Laser Beam Expander
Calculate laser beam expander magnifying power, output diameter, and output divergence from lens focal lengths and input beam parameters. Free online optics calculator with charts and formula breakdowns for physics, engineering, and laser applications.
About This Calculator
The Laser Beam Expander Calculator helps optics engineers, physics students, and laser professionals design and analyze beam expansion systems. A laser beam expander takes a collimated laser beam with a small diameter and expands it by a factor determined by the ratio of the objective and image lens focal lengths, while simultaneously reducing the beam divergence to maintain beam quality over longer distances.
The calculator uses the standard beam expander formulas: magnifying power MP = fO / |fI|, magnification m = 1 / MP, output beam diameter DO = MP × DI, output divergence θO = θI / MP, and beam diameter at distance L as DL = DO + L × tan(2θO). These apply to both Galilean and Keplerian beam expander designs — the only difference is the sign convention of the image lens focal length.
Two classic optical designs exist for beam expanders. The Galilean design uses a negative (diverging) lens as the image element and a positive objective lens, creating a compact device with no internal focus — ideal for continuous-wave high-power lasers. The Keplerian design uses two positive lenses with a real internal focus, allowing spatial filtering to clean the beam profile, but is better suited for pulsed lasers to avoid thermal effects at the focus.
Regional Notes
Laser beam expander specifications follow international standards regardless of region. Focal lengths are universally specified in millimeters (mm), beam diameters in millimeters (mm), divergence in milliradians (mrad), and distances in meters (m). The underlying physics and formulas are identical worldwide — this calculator serves the global scientific and engineering community without region-specific adjustments.
Frequently Asked Questions
What is a laser beam expander and how does it work?
A laser beam expander is an optical device that increases the diameter of a collimated laser beam while reducing its divergence. It works by passing the beam through two lenses — an image lens (input) and an objective lens (output) — placed at a distance equal to the sum of their focal lengths. The ratio of objective to image focal length gives the magnifying power (MP = fO / |fI|), which determines how much the beam expands and how much divergence decreases.
What are the Galilean and Keplerian beam expander designs?
The Galilean design uses a negative (diverging) image lens and a positive objective lens, creating a compact expander with no internal focus point — ideal for continuous high-power lasers as there is no risk of air breakdown at the focus. The Keplerian design uses two positive lenses with an internal focus point, allowing spatial filtering to clean the beam but risking thermal effects at the focus — better suited for pulsed lasers where the brief pulse duration prevents heating.
What is the difference between magnifying power and magnification in a beam expander?
Magnifying power (MP) is the ratio of the objective lens focal length to the image lens focal length (MP = fO / |fI|), representing how many times the beam diameter increases. Magnification (m) is the reciprocal of magnifying power (m = 1 / MP). For example, a beam expander with fO = 150 mm and fI = 25 mm has MP = 6X and m = 0.167 — the beam diameter grows 6 times while the divergence shrinks to 1/6 of the input value.
How does a beam expander affect laser beam divergence?
A laser beam expander reduces the output beam divergence by the same factor as the magnifying power. The output divergence is calculated as θO = θI / MP, where θI is the input divergence. Since divergence is inversely proportional to beam waist diameter, expanding the beam reduces the angular spread, maintaining beam quality over longer distances — a key advantage in applications like LIDAR, laser communication, and long-range targeting.
How do I calculate the beam diameter at a specific distance after the expander?
Beam diameter at a distance L from the expander is calculated as DL = DO + L × tan(2θO), where DO is the output beam diameter at the objective lens and θO is the output divergence in radians. For typical laser beams with small divergence (milliradians), tan(2θ) ≈ 2θ, so the beam grows approximately linearly with distance. Enter the distance in meters to see the beam spot size at your target.
What are typical specifications for laser beam expanders?
Common laser beam expander specifications include magnifying powers from 2X to 20X, input beam diameters from 1 mm to 10 mm, output diameters up to 100 mm, and wavelength ranges from UV (193 nm) to IR (10.6 µm). Galilean expanders are preferred for high-power CO2 lasers (10.6 µm) due to their air-spaced design without internal focus, while Keplerian expanders with pinhole spatial filters are common in Nd:YAG and excimer laser systems requiring high beam quality.
Can a laser beam expander be used in reverse as a beam reducer?
Yes, a laser beam expander can be used in reverse as a beam reducer (or beam compressor) by sending the beam through the objective lens first. In this configuration, the beam diameter is reduced by the magnifying power, but the divergence increases by the same factor. This is useful when coupling a large-diameter beam into a fiber optic cable or when matching beam sizes to downstream optics. The same formulas apply swapping the input and output parameters.
What factors affect the choice between Galilean and Keplerian beam expanders?
The choice depends on laser power, pulse duration, beam quality requirements, and spatial constraints. Galilean expanders are shorter (ideal for compact setups), have no internal focus (preventing air breakdown with CW lasers), and cause less wavefront distortion. Keplerian expanders allow spatial filtering via a pinhole at the internal focus (improving beam quality), support higher expansion ratios, and enable more precise alignment but are longer and risk thermal lensing at the focus for high average power beams.