Abstract

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 OSA

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, 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 [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 micro-disk 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

2πn0a=lλ
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 representing the circumferential mode number. Equation (1) is a first order approximation and holds for a >>λ. A minute change in the size or the refractive index of the microsphere will lead to a shift in the resonance wavelength as
dλλ=dn0n0+daa
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. Electrostatic Field-Induced Stress in a Solid Dielectric Sphere

We first consider an isotropic solid dielectric sphere of radius a and inductive capacity ε1 , embedded in an inviscid dielectric fluid of inductive capacity ε2. The sphere is subjected to a uniform electric field E0 in the direction of negative z as shown in Fig. 1 .

 

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

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

2u+112νu+fG=0
where u is the displacement of a given point within the dielectric sphere, ν, 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=12E2ε114(a1+a2)E2
where Eis the electric field within the sphere, ε is the inductive capacity, a1 and a2 are coefficients that describe the dielectric properties. Physically, the parameter a1 represents the change of inductive capacity ε 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 [18]:
E=3ε2ε1+2ε2E0
Therefore, the second term on the right hand side of Eq. (4) is also zero. Thus, Eq. (3) becomes:
2u+112νu=0
The solution of this equation in spherical coordinates is given by [14]:
ur=[An(n+1)(n2+4ν)rn+1+Bnnrn1]Pn(cosϑ)uϑ=[An(n+1)(n+54ν)rn+1+Bnnrn1]dPn(cosϑ)dϑ}
where ur and uϑ are the components of displacement in the radial, r, and polar, ϑ 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:

σrr=2G[An(n+1)(n2n22ν)rn+Bnn(n1)rn2]Pn(cosϑ)
σϑϑ=2G{[An(n2+4n+2+2ν)(n+1)rn+Bnn2rn2]Pn(cosϑ)+[An(n+54ν)rn+Bnrn2]cot(ϑ)dPn(cosϑ)dϑ}
σϕϕ=2G{[An(n+1)(n22ν4nν)rn+Bnnrn2]Pn(cosϑ)+[An(n+54ν)rn+Bnrn2]cotϑdPn(cosϑ)dϑ}
σrϑ=2G[An(n2+2n1+2ν)rn+Bn(n1)rn2]Pn(cosϑ)ϑ
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=[αE(En)]2+[αE(En)]1[βE2n]2[βE2n]1
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 medium) The constants α and β are given as [18]:
α=ε+a2a12   ,       β=ε+a22
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:
α=ε       ,       β=ε06(k22k2)
for the fluid (medium 2). Here, ε0 is the inductive capacity of vacuum, and k (k = ε/ε0) is the dielectric constant. Using Eq. (5) and Eq. (12) the pressure acting at the dielectric interface is given by:
P=(A'B')Cos(ϑ)2+B'
where A’ and B’ are defined as:
A'=(3ε2ε1+2ε2E0)2[(ε1ε2)2(α2β2)α1+β1]
B'=(3ε2ε1+2ε2E0)2(β1β2)
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
P=ε06E2(k21)(k2+2)
For gas media, k≈1, thus δP 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

σrr(a)=Pσrϑ(a)=0
The coefficient An and Bn are determined by expanding the pressure P in terms of Legendre series as follows:
P=ZnPn(cosϑ)
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=13(A'+2B')   ,   Z2=23(A'B')
Plugging Eq. (8) and (11) and Eq. (20) and (21), into Eq. (19), the coefficients An and Bn are determinate as follows:
A0=(A'+2B')12G(1+ν),A2=(A'B')6Ga2(5ν+7),B2=(A'B')(2ν+7)6G(5ν+7)
The radial deformation can be determined by using Eq. (7):

ur=2A0(2ν1)r+(12A2νr3+2B2r)12(3cos(ϑ)21)

3. 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 ϑ = π/2) by plugging Eq. (22) into Eq. (23):

daa=(3ε2ε1+2ε2E0)2{(12ν)6G(1+ν)[((ε1ε2)2(α2β2)α1+3β12β2)]+(4ν7)3G(5ν+7)[(ε1ε2)2(α2β2)+α1β2]}
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+Cσ1rr+C2(σϑϑ+σφφ)nϑ=noϑ+Cσ1ϑϑ+C2(σrr+σφφ)nφ=noφ+Cσ1φφ+C2(σϑϑ+σrr)
Here nr,nϑ,nφ are the refractive indices in the direction of the three principle stresses and n0r,n0ϑ,n0φ are those values for the unstressed material. Coefficients C1 and C2 are the elasto-optical constants of the material. In our analysis we consider PDMS microspheres that are manufactured as described in Ref [15]. For PDMS these values are C1 = C2 = C = −1.75x10−10 m2/N [20]. Thus, for a spherical sensor, the fractional change in the refractive index due to mechanical stress is reduced to:
dnono=nrnornor=nϑnoϑnoϑ=nrnoφnoφ=C(σrr+σϑϑ+σφφ)n
Thus, evaluating the appropriate expressions for stress in Eq. (8), 9, 10) at ϑ = π/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 no experimental measurements of a1 and a2 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 ∆λ = λ/Q, the measurement resolution is defined as δE=(λ/Q)(dλ/dE)1 . The results of Fig. 2 indicate that for a quality factor of Q~107 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).

