1642 J. Opt. Soc. Am. B/Vol. 16, No. 10 /October 1999 Seve et al.

Generation of vector dark-soliton trainsby induced modulational

instability in a highly birefringent fiber

E. Seve, G. Millot, and S. Wabnitz

Laboratoire de Physique, Universite de Bourgogne, B.P. 47 870, 21078 Dijon, France

T. Sylvestre and H. Maillotte

Laboratoire d’Optique P. M. Duffieux, Unite Mixte de Recherche, Centre National de la Recherche Scientifique/Universite de Franche-Comte No. 6603, 25030 Besancon Cedex, France

Received April 1, 1999

We present a set of experimental observations that demonstrate the generation of vector trains of dark-solitonpulses in the orthogonal axes of a highly birefringent optical fiber. We generated dark-soliton trains withterahertz repetition rate in the normal group-velocity dispersion regime by inducing a polarization modula-tional instability by mixing two intense, orthogonal continuous laser beams. Numerical solutions of thepropagation equations were used to optimize the emission of vector dark pulses at the fiber output. © 1999Optical Society of America [S0740-3224(99)02110-4]

OCIS codes: 060.4510, 060.5530, 190.4370, 190.4380, 270.3100.

1. INTRODUCTIONOptical fibers have the remarkable property of being ableto support the propagation of several types of nonlinearlight-wave packet that may propagate undistorted inspite of group-velocity dispersion (GVD) and that emergeintact after a collision. The best known of these wavesare the bright solitons in the anomalous GVD regime,which have great potential for use as information carrierbits in long-haul communication systems.1 Extensive re-search is under way to implement soliton transmissionsover transoceanic distances by use of a combination ofwavelength-division multiplexing and GVD management.From the spectrum of research on optical solitons, experi-ments have shown that it is possible to launch in a fiber avariety of dark-type2 or domain wall3 solitons: An inter-esting feature of these light structures is their reducedsensitivity (with respect to bright pulses) to mutual inter-actions and amplifier-noise timing jitter.

In optical fibers, the modulational instability (MI) ofcontinuous waves (cw’s) is closely linked to the possibilityof generating soliton trains by imposing a weak modula-tion on an intense laser beam.4 Indeed, MI (orBenjamin–Feir instability) is a ubiquitous nonlinear pro-cess whereby weak modulations of a cw with a frequencyin a given bandwidth are amplified by parametric mixingin a nonlinear dispersive medium. MI is found to occurin a variety of physical contexts, such as in plasmas,5 influids,6 in solid-state lattices,7 in electrical circuits,8 andof course in nonlinear optics.9,10 The first experimentaldemonstration of the generation of bright solitons by in-duced MI in the anomalous GVD regime of an optical fiberwas achieved by Tai et al.10 In the normal GVD regime,however, the propagation of an individual intense beam isnot subject to MI. Nevertheless, as was first pointed out

0740-3224/99/101642-09$15.00 ©

by Berkhoer and Zakharov,11 the nonlinear coupling be-tween two different (e.g., polarization) modes may extendthe domain of MI into the normal GVD regime.12–14 Infact, in birefringent optical fibers the nonlinear propaga-tion of the field envelope is governed by a set of twocoupled nonlinear Schrodinger (CNLS) equations.1

Moreover, families of vector dark solitons solutions werefound for the CNLS equations that apply to highly bire-fringent (hi-bi) fibers.15

As we point out and experimentally demonstrate here,by properly adjusting the frequency spacing and the in-tensities of two input cw laser beams one may generate atrain of vector dark solitons through cross-phasemodulation- (XPM-) induced MI in hi-bi fibers. In our ex-periments we obtained a dark-vector soliton train repeti-tion rate (which is fixed by the modulation frequency) ashigh as 2.5 THz. The observations are in excellent agree-ment with theoretical predictions from a numerical simu-lation of the CNLS equations.

This paper is structured as follows: In Subsection 2.Awe present the incoherent CNLS equations that governthe propagation of an intense beam in a hi-bi fiber. Themain results of the linear stability analysis of the cw so-lutions of the CNLS equations are recalled in Subsection2.B. Subsection 2.C is devoted to illustrating a particu-lar solution of CNLS equations: symmetric vector or po-larization dark solitons. Interestingly, the formation ofsuch vector dark waves in spite of the considerable group-velocity walk-off in the hi-bi fiber may be thought of as aresult of the mutual nonlinear trapping between the twopolarization components of the dark soliton. The finetuning of the input beam parameters that permits a linkto be established between induced MI and vector dark-soliton generation is clearly illustrated in Subsection 2.D.

