Can expanding a laser beam effectively create a smaller beam size?

Since a beam expander converts an input beam with a smaller waist and larger divergence into an output beam with a larger waist and smaller divergence, the output beam can have a smaller diameter far from the beam expander when compared with the input beam. The beam diameter does not remain constant with distance due to the effects of diffraction. As the beam divergence is higher for beams with smaller waists, using a beam expander to increase the waist diameter is a technique for reducing the rate at which the beam diameter increases as the beam propagates away from its waist. Beam expanders are often used to reduce beam divergence and ensure the beam diameter does not exceed a specified limit at distances far from the output beam waist.

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Figure 1: The input beam has a smaller waist diameter than the output beam. However, the diameter of the smaller input beam changes at a substantially higher rate than the larger output beam. Within the limited range shown above, the diameter of the input beam exceeds that of the output beam.

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Figure 2: A beam has a nearly constant divergence angle far from the beam waist. As described in the text, this angle (θ ) can be estimated using two measurements (D₂ and D₁) of beam diameter, taken far from the waist and separated by a known distance (Δz ). 

Beam Expanders (and Reducers)
Beam expanders (and beam reducers) accept a collimated beam with one waist diameter and provide a collimated beam with a different waist diameter. The expansion ratio, or magnification (m ),

relates the diameters of the beam waists before (2W_o ) and after (2W'_o) the beam expander (Figure 1). The device is a beam expander and provides a larger output beam waist when m > 1. When the output beam waist is smaller, the device is a beam reducer and m < 1.

Measuring the beam cross sections at the input and output beam waist locations can provide estimates of the waist diameters. The manufacturer often specifies the output beam waist location, but if it is not known, a measurement of the beam diameter close to the output of the beam expander can be used to approximate the waist diameter.

Beam Divergence
A laser beam's diameter is always smallest at its waist. Away from the waist, the beam diameter increases due to the effects of diffraction, and this rate of increase is the beam's angular divergence. Far from the beam waist, the divergence angle is approximately constant (Figure 2). An estimate of this divergence angle (θ ) in radians,

can be calculated from the wavelength () and beam waist diameter, or from two measurements of the beam diameter (D₁ and D₂ ) separated by a known distance (Δz ). The two beam diameter measurements should be made far from the beam waist, well outside of the Rayleigh range.

The difference in the angular divergence of the beams input to and output from a beam expander or reducer can be estimated using the beam expansion ratio (m ). While a beam expander changes the waist diameter by a factor of m, the divergence angle far from the beam waist is scaled by a factor equal to 1/m,

in which θ_in  and θ_out are the divergence angles of the input and output beams, respectively, far from their beam waists.

In the case of a beam expander (m > 1), the output beam waist diameter is larger (W_o' > mW_o ) and the output divergence angle is smaller (θ_out < θ_in / m ) than the input beam. Due to the output beam's reduced divergence angle, beam expanders are often used to enable free-space beams to propagate over longer distances without exceeding the maximum beam diameter specified by the application.

A beam reducer (m < 1) provides an output beam with a smaller waist diameter (W_o' < mW_o ) and a larger divergence angle (θ_out > θ_in / m ) than the input beam. As a result, the output beam diameter will be smaller close to the output of the beam reducer, but as the light propagates away, the beam size will increase at a faster rate than the input beam. When using a beam reducer, it is critical to check the output beam diameter along the optical axis to ensure the beam does not exceed the application's maximum acceptable beam size.

Avoid Beam Clipping
The diameter of the output beam close to the beam expander is often a good approximation of the output beam waist diameter. Farther away, the beam diameter may have increased enough to cause beam clipping or other unintended effects. Calculating estimates of the beam diameter (D ),

at critical distances (z ) from the output beam waist can be useful during system design.