How to Calculate Beam Waist
Before 1960, Gaussian beams were a curiosity, a mathematical solution to the propagation equations with no practical application. Then came the laser. The fundamental operating mode of nearly all lasers is a Gaussian beam, vaulting the mathematics of Gaussian beam propagation from obscurity to importance in a flash. Lasers aren’t perfect, and they don’t produce “perfect” Gaussian beams, but many of them are close enough that the difference is unimportant; so they can be treated as Gaussian. The behavior of Gaussian beams can be predicted once the beam waist is known.
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Instructions
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Direct Beam Waist Measurement
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1
Place neutral density filters in the laser beam.
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2
Insert the beam profiler measurement head in the beam. Remove neutral density filters one by one until the beam profiler registers a signal.
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3
Set the intensity markers on the beam profiler to the 1/e^2 point, where the intensity has fallen to 13.5 percent of its maximum value. Record the distance between the two points. That is the beam diameter.
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4
Move the profiler measurement head along the beam to find the minimum beam diameter. The minimum beam diameter is at the beam waist. Divide this number in half to get the radius, which is the standard measurement of the beam waist.
Indirect Measurement of the Waist
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5
Laser light spreads out faster if it starts out at a small waist. Measure the beam diameter at a distance from the laser. Record it as Diameter 1.
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6
Move a distance, d, farther from the laser and measure the beam diameter again. Record the measurement as Diameter 2.
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7
Calculate the beam divergence angle. For relatively small angles, accurate for almost all laboratory-scale lasers, the angle is given by
divergence = (diameter 2 - diameter 1) / (2 * d).For example, if diameter 1 is 3 mm, and diameter 2 is 4 mm, with the second measurement taken 3 meters farther away from the laser, then the angle is
(4 mm - 3 mm) / (2 * 3000 mm) = .00017 -
8
Calculate the beam waist. The waist is given by wavelength / (pi * divergence).
If the laser measured in the example of Step 3 emits 532 nm light (that is, light with a wavelength of 532 billionths of a meter), then the waist is given by
waist = (532 x 10^-6) / (pi * .00017) = 1 mm.
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Tips & Warnings
One interesting characteristic of Gaussian beams is that they keep the same intensity profile — the same shape — as the go through lenses, travel distances, bounce off mirrors. This means putting the beam through a pair of lenses configured as a beam expander can create another beam waist.
Most beam profilers automatically measure the beam diameter — just select the definition of beam diameter you'd like to work with.
Gaussian beam propagation is a complex subject, so consult the additional resources to extend the analysis to other situations.
Remember that the coherent energy of lasers can easily cause eye damage. Avoid direct or indirect exposure to laser light.
References
Resources
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