1999 Optical Society of America

Seve et al. Vol. 16, No. 10 /October 1999 /J. Opt. Soc. Am. B 1643

The experimental results for dark-solitonlike generationwith nanosecond pump pulses are discussed in detail inSection 3. In Section 4 we present further numerical andexperimental results obtained with picosecond inputpulses. Finally, Section 5 is devoted to our conclusions.

2. THEORYA. Coupled Nonlinear Schrodinger EquationsIn hi-bi optical fibers the propagation of a quasi-monochromatic field is governed by a set of CNLSequations1,16:

]Ex

]z1 d

]Ex

]t1

1

2ib2

]2Ex

]t 2 5 igS uExu2 12

3uEyu2DEx ,

]Ey

]z2 d

]Ey

]t1

1

2ib2

]2Ey

]t 2 5 igS uEyu2 12

3uExu2DEy ,

(1)

where Ex(z, t) and Ey(z, t) are the slowly varying ampli-tude of the field components, b2 is the GVD, d 5 (Vgx

21

2 Vgy21)/2 is the group-velocity mismatch or walk-off,

and g is the nonlinear coefficient. In these CNLS equa-tions, dissipative terms such as linear fiber loss andstimulated Raman scattering were not included. Thisapproximation is valid when we consider the relativelyshort fiber lengths (less than 2 m) and low powers used inour experiments. The second and third terms on the left-hand sides of Eqs. (1) represent the birefringent and dis-persive contributions to the propagation, whereas the twoterms on the right-hand sides describe the action of theself-phase modulation and the degenerate XPM, respec-tively. As is well known, in birefringent fibers XPM oc-curs because the effective refractive index of a polariza-tion component of the field depends not only on theintensity of that component but also on the intensity ofthe other copropagating, orthogonally polarized wave.

B. Modulational Instability AnalysisWhereas the modulational instability of a single cw beamin fibers requires anomalous GVD,4 in the normal GVDregime the coupling between two polarization modes maylead to MI in the normal GVD regime also.11,13,14,17 Inthis subsection we recall the principal results of the linearstability analysis of MI in the normal dispersion regime ofa birefringent fiber. In hi-bi fibers,13,14 cw or quasi-cwbeams are unstable only whenever their polarization isintermediate between the fast and the slow axes of the fi-ber, whereas a beam that is exactly aligned along one ofthe two birefringence axes is modulationally stable. TheMI gain for a given modulation frequency V and orienta-tion of the initial pump polarization is obtained from theeigenvalues and the eigenvectors of the following 43 4 matrix13,14:

where Px 5 P cos2(u) and Py 5 P sin2(u) is the power ofthe pump component along the slow or the fast axis, u rep-resents the orientation of the input pump with respect tothe slow axis, and P is the total pump power. A simpleanalytical formula for the MI gain with u 5 45° (this caseleads to maximum power gain) reads as13

G~V! 5 2 uIm$r 1 j2 6 @~r 1 j2!2

1 C2 2 ~r 2 j2!#1/2%1/2u, (2)

r 5 1/2b2V2~1/2b2V2 1 gP !,

C 5 1/3V2Pgb2 , j 5 Vd. (3)

In a single-pump frequency configuration, and in thenormal dispersion regime of propagation (i.e., b2 . 0),the Stokes and anti-Stokes sidebands are polarized alongthe slow and fast axes of the fiber, respectively. More-over, the absolute frequency shift (with respect to thepump) of these unstable sidebands extends from the low-frequency value f1 5 (1/2p)@(2d/b2)2 2 10gP/3b2#1/2 upto the high-frequency value f2 5 (1/2p)@(2d/b2)2

2 2gP/3b2#1/2. This is the spectral band about thephase-matched frequency for which the nonlinear phaseshift and the GVD effect between the pump and the para-metric sidebands are compensated for by the group-velocity mismatch between the two orthogonal modes.

