Numerical modeling of a four-wave-mixing-assisted Ramanfiber laser
Frédérique Vanholsbeeck,* Stéphane Coen,* and Philippe Emplit
Service d’Optique et Acoustique, Université Libre de Bruxelles, 50 Avenue F. D. Roosevelt, CP 194/5, B-1050 Brussels, Belgium
Catherine Martinelli and Florence Leplingard
Alcatel Research and Innovation, Route de Nozay, F-91461 Marcoussis Cedex, France
Thibaut Sylvestre
Département d’Optique P. M. Duffieux, Institut Franche-Comté Electronique, Mécanique, Thermique,Optique—Sciences et Technologies, Université de Franche-Comté, Unité Mixte de Recherche 6174, 16 Route de Gray,
Raman fiber lasers (RFLs) are typically all-f ibernested Fabry–Perot resonators with fiber Bragggrating (FBG) ref lectors. In these resonators ahigh-power continuous pump wave near 1100 nmis converted into a single or multiple high-powerwave(s) in the 1200–1600-nm range through cascadedRaman generation. This Raman cascade can beunderstood as an iteration of fundamental stimulatedRaman scattering (SRS) processes that lead to powertransfer from a shorter-wavelength pump wave to alonger-wavelength Stokes wave.1
We must point out, however, that, since the pioneer-ing theoretical study of Bloembergen and Shen,2 SRShas been known to be strongly affected by paramet-ric four-wave mixing (FWM). In particular, when thedispersion of the propagating medium is suff icientlylow, even non-phase-matched FWM interactions canplay an important role by seeding waves that are sub-sequently amplif ied through SRS, a process knownas Raman-assisted FWM.3,4 Although this phenome-non has been observed, its role in RFLs has so far toour knowledge not been considered in detail. The re-cent experimental demonstration of a FWM-assistedRaman laser with tailored dispersion characteristics,however, revealed how FWM can significantly affectthe dynamics of RFLs.5 At this stage, to the best ofour knowledge, there is no numerical model capableof reproducing these experimental results. Actually,among the numerous RFL models developed, only oneincludes chromatic dispersion.6 However, that modeldoes not take into account FWM processes involvingsuccessive Stokes orders of the RFL, as in the FWM-assisted RFL,5 but considers only FWM effects that oc-cur independently within each order. The aim of ourwork is to study numerically the role of FWM in RFLsso as to describe, for example, the remarkable proper-
0146-9592/04/232719-03$15.00/0
ties of the RFL of Ref. 5. Together with that experi-ment, our investigations reveal the importance of chro-matic dispersion in RFLs.
For simplicity, we consider a unidirectional ringcavity RFL model. Over one cavity round trip, theevolution of the electric f ield envelope A�z, t� issimulated with a generalized nonlinear Schrödingerequation1: ≠zA � 2ib2≠2
ttA�2 1 b3≠3tttA�6 1 · · · 1
igARR�t0� jA�z, t 2 t0�j2dt0. Here z is the longitu-
dinal fiber coordinate; t is the retarded time; andb2,3, ... and g are, respectively, the dispersion andthe nonlinearity coeff icients of the f iber. R�t� is thenonlinear response of silica that includes instan-taneous Kerr and delayed Raman nonlinearities.After each round trip, we apply ref lecting boundaryconditions to represent the role of the FBGs and to su-perimpose the incoming pump wave. Note that onlyforward-propagating waves are taken into account.Regarding FWM, this approximation is fairly goodsince parametric effects between counterpropagatingwaves are normally not phase matched. Moreover,even if backward SRS plays an important role inpractical RFLs based on nested Fabry–Perot cavities,unidirectional models have been shown to provide goodqualitative predictions.7 We must stress, however,that, in contrast with previous numerical studiesbased on coupled power-mode equations (see, e.g.,Ref. 7), our nonlinear Schrödinger model completelydescribes the spectrum of the intracavity field andtakes into account the exact shape and bandwidth ofthe Raman spectrum.4 It is therefore able to predictthe spectral width of the generated waves and thenoise in between.
To illustrate the interest of our study, we simulatea RFL similar to that of Ref. 5. This laser is actuallydedicated to second-order Raman pumping, a pumping
scheme requiring low power emission in a Stokesorder at 14xx nm and, simultaneously, a powerfulorder at 13xx nm. This dual-wavelength RFL isbuilt around a cw pump source at lp � 1117 nmlaunched into a cavity of length L � 500 m, whichleads, after a five-step cascade, to two useful out-puts: lS4 � 1351 nm and lS5 � 1427 nm. Aftereach round trip, we extract 96% and 96.6% of fourthand fifth Stokes orders S4 and S5, respectively, whilerecirculating 99% of the three intermediate Stokes or-ders �lS1 � 1168 nm, lS2 � 1223 nm, lS3 � 1284 nm�.For completeness the simulated spectrum includes thesixth Stokes order at lS6 � 1511 nm, which does notresonate in the cavity, as well as all the anti-Stokesbands. We also include the losses of the FBGs as wellas the wavelength-dependent losses of the cavity fiber.
To assess the role of FWM, we made two sets of simu-lations differing only in the dispersion profile of theintracavity f iber. First, we considered a “classical”laser with a zero-dispersion wavelength (ZDW) farfrom all the generated Stokes bands �l0 � 1697 nm�.A “FWM-assisted” laser with a ZDW just above thefourth Stokes order �l0 � 1385 nm� was studied in thesecond case. The dispersion parameters used hereare those of the RFL of Ref. 5: b2 � 4.82 ps2�km,b3 � 0.0623 ps3�km, and b4 � 1.18 3 1024 ps4�km atl � 1310 nm.
