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Why Integral Equations?

The differential (Helmholtz) equation from Section 5.1 requires discretizing the entire computational domain. Integral equation formulations reformulate the scattering problem over only the object domain $D$ (volume integrals) or its boundary $\partial D$ (surface integrals), reducing dimensionality and automatically incorporating the radiation condition through the Green's function.

Theorem: The Lippmann-Schwinger Equation

The total field $u(\mathbf{r})$ in the presence of a scatterer with contrast $\chi(\mathbf{r})$ satisfies the volume integral equation (Lippmann-Schwinger equation):

$u(\mathbf{r}) = u^{\text{inc}}(\mathbf{r}) + \kappa_{0}^{2} \int_D G(\mathbf{r}, \mathbf{r}')\, \chi(\mathbf{r}')\,u(\mathbf{r}')\,d\mathbf{r}'.$

Equivalently, defining the volume scattering operator $\mathcal{G}: f \mapsto \kappa_{0}^{2} \int_D G(\mathbf{r}, \mathbf{r}')f(\mathbf{r}')d\mathbf{r}'$ :

$u = u^{\text{inc}} + \mathcal{G}[\chi \cdot u].$

Show Hint

This is a Fredholm integral equation of the second kind in $u$ .

Proof

From Helmholtz to integral form

The scattered field satisfies $\nabla^2 u^{\text{sca}} + \kappa_{0}^{2} u^{\text{sca}} = -\kappa_{0}^{2}\chi\,u$ . Since $G$ is the Green's function for $\nabla^2 + \kappa_{0}^{2}$ , the solution is the convolution:

$u^{\text{sca}}(\mathbf{r}) = \kappa_{0}^{2} \int_D G(\mathbf{r}, \mathbf{r}')\,\chi(\mathbf{r}')\,u(\mathbf{r}')\,d\mathbf{r}'.$

Adding $u^{\text{inc}}$ gives the Lippmann-Schwinger equation. $\blacksquare$

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The Nonlinearity of Inverse Scattering

The unknown $u$ appears both on the left and inside the integral (through the total field), making the Lippmann-Schwinger equation nonlinear in $\chi$ when $\chi$ is also unknown. This nonlinearity is the fundamental challenge of inverse scattering: we seek $\chi$ from measurements of $u^{\text{sca}}$ , but the mapping $\chi \mapsto u^{\text{sca}}$ involves solving the integral equation with $\chi$ in the kernel.

Definition:
The Data Equation (Observation Equation)

At receiver locations $\mathbf{r}_{j}$ outside the scatterer, the scattered (measured) field is:

$u^{\text{sca}}(\mathbf{r}_{j}) = \kappa_{0}^{2} \int_D G(\mathbf{r}_{j}, \mathbf{r}')\, \chi(\mathbf{r}')\,u(\mathbf{r}')\,d\mathbf{r}'.$

This is the data equation — it relates measurements $y_j = u^{\text{sca}}(\mathbf{r}_{j})$ to the unknown contrast $\chi$ and the (also unknown) total field $u$ inside $D$ .

The scattering problem thus consists of two coupled equations:

Domain equation: $u = u^{\text{inc}} + \mathcal{G}[\chi u]$ (field inside $D$ ).
Data equation: $\mathbf{y} = \mathcal{G}_S[\chi u]$ (field at sensors).

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Theorem: The Born Series (Neumann Series for Scattering)

The Lippmann-Schwinger equation $u = u^{\text{inc}} + \mathcal{G}[\chi u]$ can be written as $(I - \mathcal{G}\chi)\,u = u^{\text{inc}}$ . If $\|\mathcal{G}\chi\|_{\text{op}} < 1$ (weak scattering), the Neumann series converges:

$u = \sum_{n=0}^{\infty} (\mathcal{G}\chi)^n u^{\text{inc}} = u^{\text{inc}} + \mathcal{G}[\chi u^{\text{inc}}] + \mathcal{G}[\chi\,\mathcal{G}[\chi u^{\text{inc}}]] + \cdots$

Each term represents an additional order of scattering: $n = 0$ (incident), $n = 1$ (single scattering = Born approximation), $n = 2$ (double scattering), etc.