C. Vector Dark SolitonsIn the normal GVD regime, exact vector dark-soliton so-lutions of Eqs. (1) can be found with the help of thechange of variables15:

U 5 Ex expF iS 2d

b2t 1

d 2

2b2z D G , (4)

V 5 Ey expF iS d

b2t 1

d 2

2b2z D G , (5)

which leads to the two coupled NLS equations in terms ofU and V:

]U

]z1

1

2ib2

]2U

]t 2 5 igF S uUu2 12

3uVu2DUG ,

]V

]z1

1

2ib2

]2V

]t 2 5 igF S uVu2 12

3uUu2DVG . (6)

Note that the envelope U (V) represents the complex am-plitude of a field whose optical carrier is frequency shiftedby dv 5 2d/b2 (dv 5 d/b2) with respect to the originalfield Ex (Ey). This formal frequency shift permits thewalk-off term in Eqs. (6) to be eliminated; clearly, in thepresence of a single-frequency input field the walk-offshould be taken into account in the initial conditions forEqs. (6). On the other hand, if one injects into the fiber

M 5 F 2dV 1 b2~V2/2! 1 gPx gPx2/3gAPxPy

2/3gAPxPy

2gPx 2dV 2 b2~V2/2! 2 gPx 22/3gAPxPy 22/3gAPxPy

2/3gAPxPy2/3gAPxPy dV 1 b2~V2/2! 1 gPy gPy

22/3gAPxPy 22/3gAPxPy 2gPy dV 2 b2~V2/2! 2 gPy

G ,

1644 J. Opt. Soc. Am. B/Vol. 16, No. 10 /October 1999 Seve et al.

two frequency-shifted (by 6dv) orthogonally polarizedfields, XPM-induced cross trapping leads to dark-vectorsolitons.15 Indeed, such waves have the property ofmaintaining unchanged along the propagation directionboth their intensity profiles and their input states of po-larization. A symmetric vector dark-soliton solution ofEqs. (6) reads as15

U 5 V 5 U0 tanh@~5g/3b2!1/2U0t#exp~5/3igU02z !. (7)

As the above vector dark-soliton solution [Eq. (7)] rep-resents the effect of mutual trapping between the two po-larization components, one may study its stability ana-lytically by means of a variational approach.18

D. Vector Dark Solitons and Modulation InstabilityMI manifests itself as a breakup of a cw or quasi-cw waveinto a train of ultrashort pulses. The repetition rate ofsuch a train is determined by the modulational frequency.The link between soliton solutions of the CNLS equationsand MI was analyzed in the case of a weakly birefringentfiber, for which the group-velocity walk-off between thetwo polarizations can be neglected.19 In this case, peri-odic stationary wave solutions of the CNLS equations canbe written in terms of Jacobian elliptic functions (sn anddn). When the modulation depth of these periodic solu-tions decreases, one may reduce the wave dynamics, inthe small-amplitude limit, to that of the three-wavemodel of MI.19,20 In a similar way, we can interpret thesolution [Eq. (7)] as the vector dark solitons that are as-sociated with the MI effect in hi-bi fibers.

In general, it should be noted that the temporal profileof ultrashort pulses that are generated at the fiber outputby induced MI does not coincide with a periodic dark-soliton train.21 In fact, one should control the outputpulse shape that is generated by induced MI by injectinginto the fiber the appropriate combination of the weaksignal and the strong pump. In particular, the signal fre-quency detuning from the pump should be properly se-lected. In the experiments we induced MI by injectinginto the fiber a pump and a single Stokes signal beam.Under these conditions the signal Stokes wave is ampli-fied by MI and the anti-Stokes wave (i.e., the conjugatedidler) is generated from the parametric coupling.Clearly, the frequency detuning between the two wavesdetermines the modulation frequency ( fmod). As weshall see, by proper adjustment of the value of fmod and ofthe power ratio signal/pump (a), it is possible to obtain atrain of dark solitons at the fiber output.21 To determinethe modulation parameters we carried out numericalsimulations of the CNLS equations [Eqs. (1)]. Thesesimulations permitted us to determine in particular theinitial parameters ( fmod and a) that yield a vector dark-soliton train at the fiber output for fixed values of the in-put pump power and fiber length.

