# Wavenumber

Author: the photonics expert Dr. Rüdiger Paschotta

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

Categories: general optics, light detection and characterization

Units: rad/m, cm^{−1}

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

Unfortunately, there are different definitions of the *wavenumber* of light in the literature, as explained in the following sections.

## Wavenumbers in Physics

In general 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.

For avoiding confusion with the other definition (see below), one may call a wavenumber as defined above an *angular wavenumber* (in analogy with *angular frequency*), but that term is not very common.

For light in a medium, the wavenumber is the vacuum wavenumber times the refractive index. It can be complex-valued for a lossy medium.

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.

## Spectroscopic Wavenumbers

The other definition

$$\tilde \nu = \frac{1}{\lambda }$$(without that factor <$2\pi$>) is widely used in the field of spectroscopy and therefore called the *spectroscopic wavenumber*. It is based on the wavelength in vacuum, rather than in air, as the refractive index of air depends on pressure, humidity, etc., and is thus harder to standardize. As a result, spectroscopic wavenumbers are always real (not complex) quantities.

Spectroscopic wavenumbers are usually used with units of cm^{−1}. If these units are found, it is usually clear that a spectroscopic wavenumber is meant.

Note that the inverse wavelength is the vacuum velocity of light times the optical frequency. In practice, however, wavenumbers are usually not derived from frequency measurements, but rather from interferometric measurements, for example with a wavemeter. This may explain why wavenumbers instead of optical frequencies have become a common standard in spectroscopy.

Spectroscopic wavenumbers are also common for specifying a bandwidth or some other kind of spectral interval width. For example, a bandwidth of 1 cm^{−1} corresponds to a frequency range of ≈30 GHz. The difference in wavelength (for small intervals) is

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