Laser Beam Expander Calculator - Magnification, diameter, divergence

A laser beam expander calculator turns two focal lengths, an input beam diameter, and an input divergence into the magnifying power, output diameter, output divergence, and downstream spot size of any Galilean or Keplerian beam expander. It applies MP = fO / fI, D_O = MP * D_I, theta_O = theta_I / MP, and D_L = D_O + L * tan(2 * theta_O) so an optical engineer or photonics student can size a collimator, predict the far-field footprint, or compare designs.

• • Sizing a beam expander: Pick two off-the-shelf lenses, enter their focal lengths, and read off the magnifying power, output diameter, and output divergence.
• • Predicting beam diameter at a target: Enter the distance from the output lens to the workpiece and the page returns the downstream beam diameter.
• • Choosing between designs: Compare the same focal-length ratio with and without a negative image lens.
• • Photonics homework: Plug in textbook values to get MP, D_O, theta_O, and D_L for the write-up.

A laser beam expander is a telescope run backwards: light enters through the image lens, exits through the objective lens, and the diameter is scaled by the focal-length ratio while the divergence is scaled by the reciprocal. The two designs differ in how that ratio is achieved, but the formulas do not change.

The input divergence that this expander shrinks by the factor MP is the same theta_I used by the laser beam divergence calculator, so the two pages read naturally as a pair during laser setup.

The calculator reads the two focal lengths and the input beam parameters, computes the magnifying power as fO / fI, then uses that MP to scale the input diameter and the input divergence. With the output divergence in hand it evaluates D_L = D_O + L * tan(2 * theta_O) to return the downstream spot size.

• fI: Absolute focal length of the image lens in millimetres. Enter 25 for a Galilean expander whose input lens has fI = -25 mm.
• fO: Focal length of the objective lens in millimetres.
• DI: Input beam diameter at the image lens in millimetres.
• thetaI: Full-angle divergence of the incoming beam in milliradians.
• L: Distance from the output lens to the target plane in metres.
• MP: Magnifying power; dimensionless ratio fO / fI.
• D_O: Output beam diameter at the objective lens in millimetres.
• theta_O: Output full-angle divergence in milliradians.
• D_L: Beam diameter at distance L from the expander in millimetres.

The factor that does the most work in all four formulas is the magnifying power MP: the same number that multiplies the input diameter divides the input divergence, so a 6X Galilean expander both enlarges the beam sixfold and tightens the divergence to one sixth of its original value.

According to Omni Calculator Laser Beam Expander, the magnifying power of a laser beam expander is MP = fO divided by fI, the output beam diameter is D_O = MP * D_I, the output divergence is theta_O = theta_I divided by MP, and the diameter at a distance L from the expander is D_L = D_O + L * tan(2 * theta_O).

According to Wikipedia Beam expander, a laser beam expander is an optical system that increases the diameter of a collimated beam, with the Galilean design using one converging and one diverging lens and the Keplerian design using two converging lenses, both spaced by the sum of their focal lengths.

Once the focal lengths fI and fO are chosen, the lensmakers equation calculator lets you back-derive the required lens curvatures and refractive index for the same MP at a fixed lens thickness.

Four ideas explain why a beam expander scales the diameter up and the divergence down by the same factor, and why the two telescope designs give the same arithmetic.

Keeping the four concepts separate prevents the most common reporting mistake: quoting MP in place of m, or assuming the two designs give different diameter scaling.

The output divergence theta_O and the new beam diameter together fix the Rayleigh criterion for downstream imaging, so the angular resolution calculator is the natural companion when the expander is used in front of a focusing lens.

Six short steps take you from a chosen pair of lenses to the output beam diameter, output divergence, and downstream spot size.

1. 1 Pick the two lenses: For a Galilean design the image lens is diverging (fI < 0); for a Keplerian design both focal lengths are positive.
2. 2 Measure the input beam: Record the beam diameter at the image lens in millimetres and the full-angle input divergence in milliradians.
3. 3 Enter the focal lengths: Type |fI| and fO in the first row of the form.
4. 4 Enter DI and thetaI: Type the input diameter and divergence in the second row.
5. 5 Set the distance L: Type the distance in metres from the output lens to the target plane.
6. 6 Read the panel: Primary outputs are MP, output diameter in mm, output divergence in mrad, and diameter at the chosen distance.

With fI = -25 mm, fO = 150 mm, DI = 2 mm, thetaI = 0.4 mrad, and L = 5 m the result panel returns MP = 6X, m = 0.1667, D_O = 12 mm, theta_O = 0.0667 mrad, and D_L = 12.67 mm, matching the Omni Calculator worked example.

When the expanded beam has to be refocused onto a target instead of sent collimated downrange, the thin lens equation calculator takes the objective focal length and object distance and returns the image distance where the spot reaches its smallest size.

Five practical benefits make this laser beam expander calculator a quick check during beam expander selection and photonics homework.

• • Both designs, one form: The MP formula is the same for Galilean and Keplerian expanders.
• • Output in mm and mrad: Hardware sizes and divergence budgets are both returned at once.
• • Downstream spot size: Entering L returns the beam diameter at the target using D_L = D_O + L * tan(2 * theta_O).
• • Compact input rows: Focal lengths sit side by side, beam parameters sit side by side.
• • Reverse-mode support: The same formulas can be reused to find one lens focal length given a desired MP.

Use the page during lens selection to compare the output divergence of two candidate expander pairs and confirm the measured bench divergence matches the predicted theta_O.

Five factors set the output diameter, output divergence, and downstream spot size, plus two limitations to keep in mind whenever the result is interpreted.

• • The formula assumes the input beam is already collimated at the image lens. A diverging or converging input beam shifts the effective waist inside the expander in ways this calculator does not predict.
• • The D_L formula treats the beam as a perfect cone. Real Gaussian or top-hat profiles have slightly different growth rates, so the result is a first approximation, not a replacement for a wave-optics simulation.

These factors are independent. When the measured output divergence on the bench lands well above the predicted value, the gap is usually dominated by lens aberrations or misalignment.

According to RP Photonics Beam Expanders, the magnification of a laser beam expander equals the ratio of output beam diameter to input beam diameter, and the output divergence is reduced by the same factor because the etendue of the beam is preserved.

After the expander the new output diameter D_O sets the clear aperture of every downstream optic, so the aperture area calculator converts that diameter into the light-collecting area used in power and irradiance budgets.