Second-order moments of the Wigner distribution function in first-order optical systems

Martin J. Bastiaans

Starting from the well-known description of a first-order optical system by means of its symplectic ABCD ray transformation matrix, a propagation law for the second-order moments of the Wigner distribution function is derived. A matrix is constructed from the second-order moments, which undergoes a similarity transformation when it propagates through first-order optical systems. This property leads to invariant expressions in terms of the second-order moments, based on the eigenvalues of the matrix mentioned before. Properties of these eigenvalues are derived. It is shown that, at least in the one-dimensional case, a relation similar to the well-known bilinear relationship which expresses the propagation of the curvature of a spherical wave through a first-order system, can be formulated in terms of the second-order moments. Finally, it is shown that in the higher-order dimensional case such a relationship only holds under the condition that the second-order moments form a matrix that is symplectic, which is the case for a Gaussian signal, for instance.

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