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The following equations describe the X and Y centroid locations from the collection of data points that
satisfy the above energy clip level criteria.
(
)
x centroid
Xz
z
=
×
∑
∑
(
)
∑
∑
×
=
z
zY
centroidy
Where:
X
=
x
locations of selected pixels.
Y
=
y
locations of selected pixels.
z
= value of selected pixels.
6.11 Beam Widths and Diameters
To some extent, beam width is a term that describes how you have decided to measure the size of your
laser beam. The LBA-PC is designed to give you a set of measurement tools that will allow you to make
this measurement as you see fit. During the past few years there has been some movement toward a
consensus regarding a standard definition of beam width. This definition has grown out of laser beam
propagation theory and is called the Second Moment, or D-4-Sigma beam width. (The D erroneously
stands for Diameter.) Sigma refers to the common notation for standard deviation. Thus an X-axis
beam Width is defined as 4 times the standard deviation of the spatial distribution of the beam’s
intensity profile evaluated in the X transverse direction. Taken in the Y transverse direction will yield
the Y-axis beam Width.
Note: For a TEM
00
(Gaussian) beam, 2-Sigma is the 1/e² radius about the centroid.
The term Diameter implies that the beam is radially symmetric or circular in shape. The term Width
implies that the beam is non-radially symmetric, but is however axially symmetric and characterized by
two principal axes orthogonal to each other. Beams that are asymmetric, distorted, or irregularly
shaped will fail to give significantly meaningful or repeatable beam width results using any of the
standard methods.
6.11.1 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
Operator’s Manual LBA-PC
132