In the simulations we took into account the experimen-tally accessible condition for the seed, namely, that ofsingle-sideband excitation (i.e., a Stokes signal is injectedalong the slow axis). The numerical solutions of theCNLS equations were based on the standard split-stepFourier method. The corresponding initial conditionswere

Ex~z 5 0, t! 5 AP/2 1 APs exp~2ipfmodt!, (8)

Ey~z 5 0, t! 5 AP/2, (9)

where P and Ps are the input pump and the signal totalpower, respectively, and fmod is the frequency detuningbetween the carrier frequency of the pump and the signalStokes beams. In the numerical simulations we consid-ered the following experimental fiber parameters: TheGVD was b2 5 60 ps2 km21, the birefringence was dn5 3.5 3 1024 (d 5 0.585 ps m21), the nonlinear coeffi-cient was g 5 0.052 m21 W21, and the fiber length wasL 5 1.8 m. Moreover, we took P 5 112 W, Ps 5 2 W,and fmod 5 2.5 THz. Figure 1 shows the evolution alongthe fiber length of the power in the two orthogonal com-ponents of the field. As can be seen, an initial sinusoidalmodulation of the cw wave in the slow axis gradually in-duces full modulation on both fiber axes. It is importantto note that, even when a single pump component ismodulated at the fiber input, as occurs in the presentcase, a dark-solitonlike train is generated on both axes atthe fiber output.

The dark-soliton nature of the pulse train that is gen-erated at the fiber output is confirmed in Fig. 2. Here wedisplay the output temporal profile and the correspondingspectrum for each polarization component of the field.As can easily be seen from this figure, a dark-solitonliketrain is generated at the fiber output on both axes. Inthe upper part of Fig. 2 we show along with the intensityprofile of the dark-soliton train (solid curves) the phaseprofile (dashed curves), which has a series of abrupt pphase shifts between any two adjacent peaks. Note thatthe phase remains constant between any two consecutivepeaks. A comparison of the phase profiles of the two po-larization components in the upper part of Fig. 2 showsthat the two coupled dark-soliton trains are phase-conjugate solitons. The power spectrum of each polariza-tion component in the lower part of Fig. 2 shows that the

Fig. 1. Theoretical evolution with distance z of the powers inthe slow- and fast-polarization components of the field versustime t. The total pump (signal) power is 112 W (2 W), the fre-quency detuning is 2.5 THz, and the fiber length is 1.8 m.

Seve et al. Vol. 16, No. 10 /October 1999 /J. Opt. Soc. Am. B 1645

two dark-soliton trains along the slow and fast axes arefrequency downshifted and upshifted, respectively, byfmod / 2 with respect to the mean frequency of the fields.Indeed, the vertical dashed lines in the spectra of Fig. 2indicate the carrier frequency of each individual dark-soliton train. In Subsection 2.C we showed that two or-thogonal dark solitons may be mutually trapped by the fi-ber nonlinearity if their carrier frequencies are separatedby (d f )trap 5 (2d/b2)/2p 5 3.1 THz. The physical inter-pretation of this fact is quite intuitive: In fact, thephase-matched Stokes signal polarized on the slow axisand the phase-matched anti-Stokes idler polarized on thefast axis are parametrically amplified during propaga-tion, whereas the two other polarization components ofthe Stokes and the anti-Stokes waves, as well as thehigher-order harmonics, are not phase matched; never-theless they are generated with reduced efficiency byfour-wave mixing processes in the fiber14 (lower part ofFig. 2). The phase-matching condition described aboveleads to a spectral distribution with an opposite frequencyshift of the mean spectral power on each axis. In fact,the slow (fast) component of the field speeds up (slowsdown) so both polarization components can attain thesame group velocity, which then permits the nonlinearcross-trapping effect.

The upper part of Fig. 3 provides a graphic illustrationof the linear phase-matching relation between the twocomponents of the pump and the sidebands. Here weshow the variation of the inverse of the group velocity ver-sus the frequency detuning for the slow and fast axes ofthe fiber. 1/b1 and vag indicate the group velocity of eachwave and the mean group velocity, respectively. The twocomponents of the pump beam are represented by the twofilled circles, whereas the filled triangles stand for theStokes and anti-Stokes phase-matched sidebands. Thefilled squares indicate the carrier frequency of each dark-

Fig. 2. Theoretical time dependence of output powers (solidcurves) and phases (dashed curves) of cw dark-soliton trains inthe slow (left) and fast (right) fiber axes, and the correspondingoutput spectra.

solitonlike train. As shown by this part of Fig. 3, the twodark-soliton trains have the same group velocity: Thefrequency shift between the carrier frequencies of thetrains in the two axes exactly compensates for the group-velocity mismatch of the hi-bi fiber. In the lower part ofFig. 3 we show the MI gain profiles for three different val-ues of the pump power drawn to the same frequency scaleas the upper diagram. Moreover, we took P 5 10 W(small solid curves), P 5 56 W (dashed curves), and P5 112 W (dotted–dashed curves). As can easily be seenfrom Fig. 3, the mutual trapping condition between thetwo polarizations corresponds to a modulational fre-quency that yields peak MI gain at low pump powers (i.e.,the linear phase-matching condition). In the presence ofa relatively intense pump wave, the nonlinear phase mis-match must be included in the phase-matching condition,which leads to a frequency shift of both Stokes and anti-Stokes components toward the pump (the lower part ofFig. 3). Indeed, the numerical solutions and the experi-ments show that a vector dark-soliton train may be gen-erated whenever the modulational frequency is smallerthan the linear phase-matching frequency (df )trap .

