Document Type
Article
Publication Date
9-1-2026
Abstract
The optimal modes for correcting atmospheric turbulence on coherent arrays are determined. These Karhunen-Loève modes are eigenvectors of a covariance matrix. Creating this covariance matrix requires knowledge of the power spectrum of the turbulence, the aperture geometry, and a basis set for the matrix. The Kolmogorov power law is ordinarily chosen here for the turbulence spectrum. By choosing the phase on each array subaperture element minus the phase averaged over all subapertures as this basis, infinite values in the variances and covariances can be avoided. The piston mode of the whole array, which would otherwise also be infinite, is thus assigned an eigenvalue of zero. The elements of this covariance matrix are demonstrated to be expressible with the expectation values of the square of the phase difference between subaperture pairs; this differential phase variance is seen to clearly remain finite for the Kolmogorov power spectrum. Each entry in the covariance matrix involves these differential phase variances for all pairs of subapertures but combined in different ways. The requisite differential phase variance between a pair of subapertures is then shown to depend upon the phase structure function. It is shown to equal the phase structure function averaged over all pairs of points between the involved subapertures minus the structure function averaged over all pairs of points on the individual subapertures. Theoretical expressions for these differential phase variances are derived for square and circular subapertures. Using these tools, the Karhunen-Loève modes are determined for some example coherent array geometries.
Source Publication
Optical Engineering (ISSN 0091-3286 | eISSN 1560-2303)
Recommended Citation
Jack E. McCrae Jr., Santasri R. Bose-Pillai, Steven T. Fiorino "Karhunen-Loève modes for coherent arrays," Optical Engineering 65(8), 081826 (1 Sept 2026) https://doi.org/10.1117/1.OE.65.8.081826
Comments
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