Laser Beam Expander Calculator — Diameter & Divergence
Enter the input beam diameter, input divergence and magnification M to calculate the expanded beam diameter and reduced divergence after a two-lens beam expander.
mm
mrad
D_out = M × D_in
- 1
Output beam diameter
M × D_in = 5 × 2 = 10A beam expander multiplies the input diameter by the magnification M. - 2
Output divergence
θ_in ÷ M = 1.5 ÷ 5 = 0.3Divergence is divided by M — the beam parameter product is conserved.
How does this calculator work?
A beam expander with magnification M multiplies beam diameter by M and divides divergence by M: D_out = M·D_in and θ_out = θ_in/M. The beam parameter product d·θ is conserved. A 5× expander that receives a 2 mm, 1.5 mrad beam delivers a 10 mm, 0.3 mrad collimated output.
Formula
How this is calculated
A laser beam expander — either Galilean (diverging then converging lens) or Keplerian (two converging lenses) — acts as a reverse telescope. The magnification M equals the ratio of the second lens focal length to the first: M = f₂/f₁. Passing the beam through it multiplies the beam diameter by M and divides the full-angle divergence by M.
The product of beam diameter and divergence (beam parameter product, BPP = d·θ/4) is conserved by an ideal expander — a fundamental consequence of the brightness theorem. Expanding the beam makes it more collimated (lower divergence), which is the main purpose in long-range free-space links, materials processing, and beam delivery for focusing optics where a larger input beam yields a tighter focused spot.
The model assumes a perfectly collimated, ideal Gaussian beam and lossless, aberration-free thin lenses. Real expanders introduce wavefront error, chromatic aberration (refractive designs) and vignetting at large diameters. The design should ensure the output beam does not exceed the clear aperture of any downstream optics.
Frequently asked questions
Divergence angle and beam diameter are coupled through the beam parameter product (BPP). An ideal optical system preserves BPP = d·θ/4 so doubling the beam diameter halves the divergence. Physically, the larger beam presents a bigger wavefront that changes direction more slowly as it diffracts.
A Galilean expander uses a diverging (negative) lens followed by a converging (positive) lens, so the beam never focuses inside — this makes it safe for high-power laser beams. A Keplerian expander uses two converging lenses with a focus between them, which allows a spatial filter (pinhole) at the internal focus to clean up the beam quality, but the focal spot can damage optical coatings or air at high power.
A focusing lens produces a spot of radius w₀ = λ·f/(π·w), where w is the input beam radius at the lens. A larger w from the expander gives a smaller w₀ — so the focused spot is tighter in proportion to the expansion ratio M. This is why beam expanders are routinely placed before objective lenses in laser machining and microscopy.
Also known as
TG we-Calculate Editorial Team. (2026). Laser Beam Expander Calculator — Diameter & Divergence [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/laser-beam-expander-calculator
TG we-Calculate Editorial Team. "Laser Beam Expander Calculator — Diameter & Divergence." TG we-Calculate. 2026. https://we-calculate.com/calculator/laser-beam-expander-calculator.
TG we-Calculate Editorial Team, "Laser Beam Expander Calculator — Diameter & Divergence," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/laser-beam-expander-calculator
@misc{wecalculate_laser_beam_expander_calculator, title = {Laser Beam Expander Calculator — Diameter & Divergence}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/laser-beam-expander-calculator}}, year = {2026}, note = {TG we-Calculate} }
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