# Effective Refractive Index

Definition: a number quantifying the phase delay per unit length in a waveguide, relative to the phase delay in vacuum

German: effektiver Brechungsindex

Category: fiber optics and waveguides

Formula symbol: *n*_{eff}

Units: (dimensionless)

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Author: Dr. RĂ¼diger Paschotta

For plane waves in homogeneous transparent media, the refractive index *n* can be used to quantify the increase in the wavenumber (phase change per unit length) caused by the medium: the wavenumber is *n* times higher than it would be in vacuum.
The *effective refractive index* *n*_{eff} has the analogous meaning for light propagation in a waveguide with restricted transverse extension: the β value (phase constant) of the waveguide (for some wavelength) is the effective index times the vacuum wavenumber:

The mode-dependent and frequency-dependent β values can be calculated with a mode solver software and depend on the refractive index profile of the waveguide.

Note that the effective refractive index depends not only on the wavelength but also (for multimode waveguides) on the mode in which the light propagates.
For this reason, it is also called *modal index*.
Obviously, the effective index is not just a material property, but depends on the whole waveguide design.
Its value can be obtained with numerical mode calculations, for example.
It can vary substantially near a mode cut-off.

The effective index may be a complex quantity. In that case, the imaginary part describes gain or loss – see the article on propagation constant for more details.

The effective refractive index contains information on the phase velocity of light, but not on the group velocity; for the latter, one can similarly define an *effective group index* in analogy to the group index for plane waves in a homogeneous medium.

A common but wrong belief is that the effective refractive index is a kind of weighted average of the refractive index of core and cladding of the waveguide, with the weight factors determined by the fractions of the optical power propagating in the core and cladding. This impression may result from the common observation that higher-order modes, e.g. of a fiber, have a lower effective index and also a lower mode overlap with the core. However, consider e.g. a step-index multimode waveguide with a high numerical aperture and large core diameter. Here, all modes overlap to nearly 100% with the core (i.e. the mode overlaps are very similar), whereas the effective indices differ substantially.

See also: propagation constant, refractive index, group index, waveguides, Spotlight article 2007-10-07

and other articles in the category fiber optics and waveguides

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