Gaussian Beam Waist Evolution Calculator
Compute the beam radius w(z) at any distance from the waist, or invert the formula to find the waist, distance, or wavelength. Includes the Rayleigh range, beam divergence, wavefront curvature, and the classic beam caustic diagram.
| z / zR | z (actual) | w(z) | w/w₀ | Wavefront R(z) | Gouy phase |
|---|
| Laser / source | λ | zR at your w₀ | Divergence θ at your w₀ | Notes |
|---|
| Quantity | Small unit | Mid unit | Large unit |
|---|
The waist is the narrowest point
The beam radius w(z) has its minimum value w₀ at z = 0 — the beam waist. Away from the waist in either direction the radius grows as a hyperbola. Because of the z² term, the beam is symmetric about the waist — the profile at −z looks identical to the profile at +z.
Rayleigh range sets the scale
zR = πw₀²/λ is the distance over which the beam area doubles. Inside ±zR the beam is roughly collimated (w < w₀√2). Outside ±zR it expands linearly with slope θ = w₀/zR = λ/(πw₀). Tight focus (small w₀) → short zR and rapid divergence.
Divergence is fundamental
A beam cannot be both narrow and well-collimated. The product w₀·θ = λ/π is fixed by the wavelength — the Gaussian-beam equivalent of the diffraction limit. Reducing w₀ by 10× increases the divergence angle by exactly 10×.
Two solutions when solving for w₀
Given w, z and λ, two values of w₀ can satisfy the propagation equation — a narrow waist (small zR, fast divergence) and a wide waist (large zR, slow divergence). This calculator returns the smaller (true waist) value; the alternative is listed in the step-by-step solution.
How to Use This Calculator
The Gaussian beam formula relates four quantities — beam radius w(z), waist radius w₀, distance z from the waist, and wavelength λ. Enter any three and the fourth is computed instantly.
1. Choose what to solve for
Solve for w(z) to find how much the beam has expanded. Solve for w₀ to back-calculate a waist from a measured beam size. Solve for z to find where the beam reaches a target size. Solve for λ to identify an unknown source from geometry.
2. Enter the three known values
Type a number, drag the slider, or tap a preset. Lengths (w, w₀) switch between µm / mm / cm / m. Distance z switches between mm / cm / m / km. Wavelength switches between nm / µm / mm / m.
3. Read the result card
The value appears large with unit, alongside the Rayleigh range, waist, and current distance for context. The step-by-step panel shows the full computation with the Rayleigh range, the ratio z/zR, and (when solving for w₀) the alternative mathematical root.
4. Check the charts
The Beam caustic tab draws the hyperbola ±w(z) with the waist, ±zR, and current z marked. The Intensity cross-section tab shows the Gaussian I(r) profile at the waist, at zR, and at your current z. The Waist scaling tab shows w(z) for several w₀ values so you can see how zR shifts.
5. Reference tables
Table 1 gives the beam parameters at each multiple of zR. Table 2 lists common laser wavelengths with their Rayleigh ranges and divergences computed for your current w₀. Table 3 (Advanced) converts every value across units.
6. Advanced mode
See sensitivity to each input, plus companion quantities: Rayleigh range, confocal parameter, divergence half-angle, wavefront curvature, Gouy phase, on-axis intensity ratio, and beam area ratio.
What This Tool Offers
A complete workbench for Gaussian beam propagation from the waist — the standard tool for laser cavity design, focusing calculations, and beam-expander layout.
Four-way solve
Solve for w(z), w₀, z, or λ from the other three. The inverse problem (solving for w₀) is handled analytically with a discriminant check.
Three interactive views
Beam caustic hyperbola with waist and Rayleigh range marked, intensity cross-sections at three key z values, and a waist-scaling parametric plot.
Length unit switching
w and w₀ between µm / mm / cm / m. Distance between mm / cm / m / km. Wavelength between nm / µm / mm / m.
Reference tables
Beam parameters at multiples of zR, common laser wavelengths with their Rayleigh ranges, and a unit conversion table for all your values.
Sensitivity check
See how a +1% change in each input moves the result — useful for understanding measurement and alignment tolerances.
Companion quantities
Rayleigh range, confocal parameter, divergence angle, wavefront curvature R(z), Gouy phase, on-axis intensity ratio, and beam area ratio.
What Is a Gaussian Beam?
A Gaussian beam is the fundamental transverse mode (TEM₀₀) of a laser resonator. Its transverse intensity profile is a circular Gaussian, its phase fronts are approximately spherical near the waist and approximately planar far from it, and its radius evolves along the propagation axis as a hyperbola. Because the paraxial wave equation has a simple closed-form solution for this mode, TEM₀₀ Gaussian beams are the workhorse model of laser optics: they describe everything from the output of a helium–neon laser to the focused spot in a laser microscope. The two parameters that fully determine the propagation are the waist radius w₀ (the narrowest radius, located at z = 0) and the wavelength λ. From these two the Rayleigh range zR = πw₀²/λ follows, and every other quantity — beam radius at any z, divergence angle, wavefront curvature, Gouy phase — is a function of w₀, λ and the propagation distance z.
