Calculation of Hyper-Gaussian Beams

Algorithm for calculation of hyper-Gaussian beams is applied to analyze single-pass systems (SP) when divergence parameter (M²) of the input beam set to a value larger than one.

Hyper-Gaussian beam

Hyper-Gaussian beam in rezonator is defined as a beam whose divergence is M² times greater than the divergence of a Gaussian beam having the same waist radius. Therefore its Rayleigh distance is M² times less.

Equivalent Gaussian beam

For describing the propagation of hyper-Gaussian beam in free space, it is enough to replace the Rayleigh distance z₀ with z₀/M² in formulas for w(z) and R(z):

It is equivalent to changing waist size from w₀ to w₀/M:

A beam having such a reduced waist size is called Equivalent Gaussian Beam:

Then we can express the waist radius of the hyper-Gaussian beam in terms of the equivalent Gaussian beam:

And substitute it into formulas for w(z) and R(z):

Since equivalent Gaussian beam has the same Rayleigh distance as the hyper-Gaussian beam, its radius is M times less than the radius of the original beam not only at the waist but anywhere.

Because equivalent Gaussian beam is a standard Gaussian beam, its complex parameter can be transformed by ray matrices as usual.

Algorithm of computing

The program uses the next algorithm for calculation of propagation of hyper-Gaussian beams:

1. Equivalent size of the input beam is calculated. It is M times less than specified in the dialog of input beam parameters.

2. Propagation of an equivalent Gaussian beam is calculated.

3. Characteristics of the resulting hyper-Gaussian beam are calculated. Its radius is M times greater than the radius of the equivalent beam and radius of wavefront curvature is equal to that.

   

See also

Gaussian Beam