Singular-value decomposition and electromagnetic coherence of optical beams

Area of Science:

• Optics and Photonics
• Electromagnetism
• Mathematical Physics

Background:

• Characterizing the spectral spatial coherence of stationary light beams is crucial for understanding their electromagnetic properties.
• Existing methods for analyzing vector-beam coherence can be complex, involving multiple correlation functions.

Purpose of the Study:

• To investigate the application of singular-value decomposition (SVD) to the cross-spectral density (CSD) matrix for describing electromagnetic spectral spatial coherence.
• To simplify the formulation of vector-beam coherence, including coherence Stokes parameters and the degree of coherence.

Main Methods:

• Applied singular-value decomposition (SVD) to the cross-spectral density (CSD) matrix of stationary light beams.
• Expressed any CSD matrix as a mixture of two fully polarized, spatially partially coherent CSD matrices.
• Established two-point analogs of spectral and characteristic decompositions of the polarization matrix.

Main Results:

• Demonstrated that the polarization and coherence structures of constituent beams are determined by the singular vectors and singular values of the CSD matrix.
• Showed that vector-beam coherence can be formulated using only two correlation functions.
• Identified the degree of cross-polarization as the key factor in the composition of decompositions for Hermitian CSD matrices.

Conclusions:

• SVD of the CSD matrix provides a powerful framework for analyzing electromagnetic spectral spatial coherence in light beams.
• The proposed method offers a significant simplification in formulating and understanding vector-beam coherence.
• The degree of cross-polarization plays a critical role in the decomposition of polarization matrices for Hermitian CSD matrices.