Group delay dispersion (GDD)

Group delay dispersion (GDD) measures how much the transit time of light through an element changes with frequency. It is the derivative of the group delay $\tau_g = d\phi/d\omega$ with respect to angular frequency, and so the second derivative of the spectral phase:

$$\mathrm{GDD} = \frac{d\tau_g}{d\omega} = \frac{d^2\phi}{d\omega^2}.$$

It is quoted in fs² (1 fs² = 10⁻³⁰ s²) for ultrafast optics and in ps² for fiber. Typical values run from a few tens of fs² for a thin window or one mirror bounce, through a few hundred fs² for the crystal and optics inside a femtosecond oscillator, to 10⁴–10⁷ fs² in the stretchers and compressors of amplifier systems.

GDD and GVD

GDD belongs to an element; group velocity dispersion (GVD, $\beta_2$) belongs to a material or waveguide and is quoted per unit length. For a uniform medium of length $L$,

$$\mathrm{GDD} = \beta_2\, L .$$

The fused silica Sellmeier equation gives $\beta_2$ = 36.16 fs²/mm at 800 nm, so a 10 mm window or lens adds

$$\mathrm{GDD} = 36.16 \times 10 = 362\ \mathrm{fs^2}.$$

The GDD values of all elements in a beam path add, which is what makes the quantity convenient for budgeting dispersion in a laser or a beamline. In fiber the same arithmetic applies at a larger scale: standard single-mode fiber at 1550 nm, with $\beta_2 = -22.27$ ps²/km, contributes −22,270 fs² per meter. The fiber community usually expresses this through the dispersion parameter $D$ of chromatic dispersion instead.

Sign convention

Positive GDD means that the group delay increases with frequency: blue components arrive after red ones. Transparent materials in their normal-dispersion region, including all common glasses and crystals in the visible and near infrared, have positive GDD. A pulse leaving such an element carries a positive (up) chirp, its instantaneous frequency rising from the leading edge to the trailing edge. Negative GDD, with blue ahead of red, comes from anomalous dispersion in a material or waveguide and from angular dispersion and multilayer coatings designed for it.

Pulse broadening

GDD leaves the spectrum unchanged and alters only the spectral phase. A transform-limited Gaussian pulse of FWHM duration $\tau_\text{in}$ that acquires a GDD emerges with duration

$$\tau_\text{out} = \tau_\text{in} \sqrt{1 + \left(\frac{4\ln 2\;\mathrm{GDD}}{\tau_\text{in}^2}\right)^{2}} .$$

For 362 fs² of fused silica:

The pulse lengthens by $\sqrt 2$ when GDD = $\tau_\text{in}^2/(4\ln 2)$: 144 fs² for a 20 fs pulse and 3,607 fs² for a 100 fs pulse. The scaling with the inverse square of duration follows from the time-bandwidth product: a 20 fs Gaussian pulse at 800 nm has a bandwidth of 22.1 THz (47 nm), and 362 fs² spreads the group delay across that bandwidth by about 50 fs, more than twice the original pulse length. A 100 fs pulse has one fifth of the bandwidth and is barely affected.

Higher orders

The spectral phase is expanded about the carrier frequency $\omega_0$:

$$\begin{aligned} \phi(\omega) = {} & \phi_0 + \tau_g\,\Delta\omega + \tfrac{1}{2}\mathrm{GDD}\,\Delta\omega^2 \\ & + \tfrac{1}{6}\mathrm{TOD}\,\Delta\omega^3 + \dots \end{aligned}$$

with $\Delta\omega = \omega - \omega_0$. The third-order dispersion (TOD), in fs³, makes the pulse asymmetric, with a tail of satellite pulses on one side. Fused silica at 800 nm has about 27.5 fs³/mm, so 275 fs³ in 10 mm. TOD and higher orders matter below about 30 fs and when large amounts of GDD are added and removed, as in chirped pulse amplification.

Compensation

Positive material GDD is canceled with elements of negative GDD:

• Prism pairs produce negative GDD from angular dispersion, tunable by moving one prism into or out of the beam, typically for hundreds to a few thousand fs².
• Grating pairs give large negative GDD, 10⁴–10⁷ fs², for amplifier systems, with a TOD that adds to that of the material.
• Chirped mirrors give a designed negative GDD per reflection, typically tens of fs², over bandwidths of hundreds of nanometers, with control of the higher orders.

The pulse compressor entry covers the designs and their trade-offs. Inside a mode-locked ultrafast laser, the net round-trip GDD is set slightly negative so that it balances the nonlinear phase of the Kerr medium.

Measurement

The GDD of an optic is measured by white-light interferometry, which records group delay against wavelength for differentiation, or inferred from the spectral phase that FROG or SPIDER retrieve before and after the element.

Common questions

What is the difference between GDD and GVD?

GVD is per unit length of material, in fs²/mm or ps²/km; GDD is the total for an element or path, in fs². GDD equals GVD multiplied by the length.

Does GDD change the pulse spectrum?

No. It changes only the spectral phase, so the pulse can be recompressed to its original duration by an element of opposite GDD, provided higher-order terms are also small.

What units is GDD expressed in?

fs² in ultrafast optics and ps² in fiber; 1 ps² = 10⁶ fs².

References: J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006); A. M. Weiner, Ultrafast Optics (Wiley, 2009); G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic Press, 2013); I. H. Malitson, "Interspecimen comparison of the refractive index of fused silica," J. Opt. Soc. Am. 55, 1205 (1965).