Laguerre-Gauss fn -- Hermite Gauss -- Gaussian Beams
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The first panel of this window explains itself very well. The second panel however needs help.
To start with, notice if n, l, z are all zero that the entry in σ will produce nearly the same result as would the same entry on panel one. The difference is that the Gauss function made by panel one has maximum value one while the Gauss function produced in this panel is normalized such that the integral over the square of the function is 1/N. Provided the function is nearly all within the boundary of the screen.. It is just a gaussian. Other and more interesting values of n & l result in Lagerre-Gauss functions. The pixel value of σ will be translated into millimeters if the pixel pitch textbox is filled in.
Non-zero entries for z give the projection of the function along the Gaussian Beam. The wave length (λ) and pixel pitch must both be non-zero to get a result in the z ≠ 0 case. Conversely if z=0 the value of the pixel pitch and λ will be ignored.
If you enter a negative z and use the result as a hologram, the appropriate LG function will be produced a distance z from the hologram.
Try looking up "Gaussian Beams" on "Wikipedia".
The tutorial number 10 describes how to use a sum of these functions to build a Hopf link of two vortex loops.

The Hermite Gauss window is simple enough. Notice that sigma does not need to be an integer in any of these panels. Also note that, many text books put an additional factor of root 2 in the definition of the Gauss fn. so that the width of the resulting power function has the appropriate sigma. You must put this in yourself if you need it.