Chapter Summary

Chapter 5 Summary: Electromagnetic Scattering Theory

Key Points

• 1.

  Time-harmonic Maxwell's equations in an inhomogeneous medium yield the Helmholtz equation with spatially varying wavenumber $\kappa^{2}(\mathbf{r}) = \kappa_{0}^{2}(1 + \chi(\mathbf{r}))$, where the contrast function $\chi(\mathbf{r})$ encodes the scattering object.

• 2.

  The free-space Green's function — $e^{-j\kappa_{0} r}/(4\pi r)$ in 3D and $(j/4)H_0^{(2)}(\kappa_{0} r)$ in 2D — is the field produced by a point source and serves as the kernel of all integral equations.

• 3.

  The Lippmann-Schwinger equation $u = u^{\text{inc}} + \mathcal{G}[\chi u]$ reformulates scattering as a volume integral over the object domain, with the Born series providing a perturbation expansion in orders of scattering.

• 4.

  The Born approximation replaces $u^{\text{tot}} \approx u^{\text{inc}}$ inside $D$, yielding the linear forward model $\ntn{img} = \mathbf{A}\ntn{refl_vec} + \mathbf{w}$, valid when $\kappa_{0} a|\chi_0| \ll 1$. The Fourier diffraction theorem shows that Born measurements sample $\hat{\chi}$ on the Ewald sphere with resolution limited to $\lambda/2$.

• 5.

  The Rytov approximation linearizes the complex phase $\phi = \log(u/u^{\text{inc}})$, requiring only $|\chi_0| \ll 1$ regardless of object size — preferred for large, weakly scattering objects.

• 6.

  When first-order approximations fail, the DBIM, BIM, and CSI methods iteratively refine $\chi$ by updating the internal field, extending imaging to higher contrast and larger objects.

• 7.

  The scattering matrix $\mathbf{S}$ with entries $f(\hat{r}_q; \hat{k}_p)$ provides a complete far-field characterization. Under Born, $\mathbf{S}$ is linear in $\chi$ — the bridge to the sensing operator in Chapter 6.