Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 70 / 4 / a / Solution

Choose the scattering-potential convention $V(\mathbf{r})=k^2[n(\mathbf{r}{)}^2-1]$. Let $G_k(\mathbf{r},{\mathbf{r}}^{\prime })=e^{ik|\mathbf{r}-{\mathbf{r}}^{\prime }|}/(4\pi |\mathbf{r}-{\mathbf{r}}^{\prime }|)$, so $(\Delta +k^2)G_k=-\delta$. Since $(\Delta +k^2)\psi =-V\psi$, the outgoing Green representation is the Lippmann-Schwinger equationIterating it produces the Born seriesFor example its first two terms areThe first term describes one scattering event driven by the incident field. The second propagates its first scattered wave to another interaction before reaching the observer. Higher terms contain more successive interactions, including repeated visits to the same region; they account for multiple scattering rather than additional incident waves.

A sufficient convergence condition on an appropriate field norm is $\Vert T\Vert =\rho <1$. Then the first-Born remainder is bounded by ${\rho }^2\Vert {\psi }_i\Vert /(1-\rho )$, while $\Vert T{\psi }_i\Vert \le \rho \Vert {\psi }_i\Vert$. A simple concrete sufficient condition on the bounded scatterer, using the supremum norm, isThis makes the field inside the inhomogeneity close to the incident field. Physically weak index contrast with small accumulated phase shift, such as $k\ell |n-1|\ll 1$ for a smooth weak scatterer of thickness $\ell$, is a common first-Born regime, provided resonant enhancement and multiple reflections are negligible. Weak local contrast alone is not sufficient for an arbitrarily thick scatterer. If the opposite sign convention for $V$ is used, the integral operator changes sign and the same consistent iteration applies.