It is important to note that the frequency detuning fmodand the signal fraction a are chosen to generate a vectordark-solitonlike train at the output of the fiber. Becauseof the spatially recurrent nature of the nonlinear wavemixing process,22,23 further propagation along the fiberleads to a periodic energy exchange between the pumpand the parametric sidebands.24 As a result, the vectordark-solitonlike train is generated only at periodicallyspaced set of points along the fiber. At the left in Fig. 4we have represented the theoretical power spectrum evo-lution along the slow axis of the fiber. The fiber length is3.5 m, and all other parameters are identical to those forthe case in Fig. 1. At the right is the contour plot for a

Fig. 3. Schematic diagram of the phase-matching condition ofthe associated four-wave mixing process. The two componentsof the pump and each sideband are represented by filled circlesand filled triangles, respectively; filled squares give the centralfrequency of the dark-soliton pulses train on each axis. Below,MI gain versus frequency detuning for three different total pumppowers: 112 W (dotted–dashed curves), 56 W (dashed curves),and 10 W (small solid curves).

1646 J. Opt. Soc. Am. B/Vol. 16, No. 10 /October 1999 Seve et al.

fiber that is twice as long (L 5 7 m). As all the spectrallines are quite sharp, the space between them was elimi-nated for clarity and the spectral width of each line wasmultiplied by 3. Note that only the four most intenselines are represented here. As can be seen, for z5 1.8 m the pump and the Stokes components have ex-actly the same power. A dark-soliton pulse train is gen-erated at this distance (see Fig. 2). On further propaga-tion, the direction of energy exchange is inverted and allthe energy flows in and out of the pump (i.e., the field re-turns to its original shape) until another dark train is cre-ated at 5.4 m. Moreover, Fig. 3 also shows that the non-phase-matched slow anti-Stokes waves execute fastoscillations, whereas the phase-matched Stokes signalevolves smoothly with distance.

3. RESULTS IN THE NANOSECONDREGIMEIn this section we describe in full detail the experimentalsetup that was employed for the observation of inducedMI in a hi-bi fiber with nanosecond pulses (see Fig. 5).Quasi-cw input pump and signal beams were obtainedfrom two different laser sources that delivered nanosec-ond pulses. Indeed, by comparing such long-pulse dura-tions with the MI period (of a few picoseconds) it is rea-sonable to consider all the input beams as cw’s. Thepump beam was obtained from a cw-tunable ring dye la-ser pumped by a cw argon laser and amplified by a three-stage dye amplifier. The dye amplifier was pumped by afrequency-doubled, injection-seeded, and Q-switchedNd:YAG laser (l 5 532.26 nm) operating at a repetitionrate of 25 Hz. The signal pulses were obtained by fre-quency shifting of the Nd:YAG laser output by means ofself-stimulated Raman scattering in a multipass carbondioxide cell.21 In all the present experiments the pumpwavelength was tuned near lp 5 572 nm and the signalwavelength was fixed to ls 5 574.72 nm. Pump and sig-nal beams were finally combined by a beam splitter and

Fig. 4. Left, theoretical power spectrum evolution along the fi-ber slow axis. The fiber length is 3.5 m, and all other param-eters are identical to those in Fig. 1. Right, contour plot for a7-m fiber. The darkness of the gray levels is proportional to theintensity.

focused with a 203 microscope objective in a 1.8-m-longfiber (HB600 Fibercore). At the fiber output, two types ofmeasurement were performed: a spectral analysis witha spectrometer and a temporal characterization of thepulse trains by means of a second-harmonic generationautocorrelator.25

