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424 Fiber Optic Communications

and
Nq 2
2 e
n = 1 + (10.34)
2
2
m( − − ir∕m)
0
2
=(n + in ) , (10.35)
i
r
where n = Re(n) and n = Im(n). From Eq. (10.34), we see that the refractive index becomes complex in the
r i
presence of damping force. The forward-propagating plane wave solution takes the form (Eq. (1.95))
E = E exp [−z∕2 − i(t − n z∕c)], (10.36)
x 0 r
where
2n i
= . (10.37)
c
When atoms absorb electromagnetic energy (which is used to increase the internal energy of the atomic
system), > 0 and the incident electromagnetic signal is attenuated. If the atoms transfer energy to the elec-
tromagnetic signal due to a mechanism such as stimulated emission (see Chapter 3), becomes negative and
the incident electromagnetic signal is amplified.

10.2.2 Nonlinear Susceptibility

So far we have assumed that the restoration force is proportional to the displacement of the electron cloud
(Eq. (10.2)), which leads to harmonic oscillation of the electron cloud when the incident electromagnetic
field is harmonic (Eq. (10.6)). This holds true when the incident field is weak. As the incident electromag-
netic field becomes intense, the assumption of the linear dependence of the restoration force on displacement
breaks down and in this case, electron cloud oscillation is not harmonic. As a result, the generated electro-
magnetic field is also not harmonic. When the incident electromagnetic field is weak, we have found that
polarization is directly proportional to incident electric field intensity (Eq. (10.21)),
(1)
P = E. (10.38)
0
If the medium is not isotropic, the susceptibility depends on direction as well and Eq. (10.38) is modified as
(1) (1) (1)
P = E + E + E , j = x, y, z (10.39)
j jx x jy y jz z
or
(1)
P = .E. (10.40)
0
Here, (1) is a 3 × 3 matrix and . denotes matrix multiplication. However, as the incident field becomes strong,
the linear dependence does not hold true and P becomes a function of E. In general, P can be expanded in
terms of increasing powers of E,
P = (1) ⋅ E + (2) ∶ EE + ⋮EEE +… , (10.41)
(3)
0 0 0
where (j) is the jth-order susceptibility and is a tensor of rank j + 1. The first-order susceptibility (1) is
related to the linear refractive index (Eq. (10.29)). The second-order susceptibility (2) is responsible for
second harmonic generation; if the incident optical wave is sinusoidal of frequency , a new optical wave of
frequency 2 is generated. Anharmonic oscillation of the electron cloud due to intense electromagnetic field
can be expanded as a Fourier series with frequency components , 2, … , n, and electron cloud oscillations