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
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 , micro-cavity laser technology , and optical communications (switching  filtering  and wavelength division and multiplexing ). For example, mechanical strain  and thermooptical  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 , impurity detection in liquids  as well as mechanical sensing including force [11,12], pressure , temperature  and wall shear stress . 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 . 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 isEquation (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 asEq. (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 .
The force exerted by the electrostatic field on the sphere will induce an elastic deformation (electrostriction) that is governed by the Navier Equation :18]:Eq. (4) is zero. The electric field inside the dielectric sphere is uniform and parallel to the z axis, with its magnitude :Eq. (4) is also zero. Thus, Eq. (3) becomes:14]:
Using the stress displacement equations, the components of stress can be expressed as:18]18]:15] leading to:Eq. (5) and Eq. (12) the pressure acting at the dielectric interface is given by: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
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 conditionsEq. (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:Eq. (8) and (11) and Eq. (20) and (21), into Eq. (19), the coefficients An and Bn are determinate as follows:Eq. (7):
3. WGM Shift in a Solid Sphere Due to Electrostriction
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 :15]. For PDMS these values are C1 = C2 = C = −1.75x10−10 m2/N . Thus, for a spherical sensor, the fractional change in the refractive index due to mechanical stress is reduced to: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 . 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 . 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).
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.
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:Eq. (7) and Eq. (12) the pressure distributions at the inner and outer interface are given as follows:18]: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: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.
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 :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 systemsEq. (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.
Next we look at the case where the fluid inside the sphere has a higher inductive capacity than that of the surrounding medium (ε3>ε2). 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.
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 , 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.
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 . 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.
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).
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 . 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.
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.
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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