Chromatic dispersion

Chromatic dispersion describes how different wavelength components of an optical pulse propagate at different group velocities, causing pulse broadening over distance.

The standard parameter for fibers is D, the dispersion coefficient in ps/(nm·km):

$$D \;=\; -\frac{2\pi c}{\lambda^2} \beta_2 \;=\; \frac{1}{c} \cdot \frac{d n_g}{d\lambda} \cdot 10^6,$$

with $\beta_2 = d^2\beta/d\omega^2$ being the group velocity dispersion parameter in ps²/km.

A pulse of spectral width $\Delta\lambda$ traveling through length $L$ acquires additional temporal width:

$$\Delta \tau \;=\; |D| \, L \, \Delta\lambda.$$

For a 10 Gb/s signal with $\Delta\lambda = 0.1$ nm propagating through 80 km of SMF-28 (D = 17 ps/(nm·km)):

$$\Delta \tau \;=\; 17 \times 80 \times 0.1 \;=\; 136 \text{ ps}.$$

A 136 ps pulse spread is significant compared to the 100 ps bit period.

Typical fiber dispersion at 1550 nm:

Fiber	$D$ at 1550 nm	$D$ at 1310 nm
SMF-28 (standard single-mode)	$+17$ ps/(nm·km)	$\sim 0$ (design zero)
Dispersion-shifted (DSF, ITU G.653)	$\sim 0$	$-17$
Non-zero dispersion-shifted (NZDSF, G.655)	$+2$ to $+6$	$-10$
Dispersion-compensating (DCF)	$-80$ to $-200$	n/a

Sign convention. Positive $D$ corresponds to longer wavelengths traveling more slowly (called anomalous dispersion for $D > 0$ in fibers — different from the spectroscopic sign convention). Negative $D$ is normal dispersion.

Dispersion compensation. Long-haul links combine positive-$D$ transmission fiber with negative-$D$ DCF or fiber Bragg grating compensators to net zero accumulated dispersion. For coherent transmission, dispersion is digitally compensated in the receiver via DSP, allowing transmission over arbitrary lengths of standard SMF without per-span optical compensation.

Material vs waveguide dispersion. Total dispersion is the sum of:

• Material dispersion — wavelength dependence of bulk refractive index (silica has $D_\text{material} \approx +20$ ps/(nm·km) at 1550 nm)
• Waveguide dispersion — wavelength dependence of guided-mode confinement (depends on fiber/waveguide design, typically $-3$ to $-5$ for standard SMF)

These add to produce the net dispersion. Engineering waveguide dispersion to cancel material dispersion is how dispersion-shifted fibers achieve zero D at desired operating wavelengths.

For SOI waveguides, waveguide dispersion typically dominates and is much larger than fiber dispersion: $|D|$ values of 1,000 to 10,000 ps/(nm·km) are common, important for short-distance integrated photonic links.