Nonlinear pulse shaping by coherent addition ofmultiple redshifted solitons

Esben Ravn Andresen,1 John M. Dudley,2 Dan Oron,3 Christophe Finot,4 and Hervé Rigneault1,*1Institut Fresnel, Aix-Marseille Université, CNRS, DomaineUniversitaire de St. Jérôme, 13397Marseille Cedex 20, France2Laboratoire d’Optique P. M. Duffieux, Institut France Comté Electronique Mécanique Thermique et Optique—Sciences

et Technologies CNRS, 16 Route de Gray, 25030 Besançon Cedex, France3Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel

4Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS, Université de Bourgogne, 21078 Dijon, France*Corresponding author: herve.rigneault@fresnel.fr

Received March 23, 2011; accepted May 19, 2011;posted May 24, 2011 (Doc. ID 144712); published June 21, 2011

The injection of a phase- and amplitude-shaped pulse into a photonic-crystal fiber provides additional degrees offreedom that can significantly influence the nature of nonlinear propagation and nonlinear and dispersiveinteractions. This strong sensitivity of nonlinear effects—particularly the Raman soliton self-frequency shift—greatly extends the parameter space available to generate tailored output fields for applications such as micro-scopic imaging. By numerical simulations, we identify the relevant interpulse interactions, andwe experimentallydemonstrate the additional capabilities of this nonlinear pulse-shaping method. © 2011 Optical Societyof America

OCIS codes: 060.4005, 060.5295, 190.4370, 260.3160, 320.5540, 320.7140.

1. INTRODUCTIONNonlinear propagation and spectral broadening in photonic-crystal fiber (PCF) has now been extensively studied andapplied for diverse applications including frequency metrol-ogy, imaging, and spectroscopy [1,2]. For some applications,the generation of broadband spectra is not itself a require-ment, and what is more important is the generation ofhigh-brightness spectral components covering particularwavelength ranges. In this case, the absolute bandwidth spanof the generated supercontinuum is less important than thecharacteristics of the spectral components of interest. Thisparameter regime can be referred to as a class of “nonlinearpulse shaping” where the preshaping of an input pulse inphase and amplitude can be used to exert control over thePCF spectral output.

The general setup for nonlinear pulse shaping is sketched inFig. 1, where a comparison with conventional phase and am-plitude shaping [3,4] is also given. This kind of setup, utilizinga phase and amplitude shaper before an optical fiber, wasused in [5–8]. Of course, numerous examples of nonlinearpulse shaping have been demonstrated, albeit under variousother names; common to all though is that the phase or am-plitude of the PCF input is controlled in some way, though notnecessarily by a phase and amplitude shaper. Examples usingonly a single input pulse are optical power control of the so-liton self-frequency shift (SSFS) [9–11] and spectral compres-sion by self-phase modulation of a negatively chirped pulse[12–14], which have found application in nonlinear micro-scopy [5,15–18]. SSFS in combination with progressive spec-tral narrowing upon redshift in a dispersion-tailored PCF wasalso demonstrated [19]. Finally, fiber-generated superconti-nua have been controlled to some degree by the shape ofthe input pulse [6,7,20–22].

Recently, the prospect of temporal compression based onthe addition of several femtosecond redshifted solitons pro-duced in a PCF was raised [23]. Two different solitons withdifferent frequencies were coherently added to yield a broad-er spectrum allowing temporal compression to a shorter dura-tion relative to the constituent solitons. Reference [5] focusedpurely on spectral shaping by addition of redshifted solitons;several solitons with different frequencies were added to forma broad PCF output spectrum. These papers underline theclear and current interest in developing a more complete pic-ture of how control of the nonlinear propagation can be usedto yield tailored output fields with characteristics specificallysuited to particular applications.

