Group velocity

Group velocity is the speed at which the envelope of an optical pulse moves through a medium. In a dispersive material it differs from the phase velocity, the speed of the carrier's phase fronts, and in the transparent regions of ordinary materials it is lower. In standard single-mode fiber at 1550 nm, with group index 1.4682, the envelope travels at $2.04\times {10}^8$ m/s and takes 4.90 μs to cross 1 km. In a silicon wire waveguide the group index is around 4.2, so light moves at under a quarter of $c$.

Definitions

The phase velocity of a single frequency is

$$v_p=\frac{\omega }{k}=\frac{c}{n(\omega )},$$

where $n$ is the refractive index. A pulse contains a band of frequencies around ${\omega }_0$; expanding $k(\omega )$ to first order about ${\omega }_0$ shows that the envelope moves at

$$v_g=\frac{d\omega }{dk}=\frac{c}{n_g},$$

with the group index

$$n_g=n+\omega \frac{dn}{d\omega }=n-\lambda \frac{dn}{d\lambda }\mathrm{.}$$

In a waveguide, $k$ is replaced by the propagation constant $\beta$ and $n$ by the effective index; the waveguide geometry then contributes to $n_g$ as well as the material. Below an absorption resonance, $dn\mathrm{/}d\lambda <0$ (normal dispersion), so $n_g>n$ and $v_g<v_p$.

Typical values at 1550 nm

The bulk values come from the Malitson Sellmeier fit for fused silica and the Li (1980) Sellmeier fit for crystalline silicon at room temperature. The SOI values are typical mode-solver results; the large gap between $n_{\text{eff}}$ and $n_g$ is mostly waveguide dispersion, since the mode shifts from silicon into the cladding as wavelength increases.

A worked contrast. Over 1 cm of that SOI waveguide a pulse is delayed by 140 ps, while a calculation using the phase index would give 81 ps. Delay-line and resonator designs must use the group index for this reason.

Measuring it

Group velocity is what time-of-flight and spectral-period measurements return.

• Resonator spectra: the free spectral range of a cavity or ring resonator is $\mathrm{\Delta }\nu =c\mathrm{/}(n_{g}L)$. A silicon ring of 10 μm radius ($L=62.8$ μm) with $n_g=4.2$ has an FSR of 1.14 THz, or 9.1 nm at 1550 nm. Measuring the FSR is the standard way to extract $n_g$ on a chip.
• Unbalanced interferometers: the fringe period of a Mach–Zehnder with path difference $\mathrm{\Delta }L$ is ${\lambda }^2\mathrm{/}(n_g\mathrm{\Delta }L)$.
• Time of flight and reflectometry: OTDR converts echo delay to distance with the group index, so an incorrect $n_g$ setting scales every reported distance.
• Low-coherence interferometry: the envelope peak of a white-light fringe pattern locates the group delay.

Interferometric phase measurements at a single wavelength, by contrast, return the phase index.

Dispersion of the group velocity

$v_g$ itself varies with wavelength. That variation, group velocity dispersion, spreads pulses: in standard fiber with dispersion $D=17$ ps/(nm·km) at 1550 nm, two components 0.1 nm apart separate by 17 ps after 10 km. The full treatment is under chromatic dispersion.

Slow and fast light

Near a sharp resonance, $dn\mathrm{/}d\omega$ can be very large. In electromagnetically induced transparency in an ultracold sodium gas, Hau and co-workers measured $v_g=17$ m/s in 1999, a reduction by a factor of $1.8\times {10}^7$. Photonic crystal waveguides near a band edge reach group indices of tens to about a hundred, which shortens delay lines and enhances light–matter interaction per unit length, at the cost of higher loss and narrow bandwidth.

With anomalous dispersion inside an absorption or gain feature, $v_g$ can exceed $c$ or become negative. This does not violate causality: the pulse is reshaped by the medium so that its peak appears early, while the front of the signal, which carries new information, still travels at no more than $c$.

Common questions

Is group velocity the speed of energy?

In a lossless medium, yes: the energy velocity equals $v_g$. In absorbing media near resonance the two can differ, and $v_g$ loses its simple meaning.

Why is the group index of a silicon waveguide larger than that of silicon?

Waveguide dispersion adds to material dispersion. As wavelength increases, more of the mode moves into the low-index cladding, so $n_{\text{eff}}$ falls steeply with $\lambda$ and the $-\lambda d{n}_{\text{eff}}\mathrm{/}d\lambda$ term is large.

Which index gives the speed of light in fiber?

For signal timing, the group index; see speed of light in fiber.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019); Agrawal, Nonlinear Fiber Optics (6th ed., 2019), Ch. 1; Hecht, Optics (5th ed., 2017), Ch. 7; L. V. Hau, S. E. Harris, Z. Dutton and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594–598 (1999).