Figure 3: Sample contour of measurement at 14cm at Using MATLAB, we wrote a program to 0.5*max level. The crossing lines connect the max batch process the image files using the following and min x and y coordinates. algorithm: Compile data into array Calculate maximum intensity value of each image Scale the maximum values to account for varying levels of neutral density filters Reduce the maximum by a coefficient as described above Find the x-y coordinates of a contour of that at the reduced maximum level Find the distance between the minimum and maximum x- and y-coordinates of the contour and average them as a measurement Figure 4: Sample contour of measurement at 14cm at of the spot diameter *max level. The crossing lines connect the max and Compute the derivative by min x and y coordinates. Compute the second derivative by Conclusions Results Figure 1: First derivative data averaged over all Based on our data, there is an readily apparent diameters. Note the near-zero value at 14cm. minimum size of the beam located around the 14 cm mark. The existence of this minimum is corroborated Figure 2: Second derivative data averaged over all by both the numerically calculated first and second diameters. Note the absolute minimum value at derivatives of the beam size over numerous 14cm. definitions of beam diameter and a two dimensional averaged diameter measurement. Since our As is evident from these two figures, our measurements were only taken at integer distances measurement of the beam at 14cm from the aperture from the laser’s aperture, there is a high likelihood exhibits precisely the two criteria we predicted the that the exact location of the waist differs from our beam waist would exhibit. We know the valley on the measurement, though it differs by no more than 1cm Figure 2 is indeed a universal extreme as this in either direction and is most likely within 0.5 cm of particular point was exhibited by graphs over all the 14cm mark. beam diameter definitions, while any other peaks were unique to one graph or another. Similarly for References Figure 1, graphs of all definitions show the value at 14 cm to be zero, despite some having zero-valued CVI Melles Griot Gaussian Beam derivatives at other locations. Propagation Newport, Gaussian Beam Optics Tutorial Klein, Deb. Learning How Laser Beams Propagate.
Using MATLAB, we wrote a program to batch process the image files using the following algorithm: Compile data into array Calculate maximum intensity value of each image Scale the maximum values to account for varying levels of neutral density filters Reduce the maximum by a coefficient as described above Find the x-y coordinates of a contour of that at the reduced maximum level Find the distance between the minimum and maximum x- and y-coordinates of the contour and average them as a measurement of the spot diameter Compute the derivative by Compute the second derivative by Results Figure 1: First derivative data averaged over all diameters. Note the near-zero value at 14cm. Figure 2: Second derivative data averaged over all diameters. Note the absolute minimum value at 14cm. As is evident from these two figures, our measurement of the beam at 14cm from the aperture exhibits precisely the two criteria we predicted the beam waist would exhibit. We know the valley on the Figure 2 is indeed a universal extreme as this particular point was exhibited by graphs over all beam diameter definitions, while any other peaks were unique to one graph or another. Similarly for Figure 1, graphs of all definitions show the value at 14 cm to be zero, despite some having zero-valued derivatives at other locations. Figure 3: Sample contour of measurement at 14cm at 0.5*max level. The crossing lines connect the max and min x and y coordinates. Figure 4: Sample contour of measurement at 14cm at *max level. The crossing lines connect the max and min x and y coordinates. Conclusions Based on our data, there is an readily apparent minimum size of the beam located around the 14 cm mark. The existence of this minimum is corroborated by both the numerically calculated first and second derivatives of the beam size over numerous definitions of beam diameter and a two dimensional averaged diameter measurement. Since our measurements were only taken at integer distances from the laser’s aperture, there is a high likelihood that the exact location of the waist differs from our measurement, though it differs by no more than 1cm in either direction and is most likely within 0.5 cm of the 14cm mark. References CVI Melles Griot Gaussian Beam Propagation Newport, Gaussian Beam Optics Tutorial Klein, Deb. Learning How Laser Beams Propagate.
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