Editing Transverse mode (section)

===Lasers===
[[File:Laguerre-gaussian.png|thumb|right|Cylindrical transverse mode patterns TEM(''pl'')]]
In a laser with cylindrical symmetry, the transverse mode patterns are described by a combination of a [[Gaussian beam]] profile with a [[Laguerre polynomials|Laguerre polynomial]]. The modes are denoted {{math|TEM<sub>''pl''</sub>}} where {{mvar|p}} and {{mvar|l}} are integers labeling the radial and angular mode orders, respectively. The intensity at a point {{math|(''r'',''φ'')}} (in [[Coordinates (elementary mathematics)#Circular coordinates|polar coordinates]]) from the centre of the mode is given by:
<math display="block">I_{pl} (\rho, \varphi) = I_0 \rho^l \left[L_p^l (\rho)\right]^2 \cos^2 (l\varphi) e^{-\rho}</math>
where {{math|1=''ρ'' = 2''r''<sup>2</sup>/''w''<sup>2</sup>}}, {{math|''L{{su|b=p|p=l}}''}} is the associated [[Laguerre polynomial]] of order {{mvar|p}} and index {{mvar|l}}, and {{mvar|w}} is the spot size of the mode corresponding to the Gaussian beam radius.
[[File:Tem p 2 l 1 plot.png|thumb|upright|Cylindrical transverse mode with ''p''=2, ''l''=1]]
With {{math|1=''p'' = ''l'' = 0}}, the TEM<sub>00</sub> mode is the lowest order. It is the fundamental transverse mode of the laser resonator and has the same form as a Gaussian beam. The pattern has a single lobe, and has a constant [[phase (waves)|phase]] across the mode. Modes with increasing {{mvar|p}} show concentric rings of intensity, and modes with increasing {{mvar|l}} show angularly distributed lobes. In general there are {{math|2''l''(''p''+1)}} spots in the mode pattern (except for {{math|1=''l'' = 0}}). The {{math|TEM<sub>0''i''*</sub>}} mode, the so-called ''doughnut mode'', is a special case consisting of a superposition of two {{math|TEM<sub>0''i''</sub>}} modes ({{math|1=''i'' = 1, 2, 3}}), rotated {{math|360°/4''i''}} with respect to one another.

The overall size of the mode is determined by the Gaussian beam radius {{mvar|w}}, and this may increase or decrease with the propagation of the beam, however the modes preserve their general shape during propagation. Higher order modes are relatively larger compared to the {{math|TEM<sub>00</sub>}} mode, and thus the fundamental Gaussian mode of a laser may be selected by placing an appropriately sized aperture in the laser cavity.

In many lasers, the symmetry of the optical resonator is restricted by [[polarizer|polarizing elements]] such as [[Brewster's angle]] windows. In these lasers, transverse modes with rectangular symmetry are formed. These modes are designated {{math|TEM<sub>''mn''</sub>}} with {{mvar|m}} and {{mvar|n}} being the horizontal and vertical orders of the pattern. The electric field pattern at a point {{math|(''x'',''y'',''z'')}} for a beam propagating along the z-axis is given by<ref name="svelto">{{cite book| author=Svelto, O.|title=Principles of Lasers |edition=5th|year=2010|page=158}}</ref>
<math display="block">E_{mn}(x, y, z) = E_0 \frac{w_0}{w} H_m\left(\frac{\sqrt{2}x}{w}\right) H_n\left(\frac{\sqrt{2}y}{w}\right)\exp\left[-(x^2 + y^2) \left(\frac{1}{w^2} + \frac{jk}{2R}\right) - jkz - j(m + n + 1)\zeta\right]</math>
where <math>w_0</math>, <math>w(z)</math>, <math>R(z)</math>, and <math>\zeta(z)</math> are the waist, spot size, radius of curvature, and [[Gouy phase shift]] as given for a [[Gaussian beam]]; <math>E_0</math> is a normalization constant; and <math>H_k</math> is the {{mvar|k}}-th physicist's [[Hermite polynomials|Hermite polynomial]]. The corresponding intensity pattern is
<math display="block">I_{mn}(x, y, z) = I_0 \left( \frac{w_0}{w} \right)^2 \left[ H_m \left( \frac{ \sqrt{2} x}{w} \right) \exp \left( \frac{-x^2}{w^2} \right) \right]^2 \left[ H_n \left( \frac{ \sqrt{2} y}{w} \right) \exp \left( \frac{-y^2}{w^2} \right) \right]^2</math>

[[File:Hermite-gaussian.png|thumb|right|Rectangular transverse mode patterns TEM(mn)]]

The TEM<sub>00</sub> mode corresponds to exactly the same fundamental mode as in the cylindrical geometry. Modes with increasing {{mvar|m}} and {{mvar|n}} show lobes appearing in the horizontal and vertical directions, with in general {{math|(''m'' + 1)(''n'' + 1)}} lobes present in the pattern. As before, higher-order modes have a larger spatial extent than the 00 mode.

The [[phase (waves)|phase]] of each lobe of a {{math|TEM<sub>''mn''</sub>}} is offset by {{math|''π''}} radians with respect to its horizontal or vertical neighbours. This is equivalent to the polarization of each lobe being flipped in direction.

The overall intensity profile of a laser's output may be made up from the superposition of any of the allowed transverse modes of the laser's cavity, though often it is desirable to operate only on the fundamental mode.