Editing Gaussian beam (section)

==Beam optics==
[[File:Gaussian Beam and Lens Diagram.svg|thumb|A diagram of a gaussian beam passing through a lens.]]

When a gaussian beam propagates through a [[thin lens]], the outgoing beam is also a (different) gaussian beam, provided that the beam travels along the cylindrical symmetry axis of the lens, and that the lens is larger than the width of the beam.  The focal length of the lens <math>f</math>, the beam waist radius <math>w_0</math>, and beam waist position <math>z_0</math> of the incoming beam can be used to determine the beam waist radius <math>w_0'</math> and  position <math>z_0'</math> of the outgoing beam.

===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 |url-access=subscription }}</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>.

===Beam focusing===
In some applications it is desirable to use a converging lens to focus a laser beam to a very small spot.  Mathematically, this implies minimization of the magnification <math>M</math>.  If the beam size is constrained by the size of available optics, this is typically best achieved by sending the largest possible collimated beam through a small focal length lens, i.e. by maximizing <math>z_R</math> and minimizing <math>f</math>.  In this situation, it is justifiable to make the approximation <math>z_R^2/(z_0-f)^2\gg 1</math>, implying that <math>M\approx f/z_R</math> and yielding the result <math>w_0'\approx fw_0/z_R</math>.  This result is often presented in the form

:<math>
\begin{align}
2w_0' &\approx \frac{4}{\pi}\lambda F_\# \\[1.2ex]
z_0' &\approx f
\end{align}
</math>

where
:<math>
F_\# = \frac{f}{2w_0},
</math>

which is found after assuming that the medium has index of refraction <math>n\approx 1</math> and substituting <math>z_R=\pi w_0^2/\lambda</math>.  The factors of 2 are introduced because of a common preference to represent beam size by the beam waist diameters <math>2w_0'</math> and <math>2w_0</math>, rather than the waist radii <math>w_0'</math> and <math>w_0</math>.