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# Zero Dispersion Wavelength

Author: the photonics expert

Acronym: ZDW

Definition: a wavelength where the group velocity dispersion of a fiber or a material is zero

Units: m

Formula symbol: <$\lambda_0$>

The zero dispersion wavelength can be defined either for an optical material or for a waveguide (e.g., an optical fiber), and there is an important difference between those areas.

## Zero Dispersion Wavelength of a Material

A zero dispersion wavelength of an optical material is a wavelength where the group velocity dispersion (second-order chromatic dispersion)

$$k'' \equiv \frac{{{\partial ^2}k}}{{\partial {\omega ^2}}}$$

is zero. Here, <$k$> is the frequency-dependent wavenumber

$$k = \frac{2\pi \: n}{\lambda} = \frac{n \: \omega}{c}$$

with the refractive index <$n$> and the vacuum wavelength <$\lambda$>.

Note that this is not the wavelength where the wavelength (or optical frequency) derivative of the refractive index vanishes, since <$k$> depends on the frequency not only through the refractive index, but also directly, as the previous equation shows.

Many materials have only one zero dispersion wavelength within the transparency region, with normal dispersion below that wavelength and anomalous dispersion for longer wavelengths. The group velocity then has its maximum at the zero dispersion wavelength. There, a light pulse travels with highest speed.

For example, fused silica has its zero dispersion wavelength at 1.27 μm. For other optical glasses, far shorter values are common (often in the visible range), and more than one zero dispersion wavelength can occur.

## Zero Dispersion Wavelength of a Waveguide (Fiber)

The concept can also be applied to optical fibers and other types of waveguides, but here one considers the phase constant <$\beta$> (imaginary part of the propagation constant) for a specific waveguide mode instead of the wavenumber <$k$>. Therefore, a zero dispersion wavelength of a waveguide is a vacuum wavelength where one has a zero crossing of <$\partial^2\beta/\partial \omega^2$>. It is generally different for each mode, but there may be only one guided mode (→ single-mode fibers).

Note that the phase constant does not only depend on the core material and the wavelength, but has some more complicated frequency dependence as a property of a waveguide mode. Therefore, the zero dispersion wavelength of a single-mode fiber, for example, may deviate substantially from that of the fiber core material.

For standard telecom fibers, which are based on germanosilicate glass, the zero dispersion wavelength is around 1.3 μm, which is close to that of the core material. However, by employing fiber designs with modified waveguide dispersion it is possible to shift the zero dispersion wavelength e.g. to the 1.5-μm region (→ dispersion-shifted fibers). Certain more sophisticated fiber designs can have two zero dispersion wavelengths – with anomalous dispersion between those and normal dispersion otherwise – or even more such wavelengths.

For photonic crystal fibers with small mode areas, which can exhibit particularly strong waveguide dispersion, the zero dispersion wavelength can be shifted e.g. into the visible spectral region, so that anomalous dispersion is obtained in the visible wavelength region, allowing for, e.g., soliton transmission. Photonic crystal fibers as well as some other fiber designs can exhibit two or even three different zero dispersion wavelengths.

Case Studies

## Case Study: Dispersion Engineering for Telecom Fibers

We explore different ways of optimizing refractive index profile for specific chromatic dispersion properties of telecom fibers, resulting in dispersion-shifted or dispersion-flattened fibers. This also involves automatic optimizations.

Achromatic optics are generally not made by operating optical elements at their zero dispersion wavelength. Instead, one usually compensates chromatic effects from different components, e.g. in achromatic doublet lenses.

## Effects of Vanishing Dispersion

When ultrashort pulses of light propagate in a medium with zero chromatic dispersion, dispersive pulse broadening is avoided. Similarly, operation of a telecom system around the zero dispersion wavelength greatly reduces dispersive broadening of optical signals. Note, however, that a short pulse or a telecom signal covers some finite wavelength range, so that one cannot have strictly have zero dispersion even if the center wavelength coincides with the zero dispersion wavelength. There is then still some amount of higher-order dispersion, which however often has only weak effects.

If the dispersive effects are weak, however, the signals become relatively sensitive to optical nonlinearities of the fiber, such as four-wave mixing, which can be phase matched under these conditions. It is therefore not always advantageous to operate in that regime; an improved approach is dispersion management in the form of alternatively using fibers with different signs of group velocity dispersion.

In other situations, phase matching of nonlinearities near the zero dispersion wavelength can be useful for nonlinear devices, such as optical parametric oscillators based on the <$\chi^{(3)}$> nonlinearity of optical fibers. Also, supercontinuum generation can lead to particularly broad optical spectra when the pump light has a wavelength near the zero dispersion wavelength.

## More to Learn

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## Questions and Comments from Users

2020-08-24

I have a waveguide with specific dimensions that has a certain chromatic dispersion. How to change this waveguide's structure in (width and height) such that the fundamental mode has a zero-dispersion wavelength in the near infrared, for example?