Constraints on spheroidal beam wavefunctions.
The oblate spheroidal wavefunctions proposed by Rodríguez-Morales and Chávez-Cerda are shown to be possible representations of physical beams only when the angular function S(mn)(β,η) has odd n-m. This condition makes S(mn) odd in η, which ensures the convergence of integrals of physical quantities over a cross section of the beam. The odd n-m condition also makes S(mn)(β,η) zero in the focal plane z=0 outside the circle ρ=b, and thus allows for the physically necessary discontinuity in phase at z=0 on the ellipsoidal surfaces of otherwise constant phase. Only a subset of the oblate spheroidal functions can be exact representations of nonparaxial scalar beams.
MacDiarmid Institute for Advanced Materials and Technology, and School of Chemical and Physical Sciences, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand. john.lekner@vuw.ac.nz
John
J
Lekner
Rufus
R
Boyack