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# Wavelength

Definition: the spatial period of a plane wave

German: Wellenlänge

Formula symbol: <$\lambda$>

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The simplest kind of wave is a monochromatic plane wave, described by the following complex wave amplitude, being a function of spatial position <$x$> and time <$t$>:

$$A(\vec r,t) = {A_0}\;\exp \left( {i\;\vec k\;\vec r - i\;\omega \;t} \right)$$

with the wave vector, the magnitude of which is the wavenumber <$k$>, and the angular frequency <$\omega$>. The wavenumber determines the wavelength, defined as the spatial period of the wave (e.g., the distance between subsequent oscillation maxima, see Figure 1):

$${\rm{GD}}{{\rm{D}}_\lambda } = - \frac{{2\pi c}}{{{\lambda ^2}}} \cdot {D_2} = - \frac{{2\pi c}}{{{\lambda ^2}}} \cdot \frac{{{\partial ^2}\varphi }}{{\partial {\omega ^2}}}$$ Figure 1: A plane wave, with the wavelength illustrated with the black ruler in the middle.

(Note that in spectroscopy the wavenumber is normally considered as the inverse of the wavelength, not involving the factor <$2 \pi$>.)

When propagating by one wavelength in <$x$> direction, the plane wave acquires a phase delay of <$2 \pi$>.

Wavelength and frequency are related to each other: as the wave propagates by one wavelength within one oscillation cycle, its phase velocity <$c$> is given by:

$$c = \lambda /T = \lambda \nu$$

For wave packets, there is another kind of velocity, called the group velocity, which can deviate from the phase velocity if there is chromatic dispersion, i.e., a dependence of the phase velocity on the frequency.

Note that for waves with a different spatial distribution – for example, for strongly focused laser beams, the amplitude is a function of position is in general not (or not exactly) periodic, and the spacing between subsequent wave maxima may somewhat deviate from the wavelength, which is always defined for a plane wave. See also the article on the Gouy phase shift, which is essentially a phase deviation of Gaussian beams against a plane wave for comparison.

## Wavelengths and Colors

Monochromatic light is seen with a certain color depending on its wavelength. Unfortunately, a computer screen cannot produce monochromatic light, but only roughly approximate the visual impression for certain wavelength. This has been done in Figure 1, which will appear somewhat differently on different computer screens.

There are objective quantitative measures for quantities related to color perception; see the article on colorimetry.

## Optical Waves: Vacuum Wavelength and the Wavelength in a Medium

If a monochromatic optical wave subsequently propagates through different transparent media, its wavelength will vary, while its optical frequency <$\nu$> stays constant. Therefore, it would be most natural to characterize such a wave by its optical frequency. For historical reasons, it is more common, however, to characterize optical waves (light) with their wavelength in air (for standard pressure and temperature: 1013.25 mbar, 15 °C, zero humidity). In some cases, however, the vacuum wavelength is specified. The wavelength in air is quite close to the vacuum wavelength, since the refractive index of air is only very slightly above 1; the small difference is not relevant for most applications. For example, a vacuum wavelength of 1000 nm would correspond to 999.7259 nm in air.

## Wavelength calculations

 Vacuum wavelength: Optical frequency: calc Refractive index of a medium: Wavelength in medium: calc

Enter input values with units, where appropriate. After you have modified some values, click a “calc” button to recalculate the field left of it.

For a given vacuum wavelength <$\lambda_0$>, the wavelength in a medium with refractive index <$n$> is <$\lambda = \lambda_0 / n$>. Generally, the refractive index depends on the optical frequency or vacuum wavelength (→ chromatic dispersion).

For visible light, the vacuum wavelength is roughly between 400 nm and 700 nm; there are no precisely defined boundaries of the visible spectral region, since the sensitivity of the human eye is a smooth function of wavelength and also differs between individuals. Light with longer vacuum wavelengths is called infrared light, while light with shorter wavelengths is ultraviolet light.

If some physical quantity depends on an optical frequency, it is very often called wavelength-dependent instead of frequency-dependent, even if the spatial aspect does not play a role in the relevant phenomena.

Also one often specifies an optical bandwidth (e.g., the gain bandwidth of a laser gain medium) in terms of the width of the wavelength range rather than the frequency range. For the conversion between wavelength and frequency ranges one needs to keep in mind that the width of a frequency interval does not only depend on the width of the corresponding wavelength interval but also on the mean wavelength: <$\delta \nu = (c / \lambda^2) \: \delta \lambda$> (assuming small intervals).

## Measurement of Optical Wavelengths

Optical wavelengths can be measured with wavemeters, which are a kind of interferometers. The indirect evaluation of the wavelength from a measured optical frequency and the vacuum velocity of light can be far more precise, since optical frequencies can be measured with extremely high precision, and the vacuum velocity of light is nowadays (in the SI system) a defined quantity, i.e., without any measurement error.

## Wavelength Standards

It is common to use certain spectral lamps as reasonably accurate wavelength standards. Besides, there are optical frequency standards relying on far more sophisticated technology, where the obtained wavelength is by many orders of magnitude more precise.

## Non-monochromatic Light

In many cases, light is not monochromatic, but rather exhibits a substantial optical bandwidth. It is then often of interest to determine either the peak wavelength or e.g. a the mean wavelength based on the “center of gravity” of the optical spectrum.

In some cases, one needs to know how the optical power is distributed over different wavelengths or optical frequencies. This can be accomplished with optical spectrum analyzers, ideally delivering a precisely defined power spectral density (PSD) versus wavelength of frequency. The units of such a quantity can be W/nm or W/THz, for example.

## Importance of Optical Wavelengths

The wavelength of light is relevant for a large number of phenomena. Some examples:

• The longer the wavelength of light, the stronger is its tendency for diffraction, i.e., for the expansion of a light beam (e.g., a laser beam). Because optical wavelength are so short, diffraction-limited beams, as are emitted by many lasers, can exhibit a rather small beam divergence, i.e., propagate over substantial distances without substantial increase in beam area. Wavelengths are also relevant in other diffraction phenomena, e.g. at diffraction gratings.
• The wavelength is an essential quantity for any interference phenomena. Because optical wavelengths are rather short, optical interferometers usually require very high mechanical stability, since even sub-micrometer changes of propagation lengths can substantially modify interference conditions.
• Many optical nonlinearities can have substantial effects only if phase matching is achieved. The phase-matching conditions contain the wavelengths of the involved optical beams, and not just the optical frequencies.

In many other cases, the actually irrelevant quantity is the optical frequency, which of course is related to the wavelength. For example, resonant effects in the optical pumping of laser-active ions cause strong frequency dependencies. The wavelengths themselves, being far larger than atoms or ions, are not relevant for those.

## Special Types of Wavelengths

In optics and photonics, there are many different terms involving the word “wavelength”. Some examples:

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