User guide

Page 98 Document No: 11294 LBA-USB User Guide
4.11.1 D4-Sigma Method
From laser beam propagation theory, the Second Moment or 4-Sigma beam
width definition is found to be of fundamental significance. It is defined as 4
times the standard deviation of the energy distribution evaluated separately in
the X and Y transverse directions over the beam intensity profile.
d
xx σ
σ
=
⋅
4
d
yy σ
σ
=
⋅
4
Where: d
σ
= The 4-Sigma beam width
σ = The standard deviation of the beam intensity
The standard deviations are derived from the variances of the energy
distributions and are equal to the standard deviations squared. The variances
are:
σ
x
yx
yx
xx Zxy
Zxy
2
2
=
−⋅
∑
∑
∑∑
()(,)
(, )
σ
y
yx
yx
yy Zxy
Zxy
2
2
=
−⋅
∑
∑
∑∑
()(,)
(, )
Where:
centroid theof scoordinate theare y and x
pixel theofintensity the=Z
Only beam propagation factors based on beam widths and divergence angles
derived from the second moments of the energy density distribution function, will
allow one to predict how a beam will propagate. Other definitions of the beam
widths and divergence angles may be used, but they must be shown to be
equivalent to the second moment definitions for computing the correct beam
propagation.
To make an accurate measurement of the beam widths with LBA-USB establish
an aperture for the beam inside a Drawn or Auto Aperture. The aperture must be
approximately 2x the size of the beam. The Auto Aperture feature will
automatically provide such an aperture under most operating conditions. It can
be used in combination with a Drawn Aperture if needed. If the beam size is
already equal to about 1/2 the beam display window, then drawing an aperture
may not be necessary. In such an event, be sure to center the beam in the
window.