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That derivative is always evaluated at a certain angular optical frequency – for example, at the center frequency of a laser pulse when considering the impact of chromatic dispersion on that pulse. If the group delay dispersion is independent of optical frequency, we have pure second-order dispersion and no higher-dispersion. Otherwise, third-order and other higher-order dispersion may be calculated via frequency derivatives of group delay dispersion.

Zenghu Chang, Li Fang, Vladimir Fedorov, Chase Geiger, Shambhu Ghimire, Christian Heide, Nobuhisa Ishii, Jiro Itatani, Chandrashekhar Joshi, Yuki Kobayashi, Prabhat Kumar, Alphonse Marra, Sergey Mirov, Irina Petrushina, Mikhail Polyanskiy, David A. Reis, Sergei Tochitsky, Sergey Vasilyev, Lifeng Wang, Yi Wu, and Fangjie Zhou Adv. Opt. Photon. 14(4) 652-782 (2022)

Bobo Gu, Chujun Zhao, Alexander Baev, Ken-Tye Yong, Shuangchun Wen, and Paras N. Prasad Adv. Opt. Photon. 8(2) 328-369 (2016)

It seems most suppliers have extensive catalogs of low-GDD mirrors, but not low-GDD lenses. Is there a reason why GDD is less of a problem in the case of lenses vs mirrors?

For example, a fiber with zero dispersion slope (wavelength-independent <$D_{\lambda }$>) would generally have some non-zero third-order dispersion.

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Mirrors have various design parameters (layer thickness parameters) which can be optimized concerning GDD, but that is not the case for lenses. By the way, for lenses it is not even straight-forward to define the GDD, as the output is distorted by chromatic aberrations.

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That also leads to a kind of pulse broadening, but more complicated than for GDD only. Such things can be most conveniently studied with suitable software, e.g. RP Fiber Power.

Basanta Bhaduri, Chris Edwards, Hoa Pham, Renjie Zhou, Tan H. Nguyen, Lynford L. Goddard, and Gabriel Popescu Adv. Opt. Photon. 6(1) 57-119 (2014)

Group velocity dispersionandgroupdelaydispersion

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The fundamental unit of group delay dispersion is s2 (seconds squared), but in practice it is usually specified in units of fs2 or ps2. (Note that 1 ps = 1000 fs, thus 1 ps2 = 1,000,000 fs2.) Positive (negative) values correspond to normal (anomalous) chromatic dispersion. For example, the group delay dispersion of a 1-mm thick silica plate is +35 fs2 at 800 nm (normal dispersion) or −26 fs2 at 1500 nm (anomalous dispersion). Another example is given in Figure 1.

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If an optical element has only second order dispersion, i.e., a frequency-independent <$D_2$> value and no higher-order dispersion, its effect on an optical pulse or signal can be described as a change of the spectral phase:

Diffraction gratings were discovered during the 18th century, and they are now widely used in spectrometry analysis, with outstanding achievements spanning from the probing of single molecules in biological samples to the analysis of solar systems in astronomy. The fabrication of high-quality diffraction gratings requires precise control of the period at a nanometer scale. The discovery of lasers in the 1960s gave birth to optical beam lithography in the 1970s. This technology revolutionized the fabrication of diffraction gratings by offering highly precise control of the grating period over very large scales. It is surprising to see that a few years after, the unique spectral properties of diffraction gratings revolutionized, in turn, the field of high-energy lasers. We review in this paper the physics of diffraction gratings and detail the interest in them for pulse compression of high-power laser systems.

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That will probably depend on the context. For example, it may mean the matching of the arm lengths in an interferometer such that pulses going through the two arms can interfere with each other when getting recombined. Another context would be a synchronously pumped optical parametric oscillator, where the group delay of one resonator round-trip needs to be matched to the spacing of the pump pulses.

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Higher-order dispersion is often specified in the form of the dispersion slope, i.e., the wavelength derivative of <$D_{\lambda }$>. From that, the third-order dispersion can be calculated as follows:

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The O/E Land’s OEDMS-100 chromatic dispersion measurement system is based on high-performance tunable laser sources integrated with a vector network analyzer (VNA). It covers a wider dispersion range than other available products in the market. OEDMS-100 is available at various center wavelengths including 1030 (or 1064), 1310, 1550 and 2000 nm. The built-in Tunable Laser Source (TLS) and power meter can also be used individually with a user-friendly interface through the USB port.

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If two optical pulses travel through an optical element with a frequency-independent group delay dispersion <$D_2$>, and their center optical frequencies differ by <$\Delta \nu$>, their group delay differs by <$2\pi \: D_2 \: \Delta\nu$>.

For Gaussian pulses a constant GDD leads to a linear frequency chirp. Is this generally the case also for other pulse shapes? Would for instance a sech2-shaped pulse also exactly have a linear chirp for constant GDD?

Definition: the frequency dependency of the group delay, or (quantitatively) the corresponding derivative with respect to angular frequency

Group velocityand phasevelocity

Group velocity dispersionderivation

That depends on the mirror design and refractive index contrast. For a given design, this could be easily calculated with suitable thin-film coating software such as RP Coating.

There are various methods for measuring the GDD of an optical element. In the case of optical fibers, one may use the pulse delay technique, based on measuring the difference in propagation time (group delay) for light pulses with different center wavelengths. There are also methods based on interferometry. For details, see the article on chromatic dispersion.

Thorlabs manufactures a robust benchtop white light interferometer for characterizing reflective and transmissive dispersive properties of optics and coatings designed for ultrafast applications. The Chromatis™ dispersion measurement system covers 500 – 1650 nm, providing a means for measuring optics used for common femtosecond systems, including Ti:Sapphire systems as well as 1 µm and 1550 nm oscillators. The Chromatis compliments our ultrafast family of lasers, amplifiers, and specialized optics including nonlinear crystals, chirped mirrors, low GDD mirrors/beamsplitters, and dispersion compensating fiber.

Note that the group delay dispersion (GDD) always refers to some optical element or to some given length of a medium (e.g. an optical fiber). The GDD per unit length (in units of s2/m) is the group velocity dispersion (GVD).

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1Aix-Marseille Université, CNRS, Centrale Marseille, Institut Fresnel, UMR 7249, Domaine Universitaire de Saint Jérôme, 13013 Marseille, France

The group delay dispersion (also sometimes called second-order dispersion) of an optical element is a quantitative measure for chromatic dispersion. It is defined as the derivative of the group delay, or the second derivative of the change in spectral phase, with respect to the angular optical frequency:

An alternative way of specifying group delay dispersion is referring to the vacuum wavelength instead of the angular optical frequency. That leads to a value in units of ps/nm (picoseconds per nanometer), for example. It can be calculated from the GDD as defined above: