## The Gaussian Beam

Wave optics as previously discussed operated under an ideal assumption that light can be confined to a uniform, rectangular shape that moves through space. A more realistic understanding of a wave that propagates through space is the goal of beam optics, which instead describes a light wave as a distribution of light.

**The Gaussian Beam**

The Gaussian beam is a common description of the distribution of a light beam which satisfies the Helmholtz equation. Light is concentrated towards the center of the beam in a Gaussian distribution.

The width of the beam is a minimum at what is termed the waist of the beam and the width increases at distances further from the waist. Eventually, the width of the beam would become very wide and the distribution of light would be wide enough, almost to approximate a spherical beam. In reference to the figure above, the leftmost distribution may for example be the distribution at the waist of the beam and the rightmost picture is the beam further from the waist. In a localized area, the beam exhibits similar characteristics to the ideal plane wave.

The width of the beam is determined by the following formula:

The complex amplitude of the Gaussian beam is described by the following formula:

Further parameters of the beam used in the above formula are the following:

- W(z): Beam width function (above)
- R(z): wavefront radius of curvature

- ξ(z): Beam center point

- W0: Minimum Beam level, found at z = 0

B. E. A. Saleh and M. C. Teich, *Fundamentals of photonics*. Hoboken: Wiley, 2019.

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