Formula and Variables
R(z) = z · [1 + (zR/z)²] · θ = λ / (π w₀) · ζ(z) = arctan(z / zR)
| Symbol | Meaning | Notes |
|---|---|---|
| w(z) | Beam radius at distance z from the waist | 1/e² intensity radius, in metres |
| w₀ | Beam waist radius | Minimum radius, at z = 0 |
| z | Distance from the waist | Can be negative (upstream of focus) |
| λ | Vacuum wavelength | In metres; for a medium with index n, replace λ by λ/n |
| zR = πw₀²/λ | Rayleigh range | Distance over which beam area doubles |
| b = 2zR | Confocal parameter | Total length of the "collimated" region |
| θ = λ/(πw₀) | Divergence half-angle | Radians; small-angle approximation w ≈ θz |
| R(z) | Wavefront radius of curvature | ∞ at the waist, minimum 2zR at z = zR |
| ζ(z) = arctan(z/zR) | Gouy phase | Extra phase shift relative to a plane wave |
| I(r, z) = I₀ (w₀/w)² exp(−2r²/w²) | Intensity profile | Circular Gaussian at every z |
Key Beam Parameters at Multiples of zR
| z / zR | w(z) / w₀ | On-axis intensity ratio | Physical meaning |
|---|---|---|---|
| 0 | 1.000 | 1.000 | Beam waist — maximum on-axis intensity |
| 0.5 | 1.118 | 0.800 | Still strongly collimated |
| 1.0 | 1.414 | 0.500 | Beam area has doubled — Rayleigh range reached |
| 2.0 | 2.236 | 0.200 | Expanding rapidly; far-field regime begins |
| 3.0 | 3.162 | 0.100 | Well into far field |
| 5.0 | 5.099 | 0.038 | Essentially a diverging spherical wave |
Frequently Asked Questions
The Gaussian beam is the paraxial solution of the Helmholtz equation. In the near field — within about ±zR — the beam is essentially collimated and the curvature of the hyperbola is small; the radius grows as √(1 + (z/zR)²). In the far field — beyond about 3zR — the 1 inside the square root becomes negligible and the radius grows linearly with z as θ·z. The hyperbola smoothly connects the two regimes, and its asymptotes are the far-field divergence lines.
The Rayleigh range zR = πw₀²/λ is the distance from the waist at which the beam area doubles. Inside ±zR the beam is usefully collimated — the radius is within a factor √2 of its minimum. Rayleigh range is the natural length scale of a Gaussian beam: everything else is measured in units of zR. For a tightly focused beam (small w₀) zR is short and the beam diverges quickly; for a large, well-collimated beam zR is long and the beam stays narrow over metres.
Because the propagation equation w² = w₀² + z²λ²/(π²w₀²) is a quadratic in w₀². Both roots give a valid Gaussian beam that passes through w at distance z — a narrow waist (small w₀, short zR, fast divergence) and a wide waist (large w₀, long zR, slow divergence). The two branches are distinct physical beams; without additional information (another w measurement, or knowledge of the divergence) you cannot distinguish between them. This calculator returns the smaller root — the one corresponding to the true minimum beam diameter — and reports the alternative in the step-by-step solution.
Real beams are not perfect TEM₀₀ Gaussians. They have an M² ≥ 1 quality factor that multiplies the effective wavelength and the divergence: w(z) = w₀√(1 + (M²z/zR)²). A diffraction-limited beam has M² = 1; a typical high-power fibre laser has M² between 1.1 and 2; a multimode laser can be M² > 20. The propagation formula here assumes M² = 1. For non-Gaussian beams, replace λ by M²λ in every formula above and the geometry still holds.
As a Gaussian beam propagates through its focus, it accumulates an extra phase relative to a plane wave. The total shift from z = −∞ to z = +∞ is π radians (180°), and it is distributed as ζ(z) = arctan(z/zR) — half the total on each side of the waist. The Gouy phase is essential for computing resonance frequencies of laser cavities, phase-matching in nonlinear optics (particularly for SHG), and mode-coupling in interferometers. It is a direct consequence of the beam's transverse confinement.
What it means
A Gaussian beam expands as it propagates, following the formula. The Rayleigh range zR = πw₀²/λ is the distance over which the beam area doubles. This is used in laser design, beam focusing, and free‑space optics. Understanding this helps engineers predict beam size and power density at various distances, which is crucial for applications like laser cutting, surgery, and microscopy.
Worked example
Gaussian Beam Waist Evolution – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| w0 (mm) | 1 |
| z (m) | 5 |
| zR (m) | 10 |
| Parameter | Value |
|---|---|
| w0 | 0.5 |
| z | 2 |
| zR | 4 |
Common mistakes
- Gaussian beam waist evolution: w(z) = w₀·√(1 + (z/z_R)²) – where z_R is the Rayleigh range.
- z_R: Rayleigh range = π·w₀² / λ – the distance over which the beam area doubles.
- Units: w₀, w(z), z, z_R all in metres – λ in metres.
- Assumes: A Gaussian beam in a homogeneous medium.
- Beyond z_R: The beam diverges linearly (far‑field approximation).
Applications
Gaussian beam waist evolution, w(z) = w₀√(1 + (z/z_R)²), describes how the beam radius w(z) changes along the propagation axis, with z_R = πw₀²/λ the Rayleigh length. This is fundamental for laser beam propagation. Engineers use it to design optical systems that require precise control of beam size, such as in laser scanning, microscopy, and material processing. By calculating the beam waist at any position, they can ensure proper focusing and collimation. This formula is also used in determining the depth of focus. Understanding Gaussian beam propagation is essential for all laser engineering applications.
- Laser scanning systems (bar code readers, confocal microscopy)
- Design of laser beam delivery systems
- Focusing and collimation of laser beams
- Optical trapping and manipulation
- Education on the properties of laser beams
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