Optimization-Free Filter and Matched-Filter Design through Spatial and Temporal Soft Switching of the Dielectric Constant

The problem of inverse design is rooted to the classical problem of inverse scattering. In general, these highly nonlinear and often ill-posed problems are solved via extensive optimization techniques. In this paper, we suggest an optimization-free method for the inverse design of a one-dimensional medium with spatially inhomogeneous dielectric constant µ(z). In addition, we derive the governing equation of an analog problem - a time-dependent homogeneous medium - and use the same technique for inverse design of the temporal profile µ(t) of a spatially homogeneous medium that is required to achieve a desired frequency response in k-space. Lastly, we use this optimization-free inversion approach to demonstrate the design of the reflection response such that the reflected wave undergoes a desired filtering, such as of Tchebychev type, differentiator, and a matched filter for chirp signal detection over additive Gaussian noise, which is of high potential significance in chirp radar and sonar applications.