Editing Gaussian beam (section)

===Lens equation===
As derived by Saleh and Teich, the relationship between the ingoing and outgoing beams can be found by considering the [[Phase (waves)|phase]] that is added to each point <math>(x,y)</math> of the gaussian beam as it travels through the lens.<ref name="fourier derivation of gaussian lens">{{cite book | title = Fundamentals of Photonics | last1 = Saleh |first1=Bahaa E. A. |last2=Teich |first2=Malvin Carl  | publisher = John Wiley & Sons | location = New York |  year = 1991 | isbn= 0-471-83965-5 }} Chapter 3, "Beam Optics"</ref>  An alternative approach due to Self is to consider the effect of a thin lens on the gaussian beam [[wavefront]]s.<ref name="wavefront derivation of gaussian lens">{{cite journal |last1=Self |first1=Sidney |date= 1 March 1983|title= Focusing of spherical Gaussian beams|url=https://doi.org/10.1364/AO.22.000658 |journal=Applied Optics |volume=22 |issue=5 |pages=658–661 |doi=10.1364/AO.22.000658|pmid=18195851 |bibcode=1983ApOpt..22..658S }}</ref>

The exact solution to the above problem is expressed simply in terms of the magnification <math>M</math>

:<math>
\begin{align}
w_0' &= Mw_0\\[1.2ex]
(z_0'-f) &= M^2(z_0-f).
\end{align}
</math>

The magnification, which depends on <math>w_0</math> and <math>z_0</math>, is given by

:<math>
M = \frac{M_r}{\sqrt{1+r^2}}
</math>

where

:<math>
r = \frac{z_R}{z_0-f}, \quad M_r = \left|\frac{f}{z_0-f}\right|.
</math>

An equivalent expression for the beam position <math>z_0'</math> is

:<math>
\frac{1}{z_0+\frac{z_R^2}{(z_0-f)}}+\frac{1}{z_0'} = \frac{1}{f}.
</math>

This last expression makes clear that the ray optics [[Thin lens|thin lens equation]] is recovered in the limit that <math>\left|\left(\tfrac{z_R}{z_0}\right)\left(\tfrac{z_R}{z_0-f}\right)\right|\ll 1</math>.  It can also be noted that if <math>\left|z_0+\frac{z_R^2}{z_0-f}\right|\gg f</math> then the incoming beam is "well collimated" so that <math>z_0'\approx f</math>.