Nonlinear Fiber Optics - 4 ed. Agrawal

10.3. Phase-Matching Techniques 383
ble to compensate it by the nonlinear contribution Δk NL in Eq. (10.3.1). The frequency
shift Ω s in that case depends on the input pump power. In fact, if we use Eq. (10.2.21)
with Δk ≈ Δk M ≈ β 2 Ω 2 s from Eq. (10.3.6), phase-matching occurs (κ = 0) when
Ω s =(2γP 0 /|β 2 |) 1/2 , (10.3.9)
where P 0 is the input pump power. Thus, a pump wave propagating in the anomalous-
GVD regime would develop sidebands located at ω 1 ± Ω s as a result of FWM that is
phase-matched by the nonlinear process of self-phase modulation. This case has been
discussed in Section 5.1 in the context of modulation instability. As was indicated
there, modulation instability can be interpreted in terms of FWM in the frequency
domain, whereas in the time domain it results from an unstable growth of weak perturbations
from the steady state. In fact, the modulation frequency given in Eq. (5.1.9) is
identical to Ω s of Eq. (10.3.9). The output spectrum shown in Figure 5.2 provides an
experimental evidence of phase matching occurring as a result of self-phase modulation.
The frequency shifts are in the range 1–10 THz for pump powers P 0 ranging from
1–100 W. This phenomenon has been used to convert the wavelength of femtosecond
pulses from 1.5- to 1.3-μm spectral region [54].
The derivation of Eq. (10.3.9) is based on the assumption that the linear phase
mismatch Δk is dominated by the β 2 term in Eq. (10.3.6). When |β 2 | is relatively
small, one should include the β 4 term as well. The frequency shift Ω s is then obtained
by solving the fourth-order polynomial
(β 4 /12)Ω 4 s + β 2 Ω 2 s + 2γP 0 = 0. (10.3.10)
Depending on the relative signs and magnitudes of β 2 and β 4 , Ω s can vary over a large
range. In particular, as mentioned in Section 5.1.2, modulation instability can also
occur for β 2 > 0 provided β 4 < 0. This situation can occur for dispersion-flattened
fibers [55]. In practice, it is easily realized in tapered or microstructured fibers [56]–
[59]. In a fiber exhibiting normal dispersion with β 4 < 0, Eq. (10.3.9) leads to the
following expression for Ω s :
Ω 2 s = 6
|β 4 |
(√β 2 2 + 2|β 4|γP 0 /3 + β 2
)
. (10.3.11)
Figure 10.9 shows how the frequency shift, ν s = Ω s /(2π), varies with β 2 and |β 4 |
for γP 0 = 10 km −1 . In the limit |β 4 |γP 0 ≪ β2 2 , this shift is given by the simple expression
Ω s =(12β 2 /|β 4 |) 1/2 . For typical values of β 2 and β 4 , the frequency shift can
exceed 25 THz, indicating that FWM can amplify a signal that is more than 200 nm
away from the pump wavelength. Such FWM is discussed in more detail in Chapter 12
in the context of microstructured and photonic-crystal fibers.
10.3.4 Phase Matching in Birefringent Fibers
An important phase-matching technique in single-mode fibers takes advantage of the
modal birefringence, resulting from different effective mode indices for optical waves
propagating with orthogonal polarizations. The index difference
δn = Δn x − Δn y , (10.3.12)