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| {{unreferenced|date=October 2011}}
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| ''For the longitudinal mode of conduction of electric currents, see [[Common mode]]''
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| [[Image:longitudinal mode v2.svg|200px|thumb|right|The first six longitudinal modes of a plane-parallel cavity.]]
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| A '''longitudinal mode''' of a [[resonant cavity]] is a particular [[standing wave]] pattern formed by [[wave]]s confined in the cavity. The longitudinal modes correspond to the [[wavelength]]s of the wave which are reinforced by constructive [[Interference (wave propagation)|interference]] after many reflections from the cavity's reflecting surfaces. All other wavelengths are suppressed by destructive interference.
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| A longitudinal mode pattern has its [[Node (physics)|nodes]] located axially along the length of the cavity. [[Transverse mode]]s, with nodes located perpendicular to the axis of the cavity, may also exist.
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| ==Simple cavity==
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| A common example of longitudinal modes are the [[light]] wavelengths produced by a [[laser]]. In the simplest case, the laser's [[optical cavity]] is formed by two opposed plane (flat) [[mirror]]s surrounding the [[gain medium]] (a plane-parallel or [[Fabry–Pérot]] cavity). The allowed modes of the cavity are those where the mirror separation distance ''L'' is equal to an exact multiple of half the wavelength, ''λ'':
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| :<math> L = q \frac{\lambda}{2} </math>
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| where ''q'' is an integer known as the mode order. | |
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| In practice, the separation distance of the mirrors ''L'' is usually much greater than the wavelength of light ''λ'', so the relevant values of ''q'' are large (around 10<sup>5</sup> to 10<sup>6</sup>). The frequency separation between any two adjacent modes, ''q'' and ''q''+1, in a material that is transparent at the laser wavelength, are given (for an empty linear resonator of length ''L'') by Δ''ν'':
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| :<math>\Delta\nu = \frac{c}{2nL} </math>
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| where ''c'' is the speed of light and n is the refractive index of the material (note: n≈1 in air).
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| ==Composite cavity==
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| If the cavity is non-empty (i.e. contains one or more elements with different values of [[refractive index]]), the values of ''L'' used are the [[optical path length]]s for each element. The frequency spacing of longitudinal modes in the cavity is then given by:
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| :<math>\Delta \nu = \frac{c}{2\sum_i n_i L_i} = \frac{c}{2}\left[ \frac{1}{n_1 L_1 + n_2 L_2 + n_3 L_3 + \ldots} \right]</math> | |
| where ''n''<sub>i</sub> is the refractive index of the i'th element of length ''L''<sub>i</sub>.
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| More generally, the longitudinal modes may be found for any type of wave in a cavity by solving the relevant [[wave equation]] with the appropriate [[boundary conditions]].
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| Both [[transverse wave|transverse]] and [[longitudinal wave]]s may have longitudinal modes when confined to a cavity.
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| The analysis of longitudinal modes is especially important in lasers with single transversal mode, for example, in single-mode [[fiber laser]]s. The number of longitudinal modes of such a laser can be estimated as ratio of the spectral width of gain to the spectral separation of longitudinal modes.
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| ==Power per longitudinal mode==
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| For lasers with single transversal mode, the power per one longitudinal mode can be significantly increased by the [[coherent addition]] of lasers. Such addition allows one to both scale-up the output power of a single-transverse-mode laser and reduce number of longitudinal modes; because the system chooses automatically only the modes which are common for all the combined lasers. The reduction of the number of longitudinal modes determines the limits of the [[coherent addition]]. The ability to coherently add one additional laser is exhausted when one longitudinal mode, common for the combined lasers, lies within the spectral width of the gain; a subsequent addition will lead to loss of efficiency of the coherent combination and will not increase the power per ''longitudinal mode'' of such a laser.
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| ==See also==
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| *[[Modelocking]]
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| *[[Normal mode]]
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| [[Category:Wave mechanics]]
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