Mathematical Model for Electric Field Sensor Based on Whispering Gallery Modes Using Navier ’ s Equation for Linear Elasticity

This paper presents and verifies the mathematical model of an electric field senor based on the whispering gallery mode (WGM). The sensing element is a dielectric microsphere, where the light is used to tune the optical modes of the microsphere. The light undergoes total internal reflection along the circumference of the sphere; then it experiences optical resonance. The WGM are monitored as sharp dips on the transmission spectrum.These modes are very sensitive to morphology changes of the sphere, such that, for every minute change in the sphere’s morphology, a shift in the transmission spectrum will happen and that is known as WGM shifts. Due to the electrostriction effect, the applied electric field will induce forces acting on the surface of the dielectric sphere. In turn, these forces will deform the sphere causing shifts in its WGM spectrum. The applied electric field can be obtained by calculating these shifts. Navier’s equation for linear elasticity is used to model the deformation of the sphere to find the WGM shift. The finite element numerical studies are performed to verify the introduced model and to study the behavior of the sensor at different values of microspheres’ Young’s modulus and dielectric constant. Furthermore, the sensitivity and resolution of the developed WGM electric filed sensor model will be presented in this paper.


Introduction
Recently, optical cavities have been used for many sensory applications due their high quality factor and resolution [1,2].These applications include wall shear stress calculation [3,4]; temperature measurement [5][6][7]; biosensing [8][9][10][11][12][13]; hemozoin nanocrystals detection in aquatic environment [14]; nanoparticle detection [15]; and acceleration measurement [16][17][18].In this paper, WGM based optical cavities are used to detect and measure the applied electric field.These cavities come in different shapes such as sphere, toroid, disk, and cylinder.The optical modes of these cavities can be excited using tunable laser.A single mode optical fiber is utilized to couple the incoming laser light into the cavity.When resonance occurs, the WGM modes are monitored as sharp dips on the transmission.WGM modes are highly responsive to changes in the morphology of the cavity and the outer medium; consequently, they are also called morphology dependent resonances (MDR) modes.Any minute external disturbance on the sphere or its surrounding environment will be detected as shift in its transmission spectrum.Consequently, these cavities are used in many sensory applications by measuring the WGM shifts corresponding to the change in the measured phenomenon.
Figure 1 presents a microsphere WGM cavity of an outer radius  subjected to an applied electric field of strength  0 .The sphere is usually fabricated from dielectric polymeric material, like polydimethylsiloxane (PDMS).At the beginning, an optical fiber (∼5 cm long) was stripped of its plastic cladding and then subjected to heat while stretching.The final diameter size after stretching is ∼50 m.Afterwards, the fiber is dipped into the PDMS polymer pool.Then, a sphere of PDMS will form at the end of the fiber due to the surface tension and the gravitational forces.The fiber and sphere assembly is heated to cure of the polymeric material as reported in [19].
The microsphere is coupled to a single mode optical fiber.A tunable laser is utilized to supply the light traveling through the optical fiber.The light is coupled to a microsphere cavity through the tapered section of the optical fiber; then it circumnavigates the microsphere undergoing total internal reflection.The condition of total internal reflection must be fulfilled, such that the refractive index of the cavity  sphere is greater than the surrounding  out .A photodiode is placed on the other side of the optical fiber to monitor the intensity of the transmitted light.
The optical resonance will only happen when the total distance traveled by the light (round trip) is almost equal to a multiple integer  of the light's wavelength as seen in 1st-order approximation of (1).Then the WGM would be presented as a sharp dips on the transmission spectrum as shown in Figure 1(c).
where  sphere is the refractive index the microsphere;  is the wavelength of the light; and  is the radius of the sphere.The change in the morphology (size or the refractive index) of the cavity will induce a shift  in the resonance wavelength as follows: In this paper, the microsphere is used to measure the applied electric field due to the electrostriction effect.When an electric field is applied on a dielectric microsphere, it induces surface and body forces across its body.In turn, theses forces will deform the sphere changing it size, causing shifts in its WGM transmission spectrum, thus creating a WGM based electric field sensor.The induced electrostrictive volumetric body force ⃗  can be written as follows [20]: where  sphere is the dielectric constant of the microsphere's material;  is the applied electric field inside the sphere; and  1 and  2 are coefficients describing the strain of the dielectric material due to the applied electric field in the directions parallel and perpendicular to the field.⃗  is the summation of the gradients of  2 and  sphere .For a homogenous dielectric material under a static uniform applied electric field, both gradients of (3) are equal to zero; thus ⃗  is also zero.However, at the interface separating the microsphere and the surrounding, a surface pressure force ⃗  exists.It exists due to the discontinuity of the dielectric constant along the media and given by and  are coefficients that depend on the electrostrictive properties of the microsphere's material.⃗  is the unit surface normal vector and the subscripts indicate the values to be taken on either side of the interface ( denotes the sphere's and  denotes the outer medium).The induced deformation due to the pressure forces can be obtained by solving Navier's equation for linear elasticity at steady state, which is given by where ⃗  is the deformation of the cavity, ] is Poisson's ratio, and  is the shear modulus of the cavity's material.Thus the deformation of the microsphere needs to be calculated to get the WGM shift due to the applied electric field.

