Tuning of whispering gallery modes of spherical resonators using an external electric field

Tindaro Ioppolo; Ulas Ayaz; M. Volkan Ötügen

doi:10.1364/OE.17.016465

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

2 π n_{0} a = l λ

where λ is the vacuum wavelength of laser, n_o 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

\frac{d λ}{λ} = \frac{d n_{0}}{n_{0}} + \frac{d a}{a}

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]:

\nabla^{2} u + \frac{1}{1 - 2 ν} \nabla \nabla \cdot u + \frac{f}{G} = 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 = - \frac{1}{2} {\vec{E}}^{2} \nabla ε_{1} - \frac{1}{4} (a_{1} + a_{2}) \nabla {\vec{E}}^{2}

where

\vec{E}

is 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 = \frac{3 ε_{2}}{ε_{1} + 2 ε_{2}} E_{0}

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

\nabla^{2} u + \frac{1}{1 - 2 ν} \nabla \nabla \cdot u = 0

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

\begin{array}{l} u_{r} = \sum_{} [A_{n} (n + 1) (n - 2 + 4 ν) r^{n + 1} + B_{n} n r^{n - 1}] P_{n} (\cos ϑ) \\ u_{ϑ} = \sum_{} [A_{n} (n + 1) (n + 5 - 4 ν) r^{n + 1} + B_{n} n r^{n - 1}] \frac{d P_{n} (\cos ϑ)}{d ϑ} \end{array}}

where u_r and u_ϑ are the components of displacement in the radial, r, and polar, ϑ directions. P_n’s represent the Legendre polynomials, and A_n and B_n are constants that are determined by satisfying the boundary conditions.

Using the stress displacement equations, the components of stress can be expressed as:

σ_{r r} = 2 G \sum_{} [A_{n} (n + 1) (n^{2} - n - 2 - 2 ν) r^{n} + B_{n} n (n - 1) r^{n - 2}] P_{n} (\cos ϑ)

σ_{ϑ ϑ} = - 2 G \sum_{} {\begin{cases} [A_{n} (n^{2} + 4 n + 2 + 2 ν) (n + 1) r^{n} + B_{n} n^{2} r^{n - 2}] P_{n} (\cos ϑ) + \\ [A_{n} (n + 5 - 4 ν) r^{n} + B_{n} r^{n - 2}] \cot (ϑ) \frac{d P_{n} (\cos ϑ)}{d ϑ} \end{cases}}

σ_{ϕ ϕ} = 2 G \sum_{} {\begin{cases} [A_{n} (n + 1) (n - 2 - 2 ν - 4 n ν) r^{n} + B_{n} n r^{n - 2}] P_{n} (\cos ϑ) + \\ [A_{n} (n + 5 - 4 ν) r^{n} + B_{n} r^{n - 2}] \cot ϑ \frac{d P_{n} (\cos ϑ)}{d ϑ} \end{cases}}

σ_{r ϑ} = 2 G \sum_{} [A_{n} (n^{2} + 2 n - 1 + 2 ν) r^{n} + B_{n} (n - 1) r^{n - 2}] \frac{\partial P_{n} (\cos ϑ)}{\partial 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 = {[α \vec{E} (\vec{E} \cdot \vec{n})]}_{2} + {[α \vec{E} (\vec{E} \cdot \vec{n})]}_{1} - {[β E^{2} \vec{n}]}_{2} - {[β E^{2} \vec{n}]}_{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 + \frac{a_{2} - a_{1}}{2}, β = \frac{ε + a_{2}}{2}

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 = - \frac{ε_{0}}{6} (k^{2} - 2 k - 2)

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^{'}) C o s {(ϑ)}^{2} + B^{'}

where A’ and B’ are defined as:

A^{'} = {(\frac{3 ε_{2}}{ε_{1} + 2 ε_{2}} E_{0})}^{2} [{(\frac{ε_{1}}{ε_{2}})}^{2} (α_{2} - β_{2}) - α_{1} + β_{1}]

B^{'} = {(\frac{3 ε_{2}}{ε_{1} + 2 ε_{2}} E_{0})}^{2} (β_{1} - β_{2})

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 = \frac{ε_{0}}{6} E^{2} (k_{2} - 1) (k_{2} + 2)

For gas media, k≈1, thus δP is negligible.

