2506 OPTICS LETTERS / Vol. 33, No. 21 / November 1, 2008

Cancellation of Raman pulse walk-off by slow light

Gil Fanjoux* and Thibaut SylvestreDépartement d’Optique P.M. Duffieux, Institut Franche-Comté Electronique, Mécanique, Thermique et Optique

Sciences et Technologies, (FEMTO-ST) Université de Franche-Comté, Centre National de la Recherche ScientifiqueUMR 6174, 16 Route de Gray, 25030 Besançon, France

*Corresponding author: gil.fanjoux@univ-fcomte.fr

Received May 27, 2008; revised July 24, 2008; accepted August 16, 2008;posted September 18, 2008 (Doc. ID 96628); published October 27, 2008

We theoretically demonstrate in a nonlinear optical fiber system with a narrowband Raman gain that pulsewalk-off between the pump and the Raman Stokes waves can be fully compensated for by Raman slow light,leading to group-velocity matching between the interacting waves, greater useful interaction length, andthereby enhanced Raman amplification efficiency. Limitations due to Kerr effect are further discussed.© 2008 Optical Society of America

OCIS codes: 190.0190, 190.5650, 190.5890, 190.3270, 260.2030, 230.2035.

Group velocity mismatch (GVM) due to chromaticdispersion is a well-known limiting factor for nonlin-ear optical frequency conversion with ultrashortpulses. As the pulses of the frequency components in-volved have different group velocities, they lose theirtemporal overlap after some propagation distance,leading to the so-called temporal or pulse walk-off[1]. If we consider the case of stimulated Raman scat-tering (SRS), the down-frequency shifted Stokespulse travels faster (slower) than the pump pulse inthe normal (anomalous) dispersion regime. This ef-fect limits the useful interaction length and therebythe Raman conversion efficiency. If the temporalwalk-off within the medium length becomes signifi-cant, it can also lead to modified pulse shapes andsometimes frequency chirp [1–3]. In addition, it hasbeen recently demonstrated that SRS can be advan-tageously used to optically control the group velocityof a signal pulse via the so-called slow-light effect[4–6]. SRS can indeed induce a strong dispersion inthe nonlinear medium only at the Raman frequencycomponents that are tunable by the Raman gain, orequivalently, the pump power.

In this Letter, based on simple considerations andnumerical simulations, we demonstrate that Ramanpulse walk-off can be fully compensated for by slowlight despite the large frequency shift between thepump and the Raman Stokes waves. More precisely,we show that group-velocity matching is made pos-sible in the normal dispersion regime of an optical fi-ber system filled with a nonlinear liquid character-ized by a Raman gain strong and sharp enough forinitiating a significant group-velocity reduction. Wefinally demonstrate that this specific regime leads toan enhanced Raman amplification efficiency.

Let us first describe the general concept of walk-offcancellation by slow light. When a pump pulse and asmall signal having different frequencies copropagatewithin a dispersive medium of length L, GVM in-duces a temporal walk-off �tL between both pulses,written as

�tL = L� 1

vgp−

1

vgs� = L�R�2, �1�

0146-9592/08/212506-3/$15.00 ©

where vgp and vgs are the group velocities of the pumpand the Raman signal, respectively, ��R /2�� is thepump-Stokes frequency detuning, or equivalently,the phonon frequency, and �2 is the group-velocitydispersion (GVD) coefficient. In addition, the opticaldelay generated by Raman slow light can be ex-pressed as [4]

�tNL = L� 1

vgp−

1

vgs� =

gRPL

��RAeff, �2�

where gR is the steady-state Raman gain (in metersper watt), P is the pump power (in watts), ���R /2��is the Raman linewidth (FWHM), and Aeff is the ef-fective core area of the fiber. From Eqs. (1) and (2),one can easily show that the delay from the Ramangain exactly compensates the delay from GVM for acritical power Pcr, given by

Pcr =�2�R��RAeff

gR. �3�

Equation (3) implies that a slight error in setting thepump power will lead to a small residual walk-off.Moreover, as the length of the nonlinear mediumdoes not appear in Eq. (3), the group-velocity match-ing is satisfied all along the propagation. However, itis important to stress that this relationship remainsvalid under the undepleted pump approximation andis verified only in the steady state regime by assum-ing that the pump is a cw field and that the signalpulse spectrum is smaller than the Raman gainbandwidth. Correspondingly, we will see in the fol-lowing that the results of our numerical simulationscan differ from our analytical predictions.

