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# Wavelength

Author: the photonics expert

Definition: the spatial period of a plane wave

Formula symbol: <$\lambda$>

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):

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

(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 <$v_{\rm ph}$> is given by:

$$v_{\rm ph} = \lambda /T = \lambda \: \nu$$

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 (at least 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 of Light and Colors

Visible light has wavelengths between roughly 400 nm and 750 nm; there is also infrared light with longer wavelengths and ultraviolet light with shorter wavelengths. As such wavelengths are rather small, light exhibits much weaker diffraction effects than sound waves, for example (disregarding sound with ultrahigh frequencies).

Monochromatic light is perceived by the human eye 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 2, 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.

## Vacuum Wavelength and the Wavelength in a Medium

When a monochromatic optical wave subsequently propagates through different transparent media, its optical frequency <$\nu$> remains constant, while its wavelength in those media will vary inversely with the refractive index of the media.

It would thus be most natural to characterize such a wave by its optical frequency. For historical reasons, however, it is more common to characterize optical waves (light) by their wavelength, since wavelengths could be measured early on with interferometers, while optical frequencies could be measured directly only much later. For a long time, optical frequencies could only be deduced from wavelengths in combination with the measured velocity of light. (Later, frequencies could be measured directly, first with frequency chains and then with frequency combs.)

Optical wavelengths can be specified for propagation in vacuum (vacuum wavelength) or in air. These two values are quite similar, since the refractive index of air is only very slightly above 1. For example, a vacuum wavelength of 1000 nm would be 999.7259 nm in standard air; there is a difference of about 0.03% (and more for much shorter wavelengths). In the case of air, factors such as barometric pressure, CO2 content, and humidity (water vapor) play a significant role. Dry air at standard pressure and temperature is often used: 1013.25 mbar, 15 °C, zero humidity, 450 ppm CO2.

Unfortunately, it is often not made clear in the literature whether a vacuum wavelength or a wavelength in air is meant, although good scientific practice would require this, at least where it is potentially relevant. (A wavelength in a solid medium, which would be much shorter, is rarely used.) In general, it makes more sense to specify vacuum wavelengths:

• In this way, the influence of various air parameters (pressure, humidity, CO2 content) is avoided.
• It is then easier to calculate the wavelength in any medium, using only its refractive index.

Interferometric wavelength measurements with wavemeters are usually performed in air. However, the reading on a wavemeter may well be a vacuum wavelength, obtained by more or less carefully compensating for the influence of air. High-precision instruments make measurements of air pressure, humidity, and CO2 content to accurately calculate the refractive index of the air in the instrument, both for the wavelength being tested and for the reference wavelength. Note that direct measurements of optical frequencies avoid these difficulties (while having others).

The small difference between wavelengths in vacuum and air is actually not relevant for many applications, but sometimes it is. For example, the 0.3 nm difference mentioned above is on the order of the gain bandwidth of Nd:YAG, so a 0.3 nm error in the gain maximum of another type of laser crystal would definitely matter.

For non-monochromatic light, the center wavelength is often specified, typically defined as the “center of gravity” rather than the peak of the optical spectrum.

## 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 directly measured with wavemeters, which are a kind of interferometers. The typical approach is to compare the the periods of interference patterns between some input light to be tested and a reference light source with accurately known optical frequency.

Although sophisticated wavemeters can be extremely accurate, the indirect evaluation of the wavelength from a measured optical frequency and the vacuum velocity of light can still 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. Essentially, phase matching means achieving a certain phase relationship between interacting waves over a substantial propagation distance. 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:

## More to Learn

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