Beam waist (w_0)

The beam waist is the position along the optical axis where a Gaussian beam reaches its minimum transverse extent. By convention, the waist radius $w_0$ is the $1/e^2$ intensity half-width (equivalently, the $1/e$ field half-width) at this position.

For a Gaussian beam propagating along $z$ with waist at $z = 0$:

$$w(z) \;=\; w_0 \sqrt{1 + \left( \frac{z}{z_R} \right)^2},$$

where $z_R = \pi w_0^2 / \lambda_0$ is the Rayleigh range. The beam radius grows from $w_0$ at the waist, doubles at $z = \sqrt{3} \, z_R$, and increases linearly far from the waist.

Relation to common beam-size conventions.

Measurement	Symbol	Relation to $w_0$
$1/e^2$ intensity radius	$w_0$	reference
$1/e^2$ intensity diameter	$2 w_0$	$2 w_0$
FWHM intensity diameter	$d_{\text{FWHM}}$	$\sqrt{2 \ln 2} \, w_0 \approx 1.18 \, w_0$
FWHM in terms of $w_0$	—	FWHM $\approx 1.18 w_0$
D4σ (ISO 11146)	$4 \sigma$	$2 w_0$ for ideal Gaussian
Knife-edge 10/90 % width	—	$\approx 1.28 \, w_0$

ISO 11146 specifies the D4σ width as the standard beam-size definition for industrial and scientific applications because it generalizes naturally to non-Gaussian beams (it is the second moment of the intensity distribution). For ideal Gaussian beams, D4σ equals $2 w_0$.

Focusing limit. A Gaussian beam of input radius $w_\text{in}$ focused by a lens of focal length $f$ reaches a waist:

$$w_0 \;\approx\; \frac{\lambda_0 \, f}{\pi w_\text{in}} \;=\; \frac{\lambda_0}{\pi \, \text{NA}},$$

where NA $\approx w_\text{in} / f$ is the focused numerical aperture. Achieving a smaller waist requires either a larger input beam or a shorter focal length.

Diffraction limit. For practical optical systems, the minimum achievable waist is set by NA and wavelength:

$$w_{0,\min} \;=\; \frac{\lambda_0}{\pi \, \text{NA}_\text{max}}.$$

For visible-wavelength microscopy with NA $= 1.4$ oil-immersion objective: $w_{0,\min} \approx 120$ nm at $\lambda = 532$ nm. For free-space laser focusing through a typical $f = 25$ mm lens with input $w_\text{in} = 1$ mm: $w_0 \approx 12$ μm at 1550 nm.

For laser diode coupling to fiber: the laser facet beam waist (after the divergent near-field has been intercepted by a coupling lens) must match the fiber mode field diameter for efficient coupling. A lensed fiber typically produces $w_0 \approx 1.5 - 2.5$ μm at $\sim 14$ μm working distance, matching standard inverse-taper edge coupler tip modes.

Beam quality. Real lasers produce beams with waist sizes larger than the diffraction limit; the beam quality factor $M^2$ quantifies the excess. For an $M^2 > 1$ beam:

$$w_0 \cdot \theta \;=\; \frac{M^2 \lambda_0}{\pi},$$

where $\theta$ is the half-angle far-field divergence. Single-mode fiber outputs have $M^2 \approx 1$; multimode fibers and broad-area lasers can have $M^2 > 100$.