To get better insight, we developed a 1D temporalnumerical model based on an extended nonlinearSchrödinger equation (NLSE) including the GVD, theKerr nonlinearity, and the delayed Raman response.In the time reference frame of the pump wave, it

reads as [1]

2008 Optical Society of America

November 1, 2008 / Vol. 33, No. 21 / OPTICS LETTERS 2507

�A

�z+

i

2�2

�2A

�t2 = i��A�2A + i�A�0

�

R�t��A�z;t − t���2dt�,

�4�

where A�z ; t� (in inverse watts) is the slowly varyingenvelope of the total electric field propagating in thez direction, �=2�n2 /�pAeff is the nonlinear coefficientwith n2 the nonlinear index and �p the pump wave-length. The first term of the right-hand side of Eq. (4)stands for the self- and cross-phase modulation(XPM) together with the four-wave-mixing (FWM)processes. The second term accounts for the delayedRaman response represented in the time domain bythe RR�t� function [1]. For the sake of simplicity, theimaginary part of the Raman function, i.e., the Ra-man gain, was numerically modeled in the Fourierdomain as a Lorentzian function whose maximum isrelated to the Raman shift �R and with a correspond-ing linewidth ��R. On the other hand, the real part,which acts on the phase of the field, was obtainedfrom the imaginary part by using the Kramers–Kronig relations. Consequently, it can reach veryhigh values for sufficient narrow Raman linewidth,thereby generating a strongly dispersive nonlinearindex with a corresponding decrease of the Ramangroup velocity with respect to that of the pump wave[4,5].

We then performed numerical simulations of theNLSE Eq. (4) by assuming picosecond pump andStokes pulses copropagating in a 1D nonlinear me-dium, e.g., a highly nonlinear liquid-filled hollow-corephotonic crystal fiber [7]. For convenience, here weconsider as a nonlinear liquid the carbon disulfideCS2, which is characterized by a strong and narrow-band Raman gain [6,8] with the aim at generating aslow-light optical delay in the picosecond range. Theinput pump pulse width (FWHM) is set to 250 ps,whereas the signal pulse duration is �S=86 ps, whichcorresponds to the duration of the same Raman pulseas if it was spontaneously generated from noise. Onthe one hand, this pulse duration prevents frompulse compression by the Raman gain during Ramanamplification. On the other hand, if the signal pulsewas shorter than �S, the signal spectrum would bewider than the Raman linewidth, leading to both sig-nal spectral filtering and pulse broadening up to �S.

The results of our numerical simulations are illus-trated in Fig. 1, which shows the pulse intensity pro-files after 3 m of propagation length for the pump(dotted curve) and for the Stokes signal pulse in thelinear dispersive regime (dashed curve) with an inputpump power P=Pcr /1000, and in the walk-off sup-pression regime (solid curve) with the critical inputpump power Pcr, respectively. First, it is clear fromFig. 1 that, in the linear regime, the signal pulse un-dergoes a large positive optical delay of 158 ps owingto normal dispersion, in good agreement with theanalytical value of 160 ps predicted by Eq. (1). By in-creasing the input pump power, the optical delay t=�tL−�tNL decreases up to be canceled. Indeed, atthe critical power P=Pcr (solid curve), Fig. 1 shows

that the slow-light process exactly counterbalances

the GVM walk-off during simultaneous Raman am-plification and without pulse distortion. This is ob-tained for an input pump power of Pcr=0.285 W andfor an effective core area equal to 10 m2. Note thatthis numerical critical pump power value is largerthan the theoretical value given by Eq. (3) at0.216 W. This difference is merely due to the factthat we are not exactly in the steady-state regime, aspreviously explained. We have also numericallychecked that the walk-off is suppressed all along thepropagation as expected from Eq. (3), thereby demon-strating that the pump and signal pulses travel inthe group-velocity matching regime in spite of thelarge Stokes frequency shift. For higher input pumppower P�Pcr, the delay t becomes negative, i.e., thegroup velocity of the signal is lower than that of thepump. Thereafter, the depletion of the pump annihi-lates the slow-light process and leads to strong dis-tortions of pump and signal pulses.

For completeness, we have also plotted in Fig. 1the Raman anti-Stokes pulse (dotted-dashed curve)which is generated by degenerated FWM involvingthe pump and the Stokes waves [9]. As can be viewed,the anti-Stokes pulse is almost synchronized with thetwo other pulses (small negative optical delay of10 ps), indicating that once generated it does not suf-fer from GVM walk-off as well. This group-velocitymatching is certainly due to the FWM process itself.Therefore, the numerical simulation of Fig. 1 demon-strates that, despite the strong GVM between thethree interacting waves, any apparent walk-off ap-pears, which is reminiscent of pulse trapping [1,10].

