# Propagation Constant

Author: the photonics expert Dr. Rüdiger Paschotta

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

Category: fiber optics and waveguides

Units: 1/m

Formula symbol: <$\gamma$>, <$\beta$>

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

$$A(z) = A(0)\;{{\mathop{\rm e}\nolimits} ^{\gamma z}}$$where *A(z)* is the complex amplitude of the light field at position <$z$>.

In lossless media, <$\gamma$> is purely imaginary; we have <$\gamma = i \beta$> with the (real) *phase constant* <$\beta$>, which is the product of the effective refractive index and the vacuum wavenumber. Optical losses (or gain) imply that <$\gamma$> 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 (but not its imaginary part itself) determines the group delay and the chromatic dispersion of the waveguide.

## Other Definitions

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., <$\beta$>. 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.

## More to Learn

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## Questions and Comments from Users

2021-12-06

How can I calculate the propagation constant for a fiber given the values of the specsheet?

The author's answer:

You would need to know the transverse refractive index profile of the fiber, which is unfortunately not usually available on specification sheets, and take that as an input for some mode solver software.

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2020-08-13

How can the effective propagation constant be written in terms of relative permittivity and permeability?

The author's answer:

For propagation in a homogeneous medium, it is <$2\pi / \lambda$> = <$2\pi n / \lambda_0$>, where the refractive index <$n$> can be written as the square root of <$\epsilon \mu$>, the product of the relative permittivity and permeability.