Gaussian beam optics: formulas and propagation

A reference for the quantities and transformations used to lay out a laser beam path — beam size, divergence, propagation through lenses, and the assumptions behind it all. Every relation here is evaluated for a full multi-element system by the calculator, which solves for whichever parameters you leave free.

Contents

Beam geometry — the summary table
Field, intensity, and the Gouy phase
Complex beam parameter and ABCD propagation
Focusing through a thin lens
The paraxial approximation and its limits
Beam quality (M²)

1 · Beam geometry

A Gaussian beam is fixed by three numbers at a given wavelength: the waist radius w₀, the axial position of that waist, and the wavelength λ. Everything else follows. The waist radius w₀ is the minimum 1/e² intensity radius; all the relations below are measured from the waist plane (z = 0).

Quantity	Symbol	Relation
Rayleigh range	zᵣ	zᵣ = π w₀² / λ
Divergence (half-angle)	θ	θ = λ / (π w₀)
Beam radius vs. z	w(z)	w(z) = w₀ √(1 + (z/zᵣ)²)
Wavefront curvature	R(z)	R(z) = z (1 + (zᵣ/z)²)
Beam parameter product	w₀ θ	w₀ θ = λ / π

Two regimes matter. Within ±zᵣ of the waist (the near field) the beam is approximately collimated and the wavefront is nearly flat. Far beyond it (the far field) the radius grows linearly, w(z) ≈ θ z, and the wavefront looks spherical, R ≈ z. The reciprocity w₀ θ = λ/π is the diffraction limit: a smaller waist always diverges faster.

2 · Field, intensity, and the Gouy phase

With the waist at z = 0, the on-axis-referenced complex field is

E(r,z) = E₀ (w₀/w(z)) · exp(−r²/w(z)²) · exp(−i[kz + kr²/2R(z) − ψ(z)])

giving the intensity I(r,z) = I₀ [w₀/w(z)]² exp(−2r²/w(z)²), with on-axis peak I₀ = 2P/(π w₀²) for total power P. The last phase term is the Gouy phase,

ψ(z) = arctan(z / zᵣ)

an extra π of axial phase accumulated across the focus relative to a plane wave. It shifts the resonant frequencies of cavity modes and matters wherever the relative phase through a focus counts (e.g. nonlinear interactions, phase-sensitive detection).

3 · Complex beam parameter and ABCD propagation

The beam's size and curvature combine into one complex number, the beam parameter q:

1/q(z) = 1/R(z) − i λ/(π w(z)²) q(0) = i zᵣ

Any paraxial element is a 2×2 ray-transfer (ABCD) matrix, and the beam transforms by the bilinear (Möbius) map

q′ = (A q + B) / (C q + D)

Chain the matrices of a system and apply this once to propagate through all of it. The two building blocks: free space of length d is (1, d; 0, 1); a thin lens of focal length f is (1, 0; −1/f, 1). This is exactly what the calculator does internally.

4 · Focusing through a thin lens

A lens does not image a Gaussian waist by the geometric lens equation — the finite Rayleigh range moves the focus. Take an input waist w₀ a distance s in front of a lens of focal length f, with zᵣ = π w₀²/λ. The output waist w₀′ forms a distance s′ behind the lens:

w₀′ = w₀ / √[ (1 − s/f)² + (zᵣ/f)² ] s′ = f + (s − f) f² / [ (s − f)² + zᵣ² ]

The waist magnification is m = w₀′/w₀ = f / √[(s−f)² + zᵣ²]. As zᵣ → 0 both expressions collapse to the geometric results (1/s + 1/s′ = 1/f, m = f/|s−f|) — the Gaussian correction is entirely in the zᵣ terms.

Waist at the front focal plane (s = f)

Setting s = f removes the first term and gives a clean, much-used pair:

w₀′ = λ f / (π w₀) = f θ s′ = f

A waist at the front focal plane images to a waist at the back focal plane, with a size set only by the focal length and the input divergence θ = λ/(πw₀). This is the Fourier relationship between the two focal planes, and the basis of fiber collimators (put the fiber tip at the focal plane and the output waist radius is f θ).

Collimated input (zᵣ » f)

For a beam that is essentially collimated at the lens with radius w, the focus lands one focal length away with spot radius

w₀′ ≈ λ f / (π w)

the familiar diffraction-limited focused-spot formula.

Open a worked fiber-to-telescope example →

5 · The paraxial approximation and its limits

Everything above is a paraxial result: it assumes small angles, so that sinθ ≈ tanθ ≈ θ and the field is a slowly varying envelope on the carrier wave. The Gaussian beam is the leading-order mode of the paraxial wave equation, so the model is self-consistent only while the divergence stays small — formally θ = λ/(π w₀) « 1.

Where it breaks down, in practical terms:

Good to ~1–2% for divergence half-angle θ ≲ 0.3 rad (~17°), i.e. numerical aperture NA ≲ 0.3, equivalently waist w₀ ≳ λ.
Degrades as the waist shrinks toward the wavelength (NA → 0.5 and beyond): the real focal spot is larger than the paraxial prediction, and a longitudinal (on-axis) field component appears, so the beam is no longer purely transverse.
Fails for tight focusing (high-NA objectives, optical tweezers, NA ≳ 0.6): polarization and vector-diffraction effects dominate. Use a vector diffraction model (Richards–Wolf) there, not Gaussian beam formulas.

Rule of thumb: if the beam's f-number is ≳ 2 (NA ≲ 0.25), the paraxial Gaussian results are accurate for design. Below that, treat them as a first-order estimate.

6 · Beam quality (M²)

Real beams diverge faster than the ideal by the beam quality factor M² ≥ 1:

θ = M² λ / (π w₀) w₀ θ = M² λ/π

An ideal TEM₀₀ beam has M² = 1. For a beam with M² > 1, the propagation relations still hold if you replace λ by M²λ — the beam propagates like an ideal one of that longer effective wavelength, so it focuses to a larger spot and has a shorter Rayleigh range.