In this paper, our purpose is to explore in detail the non-linear shaping processes relevant to experiments such asthose described in [5,23]. Here it is soliton formation andthe subsequent SSFS that are the primary nonlinear effectstaking place in the PCF. A particular aim is to expand theprevious, mainly theoretical, work on interactions betweensolitons [24–39] to a predominately experimental and appliedregime. This topic has, as far as we can see, not yet been fullytreated in the literature on fiber solitons, which tends tofocus more on soliton collisions [33,40–49] or the so-calledbound pairs, bound states of solitons, or soliton molecules[28,34,35,37–39]. In our case, we aim to focus specificallyon realistic conditions where the initial pulses are not perfectsolitons, in which case two new interactions must be consid-ered, namely the interaction between a decelerating, leadingsoliton and the dispersive residues shed by trailing solitonsand the interaction between the pulses before they have sheda fundamental soliton. By numerical simulations and experi-ments, we will consider in detail the possible interactionsinvolved in realistic experiments. Our results demonstratethe additional frequency-conversion capabilities of nonlinear

1716 J. Opt. Soc. Am. B / Vol. 28, No. 7 / July 2011 Andresen et al.

0740-3224/11/071716-08$15.00/0 © 2011 Optical Society of America

pulse shaping compared to traditional phase and amplitudeshaping.

2. MATERIALS AND METHODS

A. Numerical CodeThe numerical simulations used a standard generalized non-linear Schrödinger equation model [1,11]. Our aim was to rea-listically describe the experiments performed using the setupin Fig. 2; thus, we used the realistic Raman response and thefull dispersion curve of the fiber as detailed in Table 1. ThePCF has a single zero-dispersion wavelength at 745 nm.

We consider initial conditions of a pair of identical pulsesE0 separated in time by Δt and with relative phase Δϕ, i.e.,

EpairðtÞ ¼ E0ðtÞ þ E0ðt −ΔtÞeiΔϕ: ð1Þ

To model nonideal soliton input fields, we consider differ-ent shapes of E0 and parameters as listed in Table 2; the tablealso lists the corresponding section below where we presentthe results. The soliton number N ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðτ2γP0Þ=ð1:7632jβ2jÞ

pis

chosen such that each E0 on its own approximately producesthe same output of a single soliton redshifted by Δλ ¼ 45nmto 1080nm at the output of the PCF. The time frame was cho-sen to be stationary at λ ¼ 1035nm and centered on E0ðtÞ.

From the PCF output field Eout resulting from the numericalpropagation of Epair, the solitonic part can be identified andthe relative delayΔTsol, defined as the peak–peak separation,of the created soliton pair be found for different values ofΔϕ,which we vary over the range 0 to 2π. The resulting ΔTsol

versus Δϕ will form the basis for the comparison of the simu-lations with the experimental measurements.

B. Experimental SetupA sketch of the experimental setup is shown in Fig. 2. Thelaser used to do the experiments is a t-pulse SESAM-mode-locked, diode-pumped ytterbium laser from Amplitude Sys-tems (50MHz, 1035nm, 150 fs). The laser pulses are sentthrough a Michelson interferometer, acting as the phase-and amplitude-shaping device, to create from the laser pulseEðexpÞpair . A mirror in one arm is mounted on a delay stage with a

piezoelectric transducer, allowing us to set the initial separa-tion Δt as well as the initial relative phase Δϕ of the pulsepair. The resulting pulse pair is then

EðexpÞpair ðtÞ ¼ E0ðtÞ þ E0

�t −ΔtþΔϕλ

2πc

�eiΔϕ: ð2Þ

A neutral-density filter in one arm allows to balance theenergy of the constituent pulses. EðexpÞ

pair is sent through a72:5 cm long PCF (NL-2.0-745-02, Blaze Photonics) in whichthe majority of the energy of each pulse goes to form a funda-mental soliton with pulse energy of around 50pJ and durationFWHM τsol ¼ 60 fs, which redshifts upon propagation dueto the SSFS. The PCF output EðexpÞ

out is characterized by a non-collinear, background-free autocorrelator and a spectrumanalyzer (Ando AQ-6315A), allowing us to measure the

Fig. 1. (Color online) Schemes (top) for phase and amplitudeshaping and (bottom) for nonlinear pulse shaping.

Table 2. Initial Conditions Used in the Simulations

Subsection E0ðtÞ τ (fs) P0 (W) N

3.AffiffiffiffiffiffiP0

psechð−1:763t=τÞ 44 1710 1.0

3.BffiffiffiffiffiffiP0

pexpð−2 ln 2 t2=τ2Þ 140 600 -

3.CffiffiffiffiffiffiP0

psechð−1:763t=τÞ 140 600 1.9

Table 1. PCF Parameters Used atλ � 1035 nm

β2 −8:18299 · 104 fs2=mβ3 1:61038 · 105 fs3=mβ4 −1:95205 · 105 fs4=mβ5 2:07369 · 105 fs5=mβ6 6:05655 · 104 fs6=mβ7 1:34857 · 106 fs7=mβ8 −7:54952 · 106 fs8=mβ9 1:70524·107 fs9=mβ10 −4:71402 · 106 fs10=mγ 0:079 ðW · mÞ−1

Fig. 2. (Color online) Sketch of the experimental setup. Inset graphsshow sketches of (left) the input pulse pair temporal envelope andspectrum and (right) similar sketches for the output soliton pair.BBO, β-barium borate; APD, avalanche photodiode; PC, personalcomputer; LP, long-pass filter; ND, neutral density filter.