 

Fig. 2 The WGM shift of a solid 1 mm diameter PDMS sphere due to applied electric field (base-to-curing-agent ratio of 60:1, a/λ = 381).

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

In this section we consider a dielectric spherical shell of inductive capacity ε1 with inner radius a and outer radius b that is placed in a uniform dielectric fluid of inductive capacity ε2 as shown in Fig. 3 . The shell is filled with a fluid of inductive capacity ε3. 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.

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

Φ(r,ϑ,ϕ)=i=0(Airi+Biri1)Pi(cosϑ)
where Ф is the potential function. From the above equation, the potential function in each medium can be written as:
Φ1=Bab(ra)cosϑ+C(ba)(ar)2cosϑΦ2=E0b(rb)cosϑ+D(br)2cosϑΦ3=A(ra)cosϑ
Constant A, B, C, D are determined by satisfying the boundary condition at each interface, which are defined as:
Φ3(a)=Φ1(a)     Φ1(b)=Φ2(b)ε3Φ3r|a=ε1Φ1r|aε1Φ1r|b=ε2Φ2r|b
The coefficients are obtained by solving the following linear system
(α12α12α13α14α21α22α23α24α31α32α33α34α41α42α43α44)(ABCD)=(γ1γ2γ3γ4)
The matrix coefficient αij and γi are presented in Appendix A. The electric field distribution in each medium is obtained by
E=Φ
From the above equation each component of the electric field can be obtained, and are listed as follows:
E1,r=[B1ab+2C(abr3)]cosϑ       E1,ϑ=[B1ab+Cabr3]sinϑE2,r=[2Db2r3+E0]cosϑ           E2,ϑ=[Db2r3E0]sinϑE3,r=Aacosϑ                     E3,ϑ=Aasinϑ
Where Er and Eϑ 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=[αE(En)]a+[αE(En)]b[βE2]a[βE2]b
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=(ZY)cos(ϑ)2+YP1,2=(KW)cosϑ2+W
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:
Z=(Aa)2[(ε3ε1)2(α1β1)α3+β3]   ,Y=(Aa)2(β3β1)
K=(B1ab2Cab2)2[(ε1ε2)2(α2β2)α1+β1],W=(Bab1a+Cab2)2(β1β2)
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=ε06A2a2(k31)(k3+2)
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=14(a1+a2){[(18C2a2b2r7+18BCabr4)Cos2(ϑ)6C2a2b2r76BCabr4]r+Sin(2ϑ)(3C2a2b2r76BCabr4)ϑ}
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 (ϑ) is compared to the distribution of radial and polar body force per unit volume times the shell thickness, Bt.

 

Fig. 4 Pressure and body force distributions for a spherical PDMS shell (base-to-curing-agent ratio 60:1, a = 300 μm, a/λ = 288) due to applied electric field.

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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=[An(n+1)(n2+4ν)Rn+1+BnnRn1]Pn(cosϑ)+[CnRn(n2+3n2ν)+Dn(n+1)(n+2)Rn+2]Pn(cosϑ)
whereas the corresponding stress components are:
σrr=2G[An(n+1)(n2n22ν)Rn+Bnn(n1)Rn2]Pn(cosϑ)+[CnnRn+1(n2+3n2ν)+Dn(n+1)(n+2)Rn+3]Pn(cosϑ)
σϑϑ=2G[An(n2+4n+2+2ν)(n+1)rn+Bnn2rn2+Cnnrn+1(n22n1+2ν)Dn(n+1)2rn+3]Pn(cosϑ)[An(n+54ν)rn+Bnrn2Cnrn+1(n+44ν)+Dnrn+3]cotϑdPn(cosϑ)dϑ
σφφ=2G[An(n+1)(n22ν4nν)rn+Bnnrn2+Cnnrn+1(n+34nν2ν)Dn(n+1)rn+3]Pn(cosϑ)+[An(n+54ν)rn+Bnrn2+Cnrn+1(n+44ν)+Dnrn+3]cotϑdPn(cosϑ)dϑ
σrϑ=2G[An(n2+2n1+2ν)Rn+Bn(n1)Rn2]Pn(cosϑ)ϑ+[CnRn+1(n22+2ν)Dn(n+2)Rn+3]Pn(cosϑ)ϑ
The constants An, Bn, Cn and Dn are determined by satisfying the boundary conditions. The boundary conditions are defined as follow:
σrr(a)=P3P1,3       σrr(b)=P1,2σrϑ(a)=0           σrϑ(b)=0
The pressure acting at the boundaries of the hollow sphere can be expanded into Fourier-Legendre series as
P1,3=EnPn(cosϑ)=(ZY)cos(ϑ)2+YP1,2=FnPn(cosϑ)=(KW)cos(ϑ)2+W
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=13(Z+2Y)   F0=13(K+2W)E2=23(ZY)     F2=23(KW)
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
(β11β12β13β14β21β22β23β24β31β32β33β34β41β42β43β44)(A0B0C0D0)=(ϕ1ϕ2ϕ3ϕ4)   (δ11δ12δ13δ14δ21δ22δ23δ24δ31δ32δ33δ34δ41δ42δ43δ44)(A2B2C2D2)=(ρ1ρ2ρ3ρ4)
The matrix coefficients βij, ϕi, δij, ρi j 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. (39). However, as discussed earlier, dn0/n0<<da/a, thus we neglect this effect on WGM shifts.