In order to define the stress and strain distributions within the sphere, coefficient A_n and B_n have to be evaluated. These coefficients are calculated by satisfying the following boundary conditions

\begin{array}{l} σ_{r r} (a) = P \\ σ_{r ϑ} (a) = 0 \end{array}

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

P = \sum_{} Z_{n} P_{n} (\cos ϑ)

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 Z_n are defined as:

Z_{o} = \frac{1}{3} (A^{'} + 2 B^{'}), Z_{2} = \frac{2}{3} (A^{'} - B^{'})

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

A_{0} = - \frac{(A^{'} + 2 B^{'})}{12 G (1 + ν)}, A_{2} = - \frac{(A^{'} - B^{'})}{6 G a^{2} (5 ν + 7)}, B_{2} = \frac{(A^{'} - B^{'}) (2 ν + 7)}{6 G (5 ν + 7)}

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

u_{r} = 2 A_{0} (2 ν - 1) r + (12 A_{2} ν r^{3} + 2 B_{2} r) \frac{1}{2} (3 \cos {(ϑ)}^{2} - 1)

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):

\frac{d a}{a} = {(\frac{3 ε_{2}}{ε_{1} + 2 ε_{2}} E_{0})}^{2} {\begin{cases} \frac{(1 - 2 ν)}{6 G (1 + ν)} [({(\frac{ε_{1}}{ε_{2}})}^{2} (α_{2} - β_{2}) - α_{1} + 3 β_{1} - 2 β_{2})] + \\ \frac{(4 ν - 7)}{3 G (5 ν + 7)} [- {(\frac{ε_{1}}{ε_{2}})}^{2} (α_{2} - β_{2}) + α_{1} - β_{2}] \end{cases}}

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]:

\begin{array}{l} n_{r} = n_{o r} + C {}_{1}σ_{r r} + C_{2} (σ_{ϑ ϑ} + σ_{φ φ}) \\ n_{ϑ} = n_{o ϑ} + C {}_{1}σ {_{ϑ}}_{ϑ} + C_{2} (σ_{r r} + σ_{φ φ}) \\ n_{φ} = n_{o φ} + C {}_{1}σ_{φ φ} + C_{2} (σ_{ϑ ϑ} + σ_{r r}) \end{array}

Here

n_{r}, n_{ϑ}, n_{φ}

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

n_{0 r}, n_{0}_{ϑ}, n_{0}_{φ}

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:

\frac{d n_{o}}{n_{o}} = \frac{n_{r} - n_{o r}}{n_{o r}} = \frac{n_{ϑ} - n_{o ϑ}}{n_{o ϑ}} = \frac{n_{r} - n_{o φ}}{n_{o φ}} = \frac{C (σ_{r r} + σ_{ϑ ϑ} + σ_{φ φ})}{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

δ E = (λ / Q) {(d λ / d E)}^{- 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) = \sum_{i = 0}^{} (A_{i} r^{i} + B_{i} r^{- i - 1}) P_{i} (\cos ϑ)

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

\begin{array}{l} Φ_{1} = B \sqrt{\frac{a}{b}} (\frac{r}{a}) \cos ϑ + C (\frac{b}{a}) {(\frac{a}{r})}^{2} \cos ϑ \\ Φ_{2} = - E_{0} b (\frac{r}{b}) \cos ϑ + D {(\frac{b}{r})}^{2} \cos ϑ \\ Φ_{3} = A (\frac{r}{a}) \cos ϑ \end{array}

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

\begin{array}{l} Φ_{3} (a) = Φ_{1} (a) Φ_{1} (b) = Φ_{2} (b) \\ {ε_{3} \frac{\partial Φ_{3}}{\partial r} |}_{a} = {ε_{1} \frac{\partial Φ_{1}}{\partial r} |}_{a} {ε_{1} \frac{\partial Φ_{1}}{\partial r} |}_{b} = {ε_{2} \frac{\partial Φ_{2}}{\partial r} |}_{b} \end{array}

