4. Dispersion: Why Pulses Spread

If a 100-ps pulse goes into one end of a fiber and a 200-ps pulse comes out the other end, the link is effectively running at half the bit rate, even if the total power budget allowed twice the distance. Dispersion is the catch-all term for any mechanism that spreads pulses in time, and after attenuation it is the second-most-important fiber property.

4.1 Modal dispersion

In multimode fiber, different modes travel different effective path lengths and arrive at slightly different times. We covered this in Section 2.5: step-index multimode is dispersion-limited to maybe 20 MHz·km, and graded-index multimode to a few GHz·km. Single-mode fiber has zero modal dispersion by definition.

4.2 Material dispersion

Glass has a wavelength-dependent refractive index, $n(\lambda)$. So the phase velocity $c/n(\lambda)$ depends on wavelength. A pulse contains many wavelengths (its Fourier transform has finite width); each wavelength travels at a slightly different speed; the pulse spreads.

The relevant quantity is the group velocity dispersion parameter, with units of ps/(nm·km):

$D_M(\lambda) = -\frac{\lambda}{c} \frac{d^2 n}{d\lambda^2}.$

For standard SMF, $D_M$ goes through zero around 1310 nm. At 1550 nm, $D_M$ is about 17 ps/(nm·km), meaning a source with 1 nm spectral width spreads its pulses by 17 ps per kilometer.

4.3 Waveguide dispersion

Even at a single wavelength of a single material, the effective propagation speed in a fiber depends on how the field is distributed between core and cladding, and that distribution itself depends on wavelength. Waveguide dispersion adds an additional, fiber-geometry-dependent term to the dispersion. In standard SMF this term partly cancels material dispersion at 1310 nm (giving the famous zero-dispersion crossing), and adds to it at 1550 nm.

4.4 Total chromatic dispersion

The sum of material and waveguide dispersion is chromatic dispersion $D(\lambda)$. For standard SMF:

   D (ps/nm/km)
        │
     20 ┤                        ╱─
        │                     ╱
     10 ┤                  ╱
        │               ╱
      0 ┤────────╳───────────────────  λ
        │     ╱   ←1310 nm zero
   -10  ┤  ╱
        │
   1200 1300 1400 1500 1600 nm

Pulse spread per kilometer is

$\Delta\tau = D(\lambda) \cdot \Delta\lambda \cdot L$

where $\Delta\lambda$ is the source linewidth and $L$ is the length. For 10 Gbps NRZ over 100 km of standard SMF at 1550 nm with a DFB laser ($\Delta\lambda \approx 0.05$ nm), the spread is $17 \cdot 0.05 \cdot 100 = 85$ ps, comparable to the bit slot of 100 ps. You are bumping up against dispersion limits.

Two ways to fight chromatic dispersion in SMF:

• Dispersion-shifted fiber (G.653) moves the zero crossing to 1550 nm. Pulses do not spread, but four-wave-mixing nonlinearities in WDM are catastrophic.
• Dispersion-compensating fiber (DCF) or fiber Bragg gratings create negative dispersion to cancel positive dispersion accumulated upstream. Standard practice in pre-coherent long-haul systems.

In modern coherent systems, the receiver does it in software (Section 9). The DSP captures the optical field and applies an inverse-dispersion filter, undoing the spread digitally. This is one of the killer features of coherent transmission, and one of the reasons all dispersion-management hardware has been ripped out of submarine cables since 2015.

4.5 Polarization-mode dispersion (PMD)

A "single-mode" fiber actually supports two degenerate modes, one for each polarization of the electric field. In an ideal cylindrically symmetric fiber, both polarizations travel at the same speed. In a real fiber, residual ellipticity, internal stress, bending, and twisting break the symmetry. The two polarizations end up traveling at slightly different speeds. A pulse launched with mixed polarization spreads in time.

PMD is small per kilometer (around 0.1 ps/$\sqrt{\text{km}}$ for modern fiber) but it scales as $\sqrt{L}$ rather than $L$ because the splitting is random along the length and partially averages out.

PMD becomes a concern at 10 Gbps over hundreds of kilometers and is the bottleneck for 40 Gbps direct-detection systems. Coherent receivers fix it the same way they fix chromatic dispersion: capture both polarizations, apply a 2x2 MIMO equalizer in DSP, and let the math sort it out.