The WGM shifts at the equatorial belt (ϑ = π/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 effect 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 elongates in the direction normal to the applied field.

 

Fig. 5 The WGM shifts of a hollow PDMS (60:1) sphere with the applied electric field due to strain effects (a/b = 0.95, b/λ = 381).

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Next we look at the case where the fluid inside the sphere has a higher inductive capacity than that of the surrounding medium 32). For 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. A comparison of Fig. 6 and 5 reveals that, filling the sphere with water increases the sensitivity significantly.

 

Fig. 6 The WGM shifts of a hollow PDMS (60:1) sphere filled with water (a/b = 0.95, b/λ = 381).

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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 contaminants in surrounding medium? 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 10k V/m and the refractive index 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 the refractive index of ~10−4 in a gas (at the wavelength λ = 1.312 μm) . Figure 7, when compared with the analysis of Ref [10], indicates a resolution improvement of at least an order of magnitude when the electric field is applied to the micro-sphere. These results shows that a sensor could be developed for the detection of contaminants both in air and in liquids.

 

Fig. 7 WGM shifts of a hollow PDMS (60:1) sphere filled with water (a/b = 0.95, b/λ = 381) under constant electric field of 10kV/m. The WGM shifts obtained here are due to the change of refractive index of the surrounding medium.

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5. Experiments

Electrostriction experiments were carried out using a solid PDMS microsphere (with base to curing agent ratio of 60:1). The diameter of the microspheres was ~900 μm and it was manufactured using the same procedure as in our earlier studies [12]. The optical setup is similar to those reported in [11,12]. Briefly, the output of a distributed feedback (DFB) laser diode (with a nominal wavelength of ~1312 nm) is coupled into a single mode optical fiber. A section of the fiber is heated and stretched to facilitate optical coupling between the microsphere and the optical fiber. Stable coupling is achieved by bringing the tapered fiber in contact with the microsphere. The DFB laser is current-tuned over a range of ~0.1 nm using a laser controller while its temperature kept constant. The laser controller, in turn, is driven by a function generator which provides a saw tooth input to the controller. A schematic of the experimental arrangement is shown in Fig. 8 . The quality factor of the WGMs were observed to be Q~106 during the experiments.

 

Fig. 8 Experimental setup.

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The PDMS microsphere is placed in between two square electrodes made of brass. The side and thickness of the electrodes are 25 mm and 0.25 mm, respectively. The gap between the two electrodes is 4.5 mm. The microsphere is held in place by a 125 µm diameter silica stem (that is fixed to the PDMS sphere during the curing process). The electrodes are connected to a dc voltage supply. As the voltage is gradually increased the, WGM shifts are recorded and analyzed on a personal computer.

The experimental results are shown in Fig. 9 along with the analytical expression of Eq. (24). As shown in the figure, the same experiment is repeated multiples times over a period of several hours. There is good agreement between the experimentally obtained WGM shifts of test 1 and those predicted by Eq. (24). Test 1 was carried out without first exposing the PDMS microsphere to an electric field for an extended period. The additional measurements were made after keeping the sphere exposed to a 200 kV/m electric field over progressively longer periods of time (two minutes, two hours and four hours for tests 2, 3 and 4, respectively).

 

Fig. 9 Experimental and analytical results for a solid PDMS (60: 1) microsphere.

