How does the coherency matrix describe partially polarized waves?

A fully polarized wave has a definite polarization state (linear, circular, or elliptical) at every instant, and its coherency matrix has rank 1 (determinant = 0), meaning one eigenvalue is zero. A fully unpolarized wave has equal power in all polarizations with no correlation, and its coherency matrix is proportional to the identity matrix: J = (P/2) * I, with rank 2 and maximum determinant for a given trace. A partially polarized wave is a mixture: it can be uniquely decomposed into a fully polarized component and a fully unpolarized component using the eigenvalue decomposition of J. The degree of polarization p = (lambda_1 - lambda_2) / (lambda_1 + lambda_2), where lambda_1 >= lambda_2 >= 0 are the eigenvalues. p = 1 for fully polarized, p = 0 for fully unpolarized, and 0 < p < 1 for partially polarized. In RF engineering, partially polarized signals arise from multipath propagation (which randomizes the polarization of the reflected components), from thermal emission (which is unpolarized), and from scattering by rough surfaces or random media (which partially depolarizes incident polarized signals).

How is the coherency matrix used in polarimetric radar?

Polarimetric radar transmits and receives on two orthogonal polarizations (typically H and V) and measures the full 2x2 scattering matrix S = [[Shh, Shv], [Svh, Svv]] for each resolution cell. The coherency matrix of the scattered field is computed by averaging the outer product of the scattering vector over many looks (independent samples): T = where k is the Pauli scattering vector [Shh+Svv, Shh-Svv, 2*Shv]^T / sqrt(2). This 3x3 coherency matrix (not to be confused with the 2x2 wave coherency matrix) contains all second-order polarimetric information about the scatterer, including the mean scattering mechanism, the degree of polarimetric purity, and the target decomposition components. The Cloude-Pottier decomposition extracts the eigenvalues and eigenvectors of T to classify scatterers into surface, volume, and double-bounce categories. This classification is fundamental to SAR remote sensing for land cover mapping, crop monitoring, forest biomass estimation, and disaster damage assessment.

What is the relationship between coherency matrix and Stokes parameters?

The four Stokes parameters and the four unique elements of the 2x2 Hermitian coherency matrix contain equivalent information and are related by linear transformations. S0 = J11 + J22 (total intensity), S1 = J11 - J22 (horizontal vs vertical preference), S2 = 2*Re(J12) = J12 + J21 (45-degree vs 135-degree preference), S3 = 2*Im(J12) = j*(J21 - J12) (right vs left circular preference). Conversely: J11 = (S0+S1)/2, J22 = (S0-S1)/2, J12 = (S2+jS3)/2, J21 = (S2-jS3)/2. The Stokes parameters are directly measurable with power detectors and polarizers (no phase measurement needed), making them preferred for radiometry and optical systems. The coherency matrix preserves the complex field information, making it preferred for coherent systems like radar and interferometry. The degree of polarization is p = sqrt(S1^2 + S2^2 + S3^2) / S0 in Stokes form, or p = sqrt(1 - 4*det(J)/tr(J)^2) in matrix form, both equivalent.

Electromagnetic Theory

Coherency Matrix

Q: How is the coherency matrix used in polarimetric radar?

Polarimetric radar transmits and receives on two orthogonal polarizations (typically H and V) and measures the full 2x2 scattering matrix S = [[Shh, Shv], [Svh, Svv]] for each resolution cell. The coherency matrix of the scattered field is computed by averaging the outer product of the scattering vector over many looks (independent samples): T = where k is the Pauli scattering vector [Shh+Svv, Shh-Svv, 2*Shv]^T / sqrt(2). This 3x3 coherency matrix (not to be confused with the 2x2 wave coherency matrix) contains all second-order polarimetric information about the scatterer, including the mean scattering mechanism, the degree of polarimetric purity, and the target decomposition components. The Cloude-Pottier decomposition extracts the eigenvalues and eigenvectors of T to classify scatterers into surface, volume, and double-bounce categories. This classification is fundamental to SAR remote sensing for land cover mapping, crop monitoring, forest biomass estimation, and disaster damage assessment.

Q: What is the relationship between coherency matrix and Stokes parameters?

The four Stokes parameters and the four unique elements of the 2x2 Hermitian coherency matrix contain equivalent information and are related by linear transformations. S0 = J11 + J22 (total intensity), S1 = J11 - J22 (horizontal vs vertical preference), S2 = 2*Re(J12) = J12 + J21 (45-degree vs 135-degree preference), S3 = 2*Im(J12) = j*(J21 - J12) (right vs left circular preference). Conversely: J11 = (S0+S1)/2, J22 = (S0-S1)/2, J12 = (S2+jS3)/2, J21 = (S2-jS3)/2. The Stokes parameters are directly measurable with power detectors and polarizers (no phase measurement needed), making them preferred for radiometry and optical systems. The coherency matrix preserves the complex field information, making it preferred for coherent systems like radar and interferometry. The degree of polarization is p = sqrt(S1^2 + S2^2 + S3^2) / S0 in Stokes form, or p = sqrt(1 - 4*det(J)/tr(J)^2) in matrix form, both equivalent.

