Longitudinal mode

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The first six longitudinal modes of a plane-parallel cavity.

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 surpressed.

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:

[ L = q \frac ]

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:

[\Delta\nu = \frac ]

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 = \frac\left[ frac + frac + frac + cdots right]]

where n_i is the refractive index of the i'th element of length L_i.

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.

Longitudinal mode

See also