In Subsection 2.D the numerical simulations of CNLSequations (1) permitted us to draw an estimate, for agiven set of values of the pump power and fiber length, ofthe optimal sideband frequency detuning and power forthe generation of a train of vector dark solitons at the fi-ber output. For example, with an input pump power ofP 5 112 W and a fiber length L 5 1.8 m, we could obtainthe optimal values for the modulational frequency ( fmod5 2.5 THz) and the input signal power (Ps 5 2 W). Fig-ures 6(c) and 6(d) show the experimental spectra as ob-served at the fiber output with the above choice of initialparameters. Figures 6(a) and 6(b) display the theoreticalspectra as they were calculated from the numerical solu-tions of Eqs. (1). Here the pulsed nature of the inputbeams was included in the simulations. As can be seen,excellent agreement is obtained between experimentaland theoretical spectra. Because the pump intensity var-ies across the profile of the long input pulses, the degreeof frequency conversion into the sidebands, which is afunction of the instantaneous pump power, is averagedacross the pump pulse. Therefore, even if at the center ofthe pump pulse the intensity of the first sideband is equalto that of the pump (as we discussed above, this conditionis obtained when a dark-soliton train is generated), thesideband intensity in the wings of the pump pulse ismuch reduced. When averaging over the pump pulseprofile, one then observes an effective reduction of theoverall MI sideband intensity, which leads to the asym-metric spectral profiles of Fig. 6.

Fig. 5. Experimental setup for observation of induced MI at theUniversity of Dijon: MPC, multiple-pass cell; ODL, optical de-lay line; DVP, direct vision prism; P’s, Glan polarizers; F’s,neutral-density filters; BS, beam splitters; MO’s, microscope ob-jectives; PM, photomultiplier; L, lens; l/2’s, half-wave plates.

Seve et al. Vol. 16, No. 10 /October 1999 /J. Opt. Soc. Am. B 1647

Indeed, the simulations show that under the conditionsspecified above the train of vector dark solitons is gener-ated in a time interval that corresponds to the center ofthe pump pulses. To show this, by means of autocorrela-tion traces we carried out a comparison between the the-oretical and experimental time-domain characteristics ofthe dark-soliton train. Figures 7(a) and 7(b) display thetheoretical and Figs. 7(c) and 7(d) show the experimentalautocorrelations traces as obtained at the fiber output forlight polarized along either the slow [Figs. 7(a) and 7(c)]or the fast [Figs. 7(b) and 7(d)] fiber axis. Here we usedthe same experimental conditions as in Fig. 6; as can beseen, again there is excellent agreement between the the-oretical and experimental autocorrelation traces. In allthe previous experiments the pump wave was polarized

Fig. 7. (a), (b) Theoretical and (c), (d) experimental autocorrela-tion traces from the slow (top) and fast (bottom) fiber axes, withthe same input conditions as in Fig. 6.

Fig. 6. (a), (b) Theoretical pulse averaged and (c), (d) experi-mental spectra from (top) slow and (bottom) fast axes, with apump power of 56 W on each axis. The signal pump power is 2W and the frequency detuning is 2.5 THz.

at 45° between the fiber axes and the MI was induced byinjection of a single Stokes signal beam that was polar-ized along the slow fiber axis. By symmetry arguments,an identical vector dark-soliton train could also have beengenerated by injection of an anti-Stokes MI seed at the fi-ber input, aligned with the fast axis.

To study the influence on dark-soliton generation of theinitial polarization of the MI seed, we carried out experi-ments with an input Stokes beam linearly polarized at45° between the fiber axes. The results of numerical so-lutions of CNLS equations (1) with such a Stokes seed areshown in Fig. 8. Here the top curves illustrate the out-put pulse trains, whereas the lower curves display thepower spectra of light emerging from the two polarizationcomponents of light along the fast (left) and slow (right)axes of the fiber, respectively. The initial modulation fre-quency was set to 2.6 THz, whereas the signal power wasPs 5 2 W. As can be seen, with such an input conditiona time-symmetric dark-soliton train is generated onlyalong the slow axis of the fiber, whereas the pulse trainthat emerges from the fast axis exhibits an asymmetrictemporal profile. Indeed, the spectra in Fig. 8 show thatalthough the intensities of the pump and the anti-Stokeslight (for the fast axis) or the Stokes light (for the slowaxis) are equal in both cases, the intensities of the othersidebands are asymmetric with respect to the center fre-quency of the spectrum. This spectral asymmetry is en-hanced for a pump along the fast axis, because in thiscase a Stokes seed is injected at the input, whereas MIleads to the development of an anti-Stokes beam on thataxis. By simple symmetry considerations it is easy toprove that, if the MI were induced by injection of an anti-Stokes signal linearly polarized at 45° between the axes,one would obtain instead a dark-soliton train along thefast axis.

