gaussian beam calculator - Aaron Graves, PhDude Replica

Gaussian Beam Parameters

Enter beam and propagation inputs, then click Calculate to get Rayleigh range, divergence, spot size, curvature, and Gouy phase.

Wavelength λ (nm)

Beam waist radius w₀ at focus (µm)

Propagation distance z (mm)

Beam quality factor M² (1 for ideal TEM₀₀)

Optical power P (mW, optional)

Conventions: w is 1/e² intensity radius, θ is half-angle divergence.

What this Gaussian beam calculator does

This tool computes the most common laser beam propagation quantities from a small set of inputs. If you know your wavelength, beam waist, and distance from focus, you can quickly estimate how your beam expands and how it behaves in free space. The calculator also supports non-ideal beams using the M² factor.

Whether you are aligning a lab setup, designing an optical path, or estimating irradiance on a sensor, these equations are the baseline model for real-world Gaussian optics.

Core equations used

1) Rayleigh range

z_R = π w₀² / (M² λ)

The Rayleigh range is the distance from the beam waist to where the beam radius grows by a factor of √2. A longer Rayleigh range means a “tighter” beam that stays narrow over more distance.

2) Beam radius at distance z

w(z) = w₀ √(1 + (z/z_R)²)

This gives the 1/e² intensity radius at any point along propagation. Beam diameter is simply 2w(z).

3) Far-field divergence (half-angle)

θ = M² λ / (π w₀)

This is the asymptotic half-angle divergence in radians. Full-angle divergence is 2θ.

4) Radius of curvature and Gouy phase

R(z) = z[1 + (z_R/z)²], ψ(z) = arctan(z/z_R)

Radius of curvature helps with mode matching and lens propagation work. Gouy phase matters in resonators, interferometry, and precise phase-sensitive optical systems.

Units and conventions

Wavelength: input in nanometers (nm).
Waist radius: input in micrometers (µm), not diameter.
Distance: input in millimeters (mm).
M²: beam quality, where 1 is ideal Gaussian.
Power: optional, used to estimate on-axis peak intensity.

If you have a diameter from your beam profiler, divide by two to get radius before entering w₀.

Practical interpretation

If your beam spreads too quickly

A smaller waist gives tighter focus but increases divergence. If downstream optics need a narrow beam over long distance, you typically increase waist size before free-space propagation (beam expansion).

If irradiance is too low at target

Peak intensity drops as beam area grows. Reducing propagation distance, improving beam quality, or increasing initial waist management can significantly affect delivered intensity.

Using M² correctly

Real laser beams are rarely perfect TEM₀₀. M² captures that imperfection: larger M² means shorter effective Rayleigh range and larger divergence for the same nominal waist.

Example workflow

Measure or obtain wavelength and beam waist near focus.
Enter estimated M² from manufacturer or profiler data.
Set target distance to your sample, detector, or aperture.
Run calculator and check beam radius and peak intensity.
Iterate optical design (lens choice, waist location, beam expansion) as needed.

Limitations

This calculator assumes paraxial Gaussian propagation in a homogeneous medium. It does not include clipping, severe aberrations, nonlinear effects, thermal lensing, turbulence, or multimode structure beyond a single M² approximation.