Chapter 2.13.2 - Chromatic Dispersion

2.13.2 Chromatic Dispersion

The refractive index and the propagation constant of the fiber medium are functions
of the wavelength. Thus, assuming no modal dispersion, the travel time of each
wavelength over a unit length of fiber depends on these two parameters. Thus, as a
pulse travels in the fiber, each wavelength travels at different speed and the pulse
broadens. This is termed chromatic dispersion (Figure 2.13).

Therefore, chromatic dispersion is the rate of change of the group delay with
wavelength. The general propagation constant β may be expanded in a Taylor series:

In the series, the term β₁ is the inverse of the group velocity, and β₂ is the group velocity
dispersion, which causes pulse broadening.

In general, the photonic impulse response, H(t), of a dispersive material can be
expressed by

where z is the optical path distance in the dispersive medium, and β' is the second
derivative of the propagation constant with respect to frequency ω (or the second-
order dispersion coefficient).

Chromatic dispersion consists of two contributions, material dispersion and
wavelength dispersion, and by some sources is also termed waveguide dispersion.
Dispersion is measured in ps/nm-km (i.e., delay per wavelength variation and per
fiber length).

Figure 2.13. Fundamentals of chromatic dispersion.

Material dispersion is due to the dependence of the dielectric constant, ε, or the
refractive index, n, on frequency, ω. Thus, the propagation characteristics of each
wavelength in a fiber are different. Different wavelengths travel at different speeds
in the fiber, which results in dispersion due to the material.

Material dispersion is most significant and is characterized by a parameter M,
defined as the derivative of group index, N, with respect to wavelength, λ:

where n is the refractive index, λ is the wavelength, c is the speed of light in free
space, and N = n – λ(dn/dλ).

Consider a medium with a region in which the distribution of n changes varies
nonlinearly; then the group and phase velocity change accordingly. Now, if a narrow
light pulse that consists of a narrow range of wavelengths is launched in a
medium, each individual wavelength arrives at the end of the fiber at a different time.
The result is a dispersed pulse due to material dispersion.

Waveguide dispersion or wavelength dispersion is the contribution due to nonlinear
dependence of the propagation constant on frequency, ω. Wavelength dispersion
is explained as follows.

Assume a narrow optical impulse that consists of a narrow spectral range. Consider
two wavelengths λ₁ and λ₂ in the same impulse. We assume that both wavelengths
travel (along the core of the fiber) in a straight path, but λ₁ travels faster
than λ₂(λ₁< λ₂) due to the nonlinear dependence of the propagation constant on
frequency ω (and on wavelength).

Wavelength dispersion has a different sign than material dispersion (Figure
2.14), although material dispersion is the major contributor. The counteracting action
of wavelength dispersion to material dispersion slightly ameliorates the com-

Figure 2.14. Two contributors to chromatic dispersion: material and wavelength.

bined effect. Negative dispersion implies that shorter wavelengths travel slower
than longer wavelengths. As a consequence, fiber with negative dispersion can be
used to compensate for positive dispersion (see Section 2.13.x).

The travel time t for a group velocity v_g over a length of fiber L is

τ = L/v_g

where β is the propagation constant, β' is the first derivative with respect to ω, and
is the partial derivative.

The variation of τ with respect to ω,, is

where β' is the second derivative with respect to ω.

For a signal with a spectral width Δω,

That is, the pulse spread Δτ (chromatic dispersion) depends on the second derivative
with respect to ω, β', and it is proportional to the length of the fiber, L, and the
spectral range Δω.

Based on this, a group velocity dispersion (GVD) coefficient, D, is defined as
the variation of travel time due to the wavelength variation per unit length of fiber,
L:

The coefficient D is also known as chromatic dispersion coefficient and it gives a
measure of the group delay rate change with wavelength. Consequently, in communications,
the chromatic dispersion of a fiber can be found by measuring the time
delay of wavelength components that constitute an optical pulse.

It follows that

However,

and

and thus

and

Finally, the pulse spread, or chromatic dispersion, is expressed by ( has been replaced
by Δ):

where Δλ is the optical spectral width of the signal (in nm units).

Clearly, the pulse spread due to dispersion imposes a limitation on the maximum
bit rate, known as the dispersion penalty, which is measured in dB. The dispersion
penalty is related to the maximum allowable delay (a fraction ε of the bit period T )
before severe signal degradation and unacceptable bit error rate (BER) occur. A 1
dB dispersion penalty corresponds to a bit-period fraction of approximately 0.5. At
ultrahigh bit rates, 0.5 T may be a few picoseconds.

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Chapter 2.13.2 - Chromatic Dispersion

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