Speed of light in fiber

Light in standard single-mode fiber travels at about 204,000 km/s, 68% of its speed in vacuum, so each kilometer of fiber adds 4.90 µs of one-way delay and 100 km adds just under 1 ms of round trip. The speed that matters for a signal, a pulse or a timing measurement is the group velocity, $v_g=c\mathrm{/}{n}_g$, where $c$ = 299,792,458 m/s exactly and $n_g$ is the group index of the guided mode, which Corning specifies as 1.4682 at 1550 nm and 1.4676 at 1310 nm for SMF-28 Ultra fiber. The group index is larger than the ordinary refractive index of silica, 1.444 at 1550 nm, because the index changes with wavelength. Hollow-core fiber, which guides light in air, comes within a fraction of a percent of the vacuum delay of 3.336 µs/km.

Speed of light in fiber: formula

A signal crossing a length $L$ of fiber arrives after

$$t=\frac{L}{v_g}=\frac{n_{g}L}{c},$$

so the delay per unit length is $n_g\mathrm{/}c$. In vacuum $1\mathrm{/}c$ is 3.336 µs per kilometer; with $n_g$ = 1.4682 the group velocity is 204,190 km/s and the delay 4.897 µs/km. The value of $c$ is exact because the SI defines the meter through it.

Latency per kilometer

For SMF-28 Ultra fiber the delay is 4.895 µs/km at 1310 nm and 4.897 µs/km at 1550 nm, 4.90 µs/km at either wavelength to three figures. The round trip is twice the one-way delay:

A 10 km data-center interconnect therefore adds 49 µs of one-way latency, and a 6,000 km transoceanic route adds 29.4 ms before any equipment delay; the rule of thumb of 1 ms of round trip per 100 km is 2% high. The figures are for the fiber alone: transceivers, amplifiers, switches and forward error correction add their own delays, and a cable route is longer than the straight line between its end points. The Fiber Latency Calculator gives the one-way and round-trip delay for any route length and group index, with the hollow-core comparison, the length an OTDR reports for a given index setting, and the spread in arrival time from dispersion.

Group index vs refractive index

The refractive index $n$ sets the phase velocity $c\mathrm{/}n$, the speed of the crests of a single-frequency wave. A signal occupies a band of frequencies, and its envelope travels at the group velocity, set by the group index

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

In silica the index falls as the wavelength rises, so $n_g$ exceeds $n$: at 1550 nm fused silica has an index of 1.4440 and a group index of 1.4626, from Malitson's dispersion formula. The guided mode of a single-mode fiber travels partly in the germanium-doped core, whose index is higher, and its effective index falls faster with wavelength than the material index does as the mode spreads into the cladding, so both its effective index and its group index exceed those of pure silica. A step-index model of the fiber with an 8.2 µm core, a 0.36% index step and the dispersion of silica gives an effective index of 1.4463 at 1550 nm and group indices of 1.4676 at 1310 nm and 1.4683 at 1550 nm, within 0.0001 of Corning's values.

Only the group velocity describes when a signal or a pulse arrives; the phase velocity governs interference and phase matching.

Hollow-core fiber

Hollow-core fiber guides light in air, whose group index at 1550 nm is 1.0003 (Ciddor 1996). Confinement in the hollow core raises the group index of the guided mode slightly above that of air, and the delay is about 3.34 µs/km, close to the vacuum value of 3.336 µs/km. Against solid fiber the saving is about 1.56 µs/km, 32% of the delay, or 156 µs one way on a 100 km route, which has made hollow-core fiber attractive for latency-sensitive links between data centers and between financial exchanges. Hollow-core fiber: below the Rayleigh floor covers its loss and deployment.

