abstract
- We establish the direct connection between the Mathieu and parabolic families of nondiffracting solutions of the two-dimensional Helmholtz equation, thereby completing the set of overlap relations among the four fundamental separable Helmholtz families. By combining the known Mathieu-Bessel connection with the recently derived Bessel-parabolic overlaps, we obtain analytical expressions for the Mathieu-parabolic overlap functions in terms of Mathieu-function Fourier coefficients and generalized hypergeometric functions. These results yield explicit bidirectional expansion formulas between parity-definite Mathieu and parabolic beams, as well as between complex helical Mathieu beams and traveling parabolic beams. We further extend the construction to the finite-energy Helmholtz¿Gauss case, deriving the corresponding relations between Mathieu¿Gauss and parabolic-Gauss beams. Numerical reconstructions are presented to verify the convergence and accuracy of the derived expansions.