Group velocity dispersion (GVD)

Group velocity dispersion (GVD) is the dependence of the group velocity of light on its frequency. A pulse is a packet of many frequencies; in a medium with GVD those frequencies travel at slightly different speeds, so the pulse spreads out in time as it propagates and, if it started with a frequency sweep, may first narrow before it spreads. In optical fiber GVD is the mechanism behind chromatic dispersion, and it sets how far a pulse of a given width can travel before neighboring bits overlap.

The parameters β₂ and D

Expanding the propagation constant $\beta (\omega )=n(\omega )\omega \mathrm{/}c$ about the carrier frequency ${\omega }_0$ gives ${\beta }_1=d\beta \mathrm{/}d\omega =1\mathrm{/}{v}_g$, the inverse group velocity, and the GVD parameter

$${\beta }_2=\frac{d^2\beta }{d{\omega }^2}=\frac{d}{d\omega }\text{\hspace{0.17em}⁣}\left(\frac{1}{v_g}\right)$$

which is the change in group delay per unit length per unit angular frequency, conventionally in ps²/km. Fiber engineers use the dispersion parameter $D=d{\beta }_1\mathrm{/}d\lambda$, the change in group delay per unit length per unit wavelength, in ps/(nm·km). The two carry the same information and convert at a given wavelength $\lambda$ by

$${\beta }_2=-\frac{D{\lambda }^2}{2\pi c}$$

so at 1550 nm $D=1$ ps/(nm·km) corresponds to ${\beta }_2=-1.275$ ps²/km. The next term of the expansion, ${\beta }_3=d^3\beta \mathrm{/}d{\omega }^3$, follows from $D$ and the dispersion slope $S=dD\mathrm{/}d\lambda$:

$${\beta }_3={\left(\frac{{\lambda }^2}{2\pi c}\right)}^2(S+\frac{2D}{\lambda })$$

It matters for pulses shorter than about a picosecond and at the zero-dispersion wavelength, where ${\beta }_2$ vanishes.

The sign carries the physics. With ${\beta }_2>0$ ($D<0$), called normal dispersion, lower frequencies travel faster; this is the case in bulk silica below 1273 nm and in standard fiber below about 1310 nm. With ${\beta }_2<0$ ($D>0$), anomalous dispersion, higher frequencies travel faster; this is the case in standard fiber across the C-band, and it is the regime in which self-phase modulation can balance dispersion to form an optical soliton.

For standard single-mode fiber the ITU-T G.652 dispersion formula with ${\lambda }_0=1310$ nm and $S_0=0.092$ ps/(nm²·km) gives, at 1550 nm, $D=17.46$ ps/(nm·km), $S=0.058$ ps/(nm²·km), ${\beta }_2=-22.27$ ps²/km and ${\beta }_3=0.131$ ps³/km. The chromatic dispersion entry gives the formula and the values for other fiber types.

Dispersion length and Gaussian pulse broadening

A Gaussian pulse with field envelope $\exp [-(1+iC)T^2\mathrm{/}2T_0^2]$ has half-width $T_0$ at the $1\mathrm{/}e$ intensity point, related to the full width at half maximum by $T_{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}}=2\sqrt{\ln 2}{T}_0\approx 1.665T_0$, and a linear frequency sweep set by the chirp parameter $C$ (see frequency chirp). Its natural length scale in a dispersive medium is the dispersion length

$$L_D=\frac{T_0^2}{\mathrm{\mid }{\beta }_2\mathrm{\mid }}$$

and after a distance $z$, with ${\beta }_3$ neglected, the width $T_1$ follows from

$$\frac{T_1^2}{T_0^2}={(1+C\xi )}^2+{\xi }^2$$ $$\xi =\frac{{\beta }_{2}z}{T_0^2}$$

For an unchirped pulse this reduces to $T_1\mathrm{/}{T}_0=\sqrt{1+(z\mathrm{/}{L}_D{)}^2}$: the width grows by $\sqrt{2}$ in one dispersion length and linearly with distance beyond it. When $C{\beta }_2>0$ the chirp adds to the dispersive spread and the pulse broadens faster. When $C{\beta }_2<0$ the pulse first compresses, reaching a minimum width $T_0\mathrm{/}\sqrt{1+C^2}$ at $z=L_D\mathrm{\mid }C\mathrm{\mid }\mathrm{/}(1+C^2)$, returns to its initial width at twice that distance, and broadens thereafter.

