On Multiple Frequency Power Density Measurements

TL;DR

This paper establishes conditions on boundary illuminations ensuring non-zero and linearly independent solutions to the Helmholtz equation inside a domain for multiple frequencies, aiding hybrid imaging techniques like microwave imaging by ultrasound deformation.

Contribution

It provides a priori conditions on illuminations that guarantee solution properties independent of coefficients, facilitating internal power density measurements in hybrid imaging.

Findings

Derived conditions for illuminations ensuring solution independence

Proved new reconstruction formulas for microwave imaging by ultrasound deformation

Applicable to multiple frequencies within a fixed range

Abstract

We shall give a priori conditions on the illuminations $\phi_i$ such that the solutions to the Helmholtz equation $-div(a \nabla u^i)-k q u^i=0$ in \Omega, $u^i=\phi_i$ on $\partial\Omega$, and their gradients satisfy certain non-zero and linear independence properties inside the domain \Omega, provided that a finite number of frequencies k are chosen in a fixed range. These conditions are independent of the coefficients, in contrast to the illuminations classically constructed by means of complex geometric optics solutions. This theory finds applications in several hybrid problems, where unknown parameters have to be imaged from internal power density measurements. As an example, we discuss the microwave imaging by ultrasound deformation technique, for which we prove new reconstruction formulae.