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These results show that the WGM shift dependence on the electric field becomes stronger after exposing the sphere to electric field. This effect is most likely due to the alignment of some of the dipoles in PDMS along the electric field. Such polarization behavior has been observed earlier in polymers [23]. With increased time, a larger number of dipoles are aligned with the electric field resulting in increased polarization of the microsphere. This in turn, leads to higher electrostatic pressure at the sphere surface and hence, increased WGM shift. Test 5 in Fig. 9 was carried out after the sphere was allowed to relax for 24 hours. Clearly, after this relaxation period, the polarization goes back to its initial level and the original WGM shift dependence on the electric field is recovered.

6. Conclusion

Electrostriction effect on the whispering gallery modes of polymeric microspheres was investigated analytically and validated experimentally. The analysis shows that the external electric field strength can be measured by monitoring the WGM shifts. Hollow PDMS spheres that are filled with air are less sensitive than their solid counterparts. However, when a hollow PDMS sphere is filled with a dielectric liquid, the sensitivity of its WGMs to electric field increases significantly. 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 a WGM-based sensor may be feasible for impurity detection in gases or liquids. Electrostatic field tuning of micro-resonator WGMs may also be exploited for fast, narrowband optical switches and filters.

Appendix

α11=α24=1,α12=ab,α13=ba,α14=α21=α34=0,α22=ba,α23=ab,α31=ε1aε2,α32=1ab,α33=2ba2,α43=2ab,α42=1ab,α44=ε3ε22b,γ1=γ3=0,γ2=E0b,γ4=ε3ε2E0
δ11=6νa2,δ12=δ22=2,δ13=2a3(102ν),δ14=12a5,δ21=6νb2,δ23=2b3(102ν),δ24=12b5,δ31=(2ν+7)a2,   δ32=δ42=1,δ33=2a3(1+ν),   δ34=4a5,   δ41=(2ν+7)b2,   δ43=2b3(1+ν),   δ44=4b5,ρ1=13G(ZY),ρ2=13G(KW),ρ3=ρ4=0
β11=β21=2(1+ν),β14=2a3,β24=2b3,β32=1a2,β12=β13=β22=β23=0β31=β41=2v1,β33=2a(ν1),β34=2a3,β42=1b2,β43=2b(ν1),   β44=2b3ϕ1=(Z+2Y)6Gε06A2a(k1)(k+2),ϕ2=(K+2W)6G,ϕ3=ϕ4=0

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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7. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002). [CrossRef]  

8. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28(4), 272–274 (2003). [CrossRef]   [PubMed]  

9. A. T. Rosenberger and J. P. Rezac, “Whispering-gallerymode evanescent-wave microsensor for trace-gas detection,” Proc. SPIE 4265, 102–112 (2001). [CrossRef]  

10. 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 Aerospace Sciences Meeting and Exhibition, Reno, Nev., January 8–11 2007.

11. T. Ioppolo, M. Kozhevnikov, V. Stepaniuk, M. V. Otügen, and V. Sheverev, “Micro-optical force sensor concept based on whispering gallery mode resonators,” Appl. Opt. 47(16), 3009–3014 (2008). [CrossRef]   [PubMed]  

12. T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105(1), 013535 (2009). [CrossRef]  

13. T. Ioppolo and M. V. Ötügen, ““Pressure Tuning of Whispering Gallery Mode Resonators,” J. Opt. Soc. Am. B 24(10), 2721–2726 (2007). [CrossRef]  

14. G. Guan, S. Arnold, and M. V. Ötügen, “Temperature Measurements Using a Micro-Optical Sensor Based on Whispering Gallery Modes,” AIAA J. 44(10), 2385–2389 (2006). [CrossRef]  

15. T. Ioppolo, U. K. Ayaz, M. V. Ötügen, and V. Sheverev, “A Micro-Optical Wall Shear Stress Sensor Concept Based on Whispering Gallery Mode Resonators,” 46th AIAA Aerospace Sciences Meeting and Exhibit, 8–11 January 2008.

16. V. M. N. Passaro and F. De Leonardis, “Modeling 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). [CrossRef]  

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. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. 96(12), 341–345 (2004). [CrossRef]  

20. J. E. Mark, Polymer Data Handbook (Oxford University Press, 1999).

21. T. Yamwong, A. M. Voice, and G. R. Davies, “Electrostrictive response of an ideal polar rubber,” J. Appl. Phys. 91(3), 1472–1476 (2002). [CrossRef]  

