Laser Beam Expander Calculator
Design a Keplerian or Galilean beam expander telescope: enter two focal lengths (or a target magnification) and get the magnification M = f₂/f₁, the expanded beam diameter, the reduced divergence, and the tube length. Everything runs locally in your browser.
Scope. This tool covers beam expander magnification (telescope optics). For standalone beam divergence math see the Laser Beam Divergence Calculator, and for how a spot grows with distance see the Laser Spot Size Calculator.
1. Choose what to solve for
2. Enter the known values
3. Results
Magnification M
— ×
Output diameter Dout
— mm
Output divergence θout
— mrad
Tube length (approx)
— mm
Beam param. product in
— mm·mrad
Beam param. product out
— mm·mrad
4. Ray diagram (schematic)
Schematic, not to scale. Keplerian: internal focus shown in amber.
Formulas used:
Tube length ≈
M = f₂ / f₁ • Dout = M × Din • θout = θin / MTube length ≈
f₁ + f₂ (Keplerian) or f₂ − f₁ (Galilean). An ideal expander conserves the beam parameter product θ × D.
Frequently asked questions
What magnification does a beam expander with f₁ = 25 mm and f₂ = 100 mm give?
M = f₂ ÷ f₁ = 100 ÷ 25 = 4, so the output beam is 4 times wider than the input beam. Magnification is simply the ratio of the objective focal length to the eyepiece focal length.
A 2 mm beam enters a 4× expander. What is the output diameter?
D_out = M × D_in = 4 × 2 mm = 8 mm. Output diameter scales linearly with magnification, so every 4× expander turns a 2 mm beam into an 8 mm beam.
How does a beam expander change beam divergence?
Divergence is divided by the magnification: θ_out = θ_in ÷ M. A 1.5 mrad beam through a 4× expander leaves at 1.5 ÷ 4 = 0.375 mrad. This divergence reduction is why expanders are used to tighten beams for long-distance pointing and laser ranging.
I want 5× magnification with an eyepiece lens of f₁ = 20 mm. What objective focal length do I need?
f₂ = M × f₁ = 5 × 20 mm = 100 mm. Use the f₂ solver mode above to compute this directly.
What is the difference between Keplerian and Galilean beam expanders?
Both use the same magnification formula (M = f₂/f₁) but differ mechanically. A Keplerian expander uses two positive lenses with a real focus between them, so its tube length is about f₁ + f₂ = 25 + 100 = 125 mm for a 4× pair. A Galilean expander pairs a negative eyepiece with a positive objective and has no internal focus, so its tube is about f₂ − f₁ = 100 − 25 = 75 mm, shorter, and safer with high-power lasers because there is no air-breakdown hotspot at an internal focus.
Can a beam expander increase my laser's power?
No. Output power equals input power minus small lens losses; the beam only gets bigger, so power density drops by M². A 4× expander spreads the same power over 16× the area, which is exactly why expanding the beam is the standard way to raise the damage threshold of downstream optics.
How do I solve for f₁ when I know M and f₂?
f₁ = f₂ ÷ M. For example, to get 3× magnification with a 60 mm objective lens, f₁ = 60 ÷ 3 = 20 mm. Select the f₁ solver mode above and enter M and f₂.
Is the beam parameter product conserved through an expander?
Yes, for an ideal expander θ × D stays constant, which is why divergence drops exactly as fast as diameter grows. Check: 2 mm × 1.5 mrad = 3 mm·mrad in, and 8 mm × 0.375 mrad = 3 mm·mrad out.
What units does this calculator use?
Focal lengths and beam diameters in millimetres (mm), divergence in milliradians (mrad). The formulas are unit-consistent, so any length unit works as long as f₁, f₂ and the diameters all use the same one.
What magnifications are typical for commercial beam expanders?
Common fixed models are 2×, 3×, 5×, 10× and 20×, while variable (zoom) expanders cover roughly 1×–8×. Very high magnification (above ~20×) makes alignment and wavefront error demanding, so most lab work stays in the 2×–10× range.
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laser beam expander calculator, beam expander magnification calculator, keplerian beam expander calculator, galilean beam expander calculator, laser beam diameter expander, telescope beam expander formula M = f2/f1, beam divergence reduction calculator
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