Unified correlation theory of nondiffracting Helmholtz waves
Phys. Rev. A 114, 053505 – Published 6 November, 2026
DOI: https://doi.org/10.1103/vk7l-9tcb
Abstract
We develop a unified theory of the mutual and cross-correlation functions of the four fundamental families of nondiffracting Helmholtz waves: Bessel, Mathieu, parabolic, and plane waves. Treating these fields as Dirac-orthonormal bases of the two-dimensional Helmholtz equation at fixed transverse wave number, we derive analytical correlation kernels for both parity-definite modes and their complex helical or traveling counterparts. A central result is that all parity-definite families possess the same universal mutual-correlation structure, given by symmetric or antisymmetric superpositions of two stationary cylindrical Bessel waves centered at the source point and at its mirror image. We also obtain the corresponding cross-correlation kernels, which describe the coupling between different symmetry classes and acquire family-dependent forms. For complex helical and traveling waves, the correlation functions become intrinsically complex: their real parts are determined by the universal mutual kernel, whereas their imaginary parts arise from the parity cross correlations. These complex kernels admit a natural interpretation as generalized outgoing or incoming traveling waves associated with real or virtual source points. Finally, the unified formalism yields interfamily identities and completeness relations that connect different nondiffracting-wave bases. The results reveal a common correlation structure underlying all separable solutions of the two-dimensional Helmholtz equation and provide a compact framework for describing spatial coherence in nondiffracting optical fields.
