Transfer matrix formulation of scattering theory in two and three dimensions

In one dimension, one can dissect a scattering potential $v\left(x\right)$ into pieces $v_i\left(x\right)$ and use the notion of the transfer matrix to determine the scattering content of $v\left(x\right)$ from that of $v_i\left(x\right)$. This observation has numerous practical applications in different areas of physics. The problem of finding an analogous procedure in dimensions larger than 1 has been an important open problem for decades. We provide 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.

Figure 1

Support $I$ of a finite-range potential (the enclosed region) sliced along the $x$ axis.

Figure 2

A slab of thickness $L$ with a surface line defect shown by the thick blue line.

Figure 3

Graph of the threshold gain coefficient $g$ as a function of $\theta$ for a generic slab laser with a surface line defect.