One may draw the general conclusion that induced MIgenerates dark-soliton trains polarized on both axes only

Fig. 8. Theoretical time dependence of output powers in the fastand slow fiber axes.

1648 J. Opt. Soc. Am. B/Vol. 16, No. 10 /October 1999 Seve et al.

if the MI seed has a particular polarization state. Thatis, the Stokes (or anti-Stokes) seed should be polarizedalong the slow (fast) axis. Figure 9 compares the theo-retical (left) and experimental (right) power spectra ob-tained with the same initial parameters as in Fig. 8. Thepump and the Stokes beams were polarized at 45° be-tween fiber axes. The corresponding autocorrelationtraces are represented in Fig. 10. The theoretical and ex-perimental autocorrelation traces are shown at the leftand at the right, respectively; the top and bottom figuresrepresent the intensities on the slow and fast compo-nents. As can be seen, a fairly good qualitative agree-ment between theoretical and experimental spectra andautocorrelations traces was obtained.

4. RESULTS IN THE PICOSECOND REGIMEIn using picosecond pulses for experiments of induced MIin a fiber, one can no longer consider the excitation con-

Fig. 9. Left, theoretical pulse averaged and right, experimentalspectra from (top) slow and (bottom) fast axes, with pump andsignal powers of 56 and 2 W, respectively, on each axis. The fre-quency detuning is 2.6 THz.

Fig. 10. Left, theoretical and right, experimental autocorrela-tion traces from the slow (top) and fast (bottom) fiber axes, withinput conditions as in Fig. 9.

dition to be quasi-cw. Indeed, self- and cross-phasemodulation–induced spectral broadening of the pumppulses and walk-off-induced separation between thefinite-duration pulses should be taken into account. Wecarried out numerical simulations and experiments werecarried out with pulses in the picosecond range to exam-ine the possibility of forming a vector dark-soliton pulsetrain in this regime. In the experimental setup of Fig.11, the 38-ps (FWHM) pump pulses were delivered by afrequency-doubled, active–passive pulsed mode-lockedNd:YAG laser operating at 10 Hz. Its spectral width is0.028 nm at 532 nm. The 20-ps signal pulses were pro-vided by an optical parametric generator–amplifierpumped by the third harmonic of the Nd:YAG laser at 355nm. The output wavelength from the optical parametricgenerator–amplifier is tunable within the visible range420–680 nm. The signal was then spectrally filtered andstretched to ;40 ps (with a time–bandwidth product closeto 1) by an auxiliary Littrow grating spectrometer (notshown in Fig. 11), and its wavelength was adjusted to533.5 nm. The pump and signal beams were recombinedby a (50/50) beam splitter and launched in a hi-bi fiber(FSPA-10, Newport Corporation) with a 103 microscopeobjective. The two pulses were synchronized at the fiberinput by means of a prism delay line: The synchroniza-tion was monitored by a streak camera with a time reso-lution of 5 ps. The pump was polarized at 45° with re-spect to the slow and fast axes of the fiber, and the Stokessignal was polarized along the slow axis. The outputspectrum was analyzed by a Littrow grating spectrometer(0.3 m, 2400 lines/mm) with a spectral resolution of4.10–3 nm, coupled to a single-shot CCD camera that al-lowed for the acquisition of a single pulse.

In the experiment, the fiber parameters were the fol-lowing: the GVD was b2 5 60 ps2 km21, the birefrin-gence was dn 5 3 3 1024 (d 5 0.5 ps m21), the nonlinearcoefficient was g 5 0.0535 m21 W21, and the fiber lengthwas L 5 3 m. The pump and seed wavelengths werelp 5 532 nm and ls 5 533.5 nm. We used the above pa-rameters in the numerical simulation of CNLS equations(1), with P 5 144 W, Ps 5 800 mW, and a pulse duration

Fig. 11. Picosecond experimental setup for observation of in-duced MI at the University of Besancon: OPG, optical paramet-ric generator–amplifier; ODL, optical delay line; P’s, Glan polar-izers; l/2’s, half-wave plates; BS, beam splitter; O1, O2,microscope objectives.

