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Gaussian beam

In optics, a Gaussian beam is a beam of monochromatic electromagnetic radiation whose transverse magnetic and electric field amplitude profiles are given by the Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse gaussian mode describes the intended output of most (but not all) lasers, as such a beam can be focused into the most concentrated spot. When such a beam is refocused by a lens, the transverse ''phase'' dependence is altered; this results in a ''different'' Gaussian beam. The electric and magnetic field amplitude profiles along any such circular Gaussian beam (for a given wavelength and polarization) are determined by a single parameter: the so-called waist ''w0''. At any position ''z'' relative to the waist (focus) along a beam having a specified ''w0'', the field amplitudes and phases are thereby determined〔Svelto, pp. 153–5.〕 as detailed below.
Arbitrary solutions of the paraxial Helmholtz equation can be expressed as combinations of Hermite–Gaussian modes (whose amplitude profiles are separable in ''x'' and ''y'' using Cartesian coordinates) or similarly as combinations of Laguerre–Gaussian modes (whose amplitude profiles are separable in ''r'' and ''θ'' using cylindrical coordinates).〔Siegman, p. 642.〕〔probably first considered by Goubau and Schwering (1961).〕 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 phase which is why the net transverse profile due to a superposition 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 modal 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 lowest-order modes using these decompositions, as the spatial extent of higher order modes will tend to exceed the bounds of a laser's resonator (cavity). "Gaussian beam" normally implies radiation confined to the fundamental (TEM00) Gaussian mode.
==Mathematical form==
The Gaussian beam is a transverse electromagnetic (TEM) mode.〔Svelto, p. 158.〕 The mathematical expression for the electric field amplitude is a solution to the paraxial Helmholtz equation.〔 Assuming polarization in the ''x'' direction and propagation in the +''z'' direction, the electric field in phasor (complex) notation is given by:
:$= E_0 \, \hat \, \frac \exp \! \left\left( \! \frac\right \right) \exp \! \! \left\left( \! \! -i \! \left\left(kz +k \frac - \psi\left(z\right) \! \right\right) \! \! \right\right) \ ,$
where〔
:$r$ is the radial distance from the center axis of the beam,
:$z$ is the axial distance from the beam's focus (or "waist"),
:$i$ is the imaginary unit,
:$k = 2 \pi/\lambda$ is the wave number (in radians per meter) for a wavelength λ,
:$E_0 = E\left(0,0\right)$, the electric field amplitude (and phase) at the origin at time 0,
:$w\left(z\right)$ is the radius at which the field amplitudes fall to 1/''e'' of their axial values, at the plane ''z'' along the beam,
:$w_0 = w\left(0\right)$ is the waist size,
:$R\left(z\right)$ is the radius of curvature of the beam's wavefronts at ''z'', and
:$\psi\left(z\right)$ is the Gouy phase at ''z'', an extra phase term beyond that attributable to the phase velocity of light.
There is also an understood time dependence $e^$ multiplying such phasor quantities; the actual field at a point in time and space is given by the real part of that complex quantity.
The wave's magnetic field has an identical form but with an orthogonal polarization (in ''y'' since the electric field polarization was stipulated to be in ''x''):
:$= \hat \frac$,
where the constant ''η'' is the characteristic impedance of the medium in which the beam is propagating. For free space, η = η0 ≈ 377 Ω.
The time-averaged intensity (or irradiance) at a location is computed using $I = \frac \mathbf\left(E \times H^$
*) which removes all phase factors (since we've averaged over time, which also results in the ½ factor):
:$I\left(r,z\right) = = I_0 \left\left( \frac \right\right)^2 \exp \left\left( \frac \right\right)\ ,$
where $I_0 = E_0^2/2 \eta$ is the intensity at the center of the beam at its waist.

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