= Born approximation for scalar wave scattering
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Using $V=k_0^2(n^2-1)$, the scalar <Lippmann-Schwinger equation> is $\psi=\psi_i+\int_DG_{k_0}V\psi$. The <Born approximation> replaces the total field inside the integral by the known incident field:
$$
\psi_B=\psi_i+\int_DG_{k_0}(\mathbf r,\mathbf r')V(\mathbf r')\psi_i(\mathbf r')\,d\mathbf r'.
$$
For a unit <plane wave>, its <far-field pattern> is $(4\pi)^{-1}\int_D V(\mathbf r')e^{ik_0(\widehat{\mathbf r}_0-\widehat{\mathbf r})\cdot\mathbf r'}\,d\mathbf r'$. Thus each incident direction samples a shifted sphere in the <Fourier transform> of the potential.
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