Editing Gaussian function (section)

=== Higher-order Gaussian or super-Gaussian function or generalized Gaussian function ===
A more general formulation of a Gaussian function with a flat-top and Gaussian fall-off can be taken by raising the content of the exponent to a power <math>P</math>:
<math display="block">f(x) = A \exp\left(-\left(\frac{(x - x_0)^2}{2\sigma_X^2}\right)^P\right).</math>

This function is known as a super-Gaussian function and is often used for Gaussian beam formulation.<ref>Parent, A., M. Morin, and P. Lavigne. "Propagation of super-Gaussian field distributions". ''[[Optical and Quantum Electronics]]'' 24.9 (1992): S1071–S1079.</ref> This function may also be expressed in terms of the [[full width at half maximum]] (FWHM), represented by {{mvar|w}}:
<math display="block">f(x) = A \exp\left(-\ln 2\left(4\frac{(x - x_0)^2}{w^2}\right)^P\right).</math>

In a two-dimensional formulation, a Gaussian function along <math>x</math> and <math>y</math> can be combined<ref>{{Cite web |url=http://www.aor.com/anonymous/pub/commands.pdf |title=GLAD optical software commands manual, Entry on GAUSSIAN command |date=2016-12-15 |website=Applied Optics Research}}</ref> with potentially different <math>P_X</math> and <math>P_Y</math> to form a rectangular Gaussian distribution:
<math display="block">f(x, y) = A \exp\left(-\left(\frac{(x - x_0)^2}{2\sigma_X^2}\right)^{P_X} - \left(\frac{(y - y_0)^2}{2\sigma_Y^2}\right)^{P_Y}\right).</math>
or an elliptical Gaussian distribution:
<math display="block">f(x , y) = A \exp\left(-\left(\frac{(x - x_0)^2}{2\sigma_X^2} + \frac{(y - y_0)^2}{2\sigma_Y^2}\right)^P\right)</math>