Radiation of new structures in paraxial Gaussian-type beams

Summary form only given. New simple explicit solutions of the paraxial parabolic equation different from the well-known Laguerre-Gauss and Hermite-Gauss types are presented. Starting with the classical fundamental axisymmetric mode g, defined by g = exp(ikρ²/2ζ)/ζ, where ζ = z ε, with ρ = √(x² + y²) and ε > 0 the arbitrary constant, we seek one class of solutions in the form of u = Ψ(X,Y)g, where X = x/ζ, Y = y/ζ, which gives that Ψ is the solution of the Laplace equation Ψ_XX + Ψ_YY = 0. Of interest in the paraxial context are only those Ψ, which grow with the transverse distance ρ is such manner that u→0 as ρ→∞. We construct another class of solutions assuming the structure of a more general form, u = Ψ(X,Y,ζ)g, which gives Ψ_XX + Ψ_YY + 2ikζ²Ψ = 0. Separation of variables provides solutions of the form Ψ = ψ (X,Y)exp(-2iK²/ζ where K is a new free constant, in general, complex, and ψ is the arbitrary solution of the Helmholtz equation ψ_XX + ψ_{YY + K}²ψ = 0. Examples of solutions from the both classes, among which are the Bessel-Gauss beams, will be discussed. Similar approach was recently employed for construction of exact (non-paraxial and non-time harmonic) solutions of the wave equation in the previous reference.