1. Introduction

The nonlinear phenomena of four-wave mixing (FWM) and parametric interactions in optical fibers have attracted considerable research interest for several decades. Investigations started as early as 1974 when Stolen et al first observed FWM in a glass fiber [1], initially referring to it as three-wave mixing. The efficiency of FWM in generating waves at new optical frequencies has then led to an enormous amount of research [2–9]. Today, FWM is the underlying process of a large number of applications ranging from parametric amplification [9, 10] to wavelength conversion [11, 12] or high repetition rate pulsed light sources [13–15].

Here we focus on two lesser known aspects of FWM, namely the interplay of multiple FWM processes and FWM based on an incoherent pump. Early theoretical and experimental studies considered only one FWM process at a time [5,8,16]. In 1991, however, Thompson et al demonstrated FWM with two pump waves, which leads to a cascaded generation of sidebands [17], and theoretically described the interplay between the multiple FWM processes involved [18]. Since then, several authors have expanded these results, predicting instabilities such as sideband oscillations along the fiber propagation [3, 19] as well as a self-stabilization effect [20]. The concept of two pump FWM was also recently expanded to very broadband cascaded FWM products [2, 21]. In contrast to the cascaded generation of FWM sidebands, we demonstrate the simultaneous occurrence of largely independent FWM processes. Two processes stand out in particular: a non-phase-matched interaction between three wavelengths and the wavelength conversion of a signal with a broadband incoherent pump beam comprised of an amplified spontaneous emission (ASE) source.

As a matter of fact, little attention has been paid to non-phase-matched FWM processes [7,22,23] and it is only recently that FWM with an incoherent pump beam has attracted some interest [24–29]. Moreover most of those previous investigations have been conducted for weakly incoherent waves, i.e. waves with a spectral width of a few hundred GHz or less, and never considered FWM in the presence of other processes. In contrast, in our experiment, we observe both non-phase-matched FWM and FWM with a broadband incoherent pump simultaneously. Parametric processes with large phase-mismatches are generally ignored due to their significantly lower efficiency [22]. They can however be strongly enhanced in the presence of additional gain mechanisms such as stimulated Raman scattering (SRS) [7,23]. Nevertheless, in our experiment this effect is pronounced enough to observe competition between a non-phase-matched product and a incoherently-pumped phase-matched one.

2. Experiment

The experimental setup is depicted in Fig. 1. It is based on a continuous-wave (cw) Raman fiber laser (RFL) with a fixed wavelength of 1455 nm and a cw Erbium-doped fiber laser (EDFL), wavelength tunable between 1535 and 1565 nm. The EDFL is followed by an Erbium-doped fiber amplifier (EDFA) generating up to 33 dBm of output power, including some amplified spontaneous emission (ASE) noise. The ASE noise appears around 1550 nm with a spectral width of approximately 50 nm (at −10 dB level). The two light sources are combined into 3.1 km of dispersion-shifted fiber (DSF) by a wavelength division multiplexer (WDM). The fiber has a nonlinear parameter γ = 2 W⁻¹ km⁻¹, a zero-dispersion wavelength (ZDW) of 1550 nm (right within the tuning range of the EDFL and the bandwidth of the ASE), and dispersion coefficients β₂ = −0.473 ps²/km, β₃ = 0.119 ps³/km, β₄ = −5.66 × 10⁻⁴ ps⁴/km (at 1555 nm). The output spectrum is recorded using an optical spectrum analyzer (OSA).

 

Fig. 1 Experimental setup. RFL: Raman fiber laser, EDFL and EDFA: Erbium-doped fiber laser and amplifier, WDM: wavelength division multiplexer, DSF: dispersion shifted fiber, OSA: optical spectrum analyzer.

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Figure 2 depicts the spectra at the output of the DSF when scanning the EDFL wavelength from 1536 to 1554 nm, both as (a) a line and (b) a color plot. Here the power of the RFL was set to 1.3 W and the EDFA was adjusted to yield approximately 500 mW of output power. The spectra reveal a number of new frequency components created by independent FWM processes. The individual processes are easily distinguishable and this is to the best of our knowledge the first observation of such a large number of independent, simultaneous FWM processes. In the following we will discuss each of these in more details.

 

Fig. 2 Output spectra for EDFL wavelengths 1536 to 1554 nm: (a) line plot, (b) color plot.

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For long EDFL wavelengths, the most obvious feature in the spectra are the two sidebands created by the spontaneous scalar modulation instability (MI) of the EDFL. These sidebands are symmetrically located around the EDFL frequency. The lower and upper limits delineating the frequency region of positive MI gain are given by the two inequalities [30]:

 

Fig. 3 (a), (b) Enlargement of Figs. 2(a) and (b) around the MI region respectively. In (b), the solid and dashed lines correspond to the boundaries of the MI gain band, i.e. solutions of Eqs. (1) and (2) respectively. The dotted vertical and horizontal lines indicate the ZDW.

