In optics, a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This fundamental (or TEM00) transverse Gaussian mode describes the intended output of many lasers, as such a beam diverges less and can be focused better than any other. When a Gaussian beam is refocused by an ideal lens, a new Gaussian beam is produced. The electric and magnetic field amplitude profiles along a circular Gaussian beam of a given wavelength and polarization are determined by two parameters: the waist w0, which is a measure of the width of the beam at its narrowest point, and the position z relative to the waist. Since the Gaussian function is infinite in extent, perfect Gaussian beams do not exist in nature, and the edges of any such beam would be cut off by any finite lens or mirror. However, the Gaussian is a useful approximation to a real-world beam for cases where lenses or mirrors in the beam are significantly larger than the spot size w(z) of the beam. Fundamentally, the Gaussian is a solution of the paraxial Helmholtz equation, the wave equation for an electromagnetic field. Although there exist other solutions, the Gaussian families of solutions are useful for problems involving compact beams.
Mathematical form The equations below assume a beam with a circular cross-section at all values of z; this can be seen by noting that a single transverse dimension, r, appears. Beams with elliptical cross-sections, or with waists at different positions in z for the two transverse dimensions (astigmatic beams) can also be described as Gaussian beams, but with distinct values of w0 and of the z = 0 location for the two transverse dimensions x and y.
The Gaussian beam is a transverse electromagnetic (TEM) mode. 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 ( r , z ) = E 0 x ^ w 0 w ( z ) exp ( − r 2 w ( z ) 2 ) exp ( − i ( k z + k r 2 2 R ( z ) − ψ ( z ) ) ) {\displaystyle {\mathbf {E} (r,z)}=E_{0}\,{\hat {\mathbf {x} }}\,{\frac {w_{0}}{w(z)}}\exp \left({\frac {-r^{2}}{w(z)^{2}}}\right)\exp \left(\!-i\left(kz+k{\frac {r^{2}}{2R(z)}}-\psi (z)\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 = 2πn/λ is the wave number for a free-space (vacuum) wavelength λ, and n is the index of refraction of the medium in which the beam propagates, E0 = E(0, 0), the electric field amplitude at the origin (r = 0, z = 0), w(z) is the radius at which the field amplitudes fall to 1/e of their axial values (i.e., where the intensity values fall to 1/e2 of their axial values), at the plane z along the beam, w0 = w(0) is the waist radius, R(z) is the radius of curvature of the beam's wavefronts at z, and ψ(z) = arctan(z/zR) is the Gouy phase at z, an extra phase term beyond that attributable to the phase velocity of light. The physical electric field is obtained from the phasor field amplitude given above by taking the real part of the amplitude times a time factor:
E phys ( r , z , t ) = Re ( E ( r , z ) ⋅ e i ω t ) , {\displaystyle \mathbf {E} _{\text{phys}}(r,z,t)=\operatorname {Re} (\mathbf {E} (r,z)\cdot e^{i\omega t}),}
where ω {\textstyle \omega } is the angular frequency of the light and t is time. The time factor involves an arbitrary sign convention, as discussed at Mathematical descriptions of opacity § Complex conjugate ambiguity. Since this solution relies on the paraxial approximation, it is not accurate for very strongly diverging beams. The above form is valid in most practical cases, where w0 ≫ λ/n. The corresponding intensity (or irradiance) distribution is given by
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