Group index (n_g)

The group index of a waveguide mode, or of a material, is the speed of light in vacuum divided by the group velocity, the speed at which a pulse, and the energy it carries, travels:

$$n_g=\frac{c}{v_g}=n_{\text{eff}}-{\lambda }_0\frac{d{n}_{\text{eff}}}{d{\lambda }_0}=n_{\text{eff}}+\omega \frac{d{n}_{\text{eff}}}{d\omega },$$

where $n_{\text{eff}}$ is the effective index and ${\lambda }_0$ the free-space wavelength; for a bulk material, $n_{\text{eff}}$ is replaced by the refractive index $n$. The group index differs from the effective index whenever the index changes with wavelength. In a 500 × 220 nm silicon strip waveguide at 1550 nm, $n_{\text{eff}}$ = 2.446 but $n_g$ = 4.18, so pulses travel at 71,700 km/s and 1 cm of waveguide delays them by 139 ps. The group index sets the spacing of resonances, the delay through a device and the conversion between a resonator's Q and its loss.

Why it differs from the effective index

Two effects add. The first is material dispersion: silicon's refractive index at 1550 nm is 3.476 but its group index is 3.59, and silica's are 1.444 and 1.463. The second is waveguide dispersion: at longer wavelengths the mode spreads further into the low-index cladding, so $n_{\text{eff}}$ falls with wavelength faster than the material index does. In a high-contrast silicon wire the two together give $d{n}_{\text{eff}}\mathrm{/}d{\lambda }_0=-1.12$ µm⁻¹, most of it from the waveguide, which lifts the group index to 4.18, above that of bulk silicon itself. In standard single-mode fiber, with an index step of only 0.36%, the mode's group index of 1.468 is close to silica's.

The group index of a bulk material follows from its Sellmeier equation; the Refractive Index Calculator gives it for six common optical materials.

Values

Computed values at 1550 nm for the fundamental TE-like mode unless stated, with oxide cladding on all sides:

The silicon and nitride rows were computed with a full-vectorial finite-difference mode solver from the bulk dispersion of silicon, silica and silicon nitride. The fiber's effective index is computed for a step-index core with a 0.36% index step, and its group index is Corning's specification for SMF-28. Group indices measured on fabricated silicon strips come out somewhat higher or lower, depending on the actual width, thickness and sidewall angle; values of 4.2 to 4.3 are commonly used for design.

What it governs

• Free spectral range. A resonator of round-trip length $L$ has resonances spaced by $c\mathrm{/}(n_{g}L)$ in frequency, or ${\lambda }_0^2\mathrm{/}(n_{g}L)$ in wavelength. A silicon ring of 10 µm radius has an FSR of 9.15 nm; using the effective index instead would predict 15.6 nm, 71% too wide, which is a common error in ring design and in loss extraction.
• Group delay. A section of length $L$ delays a signal by $n_{g}L\mathrm{/}c$: 139 ps per cm in the silicon strip above and 4.9 µs per km in standard fiber, the figure behind latency in fiber. The Fiber Latency Calculator gives the one-way and round-trip delay of a fiber link from its length and group index.
• Q and loss. The intrinsic quality factor of a ring converts to a propagation loss $\alpha =2\pi n_g\mathrm{/}(Q_i{\lambda }_0)$: an intrinsic Q of $3\times {10}^5$ in the silicon strip means 0.56 cm⁻¹, or 2.5 dB/cm. The photon lifetime of a cavity likewise depends on $n_g$.
• Dispersion. The change of $n_g$ with wavelength is the group velocity dispersion that broadens pulses and limits the bandwidth of delay lines.

Measuring it

The group index is usually measured from a free spectral range. An unbalanced Mach-Zehnder interferometer with an arm-length difference $\mathrm{\Delta }L$ has fringes spaced by ${\lambda }_0^2\mathrm{/}(n_g\mathrm{\Delta }L)$, so a measured spacing of 5.75 nm for $\mathrm{\Delta }L$ = 100 µm gives $n_g$ = 4.18. A ring of known circumference gives it in the same way, and so do the Fabry-Perot fringes of a waveguide between two reflecting ends, spaced by ${\lambda }_0^2\mathrm{/}(2n_{g}L)$; the same ripple, when it appears in a coupler test structure, is what de-embedding has to remove. Optical frequency-domain and low-coherence reflectometry measure the group delay, and so $n_g$, directly.

Measuring the group index from fringe spacing, and how it differs from the phase index other methods give, is covered in How to Measure Refractive Index.

Slow light

Structures with strong dispersion near a band edge, such as photonic crystal waveguides and chains of coupled resonators, reach group indices far above any material's. A high group index compresses a pulse in space and lengthens the interaction of light with the material, which is used to shorten modulators and enhance nonlinear effects, at the cost of narrow bandwidth, strong dispersion and higher loss per unit length.

Common questions

What is the group index?

The ratio of the speed of light in vacuum to the speed at which a pulse travels in a material or waveguide, $c\mathrm{/}{v}_g$. It equals the refractive or effective index minus the wavelength times its rate of change with wavelength.

Why is the group index larger than the refractive index?

Because in transparent materials at optical wavelengths the index falls as the wavelength rises (normal dispersion), which makes $-{\lambda }_{0}dn\mathrm{/}d{\lambda }_0$ positive. In a waveguide, confinement adds a further contribution of the same sign.

How do you calculate the group index from the FSR?

It is the square of the wavelength divided by the FSR times the round-trip length: $n_g={\lambda }_0^2\mathrm{/}({\text{FSR}}_{\lambda }L)$, with $L$ the circumference of a ring or the arm-length difference of an unbalanced Mach-Zehnder interferometer.

What is the group index of optical fiber?

About 1.468 at 1550 nm for standard single-mode fiber, which gives a delay of 4.9 µs per kilometer.

References: K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006). L. Chrostowski and M. Hochberg, Silicon Photonics Design (Cambridge University Press, 2015). B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019). Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Optics Express 10, 853 (2002), for the mode solver. T. Baba, "Slow light in photonic crystals," Nature Photonics 2, 465 (2008).