Nonlinear Fiber Optics - 4 ed. Agrawal

240 Chapter 7. Cross-Phase Modulation
0.06
Pulse 1
0.1
0.08
Pulse 2
Intensity
0.04
Intensity
0.06
0.04
0.02
0.02
0
0
−8 −4 0 4 8 −8 −4 0 4 8
Frequency, (ν − ν 1
)T 0
Frequency, (ν − ν 2
)T 0
Figure 7.2: Spectra of two pulses exhibiting XPM-induced asymmetric spectral broadening. The
parameters are γ 1 P 1 L = 40, P 2 /P 1 = 0.5, γ 2 /γ 1 = 1.2, τ d = 0, and L/L W = 5.
where erf(x) stands for the error function and
τ = T /T 0 , τ d = T d /T 0 , δ = dL/T 0 . (7.4.10)
A similar expression can be obtained for φ 2 (T ) using Eq. (7.4.7).
As discussed in Section 4.1, the time dependence of the phase manifests as spectral
broadening. Similar to the case of pure SPM, the spectrum of each pulse is expected
to broaden and develop a multipeak structure. However, the spectral shape is now
governed by the combined contributions of SPM and XPM to the pulse phase. Figure
7.2 shows the spectra of two pulses using γ 1 P 1 L = 40, P 2 /P 1 = 0.5, γ 2 /γ 1 = 1.2, τ d = 0,
and δ = 5. These parameters correspond to an experimental situation in which a pulse
at 630 nm, with 100-W peak power, is launched inside a fiber together with another
pulse at 530 nm with 50-W peak power such that T d = 0, T 0 = 10 ps, and L = 5m.
The most noteworthy feature of Figure 7.2 is spectral asymmetry that is due solely to
XPM. In the absence of XPM interaction the two spectra would be symmetric and
would exhibit less broadening. The spectrum of pulse 2 is more asymmetric because
the XPM contribution is larger for this pulse (P 1 = 2P 2 ).
A qualitative understanding of the spectral features seen in Figure 7.2 can be developed
from the XPM-induced frequency chirp using
Δν 1 (τ)=− 1 ∂φ 1
2π ∂T = γ [
1L
P 1 τe −τ2 − P 2
πT 0 δ
(
e −(τ−τ d) 2 − e −(τ−τ d−δ) 2)] , (7.4.11)
where Eq. (7.4.9) was used. For τ d = 0 and |δ|≪1 (L ≪ L W ), the chirp is given by
the simple relation
Δν 1 (τ) ≈ γ 1L
πT 0
e −τ2 [P 1 τ + P 2 (2τ − δ)]. (7.4.12)
The chirp for pulse 2 is obtained following the same procedure and is given by
Δν 2 (τ) ≈ γ 2L
πT 0
e −τ2 [P 2 τ + P 1 (2τ + δ)]. (7.4.13)