Seve et al. Vol. 16, No. 10 /October 1999 /J. Opt. Soc. Am. B 1649

(1/e half-width) of T0 5 20 ps. Figures 12(a), 12(b), and12(c) display the temporal profile of the optical power oneach polarization axis in the central part of the pulse, the(incoherently) recombined total power, and the globalspectrum that results from the intensity superposition ofthe slow and fast spectra, respectively. The modulationperiod in Figs. 12(a) and 12(b) is 630 fs, which corre-sponds to the frequency separation between the pumpand Stokes signal pulses ( fmod 5 1.59 THz). The pulsetrains that are formed on each axis of the fiber are notquite dark-soliton trains in this case, because the modu-lation frequency fmod and the signal/pump power ratio ahave not been optimized. As a consequence, in Fig. 12(a)the pulse trains on orthogonal axes exhibit a slight tem-poral asymmetry, which results from the phase-conjugatenature of the two coupled pulse trains. Moreover, thecontrast of the pulse trains is slightly less than unity.Figure 12(a) also shows how the finite picosecond dura-tion of the pulse envelope affects the overall profile of thequasi-dark-soliton trains. As can be seen, the peak in-tensity at each modulation period on the slow (fast) axisincreases (decreases) from negative to positive times; theintensities on both axes are the same at t 5 0. Such asmooth variation follows from that of the two pulse enve-lopes on each axis as they separate from each other by;1.5 ps because of the pulse walk-off. Nevertheless, itcan be seen that the cross-trapping action between theslow and fast near-dark-soliton trains remains excellentin spite of the slow separation of the pulse envelopes.Hence the two trains propagate with exactly the samegroup velocity, and the recombined temporal power pro-file in Fig. 12(b) is time symmetric. The slight Gaussianshape, centered at t 5 0, of the peak power envelope [seeFig. 12(b)] shows that the vector nature of the dark-soliton train is maintained in the presence of a slow varia-tion of the peak power of the pump and signal fields.

The corresponding spectrum in Fig. 12(c), which is dis-played on a linear power scale as opposed to the spectra ofthe nanosecond experiments, shows the various gener-

Fig. 12. (a) Theoretical temporal power profile at the fiber out-put on the slow (dotted–dashed curve) and fast (solid curve) axes.(b) Theoretical recombined power profile. (c) Corresponding the-oretical global spectrum. (d) Experimental global spectrum.

ated frequency components, all shifted by fmod5 1.59 THz. The global spectrum of Fig. 12(c) is sym-metric with respect to the pump frequency (even thoughthe slow and fast spectra, not shown here, are asymmet-ric), and illustrates, as in the nanosecond experiments,the averaging of the power distribution within the pumppulses. Figure 12(c) also shows the large spectral broad-ening of the various frequency components, which is typi-cal of the picosecond regime, owing to the self-phasemodulation of the pump and of the phase-matched in-tense waves. On the other hand, the less intense non-phase-matched waves and the higher-order harmonicsare broadened through XPM. The experimental spec-trum of Fig. 12(d), obtained with P 5 160 W, Ps5 500 mW, and the other parameters as above, is in goodagreement with the theoretical spectrum, although theself-phase modulation and XPM spectral broadening of allfrequency components is less resolved. This result indi-cates that a vector near-dark-soliton train is generated inthe central part of the picosecond pulse.

5. CONCLUSIONSIn conclusion, we have presented an experimental andtheoretical study of induced modulational instability in anormally dispersive highly birefringent fiber. We haveshown that with the proper choice of pump-signal fre-quency detuning and signal-pump power ratio (as deter-mined by numerical simulations of the coupled nonlinearSchrodinger equations), the induced MI process in ahighly birefringent optical fiber may lead to coupled dark-soliton trains with terahertz repetition rates at the fiberoutput. Several experiments with various polarizationarrangements of the pump and Stokes input signal beamwere carried out. Whenever the Stokes beam was polar-ized along the slow axis, a dark-solitonlike train was gen-erated on both axes (slow and fast). In this situation, thetwo coupled trains are mutually trapped by the fiber non-linearity and form a vector dark-soliton train at a givenset of positions along the fiber. On the other hand, witha Stokes beam polarized at 45° from the fiber axes, adark-soliton train is created along the slow axis only.

ACKNOWLEDGMENTSThis research was supported by the Conseil Regional deBourgogne, the Centre National de la Recherche Scienti-fique, and the Ministere de la Recherche.

G. Millot’s e-mail address is millot@jupiter.u-bourgogne.fr.

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