Andresen et al. Vol. 28, No. 7 / July 2011 / J. Opt. Soc. Am. B 1717

dependent parameter ΔTsol. The independent parameters Δtand Δϕ can be found by bypassing the PCF and sending thePCF input directly into the spectrum analyzer.

3. RESULTS: SIMULATIONSA. Soliton PairWe now consider the case where a soliton pair is propagatedthrough the PCF. Initially, we set the Raman response of thePCF to zero and ran the simulation. The obtained resultsclosely reproduced the analytical model of the soliton force(strictly speaking, the Kerr-mediated interaction between in-terfering solitons) in [24,25]. The simulation predicted attrac-tion for Δϕ ¼ 0 and repulsion for Δϕ ¼ π. In this case, therelative phase of the soliton pair does not change along thelength of the PCF, so that the integrated interaction observedafter the PCF (ΔTsol −Δt) is directly proportional to the so-liton force. So the attractive soliton force forΔϕ ¼ 0 leads to aΔT sol-decreasing interaction and vice versa. The recordedΔT sol versus Δϕ curve is displayed in Fig. 3(e) and will beused for comparison later.

We proceed to the case of realistic Raman response of thePCF. If there were no interaction between the constituent so-litons of Epair, Eout would still comprise identical solitons red-shifted relative to the input, and there would be a correction tothe relative phase due to the phase accumulated by thesolitons while undergoing the SSFS, ΔΦsol ¼ ΔϕþΔΦSSFS,ΔΦSSFS ¼ 2πcΔλΔt=λ2.

However, the redshifting solitons do interact, but there stillis a region where the solitons can be assumed to be noninter-acting. This is the case in Figs. 3(a) and 3(b), whereΔt ¼ 500 fs. As can be seen in Fig. 3(a), ΔT sol is unchangedwhen Δϕ is changed, and ΔΦsol (not shown) is a linear func-tion ofΔϕ, completely analogous to the case of noninteractingsolitons.

The situation changes whenΔt decreases. In Figs. 3(c) and3(d) we present simulations of a soliton pair withΔt ¼ 300 fs.As can be seen from Fig. 3(d), ΔT sol now varies with Δϕ,min½ΔTsol� is at Δϕ ¼ 1:3π, and max½ΔTsol� is at Δϕ ¼ 0:2π.For these parameters,ΔΦSSFS ≈ 3:6 · ð2πÞ, so the soliton forcechanges sign repeatedly along the PCF, which means thatðΔTsol −ΔtÞ is no longer proportional to the soliton force;the local, instantaneous attraction or repulsion cannot betaken as representative for the integrated interaction. Thiswas pointed out in [36].

We perform a series of simulations like the one above for anumber of Δt and Δϕ in the range 0 to 2π. The results aresummarized in Fig. 3(e). The points plotted are min½ΔTsol�and max½ΔT sol� (when mentioning these, we assume fixedΔt). The differences between the ΔTsol in the presenceand absence of SSFS are introduced by the SSFS. In presenceof SSFS, the interaction becomes ΔTsol-decreasing atΔt < 270 fs. In addition, (max½ΔTsol� −min½ΔT sol�) is smallerthan was the case when the Raman response was set to zero.These observations are consistent with the literature. Aspointed out in [49], the SSFS can force solitons into an attrac-tive state. Also, in [36] it was shown that the Δϕ dependenceof the soliton force decreases with increasing group velocitydifference, eventually becoming attractive for all Δϕ.

In Figs. 4(a) and 4(b) we show jEoutðtÞj2 and the instanta-neous angular frequency when Δt ¼ 500 fs for two differentΔϕ separated by π. Here the result is independent of Δϕ,as expected, as the soliton force is negligible at this Δt.In Figs. 4(c) and 4(d), Δt ¼ 260 fs, and curves are given forΔϕ ¼ 1:6π and Δϕ ¼ 0:5π, corresponding to min½ΔT sol� andmax½ΔTsol�, respectively. These show that the two solitons,

(a) ∆t = 500 fs

|Eou

t|2 [a.u

.]

4500 4800 5100

(b)

δω(t

) [a

.u.]

Time [fs]

(c) ∆t = 260 fs

4500 4800 5100

(d)

Time [fs]

Fig. 4. Simulation; soliton pair with τ ¼ 44 fs, λ ¼ 1035nm, andP0 ¼ 1710W. Output temporal envelopes and instantaneous angularfrequencies for (a), (b) Δt ¼ 500 fs, (c), (d) Δt ¼ 260 fs. Δϕ ¼ 1:3π(gray), corresponding to min½ΔTsol�; Δϕ ¼ 0:2π (black), correspond-ing tomax½ΔTsol�. Soliton positions in the noninteracting case (verticaldashed) and instantaneous angular frequency in the noninteractingcase (horizontal dashed), for reference.

1020 1070 1120Wavelength [nm]

|Eou

t|2 [a.u

.] (a)∆t = 500 fs

0 2 4 6450

500

550

∆φ [rad]

∆Tso

l [fs]

(b)

1020 1070 1120Wavelength [nm]

(c)∆t = 300 fs

0 2 4 6250

300

350

∆φ [rad]

(d)

0 100 200 300 400 5000

200

400

∆t [fs]

∆Tso

l [fs]

(e)

Fig. 3. Simulation; soliton pair with τ ¼ 44 fs, λ ¼ 1035nm, andP0 ¼ 1710W. Sample output spectra and output relative delayΔTsol versus initial relative phase Δϕ for initial relative delay (a),(b) Δt ¼ 500 fs, (c), (d) Δt ¼ 300 fs, (e) min½ΔTsol� (dots) andmax½ΔTsol� (crosses) versus Δt, min½ΔTsol� (solid) and max½ΔT sol�(dotted) for Raman response set to zero, and the noninteracting case(dashed).

1718 J. Opt. Soc. Am. B / Vol. 28, No. 7 / July 2011 Andresen et al.

though initialized as identical, do not stay identical. Rather,the leading soliton slows down and frequency downshifts re-lative to the trailing pulse with the frequency difference beingslightly larger when ΔT sol ¼ min½ΔT sol�.

B. Pair of Imperfect SolitonsOne additional effect becomes significant when E0 is not a so-liton, in which case the initial steps of propagation involve thesplitting of the pulse into a soliton and a dispersive residue,the former redshifting due to the SSFS and slowing downdue to dispersion and the latter staying around the initialwavelength. The leading soliton will then cross the residueof the trailing pulse. For this series of simulations, we initializeE0 as Gaussians with duration FWHM 140 fs. A few descriptiveresults are presented in Figs. 5(a)–5(d). We see from Figs. 5(b)and 5(d) that the interaction is now in general ΔT sol-decreasing. This is due to interaction through stimulated-Raman scattering (SRS); when the leading soliton passesthrough the residue, they are close enough in frequency thatthey can interact through the Raman response of the PCF. Be-cause the soliton is at lower frequency, it will act as the RamanStokes, being amplified while the residue acts as Raman pump,being depleted. This causes an increase in the leading soliton’senergy, leading to a frequency downshift through the SSFS anddeceleration, and hence, in effect, it is a ΔTsol-decreasing in-teraction. Only whenΔTsol gets close to 300 fs, which happensfor Δt ¼ 500 fs, Figs. 5(c) and 5(d), the Δϕ-dependent solitonforce is significant in the last part of the propagation in thePCF.Wenote thatmin½ΔTsol�occurs atΔϕ ¼ 0:3π andmax½ΔT sol� atΔϕ ¼ 1:3π. That min½ΔT sol� does not occur atΔϕ ¼ 0was also

seen in Subsection 3.A for perfect solitons. Here there couldalso be contribution from a small phase shift that happensdue to the frequency shift the soliton acquires in crossingthe residue. We summarize the results of several simulationsfor different values of Δt in Fig. 5(e).

Figures 6(a) and 6(b) show jEoutðtÞj2 and the instantaneousangular frequency for a Δt ¼ 600 fs where the soliton force isnegligible. Nevertheless, although the two pulses were identi-cal at the onset, when leaving the PCF, the leading pulse hasacquired an increase in peak power and a frequency down-shift. This is compatible with the explanation that it has gainedenergy from interacting with the residue of the trailing soliton.Traces for Δϕ ¼ 0:3π as well as for Δϕ ¼ 1:3π are presentedin Fig. 6, demonstrating that this contribution to the integratedinteraction is phase independent. The trailing soliton is unaf-fected, which is expected because it never crosses any otherpulse. In Figs. 6(c) and 6(d) are shown jEoutðtÞj2 and the in-stantaneous angular frequency for Δt ¼ 500 fs, where the so-liton force is significant. (max½ΔT sol� −min½ΔTsol�) is lessthan in Subsection 3.A. Whereas in the previous case, the in-teraction is sufficient at Δt ¼ 250 fs to induce collision, in thepresent case, no collision takes place atΔTsol ¼ 200 fs (whichhappens for Δt ¼ 500 fs). This be seen by comparing Fig. 5(e)with Fig. 3(e) or Fig. 6(c) with Fig. 4(c). We ascribe this to theincreased difference in soliton group velocities caused by thesoliton–residue crossing; as pointed out in [36], theΔϕ depen-dence of the soliton force diminishes with increasing groupvelocity difference, eventually becoming attractive for all Δϕ.

C. Pair of Overlapping PulsesIf there is significant temporal overlap between the two inputpulses, the interference will significantly influence the energydeposed into each formed soliton. To simulate this effect, weinitialize the simulation with a pair of pulses with sech2 envel-opes and FWHM 140 fs. Sech2 pulses are characterized by ap-proaching zero slower with t than do Gaussians, which is whywe chose this shape to highlight the effect of interferencebetween the initial pulses. The FWHM and peak power wereset equal to those used in the previous case, facilitating

1020 1070 1120Wavelength [nm]

|Eou

t|2 [a.u

.] (a)∆t = 600 fs

0 2 4 6

200

300

400

500

600

∆φ [rad]

∆Tso

l [fs]

(b)

1020 1070 1120Wavelength [nm]

(c)∆t = 500 fs

0 2 4 6∆φ [rad]

(d)

0 200 400 600 800 10000

500

1000

∆t [fs]

∆Tso

l [fs]

(e)

Fig. 5. Simulation; Gaussian pulse pair with τ ¼ 140 fs atλ ¼ 1035nm, P0 ¼ 600W. Sample output spectra and output relativedelay ΔTsol versus initial relative phase Δϕ for initial relative delay(a), (b) Δt ¼ 600 fs, (c), (d) Δt ¼ 500 fs, (e) min½ΔTsol� (dots) andmax½ΔTsol� (crosses) versusΔt, and the noninteracting case (dashed).

(a) ∆ t = 600 fs

|Eou

t|2 [a.u

.]

4100 4500 4900

(b)

δω(t

) [a

.u.]

Time [fs]

(c) ∆ t = 500 fs

4100 4500 4900

(d)

Time [fs]

Fig. 6. Simulation; Gaussian pulse pair with τ ¼ 140 fs atλ ¼ 1035nm, P0 ¼ 600W. Output temporal envelopes and instanta-neous angular frequencies for (a), (b) Δt ¼ 600 fs, (c), (d) Δt ¼500 fs. Δϕ ¼ 0:3π (gray), corresponding to min½ΔTsol�; Δϕ ¼ 1:3π(black), corresponding to max½ΔTsol�. Soliton positions in the nonin-teracting case (vertical dashed) and instantaneous angular frequencyin the noninteracting case (horizontal dashed), for reference.

Andresen et al. Vol. 28, No. 7 / July 2011 / J. Opt. Soc. Am. B 1719

comparison. Representative results in similar form as pre-vious are given in Figs. 7(a)–7(d), and the outcomes of simu-lations for a range of Δt are summarized in Fig. 7(e). Theconclusions to be drawn here are generally the same as for theGaussian case; the soliton–residue crossing also here leads toa ΔTsol-decreasing interaction. Here on the other hand,(max½ΔT sol� −min½ΔTsol�) is larger than in Subsections 3.Aand 3.B, as seen from the increased separation of dots andcrosses in Fig. 7(e) compared to Figs. 3(e) and 5(e). Thus,there is an additional contribution to the Δϕ-dependent inter-action. This contribution stems from the interference of theconstituent pulses in EpairðtÞ in interplay with the nonlineardynamics leading up to soliton formation. The interference be-tween the two pulses leads to two symmetrically perturbedpulse envelopes that are mirror images of each other, theirshape dependent on Δϕ. Because the SSFS is not symmetricin frequency and time, the nonlinear evolutions of the twopulses are different and the perturbed pulses form solitonswith different energies.

