Gaussian beam

A Gaussian beam has a transverse intensity profile:

$$I(r, z) \;=\; I_0 \left( \frac{w_0}{w(z)} \right)^2 \exp\!\left( -\frac{2 r^2}{w(z)^2} \right),$$

where $w_0$ is the beam waist radius (at $1/e^2$ intensity) at the focal point, $w(z)$ is the beam radius at distance $z$ from the waist, and $r$ is the transverse distance from the propagation axis.

Key parameters:

Beam waist radius $w_0$ — minimum radius at $1/e^2$ intensity at the focal plane.

Rayleigh range — distance over which the beam stays approximately collimated:

$$z_R \;=\; \frac{\pi w_0^2}{\lambda}.$$

Beam radius vs propagation distance:

$$w(z) \;=\; w_0 \sqrt{1 + (z/z_R)^2}.$$

At $z = z_R$, the radius is $\sqrt{2} \, w_0$; the cross-sectional area has doubled.

Far-field divergence half-angle ($z \gg z_R$):

$$\theta \;=\; \frac{\lambda}{\pi w_0}.$$

Small waists produce strong divergence; large waists stay collimated longer.

Typical Gaussian beam dimensions:

Source	Wavelength	$w_0$	$z_R$	$\theta$
HeNe laser, free-space	633 nm	0.4 mm	0.79 m	0.5 mrad
Single-mode fiber output (SMF-28)	1550 nm	5.2 μm	55 μm	95 mrad (5.4°)
Focused beam from 0.1 NA lens	1550 nm	4.9 μm	49 μm	100 mrad
Focused beam from 0.5 NA microscope objective	633 nm	0.4 μm	0.8 μm	0.5 rad

The $\pi w_0 \theta$ product equals $\lambda$ for an ideal Gaussian beam. Real beams produce a larger product by the beam quality factor $M^2$:

$$M^2 \;=\; \frac{\pi w_0 \theta}{\lambda}.$$

$M^2 = 1$ for an ideal Gaussian; $M^2 > 1$ for any real beam.

In the paraxial approximation, the complex beam parameter $q(z) = z + i z_R$ governs propagation through optical systems via ABCD matrix methods: $q_2 = (Aq_1 + B)/(Cq_1 + D)$. This is the standard formalism for ray tracing Gaussian beams through lenses, mirrors, and free-space sections.