Sigma LBA-708, LBA-710, LBA-714PC, LBA-300, LBA-712 Beam Widths and Diameters, 11.1 D4-Sigma Method

The following equations describe the X and Y centroid locations from the collection of data points that satisfy the above energy clip level criteria.

xcentroid = ∑( X ⋅ z)

∑z

ycentroid = ∑(Y ⋅ z)

∑z

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 TEM00 (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.

6.11.1D4-Sigma Method

From laser beam propagation theory, the Second Moment or 4-Sigmabeam 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.

dσx = 4 ⋅σ x

dσy = 4 ⋅σ y

Where:

dσ = The 4-Sigma beam width

σ = The standard deviation of the beam intensity

Operator’s Manual 132 LBA-PC