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ISO 11146 Calculation of Laser Beam Center, Diameter, and M²

Project description

Simple and fast calculation of beam sizes from a single monochrome image based on the ISO 11146 method of variances. Some effort has been made to make the algorithm less sensitive to background offset and noise.

Extensive documentation about backgrounds, integrations, and other issues can be found at at <>


Just use pip:

pip install laserbeamsize

Determining the beam size in an image

Finding the center and dimensions of a good beam image:

import imageio
import numpy as np
import matplotlib.pyplot as plt
import laserbeamsize as lbs

beam = imageio.imread("t-hene.pgm")
x, y, dx, dy, phi = lbs.beam_size(beam)

print("The center of the beam ellipse is at (%.0f, %.0f)" % (x,y))
print("The ellipse diameter (closest to horizontal) is %.0f pixels" % dx)
print("The ellipse diameter (closest to   vertical) is %.0f pixels" % dy)
print("The ellipse is rotated %.0f° ccw from horizontal" % (phi*180/3.1416))

to produce:

The center of the beam ellipse is at (651, 491)
The ellipse diameter (closest to horizontal) is 334 pixels
The ellipse diameter (closest to   vertical) is 327 pixels
The ellipse is rotated 29° ccw from the horizontal

A visual report can be done with one function call:


produces something like



lbs.beam_size_plot(beam, r"Original Image $\lambda$=4µm beam", pixel_size = 12, units='µm')

produces something like


Non-gaussian beams work too:

# 12-bit pixel image stored as high-order bits in 16-bit values
tem02 = imageio.imread("TEM02_100mm.pgm") >> 4
lbs.beam_size_plot(tem02, title = r"TEM$_{02}$ at z=100mm", pixel_size=3.75)



Determining M²

Determining M² for a laser beam is also straightforward. Just collect beam diameters from five beam locations within one Rayleigh distance of the focus and from five locations more than two Rayleigh distances:

lambda1=308e-9 # meters
lbs.M2_radius_plot(z1_all, d1_all, lambda1, strict=True)



Here is an analysis of a set of images that were insufficient for ISO 11146:

lambda0 = 632.8e-9 # meters
z10 = np.array([247,251,259,266,281,292])*1e-3 # meters
filenames = ["sb_%.0fmm_10.pgm" % (number*1e3) for number in z10]

# the 12-bit pixel images are stored in high-order bits in 16-bit values
tem10 = [imageio.imread(name)>>4 for name in filenames]

# remove top to eliminate artifact
for i in range(len(z10)):
    tem10[i] = tem10[i][200:,:]

# find beam in all the images and create arrays of beam diameters
options = {'pixel_size': 3.75, 'units': "µm", 'crop': [1400,1400], 'z':z10}
dy, dx= lbs.beam_size_montage(tem10, **options)  # dy and dx in microns



Here is one way to plot the fit using the above diameters:

lbs.M2_diameter_plot(z10, dx*1e-6, lambda0, dy=dy*1e-6)

In the graph on the below right, the dashed line shows the expected divergence of a pure gaussian beam. Since real beams should diverge faster than this (not slower) there is some problem with the measurements (too few!). On the other hand, the M² value the semi-major axis 2.6±0.7 is consistent with the expected value of 3 for the TEM₁₀ mode.



laserbeamsize is licensed under the terms of the MIT license.

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