Gaussian Beam Propagation Calculator Equation

A Gaussian beam propagation calculator uses the complex beam parameter to predict how a laser beam’s waist and curvature change through free space or an ideal optical system. Enter the wavelength, initial waist radius, and propagation distance to calculate the beam radius, wavefront curvature, and Gouy phase at the destination.

Formula

For a circular, fundamental Gaussian beam traveling through a uniform medium, its radius at distance z from the waist is:

w(z) = w₀√[1 + (z/z_R)²], with z_R = πw₀²/λ.

Here, w(z) is the beam radius at the specified plane, w₀ is the waist radius, z is the axial distance from the waist, λ is the wavelength in the medium, and z_R is the Rayleigh range. All lengths in the equations must use the same unit. The calculator accepts wavelength in nanometers and waist radius in millimeters, then converts them to meters inside its calculations. Its first result converts the calculated radius back to millimeters.

The associated complex beam parameter is q(z) = z + iz_R when the waist is at z = 0. Equivalently, 1/q = 1/R − iλ/(πw²), under a common sign convention for the phase of a forward-traveling wave. R is the wavefront radius of curvature. The calculator reports beam radius and Rayleigh range, but not R; away from the waist, R(z) = z[1 + (z_R/z)²]. The Gouy phase relative to the waist is ψ(z) = arctan(z/z_R), which the calculator reports in radians.

Worked Example

Use the exact default values: wavelength λ = 632.8 nm, waist radius w₀ = 0.5 mm, and distance z = 1 m. Convert to SI units: λ = 632.8 × 10⁻⁹ m and w₀ = 0.5 × 10⁻³ m.

1. Calculate the Rayleigh range: z_R = π(0.5 × 10⁻³)²/(632.8 × 10⁻⁹) ≈ 1.2418 m.
2. Calculate the radius: w(1) = 0.5√[1 + (1/1.2418)²] mm ≈ 0.6404 mm.
3. Calculate the Gouy phase: ψ = arctan(1/1.2418) ≈ 0.6779 rad.

The calculator therefore gives a beam radius of 0.6404 mm at 1 m from the waist, a Rayleigh range of 1.2418 m, and a Gouy phase of 0.6779 rad.

What the Equation Describes

A Gaussian beam is an idealized laser mode whose transverse intensity has a Gaussian profile. The radius w is conventionally the distance from the beam axis where intensity has fallen to 1/e² of its on-axis value. It is not the full beam diameter: the 1/e² diameter is 2w. Because instruments and manufacturers may use other width definitions, check how a measured or specified beam size is defined before using it as w₀.

The waist is the plane where the beam is narrowest and the wavefront is locally flat. As the beam propagates away from that plane, it expands. The Rayleigh range marks the distance at which the radius has grown to √2 times its waist value. At z = z_R, the beam area is twice its waist-plane area, since area scales with radius squared.

The wavelength in the formulas is the wavelength in the propagation medium. In air, the difference from the vacuum wavelength is usually small for many practical estimates, but precision work should account for refractive index: approximately λ = λ₀/n, where λ₀ is vacuum wavelength and n is the medium’s refractive index. The simple calculator uses the entered wavelength directly and assumes free-space propagation.

How to Use the Calculator

• Enter the wavelength in nanometers, not meters or micrometers.
• Enter the waist radius in millimeters, not the diameter. Divide a 1/e² diameter by two first.
• Enter the distance from the waist in meters. The distance is measured along the beam axis, not from an arbitrary lens or source.
• Read the first result as a radius in millimeters. Double it for the 1/e² diameter. The other outputs give the Rayleigh range in meters and Gouy phase in radians.

The distance input is restricted to zero or greater because this version describes propagation forward from the waist. The Gaussian radius equation itself is symmetric about the waist, so a negative z gives the same radius as the corresponding positive distance; the Gouy phase changes sign. Use a signed-distance calculation or the underlying equation directly if you need the upstream phase.

Using q Through Optical Systems

The q parameter is especially useful when a beam passes through lenses or other paraxial optical elements. For a system represented by an ABCD ray-transfer matrix, the input and output parameters are related by q₂ = (Aq₁ + B)/(Cq₁ + D). The matrix elements describe the optical system under the chosen propagation convention. After calculating q₂, its real and imaginary parts can be used to recover the wavefront curvature and beam radius.

This method is more flexible than treating every section as an independent free-space distance, but the ABCD approach assumes paraxial propagation and idealized elements. A lens changes the beam’s curvature and can create a new waist; it does not generally reset the beam to a waist at the lens. Track q through each element and free-space gap to locate the resulting waist correctly.

Assumptions and Common Errors

• Radius versus diameter: Substituting a diameter as w₀ makes the predicted divergence and Rayleigh range wrong. The Rayleigh range depends on w₀ squared.
• Mixed units: The equation requires consistent units. The calculator handles its stated nm, mm, and m inputs internally, but manually using a wavelength in nm with a waist in m without conversion will produce an invalid result.
• Waist location: Distance must be measured from the beam waist, not automatically from the laser output face or a lens.
• Real beams: A real beam may not be a perfect fundamental Gaussian. Beam quality is often represented by M²; for a non-ideal beam, divergence and propagation differ from the ideal M² = 1 model. This calculator does not include M².
• Rounding: Keep full precision in intermediate steps, then round only the displayed values. Small rounding differences can appear if a calculator rounds the Rayleigh range before computing the radius.

The calculation describes an ideal, circular, astigmatism-free beam in a uniform medium. It does not account for clipping, aberrations, thermal effects, scattering, strong focusing beyond paraxial assumptions, or changes in refractive index. For beams with different horizontal and vertical waists, calculate each axis separately; an astigmatic beam can have distinct waist positions and Rayleigh ranges.

Frequently Asked Questions

What is the Gaussian beam propagation equation?
The beam radius at distance z from its waist is w(z) = w₀√[1 + (z/z_R)²], where z_R = πw₀²/λ.

What does the Rayleigh range tell me?
It is the distance from the waist at which the beam radius increases by √2. It indicates the scale over which the beam remains relatively close to its minimum width.

Is beam radius the same as beam diameter?
No. Under the common 1/e² convention, diameter is twice the radius. Make sure the beam-width definition matches before entering a measurement.

Can I use this equation after a lens?
Yes, if you know the beam’s waist and measure propagation from it. To model the lens itself and subsequent propagation, transform the complex q parameter through the optical system.

Does this calculator include M²?
No. It models an ideal Gaussian beam with M² = 1. Real beams with M² greater than 1 generally diverge more than the ideal model predicts; use an M²-aware calculation for those beams.

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in lam | Wavelength | nm | 632.8 | 0 | 1000000000
in w0 | Waist radius | mm | 0.5 | 0 | 1000000
in z | Distance from waist | m | 1 | 0 | 1000000000
out w | Beam radius at distance | mm | w0 * sqrt(1 + (z * lam * 0.000001 / (pi * w0^2))^2) | 4
out zray | Rayleigh range | m | pi * (w0 * 0.001)^2 / (lam * 0.000000001) | 4
out gouy | Gouy phase | rad | atan(z / zray) | 4