A very interesting question to be asked here con-cerns the Raman gain efficiency in the walk-off com-pensation regime. Indeed, we may expect enhancedRaman gain efficiency because the interaction lengthand the pulse overlap can be greatly improved by theslow light. In Fig. 2 we plotted the Raman gain perunit length calculated as G=log�IS�L� /IS�0�� /L andthe optical delay t in function of the GVD coefficientand under the same slow-light condition. The propa-gation length has deliberately been limited to 1 m inorder to avoid limiting effects discussed thereafter.One can clearly see in Fig. 2 that the Raman gain

Fig. 1. Pulse profiles after 3 m of propagation length forthe pump pulse (dotted curve, intensity�1) and for theStokes signal pulse in the linear dispersive regime (dashedcurve, �5.108), and in the walk-off compensation regime(solid curve, �200), respectively. The dashed-dotted curveindicates the anti-Stokes pulse ��3�108�. The parametersare ��R /2��=20 THz, ���R /2��=15 GHz, gR=23.2�10−11 m W−1, �2=4.24�10−25 s2 m−1, and �=4.134 m−1 W−1 at the pump wavelength of 532 nm.

reaches its maximum for zero optical delay, i.e., when

2508 OPTICS LETTERS / Vol. 33, No. 21 / November 1, 2008

the walk-off is exactly compensated by the slow-lightprocess, and rapidly drops elsewhere.

Let us now examine the impact of other nonlineareffects on the removal of the Raman pulse walk-off.Specifically, the combined action of XPM and pulsewalk-off can induce a substantial frequency shift ofthe Raman Stokes pulse [11,12]. Since the Ramanlinewidth is narrow and the signal spectral width isof the same order of magnitude, a small signal fre-quency shift can lead to the decrease of the Ramangain, thereby reducing the slow-light effect and theGVM walk-off cancellation. Figure 3 shows the Ra-man pulse walk-off (solid curve, left axis) and the sig-nal frequency shift (dashed curve, right axis) in func-tion of the propagation distance, with the sameparameters as in Fig. 1. We can see that after about1.5 m propagation, the Raman pulse walk-off is nolonger compensated by the slow-light delay, leadingto a small optical delay. This is due to the fact thatthe Stokes pulse continuously undergoes a frequencyshift up to +2.2 GHz, as shown by the dashed curvein Fig. 3. The corresponding normalized intensityspectrum of the Raman Stokes pulse at the input and

Fig. 2. Raman gain per unit length (squares, right axis)and optical delay (circles, left axis) in function of the group-velocity dispersion.

Fig. 3. Pulse walk-off (left, solid curve) and signal fre-quency shift (right, dashed curve) versus the propagationlength in the walk-off compensation regime. Parametersare the same as in Fig. 1 and without the Kerr effect(dashed-dotted straight line). Inset, normalized intensityspectra of the Raman Stokes signal at the input (solidcurve), and the output (dotted curve). The Raman gain

curve is also drawn in gray.

output are plotted in the inset of Fig. 3, together withthe Raman gain curve in gray. We have finallychecked that, by removing the Kerr effect in theNLSE Eq. (4), the residual walk-off completely disap-pears as indicated by the dashed-dotted straight linein Fig. 3.

In conclusion, we have theoretically studied thecompetition between pulse walk-off and slow light inSRS and show that, for a critical pump power, theycan cancel each other out. A 160 ps Raman pulsewalk-off induced by GVM has been fully suppressedby Raman slow light. With a view to generalizing thisconcept to nonlinear media other than CS2, we havenumerically checked that picosecond pulse walk-offsuppression is still possible by considering a Ramanbandwidth ten times larger than for CS2. We believethat this method could be thus applied to futurehighly efficient narrowband Raman amplifiers, suchas liquids or gases in hollow-core fibers. In conven-tional silica-based optical fibers however, we may ex-pect much smaller optical delay in the femtosecondrange because of the broad Raman gain bandwidth offused silica (�5 THz) [2,5]. Because of the largepump-Stokes frequency, the slow light will probablybe efficient to cancel out the pulse walk-off in thesmall dispersion ring only, i.e., by working almostsymmetrically around the zero-dispersion wave-length of the optical fiber [13]. Finally, this regime ofgroup-velocity matching could also find potential ap-plication to all-optical 3R regeneration, since the Ra-man pulse is simultaneously reshaped, reamplifiedand retimed [5].

The authors thank E. Lantz for helping in the nu-merics.

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