22. A. E. H. Love, The Mathematical Theory of Elasticity (Dover, 1926).

23. K. C. Kao, Dielectric Phenomena in Solids (Elsevier Academic Press, 2004).

References

  • View by:
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  • |

  1. W. von Klitzing, “Tunable whispering modes for spectroscopy and CQED Experiments,” New J. Phys. 3, 14.1–14.14 (2001).
    [CrossRef]
  2. M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. 25(19), 1430–1432 (2000).
    [CrossRef]
  3. H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photon. Technol. Lett. 14(8), 1118–1120 (2002).
    [CrossRef]
  4. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
    [CrossRef]
  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).
    [CrossRef]
  6. 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. Commun. 145(1-6), 86–90 (1998).
    [CrossRef]
  7. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
    [CrossRef]
  8. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28(4), 272–274 (2003).
    [CrossRef] [PubMed]
  9. A. T. Rosenberger and J. P. Rezac, “Whispering-gallerymode evanescent-wave microsensor for trace-gas detection,” Proc. SPIE 4265, 102–112 (2001).
    [CrossRef]
  10. 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 Aerospace Sciences Meeting and Exhibition, Reno, Nev., January 8–11 2007.
  11. T. Ioppolo, M. Kozhevnikov, V. Stepaniuk, M. V. Otügen, and V. Sheverev, “Micro-optical force sensor concept based on whispering gallery mode resonators,” Appl. Opt. 47(16), 3009–3014 (2008).
    [CrossRef] [PubMed]
  12. T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105(1), 013535 (2009).
    [CrossRef]
  13. T. Ioppolo and M. V. Ötügen, “Pressure Tuning of Whispering Gallery Mode Resonators,” J. Opt. Soc. Am. B 24(10), 2721–2726 (2007).
    [CrossRef]
  14. G. Guan, S. Arnold, and M. V. Ötügen, “Temperature Measurements Using a Micro-Optical Sensor Based on Whispering Gallery Modes,” AIAA J. 44(10), 2385–2389 (2006).
    [CrossRef]
  15. T. Ioppolo, U. K. Ayaz, M. V. Ötügen, and V. Sheverev, “A Micro-Optical Wall Shear Stress Sensor Concept Based on Whispering Gallery Mode Resonators,” 46th AIAA Aerospace Sciences Meeting and Exhibit, 8–11 January 2008.
  16. V. M. N. Passaro and F. De Leonardis, “Modeling 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).
    [CrossRef]
  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. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. 96(12), 341–345 (2004).
    [CrossRef]
  20. J. E. Mark, Polymer Data Handbook (Oxford University Press, 1999).
  21. T. Yamwong, A. M. Voice, and G. R. Davies, “Electrostrictive response of an ideal polar rubber,” J. Appl. Phys. 91(3), 1472–1476 (2002).
    [CrossRef]
  22. A. E. H. Love, The Mathematical Theory of Elasticity (Dover, 1926).
  23. K. C. Kao, Dielectric Phenomena in Solids (Elsevier Academic Press, 2004).

2009

T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105(1), 013535 (2009).
[CrossRef]

2008

2007

2006

G. Guan, S. Arnold, and M. V. Ötügen, “Temperature Measurements Using a Micro-Optical Sensor Based on Whispering Gallery Modes,” AIAA J. 44(10), 2385–2389 (2006).
[CrossRef]

V. M. N. Passaro and F. De Leonardis, “Modeling 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).
[CrossRef]

2004

F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. 96(12), 341–345 (2004).
[CrossRef]

2003

2002

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[CrossRef]

T. Yamwong, A. M. Voice, and G. R. Davies, “Electrostrictive response of an ideal polar rubber,” J. Appl. Phys. 91(3), 1472–1476 (2002).
[CrossRef]

H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photon. Technol. Lett. 14(8), 1118–1120 (2002).
[CrossRef]

2001

W. von Klitzing, “Tunable whispering modes for spectroscopy and CQED Experiments,” New J. Phys. 3, 14.1–14.14 (2001).
[CrossRef]

A. T. Rosenberger and J. P. Rezac, “Whispering-gallerymode evanescent-wave microsensor for trace-gas detection,” Proc. SPIE 4265, 102–112 (2001).
[CrossRef]

2000

1999

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).
[CrossRef]

1998

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. Commun. 145(1-6), 86–90 (1998).
[CrossRef]

1997

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Agan, S.

F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. 96(12), 341–345 (2004).
[CrossRef]

Arnold, S.

G. Guan, S. Arnold, and M. V. Ötügen, “Temperature Measurements Using a Micro-Optical Sensor Based on Whispering Gallery Modes,” AIAA J. 44(10), 2385–2389 (2006).
[CrossRef]

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28(4), 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[CrossRef]

Ay, F.

F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. 96(12), 341–345 (2004).
[CrossRef]

Ayaz, U. K.

T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105(1), 013535 (2009).
[CrossRef]

Aydinli, A.

F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. 96(12), 341–345 (2004).
[CrossRef]

Beyeler, R.

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).
[CrossRef]

Bona, G. L.

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).
[CrossRef]

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[CrossRef]

Cai, M.

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Davies, G. R.

