Mathematical Optics

4.9.2    LAGUERRE–GAUSSIAN BEAMS

Let us now look for modes that are separable in cylindrical coordinates (ρ, ϕ, z). Let us assume that at the plane z = 0, the dependence of the beam on the azimuthal angle is of the form exp(imϕ), where m is an integer. The origin is then the location of a phase vortex except when m = 0; hence, in order for the field to be well behaved there, the radial dependence must be proportional to ρ^|m| times an analytic function of ρ. Let this radial dependence be given by a Laguerre–Gaussian function like those in Equation 4.33, that is,

$\begin{array}{ll}{U}_{n,m}^{\mathrm{L}\mathrm{G}}(\rho ,\varphi ,0;w_0)& =\frac{\exp \hspace{0.17em}(\mathrm{i}m\varphi )}{\sqrt{2\pi }}{\rho }^{|m|}{u}_n^{\mathrm{L}\mathrm{G}}(\rho ;|m|,w_0)\\ =\sqrt{\frac{2^{|m|+1}n!}{\pi (n+|m|)!}}\frac{\exp \hspace{0.17em}(\mathrm{i}m\varphi )}{w_0^{|m|+1}}{\rho }^{|m|}{L}_n^{(|m|)}\left(2\frac{{\mathrm{p}}^2}{w_0^2}\right)\exp \left(-\frac{{\mathrm{p}}^2}{w_0^2}\right).\end{array}$

(4.92)

Notice that these beams are orthonormal under the inner product corresponding ...