Dispersion compensation

Dispersion compensation is the cancellation of the chromatic dispersion a signal accumulates along a fiber link. Dispersion is linear in the optical field and fixed by the fiber, so its effect on the field can be undone by any element with equal and opposite accumulated dispersion: a length of fiber with negative $D$, a chirped grating, or, when the receiver captures the full field, a digital filter. The quantity to cancel is the accumulated dispersion $DL$ in ps/nm, which for an 80 km span of G.652 fiber at 1550 nm ($D=17.46$ ps/(nm·km) from the ITU-T formula with ${\lambda }_0=1310$ nm and $S_0=0.092$) is 1,397 ps/nm.

Dispersion-compensating fiber

Dispersion-compensating fiber (DCF) has a small core and a steep index profile that give it a large negative waveguide dispersion. The length that nulls a span of transmission fiber with dispersion $D$ and length $L$ is

so with $D_{\mathrm{D}\mathrm{C}\mathrm{F}}=-100$ ps/(nm·km), a round value for common designs, the 80 km span above needs 13.97 km of DCF, and with $-120$ it needs 11.64 km. The DCF is wound on a spool in a module at the amplifier site, usually between the two stages of an EDFA, whose gain has to make up the module's loss.

Canceling at one wavelength is not enough for a WDM system, because the transmission fiber's dispersion changes across the band. For the G.652 fiber above the slope at 1550 nm is $S=0.058$ ps/(nm²·km), and a DCF with no slope leaves a residual of −94.6 ps/nm at 1530 nm and +69.0 ps/nm at 1565 nm after each span, which grows to about 950 ps/nm over ten spans at the band edge. Full compensation across the band needs a DCF with the same relative dispersion slope $S\mathrm{/}D$, here 0.0033 nm⁻¹, which for $D_{\mathrm{D}\mathrm{C}\mathrm{F}}=-100$ means $S_{\mathrm{D}\mathrm{C}\mathrm{F}}=-0.33$ ps/(nm²·km); with that DCF the computed residual over 1530–1565 nm stays within 1.5 ps/nm per span. DCF also has a small effective area, which makes it more nonlinear per unit power than the transmission fiber, and in the 10 and 40 Gb/s era systems were designed around dispersion maps, a deliberate schedule of per-span under- or over-compensation chosen to limit nonlinear interaction between channels.

Chirped fiber Bragg gratings

A fiber Bragg grating whose period varies along its length reflects each wavelength from a different position, so the group delay of the reflection varies with wavelength. Used with an optical circulator, a grating whose delay changes by $-DL$ per nanometer compensates a span over that bandwidth. The grating length scales with the delay range: a round-trip delay of 1,397 ps over 1 nm of bandwidth needs about 14.3 cm of grating at a group index of 1.468. Gratings have low loss and negligible nonlinearity; their characteristic impairment is group-delay ripple, the small periodic error in the delay curve left by fabrication.

Electronic compensation

A coherent receiver recovers the optical field in amplitude and phase, and chromatic dispersion acts on that field as an all-pass filter with a phase quadratic in frequency. The receiver's DSP applies the inverse phase with a fixed digital filter whose length is set by how many symbols the dispersion spreads each symbol across. At a symbol rate $R_s$ the signal occupies about $\mathrm{\Delta }\lambda ={\lambda }^2{R}_s\mathrm{/}c$, so the spread in symbols is roughly $\mathrm{\mid }D\mathrm{\mid }L{\lambda }^2{R}_s^2\mathrm{/}c$: 46 symbols for 80 km at 64 GBd and 573 symbols for 1000 km. The article What the Coherent DSP Actually Does describes the filter and its cost. Because this compensation is exact and costs no optical loss, new coherent links are built without DCF, and running the fiber uncompensated also reduces nonlinear interaction between channels, which makes large-dispersion G.652 fiber preferable to low-dispersion fiber for coherent transmission.

Intensity-modulated links with direct detection cannot use the same trick, because the photodiode discards the phase and square-law detection makes dispersion a nonlinear impairment of the detected signal. They rely on staying within a dispersion budget: working near the zero-dispersion wavelength in the O-band, as datacenter optics do, keeping the source's chirp small, using feed-forward or decision-feedback equalizers that recover part of the penalty, and, for longer reach, optical compensation. For a direct-detection link the practical procedure is unchanged from the DCF era: total the accumulated dispersion in ps/nm at the worst-case wavelength, compare it with the transceiver's dispersion tolerance, and compensate or shorten the difference.

The Dispersion and Pulse Broadening Calculator gives the accumulated dispersion $DL$ of a chosen fiber and length at a chosen wavelength and the DCF length that nulls it, for a DCF whose dispersion is set in the calculator.

References: G. P. Agrawal, Fiber-Optic Communication Systems, 5th ed. (Wiley, 2021), chapter on dispersion management. ITU-T Recommendation G.652, Characteristics of a single-mode optical fibre and cable. S. J. Savory, "Digital filters for coherent optical receivers," Optics Express 16, 804 (2008). K. O. Hill et al., "Chirped in-fiber Bragg gratings for compensation of optical-fiber dispersion," Optics Letters 19, 1314 (1994).