Parameter Identification for Dispersive Dielectrics Using Pulsed Microwave Interrogating Signals and Acoustic Wave Induced Reflections in Two and Three Dimensions

In this report we consider an electromagnetic interrogation technique for identifying the dielectric parameters of a Debye medium in two and three dimensions. These parameters include the dielectric permittivity, the conductivity and the relaxation time of the Debye medium. In this technique a travelling acoustic pressure wave that is generated in the Debye medium is used as a virtual reflector for an interrogating microwave electromagnetic pulse that is generated in free space and impinges on a planar interface that separates air and the Debye medium. The reflections of the microwave pulse from the air-Debye interface and from the acoustic pressure wave are recorded at an antenna that is located in air. These reflections comprise the data that is used in an inverse problem to estimate the dielectric parameters of the Debye medium. We assume that the dielectric parameters of the Debye medium are locally pressure dependent. A model for acoustic pressure dependence of the material constitutive parameters in Maxwell’s equations is presented. As a first approximation, we assume that the Debye dielectric parameters are affine functions of pressure. We present a time domain formulation that is solved using finite differences in time and in space using the finite difference time domain method (FDTD). Perfectly matched layer (PML) absorbing boundary conditions are used to absorb outgoing waves at the finite boundaries of the computational domain and prevent excessive spurious reflections from reentering the domain and contaminating the data that is collected at the antenna placed in air. Using the method of least squares for the parameter identification problem we solve an inverse problem by using two different algorithms; the gradient based Levenberg-Marquardt method, and the gradient free, simplex based Nelder-Mead method. We solve inverse problems to construct estimates for two or more dielectric parameters. Finally we use statistical error analysis to construct confidence intervals for all the presented estimates. These confidence intervals are a probabilistic statement about the procedure that we have used to construct estimates of the various parameters. Thus we naturally combine the deterministic nature of our problem with uncertainty aspects of estimates.