To get the beam thin over a larger range a long focal length lens may be used. The region close to the focal point of the lens will then be thin as shown in thinregion.
Because as homogeneous thickness as possible is preferable plano-convex lenses
should be used (See planoconvex).
Even though the paraxial region is used, the thickness will be more homogeneous
using plano-convex instead of convex-convex lenses. A limiting factor
for the thickness is diffraction. Improvements can not be made here except for
keeping the focal length as small as possible and the effective diameter of the
lens as large as possible. The length of the thin region, , determines the focal
length. Using the geometry in thinregion the thickness is given by,
d L/2 & = & Df
d & = & D L2 f. To achieve a small thickness a lens with very long focal length can be used. However the limited space available in the lab and divergence effects will cause trouble. Mirrors may be used to increase the distance, but it is preferable to use two lenses instead. We then get more parameters to tune as well as a more adjustable set-up. Still the number of components are only increased by one (the number should always be kept as small as possible because of losses and uncertainties). If the two lenses both are plano-convex the homogeneity is also improved. This is shown in twolenses.
If formula is applied to the second lens of the
geometry in twolenses is found,
1a-f_1 + 1f & = & 1f_2
f & = & f_2(a-f_1)a - f_1 - f_2. Due to diffraction the focal point will have a finite size. If the Gaussian shape of the intensity distribution and the fact that the laser beam diverge already when leaving the laser are neglected an approximation for the spot size is given by equation . This is, & = & 2 1.22 f_eff D \;. Here is the effective focal length of the two lenses for which is substituted. This is not correct, but it will give an approximation of the spot size. & = & 2.44 (a+f) D The thickness of the laser sheet should be as homogeneous as possible. To obtain this we say that the diffraction limitation of the spot size should be equal to the width of the sheet at the edge (, see twolenses). The equation giving the approximate distance to the thin region then is, = d & & D L2f = 2.44 (a+f) D
& & f^2 + a f -D^2 L2.44 \; = \; 0 This equation can easily be fulfilled by choosing appropriate lenses () and adjusting them correctly. The distance to the thin region will be a little longer than this because of the Gaussian shape of the laser's irradiance. For the same reason the thickness of the laser sheet will be smaller than in equation .