Karhunen-Loève modes of laser arrays

The optimal modes for correcting atmospheric turbulence on laser arrays are determined. These Karhunen-Loève modes are eigenvectors of a covariance matrix. Creating this covariance matrix requires the power spectrum of the turbulence and a basis set for the matrix. The Kolmogorov power law is chosen here for the turbulence spectrum. By choosing the phase on each array sub-aperture minus the phase averaged over all sub-apertures as this basis set, infinite values in the variances and covariances are 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 sub-aperture 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 sub-apertures but combined in different ways. The requisite differential phase variance between a pair of sub-apertures 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 sub-apertures minus the structure function averaged over all pairs of points on the individual sub-apertures. Theoretical expressions for these differential phase variances are derived for square and circular sub-apertures. Using these tools, the Karhunen-Loève modes are determined for some example laser array geometries.