Science calculator

Gaussian beam waist.

For a fundamental (TEM₀₀) Gaussian beam, enter the waist radius, wavelength and axial distance to get the beam radius at that point, the Rayleigh range, the wavefront's radius of curvature and the far-field divergence. The model is stated in full below — an instrument that says what it assumes.

Waist radius w₀ (1/e² intensity)

Wavelength λ

Distance from waist z

Beam radius w(z)

—

at the chosen distance

Show w(z) in Display precision

Rayleigh range z_R: —
Wavefront curvature R(z): —
Divergence half-angle θ: —

w (z) = w_{0} \sqrt{1 + {(\frac{z}{z_{R}})}^{2}}

z_{R} = \frac{π w_{0}^{2}}{λ}

R (z) = z [1 + {(\frac{z_{R}}{z})}^{2}]

θ = \frac{λ}{π w_{0}}

w₀ waist radius (1/e² intensity) · λ wavelength · z axial distance from the waist
z_R Rayleigh range · R(z) wavefront radius of curvature · θ far-field divergence half-angle

Assumptions

Fundamental Gaussian mode, TEM₀₀ (M² = 1). A real beam diverges M² times faster — use the beam-quality page for that.
Paraxial: the divergence half-angle is small, so θ ≈ λ/(πw₀).
A single homogeneous medium. Use the wavelength in that medium (λ = λ₀/n) if the beam is not in vacuum or air.
All radii are the 1/e² intensity radius; z is measured from the waist, where the wavefront is flat (R → ∞).

How it works

A Gaussian beam is narrowest at its waist and spreads as it propagates. The Rayleigh range z_R = πw₀²/λ is the distance over which the radius grows by √2; twice it is the usual depth of focus. Tighter waists reach the far field sooner and diverge faster.

Close to the waist the wavefront is nearly flat, so R(z) is very large; it passes through a minimum near z = z_R and then approaches R ≈ z far away, where the beam looks like it came from a point at the waist and opens into a cone of half-angle θ.

Results use up to four significant figures by default — enough for design and a quick bench check without claiming precision the inputs don't have. Trailing zeroes are not added to fill the chosen limit. The arithmetic runs at full double precision; only the display is rounded, and very large or small values use a proper ×10ⁿ.

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