Optical Bit Error Rate

Chapter 2.7.2 - Phase and Group Velocity

2.7.2   Phase and Group Velocity

A monochromatic (single ω or λ) wave that travels along the fiber axis is described
by

 E(t, x) = A exp[j(ωtβx)]

where A is the amplitude of the field, ω = 2πf, and β is the propagation constant.

Phase velocity, vφ, is defined as the velocity of an observer that maintains constant
phase with the traveling field, that is, ωtβx = constant.

Replacing the traveled distance x within time t, x = vφt, then the phase velocity
of the monochromatic light in the medium is

 vφ = ω/β

When a signal is transmitted in a dispersive medium, it is necessary to know its
speed of propagation. A continuous sine wave does not provide any meaningful
information, as a real optical signal consists of a band of frequencies in a narrow
spectrum. Moreover, each frequency component in the band travels (in the medium)
with slightly different phase. This is explained mathematically as follows.

Consider an amplitude-modulated optical signal traveling along a fiber:

 eAM(t) = E[1 + m cos(ω1t)]cos(ωct)

Where E is the electric field, m is the modulation depth, ω1 is the modulation frequency,
ωc is the frequency of light (or carrier frequency), and ω1 << ωc.

Trigonometric expansion of the above expression results in three frequency
components with arguments:

 ωc,ωcω1,andωc + ω1

Each component travels along the fiber at slightly different phase velocity (βc, βc
Δβ, βc + Δβ, respectively), accruing a different phase shift. Eventually, all three
components form a spreading envelope that travels along the fiber with a phase velocity:

 

where denotes a partial derivative.

Group velocity, vg = c/ng, considers a group of frequencies within a band, ω0
Δω/2 < ω < ω0 + Δω/2. The frequency band Δω is considerably smaller than the
carrier frequency ω0. The group velocity is defined as the velocity of an observer
that maintains constant phase with the group traveling envelope. Mathematically,
the group velocity is defined as the derivative of the frequency ω with respect to the
propagation constant β:

 

where β′ is the first partial derivative with respect to ω (remember that in this case
ω in the frequency band is not purely monochromatic). Notice that the group velocity
is a function of frequency; this will be useful in the discussion on dispersion.


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