1,662
edits
Leberwurscht (talk  contribs) (add exp(ikz) to be consistent with definition of E(x,y,z) from section "Mathematical form") 
m (added link) 

The equations below assume a beam with a circular crosssection at all values of ''z''; this can be seen by noting that a single transverse dimension, ''r'', appears. Beams with elliptical crosssections, or with waists at different positions in ''z'' for the two transverse dimensions ([[Astigmatism (optical systems)astigmatic]] beams) can also be described as Gaussian beams, but with distinct values of ''w<sub>0</sub>'' and of the {{nowrap1=''z'' = 0}} location for the two transverse dimensions ''x'' and ''y''.
Arbitrary solutions of the [[Helmholtz equation#Paraxial approximationparaxial Helmholtz equation]] can be expressed as combinations of [[#HermiteGaussian modesHermite–Gaussian modes]] (whose amplitude profiles are separable in ''x'' and ''y'' using [[Cartesian coordinates]]) or similarly as combinations of [[#LaguerreGaussian modesLaguerre–Gaussian modes]] (whose amplitude profiles are separable in ''r'' and ''θ'' using [[cylindrical coordinates]]).<ref name="siegman642">Siegman, p. 642.</ref><ref name="goubau">probably first considered by Goubau and Schwering (1961).</ref> At any point along the beam ''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 phaseGouy phase]] which is why the net transverse profile due to a [[Superposition principlesuperposition]] of modes evolves in ''z'', whereas the propagation of any ''single'' Hermite–Gaussian (or Laguerre–Gaussian) mode retains the same form along a beam.
Although there are other possible [[Transverse modemodal decompositions]], these families of solutions are the most 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 lowestorder modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's [[laser resonatorresonator]] (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM<sub>00</sub>) Gaussian mode.
