Group delay

Group delay is the time a signal takes to pass through something: a length of fiber, a waveguide, a filter, an amplifier. Precisely, it is the transit time of the envelope of a narrowband pulse or of a modulation impressed on a carrier, and it equals the rate at which the transmission phase changes with angular frequency. For a uniform guide of length $L$ it is $L n_g/c$, where $n_g$ is the group index. One kilometer of standard single-mode fiber ($n_g = 1.4682$ at 1550 nm) delays a signal by 4.90 µs; one centimeter of a 500 × 220 nm silicon strip ($n_g = 4.18$) by 139 ps.

Definition

If a component multiplies a field at angular frequency $\omega$ by $|t|e^{-i\phi(\omega)}$, the group delay is

$$\tau_g = \frac{d\phi}{d\omega}.$$

For a guide with propagation constant $\beta(\omega)$, $\phi = \beta L$, so

$$\tau_g = L\,\frac{d\beta}{d\omega} = \frac{L}{v_g} = \frac{L\,n_g}{c},$$

with $v_g$ the group velocity. In terms of wavelength, $\tau_g = -(\lambda^2/2\pi c)\,d\phi/d\lambda$, which is how it is computed from a phase measured on a wavelength sweep. The phase delay $\phi/\omega = L n_\text{eff}/c$ is a different quantity: for the same centimeter of silicon wire, with $n_\text{eff} = 2.446$, it is 82 ps against a group delay of 139 ps. The arrival time of a pulse envelope is given by the group delay.

Measurement

The direct method is time of flight: launch a short pulse and time its arrival, as an OTDR does for reflections along a link. The modulation phase-shift method is more precise. A carrier is intensity-modulated at frequency $f_m$ and the RF phase $\theta$ of the detected modulation is compared with that of the drive; then $\tau_g = \theta/(2\pi f_m)$, plus an integer number of modulation periods. At $f_m = 1$ GHz a change of 1 ps in group delay shifts the RF phase by 0.36°, so stepping the carrier wavelength and recording the phase change yields relative group delay versus wavelength with sub-picosecond resolution. Swept-laser interferometric instruments, often called optical vector analyzers, measure $\phi(\omega)$ of a device directly and differentiate it; optical frequency domain reflectometry applies the same principle in reflection.

Dispersion as a slope of group delay

When the group delay varies with wavelength, different spectral components of a pulse arrive at different times. Fiber engineers express this as the dispersion parameter

$$D = \frac{1}{L}\,\frac{d\tau_g}{d\lambda},$$

in ps/(nm·km). Standard fiber has $D \approx 17$ ps/(nm·km) at 1550 nm, so over 100 km two components 1 nm apart arrive 1.7 ns apart. Chromatic dispersion and group velocity dispersion cover the material and waveguide contributions and the conversion to $\beta_2$. The difference in group delay between the two polarization states of a fiber, the differential group delay, is the basis of polarization mode dispersion.

Filters, gratings and resonators

Any filter with a sharp amplitude response has a strongly varying phase near its edges, so group delay peaks near the band edges of thin-film and grating filters, and a signal passing close to an edge is distorted even when its amplitude is barely attenuated. A chirped fiber Bragg grating is designed to have a group delay that varies linearly with wavelength, opposite to that of the fiber, for dispersion compensation. Its quality is judged by the group-delay ripple: the deviation of measured $\tau_g(\lambda)$ from the ideal straight line, specified in picoseconds peak to peak, which arises from imperfections in the grating writing and causes signal-dependent penalties.

A ring resonator stores light near resonance, and its group delay there is many round-trip times. For a lossless all-pass ring the delay at resonance is about $4Q/\omega_0$, with $Q$ the loaded quality factor: $Q = 10^4$ at 1550 nm gives 33 ps, compared with a round-trip time of 0.88 ps for a 10 µm radius silicon ring. With loss, an overcoupled ring delays the signal at resonance by more than this, and the delay grows without limit as the coupling approaches critical coupling, where the transmission at resonance falls to zero; in an undercoupled ring the delay at resonance becomes negative.

Pitfalls

Group delay measured by the phase-shift method is ambiguous by whole modulation periods; absolute delay needs either a low modulation frequency or a separate time-of-flight measurement. Delay through electronics, connectors and patch cords must be calibrated out with a reference path. And near a sharp resonance the group delay can change by tens of picoseconds within a fraction of a nanometer, so the wavelength step must be smaller than the feature being measured.

Common questions

What is the group delay of optical fiber per kilometer?

About 4.90 µs/km for standard single-mode fiber at 1550 nm, slightly less at 1310 nm, where Corning quotes $n_g = 1.4676$. The speed of light in fiber entry covers latency and hollow-core fiber.

Can group delay be negative?

Yes, in narrow spectral regions such as an undercoupled resonance or an absorption line, where the phase slope reverses. The envelope peak of a smooth pulse can then appear early, but no information travels faster than light.

References: B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2010); A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007); C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).