Laser beam expander design: Galilean and Keplerian telescopes

A beam expander is a two-lens afocal telescope that magnifies a collimated laser beam — most often to reduce far-field divergence, fill the aperture of downstream optics, or lower the intensity on sensitive components. The design rules are simple; the work is in choosing focal lengths you can actually buy and keeping the beam collimated. This guide covers the formulas, the two layouts, and a worked fiber-laser example you can open directly in the free Gaussian beam calculator.

The two formulas that matter

For lenses with focal lengths f₁ (input side) and f₂ (output side):

• Magnification: M = −f₂/f₁ — the output beam radius is |M| times the input radius, and the divergence drops by the same factor.
• Lens spacing (afocal condition): d = f₁ + f₂ — the focal points coincide, so collimated in means collimated out.

Because divergence scales as θ = λ/(π·w₀), expanding a beam 3× makes it 3× less divergent — the main reason telescopes precede long free-space paths.

Galilean vs Keplerian

The same magnification can be built two ways, and the difference is the internal focus:

• Keplerian — two positive lenses, spacing f₁ + f₂, with a real focus between them.
• Galilean — a negative input lens and a positive output lens, spacing f₂ − |f₁|, with no real intermediate focus (the intermediate image is virtual).

What it means for imaging

As afocal telescopes the two also image differently. The Keplerian forms a real intermediate image at the shared focal plane and delivers an inverted final image; because that image plane is physically accessible, you can drop a field stop, reticle, or spatial-filter pinhole there. That real focus is exactly what makes it the layout for spatial filtering and beam cleanup. The Galilean has only a virtual intermediate image, gives an erect final image, is shorter, but offers no accessible focal plane — you cannot field-stop or spatially filter it, and its usable field of view is smaller (this is the "opera-glass" telescope).

Which is better for aberrations and beam quality

For a laser beam expander, Galilean is usually the better choice. Two reasons dominate:

• No real focus. The Keplerian's internal focus concentrates the full beam power into a tiny spot — at high peak power that ionizes air and can damage coatings. The Galilean avoids it entirely.
• Lower aberration, shorter path. Splitting the bend between a diverging and a converging element tends to reduce net spherical aberration versus two strong positive lenses, and the Galilean's shorter rail leaves less room for pointing drift. Fewer aberrations means the expanded beam stays closer to an ideal Gaussian (M² near 1).

Choose Keplerian when you specifically need that internal focal plane — for a spatial filter (pinhole to clean up the mode) or a field stop. Otherwise, for pure expansion, prefer Galilean. In either case, at demanding conjugates use achromatic doublets rather than singlets to keep spherical and chromatic aberration in check.

Worked example: 852 nm fiber to an expanded collimated beam

A typical cold-atom / spectroscopy layout:

1. Single-mode fiber at 852 nm with 5.3 µm mode field diameter.
2. Collimation lens one focal length from the fiber tip (Δz = f).
3. Two-lens telescope with the spacing tied by the formula d = f₂ + f₃, expanding to the target waist.

In the calculator you fix what you know — the fiber MFD, the wavelength, the final waist you want — and leave the focal lengths free. The solver returns the f values that satisfy the constraints; snap them to the nearest stock lenses and it re-solves the spacings to compensate.

Practical tips

• Keep the beam diameter under ~⅔ of each lens's clear aperture to avoid clipping the Gaussian tails.
• Very short focal lengths (≤ 10–20 mm) are aberration-sensitive — prefer achromats or aspheres there.
• A slightly non-afocal spacing lets you place the output waist at a chosen distance instead of at infinity — fix the waist position in the calculator and let the solver detune the spacing for you.