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The coherency matrix J is a 2×2 Hermitian matrix describing the full second-order polarization state of an EM wave, including partially polarized signals. J_ij = ⟨E_iE_j*⟩. Degree of polarization p = √(1 - 4 det(J)/tr(J)²). Related to Stokes: S₀ = J₁₁+J₂₂, S₁ = J₁₁-J₂₂, S₂ = 2Re(J₁₂), S₃ = 2Im(J₁₂). Used in polarimetric radar for Cloude-Pottier target decomposition.

Category: Electromagnetic Theory

Size: 2×2 Hermitian

Parameters: 4 real (= Stokes)

Understanding the Coherency Matrix

Polarization is a fundamental property of electromagnetic waves, and its complete description requires more than just stating "horizontal" or "circular." Real-world RF signals are often partially polarized: a combination of a deterministic (polarized) component and a random (unpolarized) component. This arises from multipath propagation, rough surface scattering, thermal emission, and mixing of independent sources. The coherency matrix captures this complexity in a compact 2×2 matrix that separates the polarized and unpolarized contributions through its eigenvalue decomposition.

The matrix framework connects to multiple representation systems used in different RF engineering disciplines. Antenna engineers use Jones vectors and matrices for fully polarized signals through deterministic systems. Radiometer engineers use Stokes parameters for power-based measurements of partially polarized thermal emission. Polarimetric radar engineers use scattering matrices and target decomposition theorems. The coherency matrix unifies these: Jones vectors are its rank-1 case, Stokes parameters are linear combinations of its elements, and target decomposition operates on its 3×3 generalization.

Coherency Matrix Equations

Coherency Matrix:
J = ⟨E E^H⟩ = [[⟨|E_x|²⟩, ⟨E_xE_y*⟩], [⟨E_yE_x*⟩, ⟨|E_y|²⟩]]

Degree of Polarization:
p = √(1 - 4 det(J) / tr(J)²) = (λ₁ - λ₂) / (λ₁ + λ₂)

Stokes from Coherency:
S = [J₁₁+J₂₂, J₁₁-J₂₂, 2Re(J₁₂), 2Im(J₁₂)]

Where λ₁ ≥ λ₂ ≥ 0 are eigenvalues. Fully polarized: det(J) = 0, p = 1. Unpolarized: J = (P/2)I, p = 0. Partially polarized: 0 < p < 1. Total power = tr(J) = S₀.

Polarization State Examples

State	Coherency Matrix	p	Stokes	Source
Horizontal linear	[[1,0],[0,0]]	1	[1,1,0,0]	Dipole antenna
RHCP	½[[1,-j],[j,1]]	1	[1,0,0,1]	Helix antenna
Unpolarized	½[[1,0],[0,1]]	0	[1,0,0,0]	Thermal emission
50% polarized H	[[0.75,0],[0,0.25]]	0.5	[1,0.5,0,0]	Ground reflection
45° linear	½[[1,1],[1,1]]	1	[1,0,1,0]	Rotated dipole

Common Questions

Frequently Asked Questions

How does it describe partial polarization?

Fully polarized: rank 1 (det = 0, one zero eigenvalue), p = 1. Unpolarized: J ∝ I (equal eigenvalues), p = 0. Partially polarized: eigenvalue decomposition uniquely separates polarized + unpolarized components. p = (λ₁-λ₂)/(λ₁+λ₂). Multipath, rough scattering, thermal emission produce 0 < p < 1.

How is it used in polarimetric radar?

Scattering matrix S → Pauli vector k → 3×3 coherency matrix T = ⟨kk^H⟩. Cloude-Pottier decomposition: eigenvalues/eigenvectors classify scatterers (surface, volume, double-bounce). Applications: SAR land cover mapping, crop monitoring, forest biomass, disaster assessment.

Relationship to Stokes parameters?

Linear transformation: S₀=J₁₁+J₂₂, S₁=J₁₁-J₂₂, S₂=2Re(J₁₂), S₃=2Im(J₁₂). Equivalent information. Stokes: measurable with power detectors (radiometry). Coherency: preserves complex field (radar, interferometry). p = √(S₁²+S₂²+S₃²)/S₀ = √(1-4det(J)/tr(J)²).

Related Terms

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