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We now examine the long wavelength side of the spectra shown in Fig. 2. Two Stokes peaks are observed above 1600 nm: one (S₁) remains stationary at 1651.5 nm upon tuning of the EDFL, while the wavelength of the other (S₂) varies. Figures 4(a) and 4(b) show details of the relevant part of the spectra both as line and color plots. In Fig. 4(b), we have also superimposed as a black line the idler wavelength calculated from the energy conservation condition of a degenerate FWM process involving the EDFL and the RFL respectively acting as pump and signal, ω_S2 = 2ω_EDFL – ω_RFL. As it perfectly overlaps with the observed peak S₂ for all EDFL wavelengths, we can safely assume that this is the likely origin of S₂. It must be clear however that this process cannot be phase-matched for all EDFL wavelengths (it is only phase-matched when S₂ matches S₁). In the general case, we therefore interpret it as resulting from a combination of non-phase-matched parametric FWM and SRS, i.e., Raman-assisted FWM [23]. The SRS gain from the EDFL pump overcomes the strict limitations imposed by the phase-matching condition and we observe the generated Stokes wave even with a relatively large phase-mismatch.

 

Fig. 4 Enlargement of Fig. 2 around the two Stokes waves, S₁ (fixed) and S₂ (variable). (a) Line plot, (b) color plot. In (b), the theoretical wavelength of S₂ calculated from energy conservation considerations (2ω_EDFL = ω_RFL + ω_S2) is superimposed as a black line.

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As regards the fixed peak S₁, as it does not vary with the EDFL wavelength, it is clear that the EDFL is not involved in its generation. Presuming a would-be degenerate FWM process involving S₁ and the RFL laser yields a pump wavelength at ω_p = (ω_S1 + ω_RFL)/2 = 1547.0 nm. This wavelength generally lies within the low spectral power ASE pedestal generated by the EDFA. A phase-matching analysis using the fiber parameters given above yields a phase-matched wavelength of 1651 nm very close to the experimentally observed position of S₁ at 1651.5 nm (the discrepancy is probably due to uncertainties in the dispersion values). S₁ thus results from phase-matched narrow-band degenerate FWM with an incoherent pump seeded by the RFL laser, i.e., induced MI. Note we have also considered the possibility of a non-degenerate FWM process with the RFL and a ASE slice acting as pump waves coupled with a lower order emission line of the RFL around 1.3 μm acting as a seed. However this process is associated with a very large phase-mismatch and thus is highly unlikely. The simultaneous generation of the two waves S₁ and S₂ is rather remarkable. It demonstrates the co-existence of a phase-matched and a non-phase-matched FWM process sharing the same RFL seed and mixed coherent-incoherent pump. The two waves only merge when the EDFL wavelength matches with the 1547.0 nm ASE slice involved in the generation of S₁, as seen in the center of Fig. 4. It should be noted that the phase-matched process generating wave S₁ seems less effective than the process generating S₂. This can be attributed to the relatively low spectral power of the ASE slice at 1547.0 nm [P_EDFL – P_ASE(λ = 1547.0 nm) > 20 dB]. Furthermore the two gaps with no power in S₁ seen in figure 4 are due to the spectral power of the ASE slice at 1547.0 nm dropping further when the EDFL wavlength is close [P_EDFL – P_ASE(λ = 1547.0 nm) > 40 dB]. In this case the spectral power in the ASE slice is not sufficient for significant conversion of light to wave S₁.

We have investigated further the mechanism leading to the generation of S₁ by conducting an additional experiment in which the EDFL was switched off. In this way, we studied the FWM interactions between the RFL and the ASE noise of the EDFA without the influence of the coherent EDFL signal. Figure 5(a) depicts output spectra for two different ASE power levels. Notice the efficiency with which the 1651.5 nm Stokes wave is generated. For the highest ASE power we considered (33 dBm), the power of the generated Stokes is significantly higher than the residual power of the RFL, the latter one appearing severely depleted. The level of depletion of the RFL at the fiber end actually increases with increasing ASE input power. This behavior is partly due to SRS which induces an asymmetry between the Stokes and anti-Stokes sides of the ASE pump by causing a net-energy transfer from lower to higher wavelengths [23]. In terms of efficiency, we must stress that we have also performed an experiment with a constant ASE power of 30 dBm while varying the power of the RFL. The Stokes wave was still generated quite efficiently (−20 dB with respect to the RFL output) with only 50 mW from the RFL.

 

Fig. 5 (a) Output spectra demonstrating FWM between the RFL and the ASE pump for a constant RFL power of 1 W and ASE powers of 28 dBm (solid) and 33 dBm (dotted). (b) Output spectra for constant ASE pump power of 30 dBm and RFL power of 100 mW (solid) and 900 mW (dotted).

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3. Conclusion

In conclusion we have demonstrated that multiple independent FWM processes can co-exist inside optical fibers. In particular, we have revealed a competition between phase-matched FWM with an incoherent pump and non-phase-matched FWM with a coherent pump. Additionally, we examined induced MI with an incoherent pump which surprisingly leads to a quite high conversion efficiency. Clearly, FWM with incoherent pumps leads to surprising new features and deserves more investigation.