Electric Field and Pressure Force Calculation.
To solve Navier's equation, the pressure forces must be obtained first; so the first step is to calculate the electric field composing the pressure forces.The applied electric field is obtained by differentiating the electric potential.When a dielectric microsphere is placed under an applied static axisymmetric electric field, the potential is defined by Laplace's equation in spherical coordinates (, , ) [21].
where  is the electric potential.Due to the symmetry of the applied electric field, the solution of Laplace's equation is also independent of the azimuthal angle .Separation of variables method is used to obtain the solution (, ).
where  is radius of the microsphere,  is the azimuthal angle between the position vector and the  axis,   is the associated Legendre polynomial of the th order, and sphere and  out are the electric potential inside and outside the microsphere, respectively. out is the dielectric constant of material outside the sphere.After the application of the boundary conditions, the coefficients are   and   are obtained.The gradient of the potential is applied to find the radial   and azimuthal   components of the applied electric field inside and outside the cavity.The electric field components are 0 is the applied electric field strength.The acting radial and azimuthal components of the pressure force (  and   ) on the interface are 0 is the dielectric permittivity of vacuum (8.854 × 10 −12 F⋅m −1 ).The boundary stresses acting on the sphere will be equated to the pressure forces.These forces will deform the microsphere; and the deformation is calculated using Navier's equation.

Solution of Navier's Equation.
Due to the absence of the body force, Navier's equation can be written as follows: Using Helmholtz theory, ⃗  can be expressed as follows [24]: where ⃗   and ⃗  ir are solenoidal and irrotational vectors, respectively; these vectors satisfy the following relations: Thus ⃗  can be written as follows: where  ⇀  is a harmonic vector; and Γ is a scalar potential function.Substitution of ( 14) in (11) yields Γ is chosen to satisfy the previous equation and given by where  0 is harmonic and satisfying the boundary conditions; and  ⇀  is the position vector.After substitution of Γ in (11), the solution to Navier's equation in spherical coordinates can be written as follows: Laplace's equation is solved again to find the harmonic vector  ⇀  , where the components of  ⇀  and  0 are represented as a series of Legendre polynomials.Thus, the radial deformation of the sphere due to the applied electric field can be written as follows: is series of the associated Legendre polynomials   () of the th order.The polynomials are multiplied by the coefficients   and   .These coefficients are also obtained using the boundary conditions imposed on the sphere.Finally the boundary stresses in the radial   and azimuthal   directions derived from the deformations are represented as follows: In electrostriction, the pressure forces are the applied boundary stresses on the microsphere and satisfy In this case the solution ⃗  can be presented as follows: is Young's modulus of the material of the microsphere.The radial deformation   is directly proportional to the square of the electric field strength; inversely proportional to   ; and dependent on the dielectric constants  sphere and  out .Navier's equation is solved at the equator ( is /2); thus, the calculated   is the same change  in the optical path of the circumnavigating light: The  is independent of the outer radius  and can be written using (2) as follows: is the wavelength of the light provided by the tunable laser.Using Navier's equation, an optical WGM based electric field sensor can be constructed.The relation between the microsphere's WGM shift and the applied electric field is investigated extensively by means of numerical studies using finite element simulation software.The studies calculate  using the deformation of the sphere due to the electrostriction effect.The purpose of the first two studies is to confirm the quadratic dependence of sphere's WGM shift on the applied electric field strength and the inverse proportionality between the sphere's WGM shift and Young's modulus of the sphere's material, while the third one aims to see the effect varying the dielectric constant of the sphere on its WGM shift.During the studies, the electric field is supplied by means of 2 copper electrodes.The microsphere is situated at the center between the plates as seen in Figure 1.
For the first study, cured PDMS [25] is used as the construction material of the microsphere.The mixture has an elastomer to curing agent ratio of 60 : 1.Under steadystate conditions,  measured against an increasing applied electric field.The simulation parameters during the first study are presented in Table 1.
As for the second study, the inverse proportionality between the sphere's WGM shift and Young's modulus of the sphere's material is investigated.The study measures  of several microspheres with different Young's moduli under the same applied electric field strength.The simulation parameters of the second study is similar to the first one, except for the varying Young's modulus value.Finally, the third study investigates the effect of varying the sphere's dielectric constant on its WGM shift.The 3rd study subjects several microspheres with different values of dielectric constant to the same applied electric field input; then it measures the value of  of the several microspheres.Also, the simulation parameters of the 3rd study is similar to the first one, except for the changing dielectric constant.The strength of the applied electric field for the 2nd and 3rd studies is 5 kV/m.

