Monomode phase matching

The phase matching condition for FWM is given by

∆β = βs+ βi− βp− βq (3.27)

where βp and βq represent the propagation constants of the pump modes at their

respective wavelengths, βs is the propagation constant of the Stokes mode, and βi

corresponds to the anti-Stokes mode.

We refer to the case where each wave travels in the same transverse fiber mode as a “monomode” process, distinct from “intermodal” processes where the waves correspond to different fiber modes. It is important to make the distinction between single-mode and monomode as well; single-mode corresponds to the case where the fiber only guides the fundamental mode, thus all FWM processes are necessarily monomode. In a multi-mode fiber (MMF), monomode processes can occur for photons in an arbitrary mode, provided that all interacting photons are within that same mode. In this section, we will consider phase matching for monomode processes.

For all of the experiments considered in this thesis, a single laser is used to create the pump waves p and q, so the pumps are necessarily frequency degenerate. In the monomode case they are also mode-degenerate, and can be treated as a single wave (as discussed in Sec. 3.1.2). In this limit, the phase mismatch is given by ∆β = βs+ βi− 2βp. The propagation constant β for the given mode can be approximated

by a Taylor expansion in frequency such that

β(ω) ≈ β  w=w0 +X k≥1 1 k!∆ω kdkβ dωk    w=w0 (3.28)

respect to frequency, and ∆ω = ω − ω0. Here we will define the center frequency such

that ω0 = ωp.

This Taylor expansion is valid for all the waves in the monomode FWM process, so we can substitute Eq. (3.28) into Eq. (3.27). The first term in the expansion is the same for each wave, and is not dependent on frequency, so it vanishes. ∆ω = 0 for the pump waves, so all pump-related terms also vanish. This leaves only the higher order terms for the Stokes and anti-Stokes waves. In order for energy conservation to be upheld (Eq. (3.10)), the frequency shift ∆ω for the anti-Stokes and Stokes waves must be equal in absolute value and opposite in sign. Thus all the odd terms will vanish leaving the following expression for the phase matching condition

∆β = 1 2β (2)_∆ω2₊ 1 4!β (4)_∆ω4₊ 1 6!β (6)_∆ω6_{+ ... (3.29)}

Near the pump frequency where ∆ω is small, only the first term will have considerable effect, given that higher order terms are proportional to 1/k!. As discussed in the previous section, maximum gain occurs for ∆β = −M, which for the monomode case is given by M = 2γPp, where γ = n2ωp/(cAef f). Therefore β(2) must be negative in

order to achieve maximum gain. As discussed in Sec. 2.2, negative β(2) _corresponds

to anomalous dispersion. In practice, 2γPp is generally a small offset to the phase

matching condition, so near-zero anomalous dispersion will lead to FWM gain. The β(4)_{-dependent term in Eq. (3.29) can also facilitate phase matching under}

some circumstances. If β(2) _{is approximately zero, then the second term has an effect}

for for large ∆ω. β(2) _{and β}(4)_{typically have opposite signs for a given mode, therefore}

at sufficient frequency detuning from the pump, the magnitude of the β(4)_-dependent

term can surpass that of the β(2)_{-dependent term, leading to phase matching far from}

the pump.

Figure 3·1: (a) Monomode phase matching for the LP0,6 mode in

a step-index fiber, pumped in the anomalous regime (red line, D = 1.8 ps/nm-km, λp = 1080 nm) and phase matching for the same mode

pumped in the normal dispersion (blue line, D = −1.9, λp = 1060 nm).

(b) Small-signal gain for each process.

FWM. The solid black lines on the phase matching plot (Fig. 3·1(a)), denote the boundaries of the gain bandwidth; ∆β = 0, and ∆β = −4γPp. The dashed black

like corresponds to the condition for which FWM gain is maximized (∆β = −2γPp).

The red lines for (a) and (b) correspond to a pump which has near-zero anomalous dispersion (D = 1.8 ps/nm-km, λp = 1080 nm). The gradient of the phase matching

curve within the gain bandwidth is fairly low, leading to broadband gain near the pump wavelength. This characteristic gain spectrum is also referred to as modulation instability (MI), a term which originates from the initial observation of temporal ripple in waves due to sidebands created as a result of anomalous dispersion in the medium (Benjamin and Feir, 1967). The blue lines correspond to a pump with near- zero normal dispersion (D = -1.9 ps/nm-km, λp = 1060 nm) which leads to steep

phase matching gradients within the gain bandwidth. The resulting FWM gain is narrowband, however, the gain bands are centered at wavelengths much farther from the pump than those for the anomalous dispersion case.

regimes highlight the inherent tradeoff for phase matching in the monomode regime: FWM gain can either be broadband but close to the pump, or far from the pump and narrowband. As discussed in the previous chapter, controlling the dispersion of higher order modes (HOMs) is possible for wavelengths below the zero-dispersion wavelength of silica, even using large Aeff modes. Hence designing systems for fixed-wavelength

conversion to novel near-infrared wavelengths is relatively simple (see Chap. 8). Com- plications arise, however, when the application demands both simultaneously broad gain bandwidth (useful for cases where wavelength-tunable output is required, for example), and a large frequency separation from the pump. Additionally, dispersion engineering becomes increasingly difficult in the visible portion of the spectrum where the material dispersion of glass becomes highly normal. In these cases, the intermodal regime can provide unique solutions (see Sec. 9.2).