The simulated output power characteristics of thetwo lasers are presented in Fig. 1. For the classicalRFL (dotted curve) we first observe that S4 and S5 ex-hibit the same growth slope. However, as soon as thelaser is driven above the S5 threshold, S4 becomes de-pleted because its energy is strongly transferred to S5through SRS. The growth of S5 is actually so strongthat a 10% pump power variation around the fifth-order threshold leads to a 500-mW increase in S5. Itis therefore not possible to get a stable emission of S5at low power levels �,100 mW�, a characteristic thatmakes this laser unsuitable for second-order pumping,as are most dual-wavelength RFLs.5,8
If we now compare these results with those of theFWM-assisted RFL (solid curve), we observe that S4is no longer depleted when S5 is generated. Moreover,S5 is generated earlier and with a significantly reducedgrowth slope. A 10% pump power variation aroundthe f ifth-order threshold leads only to an S5 power in-crease smaller than 100 mW. Therefore we note thata difference in the dispersion profile of the f iber cavityleads to dramatic changes in the RFL dynamics. Wemust point out that none of these phenomena wouldappear if the FWM processes between adjacent Stokesorders were neglected. As illustrated in Fig. 2, thesesimulations agree well with the experimental resultsobtained with a FWM-assisted laser similar to that ofRef. 5. Note that the input and output power scalesdiffer between the simulations and the experiment be-cause our model is based on a ring cavity conf igu-ration instead of a Fabry–Perot configuration. Thedifferences between these two models were studied inRef. 7, which predicted that, for a 500-m-long cavity,the unidirectional model overestimates the input andoutput power levels of a Fabry–Perot laser by a fac-tor of �3. Given this correction, we can conclude that
our model yields both good qualitative and quantita-tive agreements with the experiment. In particular,the reduced growth rate of S5 is correctly reproduced,a feature that no other RFL models can account for.
The role of FWM in the dynamics of the FWM-assisted RFL can be clarified by analysis of the gener-ated spectra (Fig. 3). Below the fifth-order threshold,we can see, in Fig. 3(a), that the spectrum of all theorders of the FWM-assisted RFL are slightly broad-ened. This broadening is due to FWM effects, whoseinf luence can be particularly appreciated by noting thehigh level of the f ifth Stokes order, which is generatedbelow threshold through Raman-assisted FWM.3 Incomparison, at that pump power level, S5 is totally ab-sent from the classical laser spectrum despite a largerpower available in S4. When the input power is abovethe fifth-order threshold [Fig. 3(c)], the spectral broad-ening of the Stokes orders of the FWM-assisted RFLincreases dramatically. A similar phenomenon is ob-served in low-dispersion single-pass cascaded Ramangeneration.4 However, in the laser configuration thisFWM-induced spectral broadening is so strong thatsome of the energy of S4 overf lows the 2-nm-wide re-f lecting band of the cavity coupler. Consequently, thecavity confinement effect is reduced at this particularwavelength, which decreases the Raman gain availablefor S5 growth. In this way FWM effects lead to a re-duced amplif ication of the f ifth Stokes order, which inturn explains its stability at a low power level. Forcompleteness we also plotted in Fig. 3(b) the spectra at
Fig. 1. Output power characteristics (integrated over12 nm) of the classical (dotted curve, l0 � 1697 nm) andthe FWM-assisted (solid curve, l0 � 1385 nm) RFLs.
Fig. 2. Experimental output power characteristics (inte-grated over 10 nm) of a RFL with parameters close to thoseof our simulations �l0 � 1385 nm�.
Fig. 3. Typical output spectra of the classical (top),low-dispersion (middle), and experimental (bottom) RFLs(a) below, (b) at, and (c) above the fifth-order FWM-assisted RFL threshold.
a pump power level of �9 W just above the fifth-orderthreshold of the FWM-assisted RFL (characterized bythe kink indicated by the arrows in Fig. 1). Togetherwith Figs. 3(a) and 3(c), this f igure reveals the progres-sive FWM-induced growth of the fifth Stokes order ofthe FWM-assisted laser in comparison with the sharpthreshold behavior of the classical laser. Once again,we must point out that the smooth threshold behav-ior and the spectral broadening of the Stokes orders
is due to FWM between adjacent orders and is not ac-counted for by standard RFL models. These resultsdemonstrate the usefulness of our numerical approachthat models the exact spectral shape of the generatedwavelengths.
In conclusion, we have implemented a new numericalmodel of RFLs that, to the best of our knowledge, is thefirst to fully take into account the effect of FWM. Oursimulations have revealed the large inf luence of chro-matic dispersion on the dynamics of RFLs, makingpossible radically new operating regimes. In particu-lar, we have shown that, when the ZDW of the cavityfiber lies in between the RFL Stokes orders, FWM cancontribute significantly to the generation of Stokes or-ders above the ZDW and alter the overall laser dynam-ics. These conclusions have been confirmed througha direct comparison with the experimental results re-cently obtained with a dual-wavelength FWM-assistedRFL.5 Our work therefore demonstrates that chro-matic dispersion should truly be considered a new de-gree of freedom in the design of cascaded RFLs.
This research was supported by the InteruniversityAttraction Pole program-Belgian Science Policy, theFonds pour la Formation à la Recherche dans l’Indus-trie et dans l’Agriculture, and the Fonds National de laRecherche Scientifique (Belgium). F. Vanholsbeeck’se-mail address is [email protected].
*Now with the Department of Physics, University ofAuckland, Auckland, New Zealand.
References
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