Editing Gaussian beam (section)

===Gaussian as a decomposition into modes===

Arbitrary solutions of the [[Helmholtz equation#Paraxial approximation|paraxial Helmholtz equation]] can be decomposed as the sum of [[#Hermite-Gaussian modes|Hermite–Gaussian modes]] (whose amplitude profiles are separable in {{mvar|x}} and {{mvar|y}} using [[Cartesian coordinates]]), [[#Laguerre-Gaussian modes|Laguerre–Gaussian modes]] (whose amplitude profiles are separable in {{mvar|r}} and {{mvar|θ}} using [[cylindrical coordinates]]) or similarly as combinations of  [[#Ince-Gaussian modes|Ince–Gaussian modes]] (whose amplitude profiles are separable in {{mvar|ξ}} and {{mvar|η}} using [[elliptical coordinates]]).<ref name="siegman642">Siegman, p. 642.</ref><ref name="goubau">probably first considered by Goubau and Schwering (1961).</ref><ref name="ince-beams">Bandres and Gutierrez-Vega (2004)</ref> At any point along the beam {{mvar|z}} these modes include the same Gaussian factor as the fundamental Gaussian mode multiplying the additional geometrical factors for the specified mode. However different modes propagate with a different [[#Gouy phase|Gouy phase]] which is why the net transverse profile due to a [[Superposition principle|superposition]] of modes evolves in {{mvar|z}}, whereas the propagation of any ''single'' Hermite–Gaussian (or Laguerre–Gaussian) mode retains the same form along a beam.

Although there are other [[Transverse mode|modal decompositions]], Gaussians are useful for problems involving compact beams, that is, where the optical power is rather closely confined along an axis. Even when a laser is ''not'' operating in the fundamental Gaussian mode, its power will generally be found among the lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's [[laser resonator|resonator]] (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM<sub>00</sub>) Gaussian mode.