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

Abstract

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.

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“…The sensing element is a micro-optical sphere that is made from the polydimethylsiloxane (PDMS) following the same procedure that was reported in [22]. Details about the developed optical sensor could be found in [23][24][25][26][27][28]. The sensitivity of the optical sensor used is 0.0344 pm/Vm-1 and its resolution is 6.7 V/m with a high quality factor ~ 107.…”

“…The measurements will be based on satisfying the optical resonance condition, which is also named as the Whispering Gallery optical Modes (WGM). The optical resonance occurs once the round trip of the light becomes equal to the multiple integers of the wavelength, , as follows [23]:…”

“…The performed experiments are based on two techniques to measure the diffusivity of the electric field in a glass plate as a function of the glass plate thickness and the back cover material. The first technique utilizes an ultra-sensitive electric field sensor, based on the Whispering Gallery Modes (WGM) [22][23][24], to measure the diffusivity of electric field in the upper glass surface as a function of the back cover material. The developed experimental setup consists of a glass plate with a back cover plate from a material, which is the medium to be examined.…”

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“…Optical microresonators are highly sensitive to the surrounding environment. Changes of physical properties such as temperature [1,2], humidity [3], electric field strength [4], or others [5,6] can be detected via changes in the refractive index on a variety of materials such as polymers, metal oxides or other semiconductors [7][8][9][10][11][12][13][14][15]. Also nanoparticle decorated resonators have been realised, exploiting plasmonic coupling [16].…”

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“…Recently, an MDR-based photonic electric field sensor was demonstrated [9][10][11][12][13]. The theory of operation is based on changing the radius of the cavity due to the electrostriction effect.…”