Editing Gaussian beam (section)

==Wave equation==
As a special case of [[electromagnetic radiation]], Gaussian beams (and the higher-order Gaussian modes detailed below) are solutions to the [[electromagnetic wave equation|wave equation for an electromagnetic field]] in free space or in a homogeneous dielectric medium,<ref name="svelto148">Svelto, pp. 148–9.</ref> obtained by combining Maxwell's equations for the curl of {{mvar|E}} and the curl of {{mvar|H}}, resulting in: 
<math display="block"> \nabla^2 U = \frac{1}{c^2} \frac{\partial^2 U}{\partial t^2},</math>
where {{mvar|c}} is the speed of light ''in the medium'', and {{mvar|U}} could either refer to the electric or magnetic field vector, as any specific solution for either determines the other. The Gaussian beam solution is valid only in the [[paraxial]] approximation, that is, where wave propagation is limited to directions within a small angle of an axis. Without loss of generality let us take that direction to be the {{math|+''z''}} direction in which case the solution {{mvar|U}} can generally be written in terms of {{mvar|u}} which has no time dependence and varies relatively smoothly in space, with the main variation spatially corresponding to the [[wavenumber]] {{mvar|k}} in the {{mvar|z}} direction:<ref name="svelto148" />
<math display="block"> U(x, y, z, t) = u(x, y, z) e^{-i(kz-\omega t)} \, \hat{\mathbf x} \, .</math>

Using this form along with the paraxial approximation, {{math|∂<sup>2</sup>''u''/∂''z''<sup>2</sup>}} can then be essentially neglected. Since solutions of the electromagnetic wave equation only hold for polarizations which are orthogonal to the direction of propagation ({{mvar|z}}), we have without loss of generality considered the polarization to be in the {{mvar|x}} direction so that we now solve a scalar equation for {{math|''u''(''x'', ''y'', ''z'')}}.

Substituting this solution into the wave equation above yields the [[Helmholtz equation#Paraxial approximation|paraxial approximation]] to the scalar wave equation:<ref name="svelto148" />
<math display="block">\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 2ik \frac{\partial u}{\partial z}.</math>
Writing the wave equations in the [[light-cone coordinates]] returns this equation without utilizing any approximation.<ref>{{Cite journal |last1=Esarey |first1=E. |last2=Sprangle |first2=P. |last3=Pilloff |first3=M. |last4=Krall |first4=J. |date=1995-09-01 |title=Theory and group velocity of ultrashort, tightly focused laser pulses |url=https://opg.optica.org/josab/abstract.cfm?uri=josab-12-9-1695 |journal=JOSA B |language=EN |volume=12 |issue=9 |pages=1695–1703 |doi=10.1364/JOSAB.12.001695 |bibcode=1995JOSAB..12.1695E |issn=1520-8540}}</ref> Gaussian beams of any beam waist {{math|''w''<sub>0</sub>}} satisfy the paraxial approximation to the scalar wave equation; this is most easily verified by expressing the wave at {{mvar|z}} in terms of the complex beam parameter {{math|''q''(''z'')}} as defined above. There are many other solutions. As solutions to a [[linear system]], any combination of solutions (using addition or multiplication by a constant) is also a solution. The fundamental Gaussian happens to be the one that minimizes the product of minimum spot size and far-field divergence, as noted above. In seeking paraxial solutions, and in particular ones that would describe laser radiation that is ''not'' in the fundamental Gaussian mode, we will look for families of solutions with gradually increasing products of their divergences and minimum spot sizes. Two important orthogonal decompositions of this sort are the Hermite–Gaussian or Laguerre-Gaussian modes, corresponding to rectangular and circular symmetry respectively, as detailed in the next section. With both of these, the fundamental Gaussian beam we have been considering is the lowest order mode.