Transfer Matrix Formulation of Scattering Theory in Two and Three Dimensions

TL;DR

This paper extends the transfer matrix method from one-dimensional scattering problems to two and three dimensions, providing exact solutions and applications including delta-function potentials and laser slabs with defects.

Contribution

It introduces a complete formulation of transfer matrices in higher dimensions, solving a long-standing open problem and enabling analysis of complex scattering scenarios.

Findings

Derived exact scattering amplitude for delta-function potentials in 2D and 3D

Analyzed a laser slab with a surface defect, showing defect-induced lasing at small gain

Provided a practical framework for multi-dimensional scattering analysis

Abstract

In one dimension one can dissect a scattering potential $ v(x) $ into pieces $ v_i(x) $ and use the notion of the transfer matrix to determine the scattering content of $ v(x) $ from that of $ v_i(x) $. This observation has numerous practical applications in different areas of physics. The problem of finding an analogous procedure in dimensions larger than one has been an important open problem for decades. We give a complete solution for this problem and discuss some of its applications. In particular we derive an exact expression for the scattering amplitude of the delta-function potential in two and three dimensions and a potential describing a slab laser with a surface line defect. We show that the presence of the defect makes the slab begin lasing for arbitrarily small gain coefficients.