Instantaneous time mirrors and wave equations with time-singular coefficients

TL;DR

This paper investigates the mathematical modeling of instantaneous time mirrors using wave equations with time-singular coefficients, providing refocusing estimates and regularity theory to quantify the effectiveness of time reversal.

Contribution

It develops a mathematical framework for wave equations with Dirac-type singularities, offering new insights into the analysis and quantification of instantaneous time mirrors.

Findings

Derived uniform estimates for wavefields with time-singular coefficients

Developed a regularity theory for wave equations with Dirac delta singularities

Quantified the quality of time reversal through refocusing estimates

Abstract

We study in this work the concept of instantaneous time mirrors that were recently introduced in the physics literature. Instantaneous time mirrors offer a new method for time reversal with a simplified experimental setup compared to classical techniques. At the mathematical level, instantaneous time mirrors are modeled by singularities in the time variable in the coefficients of a wave equation, and a prototype of such singularity is a Dirac delta. Our main goal in this work is to obtain refocusing estimates for the wavefield that quantify the quality of time reversal. This amounts to analyze the wave equation with Dirac-type singularities and develop a proper regularity theory as well as derive uniform estimates.