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The Transforming Lens 5.4.1
In the figure below, assume the input beam shown to the left of the focusing optic is a fundamental mode
Gaussian beam. The beam is shown propagating to the right. The beam’s wavefront radius of curvature is
becoming shorter as the beam propagates towards the waist until it reaches Z
R
, the Rayleigh length. After the
Rayleigh length, the radius of curvature begins to increase until at the waist location, Z
0
, where the radius of
curvature is infinite. As the beam passes through the waist the radius of curvature shortens until it reaches Z
R
on the opposite side of the waist. Past Z
R
the radius of curvature once again begins to increase towards
infinity.
The changing radius of curvature of a propagating laser results in some non-intuitive behavior when applied to
a measurement device that transforms with a focusing lens. The position of the focusing lens, relative to the
input beam’s waist, can alter the location of the waist formed after the lens, sometimes dramatically.
ISO 11146 instructs that at least 11 samples are required to make an accurate M
2
measurement, with a
minimum of 5 taken within ±Z
R
from the waist and 5 taken beyond ±2Z
R
from the waist.
The figures below show how the focused waist location moves relative to the input beam waist location for a
range of input beam waist widths. Preferable placement of the input beam’s waist is where Z
01
is at least equal
to the focal length of the lens. Observe that for small beams in this region, there is a drastic change in Z
02
with
small changes in Z
01
. In these scenarios, Z
01
is better situated farther from the lens.
Under most circumstances Z
01
should avoid being less than the focal length of the lens.
Beams with a waist width ≥2mm the placement will almost always yield a focus near the lens focal length,
thus placement is not critical.