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Beam Expander Magnification and Divergence Calculator

A beam expander enlarges a collimated beam and reduces its divergence by the same ratio. This calculator shows what magnification does to beam size at your working distance, and why bigger is not always better.

Your numbers

Input beam diameter

Input full-angle divergence

mrad

Input lens focal length

Output lens focal length

100 mm

Working distance

Your results

Magnification

Beam diameter at working distance

20.5

mm, geometric estimate.

Output beam diameter

mm.

Output divergence

0.25

mrad, full angle. Divergence falls as magnification rises.

Keplerian length (f1 + f2)

125

mm between lenses; a Galilean design is f2 - f1 long.

Without expander

mm at the same distance.

Geometric model with divergence above the diffraction limit. Real minimum divergence is bounded by M squared and wavelength.

Save results:

Beam diameter at working distance (mm) vs output focal length (mm)

Sweeps Output lens focal length from 5 to 250, holding your other inputs fixed.

Keep optical assembly records together

Optional expert review of how to track optical subassemblies and alignment data in your ERP.

Optional. The tool and every export work without this. No spam.

How to use the calculator

Enter the input beam diameter and full-angle divergence, the two focal lengths and the distance at which you need the beam. Magnification is the ratio of the output to input focal lengths. The output beam is larger by that ratio and diverges less by the same ratio. The chart sweeps the output focal length so you can find the magnification that gives the smallest beam at your distance. A Keplerian design has a real internal focus, so avoid it with high-power pulsed beams.

Use a Galilean design for high peak power to avoid an internal focus in air.
Make sure downstream optics have clear aperture above 1.5 times the output diameter.
Adjustable-focus expanders let you tune divergence compensation in the field.

The formulas

Magnification M equals f2 divided by f1. The output diameter is M times the input diameter, and the output full-angle divergence is the input divergence divided by M, from conservation of the beam parameter product. The geometric beam diameter at distance L is D plus theta times L. Minimizing M times Din plus theta L divided by M gives an optimum magnification equal to the square root of theta L over Din, which is 5 for the defaults here.

Worked example

A 2 mm beam with 1 mrad divergence grows to 52 mm after 50 m without an expander. A 4x expander gives an 8 mm beam and 0.25 mrad, which is about 20.5 mm at 50 m. At 5x the beam is 10 mm at the expander and about 20 mm at 50 m, the minimum. Going to 10x makes the beam larger at distance again, since the starting diameter is bigger and the gain in divergence no longer pays for it.

Common mistakes

Ignoring the diffraction limit leads to promising divergence below what the wavelength and beam size allow. For a Gaussian beam the minimum full-angle divergence is about 4 M2 lambda divided by pi D. Another mistake is forgetting that the input beam must be collimated and well centered, since lens decentration and aberration degrade the expanded beam. Track expander serial numbers and alignment checks in your maintenance system for lines that use them continuously.

Frequently Asked Questions

Why does expansion reduce divergence?

The product of beam diameter and divergence is conserved through ideal optics. Making the beam M times wider therefore makes the divergence M times smaller, which means it grows more slowly with distance.

Keplerian or Galilean?

Keplerian uses two positive lenses and has an internal focus, which allows spatial filtering but can ionize air with pulsed lasers. Galilean uses a negative input lens, has no internal focus and is shorter, so it suits high-power beams.

What is the diffraction limit here?

A Gaussian beam with M squared and wavelength lambda cannot have full-angle divergence below 4 M2 lambda / (pi D). If the calculated output divergence is below that, the geometric model is not valid and the diffraction value applies.

Can I use a zoom expander?

Yes. Zoom expanders vary the magnification by moving an element, which is equivalent to changing f2 over f1. Enter the effective ratio for each setting and compare the beam size at your working distance.

Netray can help link optical assembly records, alignment checks and spares to your ERP so recurring beam delivery issues are easy to trace.

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