Gaussian Beam

A Gaussian beam is a beam of electromagnetic radiation whose transverse electric field and intensity profiles are described by Gaussian functions. Laser beams are well-approximated as Gaussian beams.

Transverse Profile

The electric field amplitude varies as:

$$ E(r) = E_0 \exp\left(-\frac{r^2}{w^2}\right) $$

where $w$ is the beam radius (where intensity falls to $1/e^2$ of the peak).

The intensity profile is:

$$ I(r) = I_0 \exp\left(-\frac{2r^2}{w^2}\right) $$

Propagation

As a Gaussian beam propagates along $z$, the beam radius evolves as:

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

where:

• $w_0$ = beam waist (minimum radius)
• $z_R = \frac{\pi w_0^2}{\lambda}$ = Rayleigh range

Key Parameters

Parameter	Definition
Beam waist $w_0$	Minimum beam radius
Rayleigh range $z_R$	Distance where $w = \sqrt{2} w_0$
Divergence angle $\theta$	$\theta = \lambda / (\pi w_0)$
Confocal parameter $b$	$b = 2 z_R$

Self-Fourier Property

The Gaussian function is an eigenfunction of the Fourier transform:

$$ \mathcal{F}\{e^{-\pi x^2}\} = e^{-\pi f^2} $$

This explains why Gaussian beams maintain their Gaussian profile as they propagate and diffract — a unique property among beam shapes.

Beam Quality Factor (M²)

Real laser beams deviate from ideal Gaussian behavior. The M² factor quantifies this:

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

For an ideal Gaussian beam, $M^2 = 1$.

Applications

• Laser systems
• Optical communications
• Laser machining and surgery
• Optical trapping

See Also

• Fourier transform
• Diffraction