= Born non-scattering contrasts at one incident direction
{c}
{title2=$V=[\Delta^2+4(\mathbf k_i\cdot\nabla)^2]h$}
For real $h\in C_c^\infty(D)$, the displayed real compactly supported contrast has <Fourier transform> $\widehat V(\boldsymbol\kappa)=[|\boldsymbol\kappa|^4-4(\mathbf k_i\cdot\boldsymbol\kappa)^2]\widehat h(\boldsymbol\kappa)$. This vanishes on the shifted <Ewald sphere> because $|\boldsymbol\kappa|^2+2\mathbf k_i\cdot\boldsymbol\kappa=0$ 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>.
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