Optical Bit Error Rate

Chapter 2.10.2 - Paradoxes

2.10.2  Paradoxes

One consequence of this analysis is that the spectral width Δω of the signal and the
duration of the signal (or period) Δt set a minimum value:

 

The latter provides an estimation of the minimum spectral content of a signal as the
data rate increases, a limit that is useful in photonic transmission engineering. A
similar consequence is

 

Thus, a disturbance within a region Δx is related to a superposition of monochromatic
waves that lie within a spectral range Δk of the order 2πx.

Another consequence is that the product of the phase velocity u = ω/k and group
velocity ug is uug = c2. Since the group velocity in a medium is always ug < c, it
turns out that u > c, a result that is contradicts many theories.

Another consequence is that the dielectric constant of the medium ε(ω) can be
expressed in terms of the charge density Ne, the electron mass, m, and the difference
of frequency squares ω02ω2, where ω0 is the oscillation of charges (assuming
elastic forces between them) and ω is the frequency of an oscillating field:

 

The latter expresses the dielectric constant in terms of tuning or detuning between
two oscillations. When ω = 0 (that is, there is no oscillating field), then the latter relation
reduces to the static dielectric constant, ε > 1. When ω > ω0, then it reduces to
a dielectric constant, ε < 1. However, according to theory, for a frequency region ω
> ω0 and under the condition ω2 < ω02 + 4πNe2/m, the second term is such that ε(ω)
becomes negative. If such conditions are achievable, that is ω > ω0 and ω02 <
4πNe2/3m or ω2< 16πNe2/3m (assuming ω = 2ω0), then the dielectric constant is
negative
, the refractive index is purely imaginary, the group velocity meaningless,
and a new chapter in photonics and optical communications opens.

 

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