Scattering potential ($V$)

For a refractive index $n(\mathbf{r})$ and background wavenumber $k_0$, one sign convention defines the scattering potential by $V=k_0^2(1-n^2)$. The total field then satisfies $(\Delta +k_0^2)\psi =V\psi$.

The opposite convention $V=k_0^2(n^2-1)$ gives $(\Delta +k_0^2)\psi =-V\psi$ and reverses the sign attached to the corresponding Green-function integral. Either convention is valid when used consistently.

Using $V=k_0^2(n^2-1)$, the scalar Lippmann-Schwinger equation is $\psi ={\psi }_i+{\int }_D{G}_{k_0}V\psi$. The Born approximation replaces the total field inside the integral by the known incident field:For a unit plane wave, its far-field pattern is $(4\pi {)}^{-1}{\int }_{D}V({\mathbf{r}}^{\prime })e^{i{k}_0({\hat{\mathbf{r}}}_0-\hat{\mathbf{r}})\cdot {\mathbf{r}}^{\prime }}d{\mathbf{r}}^{\prime }$. Thus each incident direction samples a shifted sphere in the Fourier transform of the potential.

For real $h\in C_c^{\infty }(D)$, the displayed real compactly supported contrast has Fourier transform $\hat{V}(\mathbf{\kappa })=[|\mathbf{\kappa }{|}^4-4({\mathbf{k}}_i\cdot \mathbf{\kappa }{)}^2]\hat{h}(\mathbf{\kappa })$. This vanishes on the shifted Ewald sphere because $|\mathbf{\kappa }{|}^2+2{\mathbf{k}}_i\cdot \mathbf{\kappa }=0$ there. Thus nonzero contrasts can have zero fixed-frequency, fixed-incidence Born approximation data. A small amplitude preserves weak contrast and a positive refractive index.

The Born series is the Neumann series expansion of the Lippmann-Schwinger equation. Its terms describe successive orders of wave scattering; convergence is ensured, for example, by a scattering operator norm smaller than one.