Gentec-EO Beamage-M2 - Manual (Page 11)

Beamage-M 2 User Manual Revision 2.0 11
Far from the beam waist, the beam expansion becomes linear and the theoretical divergence half-angle θ
th
(half
of the angle shown in figure 2-1) can be obtained by evaluating the limit of the beam radius’ first derivative as
the position tends towards infinity:
For a laser beam that passes through a focusing lens of focal length f, the theoretical radius of the beam w
fth
at
the focal spot of the lens can be obtained by multiplying the beam divergence half-angle with the focal length f:
As mentioned, all of the equations above describe theoretical ideal Gaussian beams. However, they can describe
the propagation of real laser beams if we slightly modify them using the M
2 factor, which can be mathematically
defined by the following equations:
With the mathematics, it is easy to understand why small M 2 values correspond to low experimental
divergences and small experimental beam waist radiuses.
The experimental beam waist radius w
exp
(z), the experimental half-angle divergence θ
exp
and the experimental
beam radius at the focal spot of the lens w
fexp
are therefore given by the following equations:
We can now easily understand why small M
2 values correspond to low divergence beams with small focus spots.