The coefficients are obtained by solving the following linear system

(\begin{array}{c} α_{12} & α_{12} & α_{13} & α_{14} \\ α_{21} & α_{22} & α_{23} & α_{24} \\ α_{31} & α_{32} & α_{33} & α_{34} \\ α_{41} & α_{42} & α_{43} & α_{44} \end{array}) \cdot (\begin{array}{c} A \\ B \\ C \\ D \end{array}) = (\begin{array}{c} γ_{1} \\ γ_{2} \\ γ_{3} \\ γ_{4} \end{array})

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

\vec{E} = - \nabla Φ

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

\begin{array}{l} E_{1, r} = [- B \sqrt{\frac{1}{a b}} + 2 C (\frac{a b}{r^{3}})] \cos ϑ E_{1, ϑ} = [B \sqrt{\frac{1}{a b}} + C \frac{a b}{r^{3}}] \sin ϑ \\ E_{2, r} = [2 D \frac{b^{2}}{r^{3}} + E_{0}] \cos ϑ E_{2, ϑ} = [D \frac{b^{2}}{r^{3}} - E_{0}] \sin ϑ \\ E_{3, r} = - \frac{A}{a} \cos ϑ E_{3, ϑ} = \frac{A}{a} \sin ϑ \end{array}

Where E_r and E_ϑ 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 = {[α \vec{E} (\vec{E} \cdot \vec{n})]}_{a} + {[α \vec{E} (\vec{E} \cdot \vec{n})]}_{b} - {[β E^{2}]}_{a} - {[β E^{2}]}_{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:

\begin{array}{l} P_{1, 3} = (Z - Y) \cos {(ϑ)}^{2} + Y \\ P_{1, 2} = (K - W) \cos ϑ^{2} + W \end{array}

Where P_1,3 is the pressure at the inner surface of the shell, while P_1,2 is the pressure on the outer surface. The constant Z, Y, K and W are defined as:

Z = {(\frac{A}{a})}^{2} [{(\frac{ε_{3}}{ε_{1}})}^{2} (α_{1} - β_{1}) - α_{3} + β_{3}], Y = {(\frac{A}{a})}^{2} (β_{3} - β_{1})

\begin{array}{l} K = {(B \sqrt{\frac{1}{a b}} - 2 C \frac{a}{b^{2}})}^{2} [{(\frac{ε_{1}}{ε_{2}})}^{2} (α_{2} - β_{2}) - α_{1} + β_{1}], \\ W = {(B \sqrt{\frac{a}{b}} \frac{1}{a} + C \frac{a}{b^{2}})}^{2} (β_{1} - β_{2}) \end{array}

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]:

P_{3} = \frac{ε_{0}}{6} \frac{A^{2}}{a^{2}} (k_{3} - 1) (k_{3} + 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 = - \frac{1}{4} (a_{1} + a_{2}) {\begin{cases} [\begin{array}{l} (- \frac{18 C^{2} a^{2} b^{2}}{r^{7}} + \frac{18 B C \sqrt{a b}}{r^{4}}) C o s^{2} (ϑ) \\ - \frac{6 C^{2} a^{2} b^{2}}{r^{7}} - \frac{6 B C \sqrt{a b}}{r^{4}} \end{array}] \vec{r} \\ + S i n (2 ϑ) (\frac{3 C^{2} a^{2} b^{2}}{r^{7}} - \frac{6 B C \sqrt{a b}}{r^{4}}) \vec{ϑ} \end{cases}}