Where fiber latency matters

Latency budgets for trading networks, 5G fronthaul and distributed data centers count fiber delay per kilometer. Reflectometers such as the OTDR convert round-trip time into distance using the group index, 102 m of fiber per microsecond of round trip at $n_g$ = 1.4682, so a wrong index setting gives a proportionally wrong length: an instrument set to 1.4700 for fiber whose group index is 1.4682 places the end of a 10 km fiber 12 m short. Pulse timing in fiber sensors, delay lines and time-transfer links depends on the group index too, and so does the temperature sensitivity of the delay, which comes mainly from the change of the group index with temperature and partly from thermal expansion. Silica's thermo-optic coefficient, $8.5\times {10}^{-6}$ K⁻¹ (Leviton and Frey 2006), adds 28 ps of delay per kilometer per kelvin, and thermal expansion about 3 ps more. The group index changes somewhat faster with temperature than the index itself, so these two terms are a lower bound: a 100 km link that warms by 10 K delays its signals by more than 30 ns, and precision time-transfer links therefore measure and compensate the delay continuously. Signals at different wavelengths also travel at slightly different speeds because of chromatic dispersion, about 17 ps/(nm·km) near 1550 nm in standard fiber (Agrawal): over 100 km, two channels 10 nm apart arrive 17 ns apart.

Measurement

The group index of a fiber is measured by timing a pulse through a known length, or by measuring the length independently and reading the delay from an OTDR, whose group-index setting is adjusted until the two agree. For a precise value the phase delay of a modulated signal through the fiber is measured with a network analyzer as the modulation frequency or the wavelength is stepped, the phase-shift method that also measures chromatic dispersion. Between the 1310 and 1550 nm windows the group index of SMF-28 Ultra fiber differs by 0.0006, which is 2.0 ns of arrival time per kilometer.

Common questions

How fast does light travel in optical fiber?

About 204,000 km/s in standard single-mode fiber, 68% of its speed in vacuum. The speed is $c\mathrm{/}{n}_g$, and the group index $n_g$ of SMF-28 Ultra fiber is 1.4682 at 1550 nm.

What is the latency of fiber per kilometer?

4.90 µs per kilometer one way, or 9.79 µs per kilometer of round trip. Over 100 km the round trip is 0.979 ms, which gives the rule of thumb of about 1 ms per 100 km, before any delay in the equipment.

Why is light slower in fiber than in vacuum?

The electric field of the light polarizes the glass, and the radiation from the oscillating dipoles combines with the incident wave into a wave whose crests travel at $c\mathrm{/}n$, slower than in vacuum by the factor $n$ = 1.444 at 1550 nm. A signal travels more slowly still, at $c\mathrm{/}{n}_g$, because the index falls with wavelength; the group index of standard fiber is 1.468.

Is light faster at 1310 nm or 1550 nm?

Slightly faster at 1310 nm, close to the zero-dispersion wavelength of standard fiber, where the group delay is smallest. With group indices of 1.4676 and 1.4682 in SMF-28 Ultra fiber, a 1310 nm signal arrives 2.0 ns per kilometer earlier, 0.2 µs over 100 km.

How much faster is hollow-core fiber?

Light in an air-guiding hollow-core fiber takes about 3.34 µs/km against 4.90 µs/km in solid fiber, a saving of about 1.56 µs/km, or 32% of the delay. Over 100 km that is 156 µs one way and 312 µs of round trip.

References: Corning SMF-28 Ultra optical fiber product information (effective group index); Corning SMF-28 optical fiber product information (core diameter and refractive index difference); BIPM, The International System of Units (SI brochure), 9th ed. (2019); I. H. Malitson, "Interspecimen comparison of the refractive index of fused silica," Journal of the Optical Society of America 55, 1205 (1965); P. E. Ciddor, "Refractive index of air: new equations for the visible and near infrared," Applied Optics 35, 1566 (1996); D. B. Leviton and B. J. Frey, "Temperature-dependent absolute refractive index measurements of synthetic fused silica," Proc. SPIE 6273, 62732K (2006); G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2010), ch. 2.