Nonlinear Fiber Optics - 4 ed. Agrawal

10.3. Phase-Matching Techniques 377
Figure 10.2: Phase-matching diagrams for (a) mixed-mode and (b) single-mode pump propagation.
Solid and dashed lines show variations of |Δk W | and Δk M with frequency shift. Dotted lines
illustrate the effect of increasing core radius by 10%. Fiber modes are indicated using the LP mn
terminology. (After Ref. [7]; c○1975 IEEE.)
10.3.2 Phase Matching in Multimode Fibers
Multimode fibers allow phase matching when the waveguide contribution Δk W is negative
and exactly compensates the positive contribution Δk M +Δk NL in Eq. (10.3.1). The
magnitude of Δk W depends on the choice of fiber modes in which four waves participating
in the FWM process propagate. The eigenvalue equation (2.2.8) of Section 2.2
can be used to calculate Δn j ( j = 1 to 4) for each mode. Equation (10.3.4) is then used
to calculate Δk W .
Figure 10.2 shows the calculated value of Δk W as a function of the frequency shift
(ν s = Ω s /2π) for a fiber with 5-μm-core radius and a core–cladding index difference
of 0.006. The dashed line shows the quadratic variation of Δk M from Eq. (10.3.6).
The frequency shift ν s is determined by the intersection of the solid and dashed curves
(assuming that Δk NL is negligible). Two cases are shown in Figure 10.2 corresponding
to whether the pump wave propagates with its power divided in two different fiber
modes or whether it propagates in a single fiber mode. In the former case, frequency
shifts are in the range 1–10 THz, while in the latter case ν s ∼100 THz. The exact value
of frequency shifts is sensitive to several fiber parameters. The dotted lines in Figure
10.2 show how ν s changes with a 10% increase in the core radius. In general, the
phase-matching condition can be satisfied for several combinations of the fiber modes.
In the 1974 demonstration of phase-matched FWM in silica fibers, pump pulses
at 532 nm with peak powers ∼100 W were launched in a 9-cm-long fiber, together
with a CW signal (power ∼10 mW) obtained from a dye laser tunable in the range of
565 to 640 nm [6]. FWM generated a new wave in the blue region (ω 4 = 2ω 1 − ω 3 ),
called the idler wave in the parametric-amplifier configuration used for the experiment.
Figure 10.3 shows the observed idler spectrum obtained by varying the signal frequency
ω 3 . The five different peaks correspond to different combinations of fiber modes for
which phase matching is achieved. Different far-field patterns for the two dominant
peaks clearly indicate that the idler wave is generated in different fiber modes. In this
experiment, the pump propagated in a single fiber mode. As expected from Figure