As a worked case, take a 20 ps FWHM pulse at 1550 nm in the G.652 fiber above. Then $T_0=12.01$ ps and $L_D=6.48$ km. Without chirp the pulse is 64.9 ps wide after 20 km. With $C=-2$, so that $C{\beta }_2>0$, it is 156 ps wide at 20 km. With $C=+2$ it compresses to 8.94 ps FWHM at 2.59 km, is back to 20 ps at 5.18 km, and reaches 121 ps at 20 km. A 10 ps pulse has a dispersion length of only 1.62 km, since $L_D$ scales as the square of the width.

Dispersion-limited bit rate

Broadening turns into intersymbol interference once the pulse spreads beyond its bit slot. A widely used criterion (Agrawal) keeps the RMS width of the broadened pulse below a quarter of the bit period $1\mathrm{/}B$. For a source whose spectrum is narrow compared with the modulation bandwidth, and an optimally sized unchirped Gaussian input, this gives

$$B\sqrt{\mathrm{\mid }{\beta }_2\mathrm{\mid }L}\le \frac{1}{4}$$

and for a source whose RMS spectral width ${\sigma }_{\lambda }$ is much larger than the modulation bandwidth, such as a multimode Fabry-Perot laser, it gives

$$BL\mathrm{\mid }D\mathrm{\mid }{\sigma }_{\lambda }\le \frac{1}{4}$$

In the G.652 fiber at 1550 nm the narrow-source limit allows 5.9 Gb/s over 80 km, and a 10 Gb/s signal reaches 28 km; a broad source with ${\sigma }_{\lambda }=1$ nm is held to a bit-rate–distance product of 14.3 (Gb/s)·km. These are criteria for uncompensated intensity-modulated links with no equalization, and real transceivers are specified by their measured tolerance to accumulated dispersion in ps/nm. Two things lift the limit in practice. Datacenter links run in the O-band near the zero-dispersion wavelength, where $\mathrm{\mid }D\mathrm{\mid }$ is a few ps/(nm·km), and the Datacenter Link Budget Explorer works out the resulting spread per lane. Coherent C-band links keep the full optical field and undo the dispersion with a digital filter, which the article What the Coherent DSP Actually Does describes; the dispersion compensation entry covers the optical methods that preceded it.

GVD in integrated waveguides

In a high-index-contrast waveguide the geometry contributes as much GVD as the material, and the waveguide term can have either sign. Silicon channel waveguides a few hundred nanometers across can be made anomalous at 1550 nm with $\mathrm{\mid }{\beta }_2\mathrm{\mid }$ far larger than in fiber (Turner et al., 2006), and thick silicon nitride waveguides are dimensioned for weak anomalous dispersion so that they support soliton microcombs and octave-spanning frequency combs (Okawachi et al., 2011). Because the dispersion length is so much shorter in these structures, dispersion engineering of the cross-section is a design step in nonlinear integrated photonics, while for data transmission over millimeters of chip it is negligible.

The Dispersion and Pulse Broadening Calculator converts $D$ and $S$ to ${\beta }_2$ and ${\beta }_3$ at a chosen wavelength, computes the dispersion length and the broadened width of a chirped Gaussian pulse (including the distance and width of maximum compression when $C{\beta }_2<0$), and evaluates both bit-rate limits above for a chosen fiber and length.

References: G. P. Agrawal, Nonlinear Fiber Optics, 6th ed. (Academic Press, 2019), chapters 1 and 3. G. P. Agrawal, Fiber-Optic Communication Systems, 5th ed. (Wiley, 2021), chapter 2. ITU-T Recommendation G.652, Characteristics of a single-mode optical fibre and cable. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019). A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Optics Express 14, 4357 (2006). Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, "Octave-spanning frequency comb generation in a silicon nitride chip," Optics Letters 36, 3398 (2011).