T. Yamwong, A. M. Voice, and G. R. Davies, “Electrostrictive response of an ideal polar rubber,” J. Appl. Phys. 91(3), 1472–1476 (2002).
[CrossRef]

De Leonardis, F.

V. M. N. Passaro and F. De Leonardis, “Modeling 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).
[CrossRef]

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Germann, R.

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).
[CrossRef]

Guan, G.

G. Guan, S. Arnold, and M. V. Ötügen, “Temperature Measurements Using a Micro-Optical Sensor Based on Whispering Gallery Modes,” AIAA J. 44(10), 2385–2389 (2006).
[CrossRef]

Haroche, S.

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. Commun. 145(1-6), 86–90 (1998).
[CrossRef]

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Holler, S.

Horst, F.

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).
[CrossRef]

Ilchenko, V. S.

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. Commun. 145(1-6), 86–90 (1998).
[CrossRef]

Ioppolo, T.

Khoshsima, M.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28(4), 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[CrossRef]

Kocabas, A.

F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. 96(12), 341–345 (2004).
[CrossRef]

Kocabas, C.

F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. 96(12), 341–345 (2004).
[CrossRef]

Kozhevnikov, M.

Laine, J. P.

H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photon. Technol. Lett. 14(8), 1118–1120 (2002).
[CrossRef]

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Lane, P. A.

H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photon. Technol. Lett. 14(8), 1118–1120 (2002).
[CrossRef]

Lefevre-Seguin, V.

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. Commun. 145(1-6), 86–90 (1998).
[CrossRef]

Libchaber, A.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[CrossRef]

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Offrein, B. J.

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).
[CrossRef]

Otügen, M. V.

Ötügen, M. V.

T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105(1), 013535 (2009).
[CrossRef]

T. Ioppolo and M. V. Ötügen, “Pressure Tuning of Whispering Gallery Mode Resonators,” J. Opt. Soc. Am. B 24(10), 2721–2726 (2007).
[CrossRef]

G. Guan, S. Arnold, and M. V. Ötügen, “Temperature Measurements Using a Micro-Optical Sensor Based on Whispering Gallery Modes,” AIAA J. 44(10), 2385–2389 (2006).
[CrossRef]

Painter, O.

Passaro, V. M. N.

V. M. N. Passaro and F. De Leonardis, “Modeling 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).
[CrossRef]

Raimond, J.-M.

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. Commun. 145(1-6), 86–90 (1998).
[CrossRef]

Rezac, J. P.

A. T. Rosenberger and J. P. Rezac, “Whispering-gallerymode evanescent-wave microsensor for trace-gas detection,” Proc. SPIE 4265, 102–112 (2001).
[CrossRef]

Rosenberger, A. T.

A. T. Rosenberger and J. P. Rezac, “Whispering-gallerymode evanescent-wave microsensor for trace-gas detection,” Proc. SPIE 4265, 102–112 (2001).
[CrossRef]

Salemink, H. W. M.

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).
[CrossRef]

Sercel, P. C.

Sheverev, V.

Stepaniuk, V.

Tapalian, H. C.

H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photon. Technol. Lett. 14(8), 1118–1120 (2002).
[CrossRef]

Teraoka, I.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28(4), 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[CrossRef]

Treussart, F.

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. Commun. 145(1-6), 86–90 (1998).
[CrossRef]

Vahala, K. J.

Velichansky, V. L.

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. Commun. 145(1-6), 86–90 (1998).
[CrossRef]

Voice, A. M.

T. Yamwong, A. M. Voice, and G. R. Davies, “Electrostrictive response of an ideal polar rubber,” J. Appl. Phys. 91(3), 1472–1476 (2002).
[CrossRef]

Volikov, P. S.

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. Commun. 145(1-6), 86–90 (1998).
[CrossRef]

Vollmer, F.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28(4), 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[CrossRef]

von Klitzing, W.

W. von Klitzing, “Tunable whispering modes for spectroscopy and CQED Experiments,” New J. Phys. 3, 14.1–14.14 (2001).
[CrossRef]

Yamwong, T.

T. Yamwong, A. M. Voice, and G. R. Davies, “Electrostrictive response of an ideal polar rubber,” J. Appl. Phys. 91(3), 1472–1476 (2002).
[CrossRef]

AIAA J.

G. Guan, S. Arnold, and M. V. Ötügen, “Temperature Measurements Using a Micro-Optical Sensor Based on Whispering Gallery Modes,” AIAA J. 44(10), 2385–2389 (2006).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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).
[CrossRef]

V. M. N. Passaro and F. De Leonardis, “Modeling 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).
[CrossRef]

IEEE Photon. Technol. Lett.

