Editing Gaussian beam (section)

===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>.