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]:

\begin{array}{l} u_{r} = \sum_{} [A_{n} (n + 1) (n - 2 + 4 ν) R^{n + 1} + B_{n} n R^{n - 1}] P_{n} (\cos ϑ) + \\ [\frac{C_{n}}{R^{n}} (n^{2} + 3 n - 2 ν) + \frac{D_{n} (n + 1) (n + 2)}{R^{n + 2}}] P_{n} (\cos ϑ) \end{array}

whereas the corresponding stress components are:

\begin{array}{l} σ_{r r} = 2 G \sum_{} [A_{n} (n + 1) (n^{2} - n - 2 - 2 ν) R^{n} + B_{n} n (n - 1) R^{n - 2}] P_{n} (\cos ϑ) + \\ [- \frac{C_{n} n}{R^{n + 1}} (n^{2} + 3 n - 2 ν) + \frac{D_{n} (n + 1) (n + 2)}{R^{n + 3}}] P_{n} (\cos ϑ) \end{array}

\begin{array}{c} σ_{ϑ ϑ} = 2 G \sum_{} [A_{n} (n^{2} + 4 n + 2 + 2 ν) (n + 1) r^{n} + B_{n} n^{2} r^{n - 2} + \frac{C_{n} n}{r^{n + 1}} (n^{2} - 2 n - 1 + 2 ν) - \frac{D_{n} {(n + 1)}^{2}}{r^{n + 3}}] P_{n} (\cos ϑ) \\ [A_{n} (n + 5 - 4 ν) r^{n} + B_{n} r^{n - 2} - \frac{C_{n}}{r^{n + 1}} (- n + 4 - 4 ν) + \frac{D_{n}}{r^{n + 3}}] \cot ϑ \frac{d P_{n} (\cos ϑ)}{d ϑ} \end{array}

\begin{array}{c} σ_{φ φ} = 2 G \sum_{} [A_{n} (n + 1) (n - 2 - 2 ν - 4 n ν) r^{n} + B_{n} n r^{n - 2} + \frac{C_{n} n}{r^{n + 1}} (n + 3 - 4 n ν - 2 ν) - \frac{D_{n} (n + 1)}{r^{n + 3}}] P_{n} (\cos ϑ) \\ + [A_{n} (n + 5 - 4 ν) r^{n} + B_{n} r^{n - 2} + \frac{C_{n}}{r^{n + 1}} (- n + 4 - 4 ν) + \frac{D_{n}}{r^{n + 3}}] \cot ϑ \frac{d P_{n} (\cos ϑ)}{d ϑ} \end{array}

\begin{array}{l} σ_{r ϑ} = 2 G \sum_{} [A_{n} (n^{2} + 2 n - 1 + 2 ν) R^{n} + B_{n} (n - 1) R^{n - 2}] \frac{\partial P_{n} (\cos ϑ)}{\partial ϑ} + \\ [\frac{C_{n}}{R^{n + 1}} (n^{2} - 2 + 2 ν) - \frac{D_{n} (n + 2)}{R^{n + 3}}] \frac{\partial P_{n} (\cos ϑ)}{\partial ϑ} \end{array}

The constants A_n, B_n, C_n and D_n are determined by satisfying the boundary conditions. The boundary conditions are defined as follow:

\begin{array}{l} σ_{r r} (a) = - P_{3} - P_{1, 3} σ_{r r} (b) = P_{1, 2} \\ σ_{r ϑ} (a) = 0 σ_{r ϑ} (b) = 0 \end{array}

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

\begin{array}{l} P_{1, 3} = \sum_{} E_{n} P_{n} (\cos ϑ) = (Z - Y) \cos {(ϑ)}^{2} + Y \\ P_{1, 2} = \sum_{} F_{n} P_{n} (\cos ϑ) = (K - W) \cos {(ϑ)}^{2} + W \end{array}

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:

\begin{array}{l} E_{0} = \frac{1}{3} (Z + 2 Y) F_{0} = \frac{1}{3} (K + 2 W) \\ E_{2} = \frac{2}{3} (Z - Y) F_{2} = \frac{2}{3} (K - W) \end{array}