H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photon. Technol. Lett. 14(8), 1118–1120 (2002).
[CrossRef]

J. Appl. Phys.

T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105(1), 013535 (2009).
[CrossRef]

F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. 96(12), 341–345 (2004).
[CrossRef]

T. Yamwong, A. M. Voice, and G. R. Davies, “Electrostrictive response of an ideal polar rubber,” J. Appl. Phys. 91(3), 1472–1476 (2002).
[CrossRef]

J. Lightwave Technol.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

J. Opt. Soc. Am. B

New J. Phys.

W. von Klitzing, “Tunable whispering modes for spectroscopy and CQED Experiments,” New J. Phys. 3, 14.1–14.14 (2001).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

The sphere in the presence of electric field.

Fig. 2
Fig. 2

The WGM shift of a solid 1 mm diameter PDMS sphere due to applied electric field (base-to-curing-agent ratio of 60:1, a/λ = 381).

Fig. 3
Fig. 3

Notation for a hollow dielectric sphere.

Fig. 4
Fig. 4

Pressure and body force distributions for a spherical PDMS shell (base-to-curing-agent ratio 60:1, a = 300 μm, a/λ = 288) due to applied electric field.

Fig. 5
Fig. 5

The WGM shifts of a hollow PDMS (60:1) sphere with the applied electric field due to strain effects (a/b = 0.95, b/λ = 381).

Fig. 6
Fig. 6

The WGM shifts of a hollow PDMS (60:1) sphere filled with water (a/b = 0.95, b/λ = 381).

Fig. 7
Fig. 7

WGM shifts of a hollow PDMS (60:1) sphere filled with water (a/b = 0.95, b/λ = 381) under constant electric field of 10kV/m. The WGM shifts obtained here are due to the change of refractive index of the surrounding medium.

Fig. 8
Fig. 8

Experimental setup.

Fig. 9
Fig. 9

Experimental and analytical results for a solid PDMS (60: 1) microsphere.

Equations (50)

