In order to understand the beam waist and Rayleigh range after the
beam travels through the lens, it is necessary to know the magnification
of the system (α), given by:
2.11
Where w0 is beam waist before the lens and w0’ is the beam waist after
the lens. The thin lens equation for Gaussian beams can then be rewritten
to include the Rayleigh range of the beam after the lens (zR'):
2.12
The above equation will break down if the lens is at the beam waist
(s = 0). The inverse of the squared magnification constant can be used
to relate the beam waist sizes and locations3:
2.13
Focusing a Gaussian Beam to a Spot
In many applications, such as laser materials processing and or surgery,
it is very highly important to focus a laser beam down to the smallest
spot possible to maximize intensity and minimize the heated area. In
cases such as these, the goal is to minimize w0' (Figure 2.6). A modified
version of Equation 2.13 can be used to identify how to minimize the
output beam waist3:
2.14
After multiplying both sides by the denominator on from the left side
of the equation and then multiplying both sides by (w0')2, Equation 2.14
becomes:
Solving for w0' results in:
2.17
2.18
The focused beam waist can be minimized by reducing the focal length
of the lens and |s|-f. The terms next to w0 in Equation 2.17 are defined
as another form of the magnification constant α in order to compare
values of the input beam to the output beam after going through the
lens (Figure 2.7).3
2.21
There are two limiting cases which further simplify the calculations
of the output beam waist size and location: when s is much less than
zR and when s is much greater than zR.3 When the lens is well within
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Figure 2.6: Focusing a laser beam down to the smallest possible size is
crucial for a wide range of applications including this laser cutting setup
s s’
wo wo’
’
a = 2
Figure 2.7: For a magnification of 2, the output beam waist will be twice
the input beam waist and the output divergence will be half of the input
beam divergence
2.15 2.16
2.19 2.20