Nonlinear Fiber Optics - 4 ed. Agrawal

380 Chapter 10. Four-Wave Mixing
10.3.3 Phase Matching in Single-Mode Fibers
In single-mode fibers the waveguide contribution Δk W in Eq. (10.3.1) is very small
compared with the material contribution Δk M for identically polarized waves, except
near the zero-dispersion wavelength λ 0 where the two become comparable. The three
possibilities for approximate phase matching consist of: (i) reducing Δk M and Δk NL
by using small frequency shifts and low pump powers; (ii) operating near the zerodispersion
wavelength so that Δk W nearly cancels Δk M +Δk NL ; and (iii) working in the
anomalous GVD regime so that Δk M is negative and can be cancelled by Δk NL + Δk W .
Nearly Phase-Matched Four-Wave Mixing
The gain spectrum shown in Figure 10.1 indicates that significant FWM can occur
even if phase matching is not perfect to yield κ = 0 in Eq. (10.3.1). The amount of
tolerable wave-vector mismatch depends on how long the fiber is compared with the
coherence length L coh . Assuming that the contribution Δk M dominates in Eq. (10.3.1),
the coherence length can be related to the frequency shift Ω s by using L coh = 2π/|Δκ|
with Eq. (10.3.6) and is given by
L coh =
2π
|Δk M | =
2π
|β 2 |Ω 2 . (10.3.7)
s
In the visible region, β 2 ∼ 50 ps 2 /km, resulting in L coh ∼ 1 km for frequency shifts
ν s = Ω s /2π ∼ 100 GHz. Such large coherence lengths indicate that significant FWM
can occur when the fiber length satisfies the condition L < L coh .
In an early experiment, three CW waves with a frequency separation in the range
1–10 GHz were propagated through a 150-m-long fiber, whose 4-μm core diameter
ensured single-mode operation near the argon-ion laser wavelength of 514.5 nm [8].
The FWM generated nine new frequencies such that ω 4 = ω i + ω j − ω k , where i, j,k
equal 1, 2, or 3 with j ≠ k. The experiment also showed that FWM can lead to spectral
broadening whose magnitude increases with an increase in the incident power. The 3.9-
GHz linewidth of the CW input from a multimode argon laser increased to 15.8 GHz
at an input power of 1.63 W after passing through the fiber. The spectral components
within the incident light generate new frequency components through FWM as the
light propagates through the fiber. In fact, SPM-induced spectral broadening discussed
in Section 4.1 can be interpreted in terms of such a FWM process [45].
From a practical standpoint FWM can lead to crosstalk in multichannel (WDM)
communication systems, where the channel spacing is typically in the range of 10
to 100 GHz. This issue attracted considerable attention during the 1990s with the
advent of WDM systems [46]–[52]. In an early experiment [25], three CW waves
with a frequency separation ∼10 GHz were propagated through a 3.5-km-long fiber
and the amount of power generated in the nine frequency components was measured
by varying the frequency separation and the input power levels. Figure 10.6 shows
measured variations for two frequency components f 332 and f 231 using the notation
f ijk = f i + f j − f k , f j = ω j /(2π). (10.3.8)