Prerequisites & Notation

Prerequisites for This Chapter

This chapter develops electromagnetic scattering theory from Maxwell's equations through to the Born and Rytov linearizations that underpin all RF imaging forward models. It assumes familiarity with the following material.

  • Linear Algebra and Operator Theory(Review ch01)

    Self-check: Can you define a bounded linear operator on a Hilbert space and state its adjoint?

  • Inverse Problems and Regularization(Review ch02)

    Self-check: Can you explain why inverse scattering is ill-posed and how regularization helps?

  • Convex Analysis and Variational Regularization(Review ch03)

    Self-check: Can you write a Tikhonov-regularized least-squares problem?

  • Optimization Algorithms for Inverse Problems(Review ch04)

    Self-check: Do you understand gradient descent and proximal operators for iterative reconstruction?

  • Fourier Analysis and Spectral Methods(Review ch05)

    Self-check: Are you comfortable with 2D/3D Fourier transforms and the convolution theorem?

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Divergence, curl, gradient, and the Gauss and Stokes theorems in 2D and 3D.

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Plane-wave representations ejk⋅re^{j\mathbf{k}\cdot\mathbf{r}} and time-harmonic conventions.

Notation and Conventions

We establish the following notation used throughout the chapter. Unless stated otherwise, we use the e−jωte^{-j\omega t} time-harmonic convention, SI units, and assume all fields are monochromatic at angular frequency ω\omega.

SymbolMeaningIntroduced
E,H\mathbf{E}, \mathbf{H}Electric and magnetic field vectors
J,M\mathbf{J}, \mathbf{M}Electric and magnetic current densities
ε(r),μ(r)\varepsilon(\mathbf{r}), \mu(\mathbf{r})Permittivity and permeability at position r\mathbf{r}
ε0,μ0\varepsilon_0, \mu_0Free-space permittivity and permeability
κ=2π/λ\kappa = 2\pi/\lambdaFree-space wavenumber
χ(r)\chi(\mathbf{r})Object contrast function: χ=εr(r)−1\chi = \varepsilon_r(\mathbf{r}) - 1
G(r,r′)G(\mathbf{r}, \mathbf{r}')Green's function of the background medium
uinc,usca,utotu^{\text{inc}}, u^{\text{sca}}, u^{\text{tot}}Incident, scattered, and total scalar fields
σsca,σext\sigma_{\text{sca}}, \sigma_{\text{ext}}Scattering and extinction cross-sections
S\mathbf{S}Scattering matrix (far-field amplitude)
DDObject domain (support of the scatterer)
s,r,p\mathbf{s}, \mathbf{r}, \mathbf{p}Transmitter, receiver, and target positions
A\mathbf{A}Sensing matrix in the linear forward model
← Ch 4Maxwell's Equations and the Helmholtz Equation