Group velocity

In a dispersive optical medium, two distinct velocities describe wave propagation:

Phase velocity is the speed at which a single-frequency wave's phase fronts propagate:

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

where $n(\omega)$ is the refractive index at frequency $\omega$. For an optical pulse with a center frequency $\omega_0$ and finite bandwidth, $v_p$ describes the propagation of the underlying carrier wave.

Group velocity is the speed at which the pulse envelope propagates:

$$v_g \;=\; \frac{d\omega}{dk} \;=\; \frac{c}{n_g} \;=\; \frac{c}{n + \omega (dn/d\omega)} \;=\; \frac{c}{n - \lambda (dn/d\lambda)},$$

where $n_g$ is the group index.

For typical optical materials in the visible/near-IR:

• $dn/d\lambda < 0$ (normal dispersion)
• $n_g > n$
• $v_g < v_p$ (pulse envelope moves slower than phase fronts)

Numerical examples at 1550 nm:

Medium	$n$	$n_g$	$v_p / c$	$v_g / c$
Vacuum	1.000	1.000	1.000	1.000
Silica fiber	1.467	1.470	0.682	0.680
Silicon (bulk)	3.480	3.700	0.287	0.270
SOI 220×500 strip (TE)	2.44 (n_eff)	4.30	0.410	0.233

Why the distinction matters.

Phenomenon	Velocity
Interferometer fringe spacing	$v_p$
Pulse propagation time	$v_g$
Resonator FSR	$v_g$ (via $n_g$)
Time-of-flight ranging	$v_g$
Refractive index measurement	$v_p$
Photon lifetime in cavity	$v_g$ (via $n_g$)

Most photonic measurements (LIV pulse timing, OTDR, resonator FSR, ring resonator round-trip time) involve $v_g$ — the envelope velocity. Refractive-index measurements (interferometric phase) involve $v_p$.

Superluminal phenomena. In some media — near absorption resonances or in metamaterials — $dn/d\omega < 0$ can produce $v_g > c$. This does not violate causality because:

1. Group velocity is not the same as signal velocity
2. The envelope of a pulse in such media is non-classical (reshaping rather than translation)
3. Information cannot propagate faster than $c$ regardless of $v_g$

Slow light. Conversely, $v_g \ll c$ has been observed in cold atomic vapors (electromagnetically induced transparency), photonic crystals near band edges, and Brillouin amplifiers near gain resonances. Reductions of group velocity by factors of $10^7$ have been demonstrated. Practical slow-light applications include compact delay lines and quantum memory.

The dispersion of $v_g$ with frequency (GVD parameter $\beta_2 = d^2 \beta / d\omega^2$) produces pulse broadening over distance — see chromatic dispersion.