Nonlinear Fiber Optics - 4 ed. Agrawal

228 Chapter 7. Cross-Phase Modulation
and combining it with the linear part so that the total induced polarization is given by
where
P(ω j )=ε 0 ε j E j , (7.1.8)
ε j = ε L j + ε NL
j =(n L j + Δn j ) 2 , (7.1.9)
n L j is the linear part of the refractive index and Δn j is the change induced by the thirdorder
nonlinear effects. With the approximation Δn j ≪ n L j , the nonlinear part of the
refractive index is given by ( j = 1,2)
Δn j ≈ ε NL
j /2n L j ≈ n 2 (|E j | 2 + 2|E 3− j | 2 ), (7.1.10)
where the nonlinear parameter n 2 is defined in Eq. (2.3.13).
Equation (7.1.10) shows that the refractive index seen by an optical field inside an
optical fiber depends not only on the intensity of that field but also on the intensity of
other copropagating fields [2]–[4]. As the optical field propagates inside the fiber, it
acquires an intensity-dependent nonlinear phase shift
φ NL
j (z)=(ω j /c)Δn j z = n 2 (ω j /c)(|E j | 2 + 2|E 3− j | 2 )z, (7.1.11)
where j = 1 or 2. The first term is responsible for SPM discussed in Chapter 4. The
second term results from phase modulation of one wave by the copropagating wave
and is responsible for XPM. The factor of 2 on the right-hand side of Eq. (7.1.11)
shows that XPM is twice as effective as SPM for the same intensity [1]. Its origin
can be traced back to the number of terms that contribute to the triple sum implied in
Eq. (2.3.6). Qualitatively speaking, the number of terms doubles when the two optical
frequencies are distinct compared with that when the frequencies are degenerate. The
XPM-induced phase shift in optical fibers was measured as early as 1984 by injecting
two continuous-wave (CW) beams into a 15-km-long fiber [3]. Soon after, picosecond
pulses were used to observe the XPM-induced spectral changes [4]–[6].
7.1.2 Coupled NLS Equations
The pulse-propagation equations for the two optical fields can be obtained by following
the procedure of Section 2.3. Assuming that the nonlinear effects do not affect significantly
the fiber modes, the transverse dependence can be factored out writing E j (r,t)
in the form
E j (r,t)=F j (x,y)A j (z,t)exp(iβ 0 j z), (7.1.12)
where F j (x,y) is the transverse distribution of the fiber mode for the jth field ( j =
1,2), A j (z,t) is the slowly varying amplitude, and β 0 j is the corresponding propagation
constant at the carrier frequency ω j . The dispersive effects are included by expanding
the frequency-dependent propagation constant β j (ω) for each wave in a way similar
to Eq. (2.3.23) and retaining only up to the quadratic term. The resulting propagation
equation for A j (z,t) becomes
∂A j
∂z + β ∂A j
1 j + iβ 2 j ∂ 2 A j
∂t 2 ∂t 2 + α j
2 A j = in 2ω j
( f jj |A j | 2 + 2 f jk |A k | 2 ), (7.1.13)
c