# Wavenumber

Definition: the phase delay per unit length, or that quantity divided by <$2\pi$>

German: Wellenzahl

Categories: general optics, light detection and characterization

Units: rad/m, cm^{−1}

Formula symbol: <$k$>, <$\nu$>

Author: Dr. Rüdiger Paschotta

Unfortunately, there are different definitions of the *wavenumber* of light in the literature. In physics, the definition

is common, where <$\lambda$> is the wavelength in the medium (not the vacuum wavelength). That (angular) wavenumber is the magnitude of the wave vector, and is the phase delay per unit length during propagation of a plane wave.

The other definition

$$\tilde \nu = \frac{1}{\lambda }$$(with units of cm^{−1}) is widely used in the field of spectroscopy and therefore called the *spectroscopic wavenumber*. The former quantity can be called *angular wavenumber* (in analogy with *angular frequency*) to avoid confusion, but that term is not very common.

For light in a medium, the wavenumber is the vacuum wavenumber times the refractive index. Spectroscopic wavenumbers are usually considered in vacuum.

The wavenumber is related to the phase change per unit length of a plane wave in a homogeneous medium. For focused beams, the phase change per unit length is modified with respect to that for a plane wave. For Gaussian beams, for example, this modification is the Gouy phase shift. For propagation of guided waves in waveguides, the imaginary part of the propagation constant <$\gamma$> (called <$\beta$>) is the relevant quantity.

See also: wave vector, plane waves, propagation constant, refractive index

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