1 · Beam geometry
A Gaussian beam is fixed by three numbers at a given wavelength: the waist radius \(w_0\), the axial position of that waist, and the wavelength \(\lambda\). Everything else follows. The waist radius \(w_0\) is the minimum \(1/e^2\) intensity radius; all the relations below are measured from the waist plane (\(z = 0\)).
| Quantity | Symbol | Relation |
|---|---|---|
| Rayleigh range | \(z_R\) | \(z_R = \pi w_0^2 / \lambda\) |
| Divergence (half-angle) | \(\theta\) | \(\theta = \lambda / (\pi w_0)\) |
| Beam radius vs. z | \(w(z)\) | \(w(z) = w_0 \sqrt{1 + (z/z_R)^2}\) |
| Wavefront curvature | \(R(z)\) | \(R(z) = z\left(1 + (z_R/z)^2\right)\) |
| Beam parameter product | \(w_0\,\theta\) | \(w_0\,\theta = \lambda / \pi\) |
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, \(w(z) \approx \theta z\), and the wavefront looks spherical, \(R \approx z\). The reciprocity \(w_0 \theta = \lambda/\pi\) 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
\[ E(r,z) = E_0\,\frac{w_0}{w(z)}\,\exp\!\left(-\frac{r^2}{w(z)^2}\right)\exp\!\left(-i\left[kz + \frac{kr^2}{2R(z)} - \psi(z)\right]\right) \]giving the intensity \(I(r,z) = I_0\left[\dfrac{w_0}{w(z)}\right]^2 \exp\!\left(-\dfrac{2r^2}{w(z)^2}\right)\), with on-axis peak \(I_0 = 2P/(\pi w_0^2)\) for total power P. The last phase term is the Gouy phase,
\[ \psi(z) = \arctan(z / z_R) \]an extra \(\pi\) 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:
\[ \frac{1}{q(z)} = \frac{1}{R(z)} - i\,\frac{\lambda}{\pi w(z)^2} \qquad q(0) = i z_R \]Any paraxial element is a 2×2 ray-transfer (ABCD) matrix, and the beam transforms by the bilinear (Möbius) map
\[ q' = \frac{A q + B}{C q + D} \]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 \(\left(\begin{smallmatrix}1 & d\\ 0 & 1\end{smallmatrix}\right)\); a thin lens of focal length f is \(\left(\begin{smallmatrix}1 & 0\\ -1/f & 1\end{smallmatrix}\right)\). 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 \(w_0\) a distance \(s\) in front of a lens of focal length \(f\), with \(z_R = \pi w_0^2/\lambda\). The output waist \(w_0'\) forms a distance \(s'\) behind the lens:
The waist magnification is \(m = w_0'/w_0 = f / \sqrt{(s-f)^2 + z_R^2}\). As \(z_R \to 0\) both expressions collapse to the geometric results (\(1/s + 1/s' = 1/f\), \(m = f/|s-f|\)) — the Gaussian correction is entirely in the \(z_R\) terms.
Waist at the front focal plane (\(s = f\))
Setting \(s = f\) removes the first term and gives a clean, much-used pair:
\[ w_0' = \frac{\lambda f}{\pi w_0} = f\,\theta \qquad s' = f \]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 \(\theta = \lambda/(\pi w_0)\). 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 \(f\,\theta\)).
Collimated input (\(z_R \gg f\))
For a beam that is essentially collimated at the lens with radius w, the focus lands one focal length away with spot radius
\[ w_0' \approx \frac{\lambda f}{\pi w} \]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 \(\sin\theta \approx \tan\theta \approx \theta\) 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 \(\theta = \lambda/(\pi w_0) \ll 1\).
Where it breaks down, in practical terms:
- Good to ~1–2% for divergence half-angle \(\theta \lesssim 0.3\ \mathrm{rad}\) (~17°), i.e. numerical aperture \(\mathrm{NA} \lesssim 0.3\), equivalently waist \(w_0 \gtrsim \lambda\).
- 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.
6 · Beam quality (M²)
Real beams diverge faster than the ideal by the beam quality factor \(M^2 \ge 1\):
\[ \theta = \frac{M^2 \lambda}{\pi w_0} \qquad w_0\,\theta = M^2 \lambda/\pi \]An ideal TEM₀₀ beam has \(M^2 = 1\). For a beam with \(M^2 > 1\), the propagation relations still hold if you replace \(\lambda\) by \(M^2\lambda\) — 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.