# Gaussian beam

In optics, a Gaussian beam, named in honor of Carl Friedrich Gauss (rhymes with house), is a beam of light whose electric field intensity distribution is a Gaussian function. Gaussian beams have applications in telecommunications.

Many lasers emit beams with a Gaussian profile, in which case the laser is said to be operating on the fundamental transverse mode, or "TEM00 mode" of the laser's optical resonator. When refracted by a lens, a Gaussian beam is transformed into another Gaussian beam (characterized by a different set of parameters), which explains why it is a convenient, widespread model in laser optics.

The electric field amplitude of a Gaussian beam a distance r from the centre of the beam is given by:

[itex]E(r) = E_0 \exp \left( \frac{-r^2}{w^2}\right), [itex]

and the corresponding intensity distribution is:

[itex]I(r) = I_0 \exp \left( \frac{-2r^2}{w^2} \right), [itex]

where w is the radius at which the field amplitude and intensity drop to 1/e and 1/e2, respectively. This parameter is called the beam radius or spot size of the beam.

For a Gaussian beam propagating in free space, the spot size w will be at a minimum value w0 at one place along the beam, known as the beam waist. For a beam of wavelength λ at a distance z along the beam from the beam waist, the variation of the spot size is given by:

[itex]w(z) = w_0 \left[ 1+ {\left( \frac{\lambda z}{\pi w_0^2} \right)}^2 \right]^{1/2}. [itex]

The divergence of a Gaussian beam, at a sufficient distance from the waist, tends to an angle:

[itex]\Theta = \frac{2\lambda}{\pi w_0}.[itex]

The confocal parameter of the beam is given by:

[itex]b = \frac{2 \pi w_0^2}{\lambda},[itex]

this is the distance between the two points on either side of the beam waist at which w = √2 w0. Half the confocal parameter is known as the Rayleigh range or depth of focus.

The propagation of Gaussian beams can be modelled using ray transfer matrix analysis.

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