Melles Griot Optics Guide - Optimum Collimation

Gaussian Beam Propagation

Optimum Collimation

Typically, one has a fixed value for w₀ and uses the expression

to calculate w(z) for an input value of z. However, one can also utilize this equation to see how final beam radius varies with starting beam radius at a fixed distance, z. The figure below shows the Gaussian beam propagation equation plotted as a function of w₀, with the particular values of l = 632.8 nm and z = 100 m. The beam radius at 100 m reaches a minimum value for a starting beam radius of about 4.5 mm. Therefore, if we wanted to achieve the best combination of minimum beam diameter and minimum beam spread (or best collimation) over a distance of 100 m, our optimum starting beam radius would be 4.5 mm. Any other starting value would result in a larger beam at z = 100 m. We can find the general expression for the optimum starting beam radius for a given distance, z. Doing so yields

Using this optimum value of w₀ will provide the best combination of minimum starting beam diameter and minimum beam spread (ratio of w(z) to w₀) over the distance z. For z = 100 m and l = 632.8 nm, w₀ (optimum) = 4.48 mm (see example above). If we put this value for w₀ (optimum) back into the expression for w(z),

Thus, for this example,

By turning this previous equation around, we can define a distance, called the Rayleigh range (z_R), over which the beam radius spreads by a factor of the square root of 2 as

with

If we use beam-expanding optics that allow us to adjust the position of the beam waist, we can actually double the distance over which beam divergence is minimized. By focusing the beam-expanding optics to place the beam waist at the midpoint, we can restrict beam spread to a factor of the square root of 2 over a distance of 2 z_R, as opposed to just z_R

Focusing a beam expander to minimize beam radius and spread over a specified distance
This result can now be used in the problem of finding the starting beam radius that yields the minimum beam diameter and beam spread over 100 m. Using 2( z_R) = 100, or z_R = 50, and l = 632.8 nm, we get a value of w( z_R) = (2l /p)^1/2 = 4.5 mm, and w₀ = 3.2 mm. Thus, the optimum starting beam radius is the same as previously calculated. However, by focusing the expander we achieve a final beam radius that is no larger than our starting beam radius, while still maintaining the square root of 2 factor in overall variation. Alternately, if we started off with a beam radius of 6.3 mm, we could focus the expander to provide a beam waist of w₀ = 4.5 mm at 100 m, and a final beam radius of 6.3 mm at 200 m.

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