We present a general framework for the Wigner distribution function (WDF) of Helmholtz–Gauss (HzG) wave fields. The WDF is written as a rapidly convergent series of modified Bessel functions and is completely determined by the complex Fourier coefficients of the spectral function of the underlying nondiffracting Helmholtz beam. Using these same coefficients, we also obtain the normalization constant of any HzG wave as a rapidly convergent series, and show how symmetry constraints on the Fourier coefficients induce corresponding symmetry relations of the WDF in phase space. We prove that the HzG WDF has the universal structure of a four-dimensional Gaussian in phase space, multiplied by a modulation that, in general, depends on the Hamiltonian, orbital angular momentum, and two angles, all expressed in terms of phase-space variables. We apply this formalism to the four fundamental HzG families, presenting explicit expressions for the WDFs of the Mathieu–Gauss, parabolic-Gauss, and cosine- and sine-Gauss beams, and confirming previous results for Bessel–Gauss beams. The formalism can be readily applied to any superposition of HzG waves. Finally, we derive a secondary representation that yields an equivalent expansion in products of Laguerre–Gauss functions, connecting the general HzG beam to the eigenmodes of the paraxial wave equation.

Close

PrevNext

View figure in article

• Light propagation, transmission & absorption
• Non-diffracting waves
• Structured light
• Phase space methods