The Scattering Matrix and Far-Field Pattern

Characterizing Scatterers by Their Far-Field Response

The scattering matrix provides a compact, frequency-dependent description of how an object transforms incident waves into scattered waves. It bridges the physics of Sections 5.1--5.4 with the signal-processing perspective of Chapter 7: the scattering matrix is what the radar "sees."

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Definition:

Far-Field Scattering Amplitude

In the far field (rβ†’βˆžr \to \infty), the scattered field takes the asymptotic form:

2D: usca(r)∼eβˆ’jΞΊ0rr f(r^;k^inc),u^{\text{sca}}(\mathbf{r}) \sim \frac{e^{-j\kappa_{0} r}}{\sqrt{r}}\, f(\hat{r}; \hat{k}^{\text{inc}}),

3D: usca(r)∼eβˆ’jΞΊ0rr f(r^;k^inc),u^{\text{sca}}(\mathbf{r}) \sim \frac{e^{-j\kappa_{0} r}}{r}\, f(\hat{r}; \hat{k}^{\text{inc}}),

where f(r^;k^inc)f(\hat{r}; \hat{k}^{\text{inc}}) is the far-field scattering amplitude β€” a function of observation direction r^\hat{r} and incidence direction k^inc\hat{k}^{\text{inc}}.

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Definition:

The Scattering Matrix

Discretizing the incidence and observation directions into PP and QQ angles respectively, the scattering matrix S\mathbf{S} has entries:

[S]qp=f(r^q;k^p),[\mathbf{S}]_{qp} = f(\hat{r}_q; \hat{k}_p),

so S∈CQΓ—P\mathbf{S} \in \mathbb{C}^{Q \times P}. For full-angle coverage (P=QP = Q, same angles), S\mathbf{S} is square.

For vector (polarimetric) scattering, each entry becomes a 2Γ—22 \times 2 matrix:

Sqp=(SvvSvhShvShh),\mathbf{S}_{qp} = \begin{pmatrix} S_{vv} & S_{vh} \\ S_{hv} & S_{hh} \end{pmatrix},

where v,hv, h denote vertical and horizontal polarizations.

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Theorem: Reciprocity of the Scattering Matrix

For a non-magnetic, reciprocal medium, the scattering matrix satisfies the reciprocity relation:

f(r^;k^inc)=f(βˆ’k^inc;βˆ’r^).f(\hat{r}; \hat{k}^{\text{inc}}) = f(-\hat{k}^{\text{inc}}; -\hat{r}).

In matrix form, S\mathbf{S} is symmetric (for scalar scattering) when transmit and receive angles are indexed consistently.

Show Hint

Reciprocity means interchanging Tx and Rx gives the same scattering coefficient β€” halving independent measurements in monostatic radar.

Proof
From Lorentz reciprocity

The Lorentz reciprocity theorem states that for two source configurations (J1,E1)(\mathbf{J}_1, \mathbf{E}_1) and (J2,E2)(\mathbf{J}_2, \mathbf{E}_2):

∫J1β‹…E2 dV=∫J2β‹…E1 dV.\int \mathbf{J}_1 \cdot \mathbf{E}_2\,dV = \int \mathbf{J}_2 \cdot \mathbf{E}_1\,dV.

Choosing J1,J2\mathbf{J}_1, \mathbf{J}_2 as point sources in the far field (plane waves) and evaluating the scattered fields yields the reciprocity of ff. β– \blacksquare

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Definition:

Radar Cross-Section (RCS)

The radar cross-section (RCS) quantifies the scattering strength of an object as seen by a radar:

ΟƒRCS(r^;k^inc)=lim⁑rβ†’βˆž4Ο€r2∣usca(r)∣2∣uinc∣2=4Ο€βˆ£f(r^;k^inc)∣2.\sigma_{\text{RCS}}(\hat{r}; \hat{k}^{\text{inc}}) = \lim_{r \to \infty} 4\pi r^2 \frac{|u^{\text{sca}}(\mathbf{r})|^2}{|u^{\text{inc}}|^2} = 4\pi |f(\hat{r}; \hat{k}^{\text{inc}})|^2.

RCS has units of area (m2^2) and is typically expressed in dBsm (dB relative to 1 m2^2).

Monostatic RCS (r^=βˆ’k^inc\hat{r} = -\hat{k}^{\text{inc}}): the backscattering strength relevant for conventional radar.

Bistatic RCS (r^β‰ βˆ’k^inc\hat{r} \neq -\hat{k}^{\text{inc}}): the scattering strength in arbitrary directions.

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Radar cross-section (RCS)

A measure of an object's electromagnetic scattering strength: ΟƒRCS=4Ο€βˆ£f∣2\sigma_{\text{RCS}} = 4\pi|f|^2, where ff is the far-field scattering amplitude. Units: m2^2 (or dBsm).

Related: Born approximation

Scattering Geometry and Far-Field Pattern

Visualizes the far-field scattering pattern for canonical objects.

Left panel: Polar plot of bistatic RCS Οƒ(ΞΈ)\sigma(\theta) in dBsm. For small ΞΊ0a\kappa_{0} a, the pattern is nearly isotropic (Rayleigh scattering). As ΞΊ0a\kappa_{0} a increases, the pattern develops lobes corresponding to diffraction maxima.