Results and Discussion
Results of the 1st numerical studies are shown in Figure 2.  is plotted against the increasing electric field strength up to 1300 kV/m.The increasing electric field will be met by the increasing shift.It can be seen that the quadratic relation between  and the electric field strength is confirmed.In Figure 2, the numerical range under study was split to three main regions (0-200 kV/m, 200-500 kV/m, and 500-1300 kV/m).It is clear that the nonlinear behavior in the response was linearized per each region to be able to find the linear relation between the output  and the input  0 .Since the proposed sensor is a linear optical sensor, so we will find out the sensitivity and the resolution for that sensor per each region.Table 2 summarized sensitivity and resolution values for each mentioned region.
Figure 3 shows the results of the second study, where the WGM shift  is modeled against the varying Young's modulus (  ) of microsphere for different values of applied electric field strength (0.5 kV/m, 1 kV/m, 1.5 kV/m, 2 kV/m, and 5 kV/m).Decreasing Young's modulus of the sphere will lead to the increase of the shift.It is clear that the WGM shift is inversely proportional with respect to Young's modulus of the sensing element material.
Finally, Figure 4 shows the results of the 3rd study, which is the WGM shift of the microsphere is directly proportional to the relative dielectric constant (  ) of the microsphere.In this study, we do find the sensor response for the same applied electric field strength mentioned in the 2nd study (0.5 kV/m, 1 kV/m, 1.5 kV/m, 2 kV/m, and 5 kV/m).
Up to now, several sensor technologies for electric field detection have been implemented.Table 3 summarizes the sensitivities of electric field from various techniques.In this paper, we show that our new modeling technique will be promising to be used in a complex sensor structure like multilayered composite sphere as reported in [26].

Conclusion
In this paper, the mathematical model of a WGM based electric field senor is presented.Navier's equation for linear elasticity is solved to find the deformation of the seeing element upon application of the electric field strength with different values (0.5 kV/m, 1 kV/m, 1.5 kV/m, 2 kV/m, and 5 kV/m).The response indicates that the WGM shift in the transmission spectrum has a quadratic dependence on the applied electric field strength.However, the linearization behavior for the sensor is needed for the practical usage.That was the reason to split the sensing range into three main regions (0-200 kV/m, 200-500 kV/m, and 500-1300 kV/m).Results show that the WGM shift is inversely proportional to Young's modulus of the microsphere's material under different electric field strength values (0.5 kV/m, 1 kV/m, 1.5 kV/m, 2 kV/m, and 5 kV/m).Furthermore, the studies also show that the WGM shift is directly proportional to the dielectric constant of the microsphere's material under the same electric filed strength values that were mentioned before.The sensor behavior and characteristics were verified by numerical finite element studies.Finally, in this paper, we show that our new promising modeling technique will be easily used in a complex sensor structure like multilayered composite sphere.

Figure 2 :
Figure 2: WGM shifts  against the applied electric field strength during the 1st study.

Figure 4 :
Figure 4: WGM shifts  against relative dielectric constant during the 3rd study.

Table 1 :
Simulation parameters for the 1st numerical study.Three numerical studies are done to investigate the relation between  and the applied electric field.

Table 2 :
Sensitivity and resolution along the different detection regions.

Table 3 :
Electric field sensor's sensitivity with different techniques.