Longitudinal mode
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A longitudinal mode of a resonant cavity is a particular standing wave pattern formed by waves confined in the cavity. The longitudinal modes correspond to the wavelengths of the wave which are reinforced by constructive interference after many reflections from the cavity's reflecting surfaces. All other wavelengths undergo destructive interference and are suppressed.
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) mirrors surrounding the gain medium (a plane-parallel or Fabry-Perot 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:
where q is an integer known as the mode order.
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 105 to 106). The frequency separation between any two adjacent modes q and q+1 are given (for an empty linear resonator of length L) by Δν:
where c is the speed of light.
If the cavity is non-empty (i.e. contains one or more elements with a non-unity refractive index), the values of L used are the optical path lengths of varying refractive indices. This is given by:
![\Delta \nu = \sum_i \frac{c}{2 n_i L_i} = \frac{c}{2}\left[ \frac{1}{n_1 L_1} + \frac{1}{n_2 L_2} + \frac{1}{n_3 L_3} + \cdots \right]](../../../math/a/b/b/abb3a8b68a5817f691bde0ab65fe02be.png)
where ni is the refractive index of the i'th element of length Li.
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.
Both transverse and longitudinal waves may have longitudinal modes when confined to a cavity.
