T-matrix method

Definition

Let $k > 0$ and let a bounded scatterer lie inside a ball of radius $R$ centred at $0$. For integers $n \ge 0$ and $\abs{m} \le n$, let $Y_n^m$ be the spherical harmonics. Let $j_n$ be the spherical Bessel functions and $h_n^{(1)}$ the spherical Hankel functions of the first kind. Set $\phi_{nm}(x) = j_n(kr) Y_n^m(\hat{x})$ and $\psi_{nm}(x) = h_n^{(1)}(kr) Y_n^m(\hat{x})$, with $r = \abs{x}$ and $\hat{x} = x / r$. An incident solution of the Helmholtz equation is expanded as $u_i = \sum_{n,m} a_{nm} \phi_{nm}$. The scattered field is expanded for $r > R$ as $u_s = \sum_{n,m} p_{nm} \psi_{nm}$. Since the scattering problem is linear, there is a linear map $T$ with $$p_{nm} = \sum_{n', m'} T_{nm, n'm'} \, a_{n'm'}.$$ The matrix $T = (T_{nm, n'm'})$ is the T-matrix (transition matrix) of the scatterer. The T-matrix method computes $T$, usually truncated at $n \le n_{\max}$, and obtains scattered fields from it.