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2πn0a=lλ
dλλ=dn0n0+daa
2u+112νu+fG=0
f=12E2ε114(a1+a2)E2
E=3ε2ε1+2ε2E0
2u+112νu=0
ur=[An(n+1)(n2+4ν)rn+1+Bnnrn1]Pn(cosϑ)uϑ=[An(n+1)(n+54ν)rn+1+Bnnrn1]dPn(cosϑ)dϑ}
σrr=2G[An(n+1)(n2n22ν)rn+Bnn(n1)rn2]Pn(cosϑ)
σϑϑ=2G{[An(n2+4n+2+2ν)(n+1)rn+Bnn2rn2]Pn(cosϑ)+[An(n+54ν)rn+Bnrn2]cot(ϑ)dPn(cosϑ)dϑ}
σϕϕ=2G{[An(n+1)(n22ν4nν)rn+Bnnrn2]Pn(cosϑ)+[An(n+54ν)rn+Bnrn2]cotϑdPn(cosϑ)dϑ}
σrϑ=2G[An(n2+2n1+2ν)rn+Bn(n1)rn2]Pn(cosϑ)ϑ
P=[αE(En)]2+[αE(En)]1[βE2n]2[βE2n]1
α=ε+a2a12   ,       β=ε+a22
α=ε       ,       β=ε06(k22k2)
P=(A'B')Cos(ϑ)2+B'
A'=(3ε2ε1+2ε2E0)2[(ε1ε2)2(α2β2)α1+β1]
B'=(3ε2ε1+2ε2E0)2(β1β2)
P=ε06E2(k21)(k2+2)
σrr(a)=Pσrϑ(a)=0
P=ZnPn(cosϑ)
Zo=13(A'+2B')   ,   Z2=23(A'B')
A0=(A'+2B')12G(1+ν),A2=(A'B')6Ga2(5ν+7),B2=(A'B')(2ν+7)6G(5ν+7)
ur=2A0(2ν1)r+(12A2νr3+2B2r)12(3cos(ϑ)21)
daa=(3ε2ε1+2ε2E0)2{(12ν)6G(1+ν)[((ε1ε2)2(α2β2)α1+3β12β2)]+(4ν7)3G(5ν+7)[(ε1ε2)2(α2β2)+α1β2]}
nr=nor+Cσ1rr+C2(σϑϑ+σφφ)nϑ=noϑ+Cσ1ϑϑ+C2(σrr+σφφ)nφ=noφ+Cσ1φφ+C2(σϑϑ+σrr)
dnono=nrnornor=nϑnoϑnoϑ=nrnoφnoφ=C(σrr+σϑϑ+σφφ)n
Φ(r,ϑ,ϕ)=i=0(Airi+Biri1)Pi(cosϑ)
Φ1=Bab(ra)cosϑ+C(ba)(ar)2cosϑΦ2=E0b(rb)cosϑ+D(br)2cosϑΦ3=A(ra)cosϑ
Φ3(a)=Φ1(a)     Φ1(b)=Φ2(b)ε3Φ3r|a=ε1Φ1r|aε1Φ1r|b=ε2Φ2r|b
(α12α12α13α14α21α22α23α24α31α32α33α34α41α42α43α44)(ABCD)=(γ1γ2γ3γ4)
E=Φ
E1,r=[B1ab+2C(abr3)]cosϑ       E1,ϑ=[B1ab+Cabr3]sinϑE2,r=[2Db2r3+E0]cosϑ           E2,ϑ=[Db2r3E0]sinϑE3,r=Aacosϑ                     E3,ϑ=Aasinϑ
P=[αE(En)]a+[αE(En)]b[βE2]a[βE2]b
P1,3=(ZY)cos(ϑ)2+YP1,2=(KW)cosϑ2+W
Z=(Aa)2[(ε3ε1)2(α1β1)α3+β3]   ,Y=(Aa)2(β3β1)
K=(B1ab2Cab2)2[(ε1ε2)2(α2β2)α1+β1],W=(Bab1a+Cab2)2(β1β2)
P3=ε06A2a2(k31)(k3+2)
f=14(a1+a2){[(18C2a2b2r7+18BCabr4)Cos2(ϑ)6C2a2b2r76BCabr4]r+Sin(2ϑ)(3C2a2b2r76BCabr4)ϑ}
ur=[An(n+1)(n2+4ν)Rn+1+BnnRn1]Pn(cosϑ)+[CnRn(n2+3n2ν)+Dn(n+1)(n+2)Rn+2]Pn(cosϑ)
σrr=2G[An(n+1)(n2n22ν)Rn+Bnn(n1)Rn2]Pn(cosϑ)+[CnnRn+1(n2+3n2ν)+Dn(n+1)(n+2)Rn+3]Pn(cosϑ)
σϑϑ=2G[An(n2+4n+2+2ν)(n+1)rn+Bnn2rn2+Cnnrn+1(n22n1+2ν)Dn(n+1)2rn+3]Pn(cosϑ)[An(n+54ν)rn+Bnrn2Cnrn+1(n+44ν)+Dnrn+3]cotϑdPn(cosϑ)dϑ
σφφ=2G[An(n+1)(n22ν4nν)rn+Bnnrn2+Cnnrn+1(n+34nν2ν)Dn(n+1)rn+3]Pn(cosϑ)+[An(n+54ν)rn+Bnrn2+Cnrn+1(n+44ν)+Dnrn+3]cotϑdPn(cosϑ)dϑ
σrϑ=2G[An(n2+2n1+2ν)Rn+Bn(n1)Rn2]Pn(cosϑ)ϑ+[CnRn+1(n22+2ν)Dn(n+2)Rn+3]Pn(cosϑ)ϑ
σrr(a)=P3P1,3       σrr(b)=P1,2σrϑ(a)=0           σrϑ(b)=0
P1,3=EnPn(cosϑ)=(ZY)cos(ϑ)2+YP1,2=FnPn(cosϑ)=(KW)cos(ϑ)2+W
E0=13(Z+2Y)   F0=13(K+2W)E2=23(ZY)     F2=23(KW)
(β11β12β13β14β21β22β23β24β31β32β33β34β41β42β43β44)(A0B0C0D0)=(ϕ1ϕ2ϕ3ϕ4)   (δ11δ12δ13δ14δ21δ22δ23δ24δ31δ32δ33δ34δ41δ42δ43δ44)(A2B2C2D2)=(ρ1ρ2ρ3ρ4)
α11=α24=1,α12=ab,α13=ba,α14=α21=α34=0,α22=ba,α23=ab,α31=ε1aε2,α32=1ab,α33=2ba2,α43=2ab,α42=1ab,α44=ε3ε22b,γ1=γ3=0,γ2=E0b,γ4=ε3ε2E0
δ11=6νa2,δ12=δ22=2,δ13=2a3(102ν),δ14=12a5,δ21=6νb2,δ23=2b3(102ν),δ24=12b5,δ31=(2ν+7)a2,   δ32=δ42=1,δ33=2a3(1+ν),   δ34=4a5,   δ41=(2ν+7)b2,   δ43=2b3(1+ν),   δ44=4b5,ρ1=13G(ZY),ρ2=13G(KW),ρ3=ρ4=0
β11=β21=2(1+ν),β14=2a3,β24=2b3,β32=1a2,β12=β13=β22=β23=0β31=β41=2v1,β33=2a(ν1),β34=2a3,β42=1b2,β43=2b(ν1),   β44=2b3ϕ1=(Z+2Y)6Gε06A2a(k1)(k+2),ϕ2=(K+2W)6G,ϕ3=ϕ4=0

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