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# Wavenumber

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

German: Wellenzahl

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

Author:

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

$$k = \frac{{2\pi }}{\lambda }$$

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.

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