Right panel: Born validity indicator based on ΞΊ0aβˆ£Ο‡0∣\kappa_{0} a |\chi_0|.

Parameters
2
0.3

Example: RCS of Canonical Objects

Compute the monostatic RCS of the following objects and identify the applicable scattering regime.

Object Parameters
PEC sphere radius aa, ΞΊ0a≫1\kappa_{0} a \gg 1
PEC sphere radius aa, ΞΊ0aβ‰ͺ1\kappa_{0} a \ll 1
PEC flat plate area AA, normal incidence
Solution
High-frequency sphere (optics regime)

ΟƒRCS=Ο€a2\sigma_{\text{RCS}} = \pi a^2 β€” the geometric cross-section. Intuition: the sphere intercepts and re-radiates an area equal to its geometric shadow.

Low-frequency sphere (Rayleigh regime)

ΟƒRCS=9πκ04a64\sigma_{\text{RCS}} = \frac{9\pi \kappa_{0}^{4} a^6}{4}. Scales as a6/Ξ»4a^6/\lambda^{4} β€” very small for sub-wavelength objects.

Flat plate

ΟƒRCS=4Ο€A2Ξ»2\sigma_{\text{RCS}} = \frac{4\pi A^2}{\lambda^{2}}. The strong Ξ»βˆ’2\lambda^{-2} dependence motivates wideband radar.

Definition:

The Optical Theorem

The total power removed from the incident field (extinction) equals the imaginary part of the forward scattering amplitude:

3D: Οƒext=4πκ0 Im{f(k^inc)}.\sigma_{\text{ext}} = \frac{4\pi}{\kappa_{0}}\, \text{Im}\{f(\hat{k}^{\text{inc}})\}.

2D: Οƒext=8πκ0 Im{ejΟ€/4 f(k^inc)}.\sigma_{\text{ext}} = \sqrt{\frac{8\pi}{\kappa_{0}}}\, \text{Im}\{e^{j\pi/4}\,f(\hat{k}^{\text{inc}})\}.

This connects a measurable far-field quantity (forward scattering) to the total interaction (scattering + absorption). It provides a consistency check for any scattering computation.

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From Scattering Matrix to Imaging

The scattering matrix S\mathbf{S} connects to the sensing matrix A\mathbf{A} of the imaging problem:

  • Under the Born approximation, S\mathbf{S} is a linear function of Ο‡\chi: S=AΟ‡\mathbf{S} = \mathbf{A}\boldsymbol{\chi} (after discretization).
  • The SVD of S\mathbf{S} reveals which spatial features of the object are observable.
  • Multi-static data matrices (Chapter 6) are rearrangements of S\mathbf{S} tailored to specific imaging geometries.

The transition from scattering physics to signal processing happens through this matrix β€” Chapter 7 develops the radar signal-processing tools, and Chapter 6 assembles the complete sensing operator.

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Why This Matters: RCS and Radar System Design

The radar cross-section directly enters the radar range equation:

Pr=PtGtGrΞ»2ΟƒRCS(4Ο€)3R4,P_r = \frac{P_t G_t G_r \lambda^{2} \sigma_{\text{RCS}}}{(4\pi)^3 R^4},

connecting the scattering physics of this chapter to radar system design (Chapter 9). A target with larger RCS is detectable at greater range. The angular dependence of RCS determines the optimal bistatic geometry for imaging.

See full treatment in Chapter 9

Historical Note: The Optical Theorem β€” From Optics to Radar

1871--1957

The optical theorem was first derived by Lord Rayleigh in 1871 for acoustic scattering, then extended to electromagnetic waves by van de Hulst in 1957. The name "optical theorem" reflects its origins in optical extinction measurements. In radar, the theorem provides a powerful consistency check: the total scattered power can be determined from a single forward-scattering measurement.

Quick Check

If the scattering matrix of an object satisfies ST=S\mathbf{S}^T = \mathbf{S} (reciprocity), what is the practical implication for a radar system?

The object must be symmetric.

Interchanging transmitter and receiver gives the same scattered field, halving independent measurements.

The object has zero absorption.

The RCS is independent of frequency.

Correction:
Interchanging transmitter and receiver gives the same scattered field, halving independent measurements.

Reciprocity means f(r^;k^)=f(βˆ’k^;βˆ’r^)f(\hat{r}; \hat{k}) = f(-\hat{k}; -\hat{r}), so Tx-Rx exchange preserves the measurement.

Key Takeaway

The far-field scattering amplitude f(r^;k^)f(\hat{r}; \hat{k}) characterizes how an object redirects waves. The scattering matrix S\mathbf{S} discretizes ff over transmit/receive angles. Reciprocity (ST=S\mathbf{S}^T = \mathbf{S}) halves the independent measurements. RCS =4Ο€βˆ£f∣2= 4\pi|f|^2 quantifies scattering strength. The optical theorem links forward scattering to extinction. Under Born, the scattering matrix is a linear function of the contrast Ο‡\chi β€” the bridge to the sensing operator.

Beyond First-Order: Iterative MethodsChapter Summary