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

(\begin{array}{c} β_{11} & β_{12} & β_{13} & β_{14} \\ β_{21} & β_{22} & β_{23} & β_{24} \\ β_{31} & β_{32} & β_{33} & β_{34} \\ β_{41} & β_{42} & β_{43} & β_{44} \end{array}) \cdot (\begin{array}{c} A_{0} \\ B_{0} \\ C_{0} \\ D_{0} \end{array}) = (\begin{array}{c} ϕ_{1} \\ ϕ_{2} \\ ϕ_{3} \\ ϕ_{4} \end{array}) (\begin{array}{c} δ_{11} & δ_{12} & δ_{13} & δ_{14} \\ δ_{21} & δ_{22} & δ_{23} & δ_{24} \\ δ_{31} & δ_{32} & δ_{33} & δ_{34} \\ δ_{41} & δ_{42} & δ_{43} & δ_{44} \end{array}) \cdot (\begin{array}{c} A_{2} \\ B_{2} \\ C_{2} \\ D_{2} \end{array}) = (\begin{array}{c} ρ_{1} \\ ρ_{2} \\ ρ_{3} \\ ρ_{4} \end{array})

The matrix coefficients β_ij, ϕ_i, δ_ij, ρ_i _j are presented in Appendix A. Once the constants A_n, B_n, C_n and D_n are known, the change in WGM due to strain (da/a) can be calculated by using Eq. (39). However, as discussed earlier, dn₀/n₀<<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 (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.

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

\begin{array}{l} α_{11} = α_{24} = - 1, α_{12} = \sqrt{\frac{a}{b}}, α_{13} = \frac{b}{a}, α_{14} = α_{21} = α_{34} = 0, α_{22} = \sqrt{\frac{b}{a}}, α_{23} = \frac{a}{b}, \\ α_{31} = \frac{ε_{1}}{a ε_{2}}, α_{32} = - \frac{1}{\sqrt{a b}}, α_{33} = \frac{2 b}{a^{2}}, α_{43} = \frac{2 a}{b}, α_{42} = - \frac{1}{\sqrt{a b}}, α_{44} = - \frac{ε_{3}}{ε_{2}} \frac{2}{b}, \\ γ_{1} = γ_{3} = 0, γ_{2} = - E_{0} b, γ_{4} = \frac{ε_{3}}{ε_{2}} E_{0} \end{array}

\begin{array}{l} δ_{11} = - 6 ν a^{2}, δ_{12} = δ_{22} = 2, δ_{13} = - \frac{2}{a^{3}} (10 - 2 ν), δ_{14} = \frac{12}{a^{5}}, δ_{21} = - 6 ν b^{2}, δ_{23} = - \frac{2}{b^{3}} (10 - 2 ν), \\ δ_{24} = \frac{12}{b^{5}}, δ_{31} = (2 ν + 7) a^{2}, δ_{32} = δ_{42} = 1, δ_{33} = \frac{2}{a^{3}} (1 + ν), δ_{34} = - \frac{4}{a^{5}}, δ_{41} = (2 ν + 7) b^{2}, \\ δ_{43} = \frac{2}{b^{3}} (1 + ν), δ_{44} = - \frac{4}{b^{5}}, ρ_{1} = \frac{1}{3 G} (Z - Y), ρ_{2} = \frac{1}{3 G} (K - W), ρ_{3} = ρ_{4} = 0 \end{array}

\begin{array}{l} β_{11} = β_{21} = - 2 (1 + ν), β_{14} = \frac{2}{a^{3}}, β_{24} = \frac{2}{b^{3}}, β_{32} = - \frac{1}{a^{2}}, β_{12} = β_{13} = β_{22} = β_{23} = 0 \\ β_{31} = β_{41} = 2 v - 1, β_{33} = \frac{2}{a} (ν - 1), β_{34} = - \frac{2}{a^{3}}, β_{42} = - \frac{1}{b^{2}}, β_{43} = \frac{2}{b} (ν - 1), β_{44} = - \frac{2}{b^{3}} \\ ϕ_{1} = - \frac{(Z + 2 Y)}{6 G} - \frac{ε_{0}}{6} \frac{A^{2}}{a} (k - 1) (k + 2), ϕ_{2} = \frac{(K + 2 W)}{6 G}, ϕ_{3} = ϕ_{4} = 0 \end{array}

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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Tuning of whispering gallery modes of spherical resonators using an external electric field

Abstract

1. Introduction

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

3. WGM Shift in a Solid Sphere Due to Electrostriction

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

5. Experiments

6. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (9)

Equations (50)

Optics Express