# Propagation Constant

Definition: a mode- and frequency-dependent quantity describing the propagation of light in a medium or waveguide

German: Propagationskonstante

Category: fiber optics and waveguides

Formula symbol: γ, β

Units: 1/m

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Author: Dr. RĂ¼diger Paschotta

The propagation constant of a mode in a waveguide (e.g. a fiber), often denoted with the symbol γ, determines how the amplitude and phase of that light with a given frequency varies along the propagation direction *z*:

where *A(z)* is the complex amplitude of the light field at position *z*.

In lossless media, γ is purely imaginary; we have γ = *i β* with the (real) *phase constant* β, which is the product of the effective refractive index and the vacuum wavenumber.
Optical losses (or gain) imply that γ also has a real part.

The propagation constant depends on the optical frequency (or wavelength) of the light. The frequency dependence of its imaginary part determines the group delay and the chromatic dispersion of the waveguide.

Note that different definitions of the propagation constant occur in the literature.
For example, the propagation constant is sometimes understood to be only the imaginary part of the quantity defined above, i.e., β.
It is then also common to introduce a *normalized propagation constant* which can only vary between 0 and 1.
Here, the value zero corresponds to the wavenumber in the cladding, and 1 to that in the core.
Modes which are mostly propagating in the cladding will have a value close to 0.

See also: modes, effective refractive index, waveguides, fibers, group delay, dispersion, propagation losses

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