In Figs. 8(a) and 8(b) are plotted jEoutðtÞj2 and instanta-neous angular frequency for Δt ¼ 800 fs, which are qualita-tively similar to Figs. 6(a) and 6(b). For Δt ¼ 500 fs,however, Figs. 8(c) and 8(d), there is a difference comparedto Figs. 6(c) and 6(d) in that the two solitons are not equallyand oppositely affected by the Δϕ dependence. We take thisas confirmation that our explanation above is correct; the so-liton force alone is not responsible for the Δϕ dependence in

the case of significantly overlapping input pulses; interferencein interplay with nonlinear dynamics gives a contribution tothe Δϕ dependence as well.

4. EXPERIMENTAL RESULTSA. Δϕ and Δt DependenceInitially, we will attempt to establish a link between the simu-lations and the experiments. To this end, we present the set ofexperiments that the simulations described above were meantto describe, launching EðexpÞ

pair into the PCF, keeping Δt fixedwhile scanning Δϕ. A representative PCF output spectrumis given in Fig. 9(a), the intensity autocorrelation of the laserused is shown in Fig. 9(b), and the summary of the experimen-tal results is presented in Fig. 9(c). It is seen that the interac-tion is in general ΔTsol-decreasing, a manifestation of theinteraction through SRS between the leading soliton withthe dispersive residue from the trailing pulse, in accordancewith the simulations. In comparing with the simulations, thebestqualitative agreement is thus with Fig. 7(e), the most generalcase treated. That (max½ΔT sol� −min½ΔTsol�) stays nonzerountil Δt ≈ 1000 fs indicates that the temporal envelope ofthe laser is not completely Gaussian but has small tails.(max½ΔT sol� −min½ΔT sol�) can be aggravated when we, asin Fig. 10, intentionally misalign the laser. The misalignmentleads to the appearance of a narrow Kelly sideband in the la-ser spectrum and a quasi-cw component in the time domain.As such, this experiment is reminiscent of [27]. As seen inFig. 10(b), although the FWHM is now slightly smaller, largewings of the order of some per mille in relative inten-sity extend way beyond the FWHM. As seen in Fig. 10(c),these small wings are sufficient to significantly increase(max½ΔT sol� −min½ΔT sol�).

B. Spectral Brightness IncreaseEven though many effects contribute to the interaction be-tween pulses and solitons in the PCF, this has only a limitednegative impact on the applicability of multisoliton superposi-

1020 1070 1120Wavelength [nm]

|Eou

t|2 [a.u

.] (a)∆t = 800 fs

0 2 4 6

200

400

600

800

∆φ [rad]

∆Tso

l [fs]

(b)

1020 1070 1120Wavelength [nm]

(c)∆t = 500 fs

0 2 4 6∆φ [rad]

(d)

0 200 400 600 800 10000

500

1000

∆t [fs]

∆Tso

l [fs]

(e)

Fig. 7. Simulation; sech2 pulse pair with τ ¼ 140 fs at λ ¼ 1035nm,P0 ¼ 600W. Sample output spectra and output relative delay ΔTsolversus initial relative phase Δϕ for initial relative delay (a),(b) Δt ¼ 800 fs, (c), (d) Δt ¼ 500 fs, (e) min½ΔTsol� (dots) andmax½ΔTsol� (crosses) versusΔt, and the noninteracting case (dashed).Note that, in (d), the flatness of the curve aroundΔϕ ¼ 0 is due to thesolitons colliding, which is also why the spectrum in (c) does not looklike the superposition of two soliton spectra.

(a) ∆t = 800 fs

|Eou

t|2 [a.u

.]

3400 3700 4000

(b)

δω(t

) [a

.u.]

Time [fs]

(c) ∆t = 500 fs

3400 3700 4000

(d)

Time [fs]

Fig. 8. Simulation; sech2-pulse pair with τ ¼ 140 fs at λ ¼ 1035nm,P0 ¼ 600W. Output temporal envelopes and instantaneous angularfrequencies for (a), (b) Δt ¼ 800 fs, (c), (d) Δt ¼ 500 fs. Δϕ ¼ 0:2π(gray), corresponding to min½ΔTsol�; Δϕ ¼ 1:2π (black) , corres-ponding to max½ΔTsol�. Soliton positions in the noninteracting case(vertical dashed) and instantaneous angular frequency in the nonin-teracting case (horizontal dashed), for reference.

