Laser Beam Expander Calculator

Use this laser beam expander calculator to find output beam diameter and divergence, with a clear switch for full-angle vs half-angle divergence. You can also solve backward for the magnifying power or one lens focal length for Keplerian or Galilean expanders.

Solve for

Pick what you want to calculate. The calculator will use the other fields as the known inputs.

Beam expander design type

Keplerian uses two positive focal length lenses. Galilean uses a negative input (image) lens and a positive output (objective) lens. This calculator uses focal length magnitudes when computing magnifying power.

Input beam diameter (mm)

Beam diameter at the expander input. Use the same diameter definition throughout your system (often 1/e^2 for Gaussian beams). Must be greater than 0.

Input divergence angle value

How fast the beam spreads. Enter the numeric value only. Use the controls below to choose units and whether this is full-angle or half-angle.

Input divergence units

Input divergence convention

Common units are mrad or degrees. Small laser divergences are often in mrad.

Full-angle is the total cone angle. Half-angle is from the centerline to one side. The calculator converts to full-angle internally for distance-based beam size math.

Advanced options

Magnification and focal lengths

Beam expander magnifying power (times)

Also called expansion ratio. If you enter this, the calculator can compute output diameter and output divergence. Must be greater than 0.

Objective (output) lens focal length (mm)

Focal length of the output lens (objective). For Galilean, this is positive. For Keplerian, this is positive. Used with the image lens to compute magnifying power.

Image (input) lens focal length magnitude (mm)

Use the magnitude (absolute value). In a Galilean expander the input lens focal length is negative, but magnifying power uses magnitudes. Must be greater than 0.

Targets (for reverse solve)

Target output beam diameter (mm)

Used when solving for magnification or lens focal lengths. Must be greater than 0.

Target output divergence angle value

Target output divergence units

Target output divergence convention

Used when solving for magnification. Value only. Use the units and convention controls below to interpret it.

Units for the target output divergence.

Full-angle vs half-angle for the target output divergence.

Beam size at a distance (simple cone model)

Distance from expander to evaluate beam size (m)

Distance after the expander where you want the beam diameter estimate. Leave blank to skip distance-based calculation.

Calculating…

Magnifying power (times)

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If this is 6x, the output diameter is 6 times larger and the divergence is about 6 times smaller (same angle convention).

Optics magnification (m)

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Some optics texts define magnification as the inverse of magnifying power for a beam expander. Many specs use magnifying power instead.

Output beam diameter at expander exit (mm)

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Simple scaling model for an ideal expander: output diameter equals (magnifying power) times (input diameter).

Output divergence angle value

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Computed as input divergence divided by magnifying power, after converting units and full-angle/half-angle consistently.

Output divergence units

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Use mrad for small angles. 1 rad = 1000 mrad.

Required magnifying power to hit target (times)

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Shown for reverse modes. If very large, you may need different optics or a larger-clear-aperture expander.

Required focal length of the unknown lens (mm)

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Uses magnifying power = f_objective / f_image (magnitudes). For Galilean, the input lens is negative in sign, but this result is the magnitude.

Notes and warnings

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If you see a warning, fix inputs before trusting the numbers.

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How to use our Laser Beam Expander Calculator

Choose Solve for to pick what you want to calculate (for example, output diameter, required magnifying power, or an unknown focal length).
Select the Beam expander design type (Keplerian or Galilean). This does not change the basic ratio math, but it helps you keep the lens signs straight in real life.
Enter Input beam diameter (mm). This must be greater than 0 and should use the same diameter definition you use elsewhere (many lasers use the 1/e^2 diameter).
Enter the Input divergence angle value, then pick the correct units (mrad or degrees) and the correct convention (full-angle or half-angle).
Open Advanced options and enter either Magnifying power (times) or both focal lengths to compute it from lenses.
If you are solving for a required magnifying power, enter a Target output beam diameter (mm) or a Target output divergence (with its units and convention).
If you want a simple distance estimate, enter Distance from expander (m). Leave it blank to skip the distance result.
Click Calculate. If you see an error, change the specific field named in the message and calculate again.

Definitions

Beam expander: An optical setup that makes a laser beam wider so it spreads (diverges) more slowly.

Magnifying power (MP) (times): Expansion ratio. If MP = 6x, the beam diameter becomes 6 times larger and the divergence becomes about 6 times smaller (same angle convention).

Optics magnification (m): A different convention used in some optics texts where m = 1/MP.

Focal length (mm): A lens property that sets how strongly it bends light. Larger focal length usually means weaker bending.

Keplerian expander: Two positive focal length lenses. It forms an internal focus.

Galilean expander: A negative input lens plus a positive output lens. It has no internal focus.

Divergence angle: How fast the beam spreads with distance.

Full-angle vs half-angle divergence: Full-angle is the whole cone angle across the beam. Half-angle is from the centerline to one side. Full-angle = 2 times half-angle.

mrad and degrees: Two angle units. 1 rad = 1000 mrad, and 1 degree = pi/180 rad.

Methodology

Overview

This calculator uses the ideal (lossless) beam expander relationships: beam diameter scales up by the expansion ratio and divergence scales down by the same ratio [1]. It also includes a simple geometric cone model to estimate beam diameter at a distance using the full-angle divergence consistently.

Angle handling (units and convention)

Internally, the calculator converts divergence to full-angle in radians before doing distance math. If you enter half-angle, it is doubled to get full-angle. If you enter degrees, it is converted to radians.

theta_full_rad = (theta_value) * unit_to_rad * convention_to_full

unit_to_rad = 0.001 for mrad, or pi/180 for deg

convention_to_full = 1 for full-angle, or 2 for half-angle

Magnifying power (expansion ratio)

If magnifying power is provided, the calculator uses it. Otherwise, if both focal lengths are provided, it computes magnifying power from their magnitudes (absolute values). This matches common beam expander ratio specs [1][2].

MP = |f_objective| / |f_image|

m = 1 / MP

Output at the expander exit

D_out_mm = MP * D_in_mm

theta_out_full_rad = theta_in_full_rad / MP

The displayed output divergence is converted back to your chosen units (mrad or deg) and shown using the same full-angle or half-angle convention you selected for output.

Beam diameter at a distance (simple cone model)

This estimate uses full-angle divergence, so the radius grows with tan(full_angle/2). Then diameter is twice the radius. This avoids mixing full-angle and half-angle in one formula.

D_at_L_mm = D_out_mm + 2 * (L_m * 1000) * tan(theta_out_full_rad / 2)

Reverse solve modes (required magnifying power)

When you solve backward, the calculator finds a feasible required magnifying power using the target you provide.

MP_required_from_diameter = D_out_target_mm / D_in_mm

MP_required_from_divergence = theta_in_full_rad / theta_out_target_full_rad

Reverse solve for one focal length

If you solve for one focal length, the calculator uses the required (or selected) magnifying power and the known other focal length magnitude.

|f_objective|_required = MP * |f_image|

|f_image|_required = |f_objective| / MP

For a Galilean expander, the input lens focal length is negative in sign in real optics, but this calculator outputs the magnitude you would shop for, as requested [2][3].

Validation and edge cases

Blocked errors: input diameter must be greater than 0; divergence values used for division must be greater than 0; magnifying power and focal length magnitudes must be greater than 0; and the calculator stops if a divide-by-zero would occur. Non-blocking notes: if you enter both magnifying power and focal lengths, magnifying power is used and the focal lengths are treated as informational unless you choose a focal-length solve mode.