Editing Gaussian beam (section)

==Complex beam parameter==
{{main article|Complex beam parameter}}
The spot size and curvature of a Gaussian beam as a function of {{mvar|z}} along the beam can also be encoded in the complex beam parameter {{math|''q''(''z'')}}<ref name="siegman638">Siegman, pp. 638–40.</ref><ref name="garg168">Garg, pp. 165–168.</ref> given by:
<math display="block"> q(z) =  z + iz_\mathrm{R} .</math>

The reciprocal of {{math|''q''(''z'')}} contains the wavefront curvature and relative on-axis intensity in its real and imaginary parts, respectively:<ref name="siegman638" />

<math display="block">{1 \over q(z)} = {1 \over R(z)} - i {\lambda \over n \pi w^2(z)} .</math>

The complex beam parameter simplifies the mathematical analysis of Gaussian beam propagation, and especially in the analysis of [[optical cavity|optical resonator cavities]] using [[ray transfer matrix analysis|ray transfer matrices]].

Then using this form, the earlier equation for the electric (or magnetic) field is greatly simplified. If we call {{mvar|u}} the relative field strength of an elliptical Gaussian beam (with the elliptical axes in the {{mvar|x}} and {{mvar|y}} directions) then it can be separated in {{mvar|x}} and {{mvar|y}} according to:
<math display="block">u(x,y,z) = u_x(x,z)\, u_y(y,z) ,</math>

where 
<math display="block">\begin{align}
u_x(x,z) &= \frac{1}{\sqrt{{q}_x(z)}} \exp\left(-i k \frac{x^2}{2 {q}_x(z)}\right), \\
u_y(y,z) &= \frac{1}{\sqrt{{q}_y(z)}} \exp\left(-i k \frac{y^2}{2 {q}_y(z)}\right),
\end{align}</math>

where {{math|''q''<sub>''x''</sub>(''z'')}} and {{math|''q''<sub>''y''</sub>(''z'')}} are the complex beam parameters in the {{mvar|x}} and {{mvar|y}} directions.

For the common case of a [[circular symmetry|circular beam profile]], {{math|''q''<sub>''x''</sub>(''z'') {{=}} ''q''<sub>''y''</sub>(''z'') {{=}} ''q''(''z'')}} and {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = ''r''<sup>2</sup>}}, which yields<ref>See Siegman (1986) p. 639. Eq. 29</ref>
<math display="block">u(r,z) = \frac{1}{q(z)}\exp\left( -i k\frac{r^2}{2 q(z)}\right) .</math>