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Mathematical Problems in Engineering
Volume 2017, Article ID 9649524, 8 pages
Research Article

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

1Applied-Science & Robotics Laboratory for Applied-Mechatronics (ARAtronics Lab.), Mechatronics Engineering Department, German University in Cairo, New Cairo, Cairo 11835, Egypt
2Micro-Sensor Laboratory, Mechanical Engineering Department, Southern Methodist University, Dallas, TX 75275, USA

Correspondence should be addressed to Amir R. Ali; ude.ums.liam@demhara

Received 30 March 2017; Revised 5 June 2017; Accepted 18 June 2017; Published 25 July 2017

Academic Editor: Constantin Fetecau

Copyright © 2017 Amir R. Ali and Mohamed A. Kamel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.