choosing different combination of sphere material and geometry. Fobserved experimentally, while analytical results show that hollow spN resolution. While the sensitivity can be improved by choosing a mtrade off in performance. For solid polymeric microspheres, the bansensitivity increases. As shown in the analysis, for PMMA and PDM

ors", Appl. Optics, Vol. 47, No:16, pp. 3009-3014 (2008).

2. Klitzing von, W., “ Tunable whispering mode copy and CQED Experiments,” New Journal of Physics, Vol. 3, pp. 14.1-14.14 (2001).

Opt. Lett.

5. Little, B. E., S. T. Chu, H. A. Haus, “Microring resonator channel dropping filters,” Journal of Lightwave

el, and J. A. Lock, “Enhanced coupling to microsphere resonances with

un. 145, 86–90 (1998).

80, 4057–4059 (2002).

by p

rospace Sciences Meeting and Exhibition,

eth

a WGM sensor isre material property and geometry. A

wide range of sensitivities can be obtained by orce resolutions of the order of nN have been phere sensors made of PDMS may reach up to aterial with lower Young's modulus, there is a dwidth of the sensor becomes narrower as the S sensors, the bandwidths are calculated to be

of the order of MHz and hundred kHz, respectively. Hollow PMMA microspheres also have narrower bandwidth compared with their solid counterpart. For PDMS microsphere sensors, however, the analysis indicates that thin walled hollow PDMS sensors have both higher force resolutions bandwidths than their solid counterpart, making them exceptionally attractive in sensor applications.

References

1. T. Ioppolo, M. Kozhevnikov, V. Stepaniuk, M. V. Ötügen, and V. Sheverev, "A Micro-Optical Force Sensor Concept Based on Whispering Gallery Mode Resonat

Figure 9: Estimated sensor bandwidth for solid PMMA, hollow PDMS and hollow PMMA

s for spectros

3. M. Cai, O. Painter, and K. J. Vahala, “Fiber-coupled microsphere laser,” 25, 1430-1432 (2000). 4. Tapalian, H. C., J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photonics Technology Letters, Vol. 14, no.8, pp. 1118-1120 (2002).

Technology, Vol. 15, pp. 998-1000 (1997). 6. A. Serpengüzel, S. Arnold, G. Griff

optical fibers,” J. Opt. Soc. Am. B., vol. 14, no. 4, pp. 790-795, Apr. 1997. 7. V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Comm8. F. Vollmer, D. Brown, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.9. S. Arnold, M. Khoshsima, I, Teraoka, S. Holler, F. Vollmer, “Shift of whispering gallery modes in microspheres

rotein adsorption,” Optics Letters, Vol. 28, no. 4, pp. 272-274 (2003). 10. A. T. Rosenberger and J. P. Rezac, “Whispering-gallerymode evanescent-wave microsensor for trace-gasdetection,” Proc. SPIE 4265, 102–112 (2001). 11. N. Das, T. Ioppolo, and V. Ötügen, “Investigation of a micro-optical concentration sensor concept based on whispering gallery mode resonators,” presented at the 45th AIAA AeReno, Nev., January 8–11 2007.

c.

Ötügen, V. Sheverev, "A Micro-Optical Wall Shear Stress Sensor Concept Based 8.

dier, Theory of Elasticity, McGraw-Hill, 1970.

tions, New York: Dover, 1970.

roidal Shells", J. of Acoustical Society of

12. T. Ioppolo and M. V. Ötügen, "Pressure Tuning of Whispering Gallery Mode Resonators",2007, J. Opt. SoAm. B, Oct. 2007, vol. 24, No. 10. 13. G. Guan, S. Arnold and M. V. Ötügen, "Temperature Measurements Using a Micro-Optical Sensor Based onWhispering Gallery Modes", 2006, AIAA Journal, Vol. 44, pp. 2385-2389. 14. T. Ioppolo, U. K. Ayaz, M.V.on Whispering Gallery Mode Resonators", 46th AIAA Aerospace Sciences Meeting and Exhibit, 8-11 January 20015. S. P. Timoshenko and J.N. Goo16. A. E. H. Love, The Mathematical Theory of Elasticity, Dover, 1926. 17. M. Abramowitz and I. A. Stegun, Eds., Handbook on mathematical func18. F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Applied Physics 96, 341-345 (2004). 19. J. E. Mark, Polymer Data Handbook, Oxford University Press, 1999. 20. R. Rand, F. DiMaggio, "Vibrations of Fluid-Filled Spherical and SpheAmerica, Vol. 462, No 6, pp. 1278-1286, 1967.