From Gaussian beams to Helmholtz waves

We identify the scale ratio between the Helmholtz transverse-wavenumber scale and the paraxial Gaussian-beam-width scale as the physical parameter connecting separable Gaussian beams with Helmholtz waves. Our optical scale ratio factors the Helmholtz-to-paraxial angular-spectrum restriction with the paraxial-to-Helmholtz large-Rayleigh-range limit. It turns the algebraic Inönü–Wigner contraction and the Cayley–Klein deformation into optical limits governed by scale. Our spectral contraction connects Hermite, Laguerre, Ince, and Boyer–Wolf Gaussian beam families with plane, Bessel, Mathieu, and Weber wave families, respectively, at the coordinate, differential wave equation, algebraic deformation, separation operator, separated differential equation, and separation-spectrum levels.