spheres.beams.laguerre_gauss_mode

spheres.beams.laguerre_gauss_mode(N, l, coordinates='cartesian')

Returns a function evaluating a Laguerre-Gauss mode, which may take cartesian/cylindrical coordinates or vectors thereof.

\[LG(r, \phi, z) = \frac{i^{|l|-N}}{w}\sqrt{\frac{2^{|l|+1}[\frac{N-l}{2}]!}{\pi[\frac{N+|l|}{2}]!}}e^{-\frac{r^2}{w^2}}(\frac{r}{w})^{|l|}e^{il\phi}L_{\frac{N-|l|}{2}}^{|l|}(\frac{2r^2}{w^2})\]

Where \(w=\sqrt{1+(\frac{z}{\pi})^2}\) and \(L_{a}^{b}\) is a generalized Laguerre polynomial.

Parameters

• N (int) – An integer specifying the Laguerre-Gauss mode (N, l).

• l (int) – An integer specifying the Laguerre-Gauss mode (N, l).

• coodinates (str) – Whether to return a function of “cartesian” or “cylindrical” coordinates.

Returns

lg – (Vectorized) function of cartesian or cylindrical coordinates.

Return type

func