Proof

Operator inversion via Neumann series

The operator equation $(I - \mathcal{G}\chi)u = u^{\text{inc}}$ has formal solution $u = (I - \mathcal{G}\chi)^{-1}u^{\text{inc}}$ . When $\|\mathcal{G}\chi\|_{\text{op}} < 1$ , the Neumann series $(I - A)^{-1} = \sum_{n=0}^\infty A^n$ converges absolutely, giving the Born series. $\blacksquare$

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Convergence of the Born Series

The Born series converges when $\|\mathcal{G}\chi\|_{\text{op}} < 1$ , which roughly requires:

$\kappa_{0}^{2} |\chi_{\max}| \cdot \text{vol}(D) \cdot \max_{\mathbf{r}} |G| < 1.$

This fails for:

High contrast $|\chi| \gg 1$ (e.g., metal objects).
Large objects $\kappa_{0} a \gg 1$ where $a$ is the object diameter.
Resonant frequencies where interior modes are excited.

When the Born series diverges, iterative methods (Section 5.6) or full numerical solvers are required.

Definition:
Surface Integral Equations

For a homogeneous scatterer with sharp boundary $\partial D$ , the scattering problem can be reformulated using surface currents. The electric field integral equation (EFIE) for a PEC body is:

$\hat{n} \times \mathbf{E}^{\text{inc}}(\mathbf{r}) = -\hat{n} \times \int_{\partial D} \overline{\mathbf{G}}(\mathbf{r}, \mathbf{r}') \cdot \mathbf{J}_s(\mathbf{r}')\,dS', \quad \mathbf{r} \in \partial D,$

where $\mathbf{J}_s$ is the surface current density and $\overline{\mathbf{G}}$ is the dyadic Green's function.

Surface formulations reduce the dimensionality by one (3D volume $\to$ 2D surface), but require the object to be piecewise homogeneous.

Volume vs Surface Integral Formulations

Aspect	Volume (Lippmann-Schwinger)	Surface (EFIE/MFIE)
Applicable objects	Arbitrary $\varepsilon(\mathbf{r})$	Piecewise homogeneous
Unknowns	Field inside $D$ ( $N^3$ or $N^2$ )	Surface currents ( $N^2$ or $N$ )
Discretization	Volume mesh	Surface mesh
Imaging relevance	Direct basis for Born approximation	Used in RCS computation
Radiation condition	Automatic via Green's function	Automatic via Green's function

Definition:
Scattering and Extinction Cross-Sections

The scattering cross-section measures the total power scattered by the object, normalized by the incident power density:

$\sigma_{\text{sca}} = \frac{P_{\text{sca}}}{|\mathbf{S}^{\text{inc}}|} = \frac{1}{|\mathbf{S}^{\text{inc}}|} \oint |u^{\text{sca}}|^2 \, dS.$

The extinction cross-section (scattering + absorption) is related to the forward scattering amplitude via the optical theorem:

$\sigma_{\text{ext}} = \frac{4\pi}{\kappa_{0}}\,\text{Im}\{f(\hat{k}^{\text{inc}})\},$

where $f(\hat{k})$ is the far-field scattering amplitude in direction $\hat{k}$ .

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Lippmann-Schwinger equation

The volume integral equation $u = u^{\text{inc}} + \mathcal{G}[\chi u]$ that reformulates the scalar Helmholtz equation as an integral equation over the scatterer domain. Named after Bernard Lippmann and Julian Schwinger who introduced it in quantum scattering theory.

Historical Note: Lippmann, Schwinger, and Quantum Scattering

1950s

Bernard Lippmann and Julian Schwinger developed their integral equation formulation in 1950 for quantum-mechanical scattering, building on earlier work by Max Born. The equation was subsequently adopted in classical electromagnetic scattering, where it provides the natural framework for inverse problems. Schwinger shared the 1965 Nobel Prize in Physics (with Feynman and Tomonaga) for quantum electrodynamics.

Key Takeaway

The Lippmann-Schwinger equation reformulates the Helmholtz equation as a volume integral equation over the object domain. Two coupled equations define the scattering problem: the domain equation (total field inside $D$ ) and the data equation (scattered field at receivers). The Born series expands the total field in orders of scattering; truncating at first order gives the Born approximation. Surface integral equations (EFIE/MFIE) are preferred for PEC or piecewise-homogeneous objects; volume formulations are essential for imaging arbitrary contrast distributions.

The Lippmann-Schwinger Equation