1720 J. Opt. Soc. Am. B / Vol. 28, No. 7 / July 2011 Andresen et al.

tion for tailoring jEðexpÞout j2 in time and frequency. To this end,

we now present some results of nonlinear pulse shaping,employing the aforementioned laser, for which all the dis-cussed interactions are at play simultaneously. Figure 11(a)shows the resulting jEðexpÞ

out ðωÞj2 when launching EðexpÞpair with

Δt ¼ 640 fs. The spectrum changes from the thick black tothe thin black by changing Δϕ by an uneven integer multipleof π. First, we emphasize that this is an example of a bright-ness increase of almost 4 compared to the constituent solitons(gray spectra), or a factor of 2 compared to the incoherentsum. This is due to interference between the two solitons,and in a way, the SSFS has shifted the initial interferencepattern at the laser wavelength to longer wavelengths. By ex-

tension, if n solitons had been launched, brightness increasescompared to the fundamental soliton of n2 would be attain-able. The comparison to the brightness of a fundamentalsoliton is justified because its brightness is inherently limitedby the fiber parameters; this limited its usefulness when usedas Stokes pulse in coherent anti-Stokes Raman scattering(CARS) microspectroscopy [15]. Figure 11(b) shows the de-pendence of ΔT sol upon Δϕ as was expected, detailing thequalitative agreement with Figs. 5(d) and 7(d). In the experi-ment, min½ΔT sol� is found at Δϕ ¼ 0:7π due to the effects de-scribed in Section 3. Though all the effects so far discussedplay a role in the interaction, the Δϕ dependence is still be-nign; the soliton–residue interaction contributes a negativeoffset to ΔT sol and decreases (max½ΔTsol� −min½ΔT sol�),which is counteracted by the initial interference of the consti-tuent pulses, which increases (max½ΔT sol� −min½ΔTsol�).

C. Phase StabilityThe solitons in the PCF add coherently, by which we meancoherent from one laser shot to the next, i.e., the phase staysunchanged from shot to shot and the output relative phase isthus a function of the input relative phase. To do this, it is suf-ficient to examine Fig. 11(a) and take notice of the visibility ofthe spectral interference fringes, which is close to 100%. Now,these spectra are acquired over many millions of laser shots(seconds), so if there were any phase fluctuation from shot toshot, the visibility would be decreased. This is in line withwhat the simulations showed, that, in the case of correlatedP0 fluctuations, the spectral interference fringes stay wherethey are.

5. CONCLUSIONWe have investigated the impact of interpulse interactionsbetween pulse pairs Epair propagating in a PCF with generalnonlinear response. When Epair is composed of solitons at the

1030 1050 1070 1090λ [nm]

(a)

−2000 0 2000Delay [fs]

× 10× 10

(b)

0 200 400 600 800 1000 1200 14000

500

1000

∆t [fs]

∆Tso

l [fs]

(c)

Fig. 10. Experiment with misaligned laser. (a) Representative PCFoutput spectrum. (b) Intensity autocorrelation of the laser; the wingsare magnified to better show the quasi-cw component arising from aKelly sideband. (c) Output relative delay ΔTsol versus input relativedelay Δt for the attractive branch (dots), the repulsive branch(crosses), and the noninteracting case (dashed).

1050 1060 1070 1080 10900

0.2

0.4

0.6

0.8

1

λ [nm]

|Eou

t(e

xp) |2 [a

.u.]

(a)

0 1 2 3 4 5 6300

400

500

600

700

∆Tso

l [fs]

∆φ [rad]

min[∆Tsol

]

max[∆Tsol

]

Ind. sol.

(b)

Fig. 11. Experiment; pulse pair,Δt ¼ 640 fs. (a) Output spectra of theindividual solitons propagating alone (gray), the soliton pair formin½ΔT sol� (thick solid), the soliton pair for max½ΔTsol� (thin solid);(b)measuredΔt (dashed) andmeasuredΔTsol (blackdots) versusΔϕ.

1030 1050 1070 1090λ [nm]

(a)

−2000 0 2000Delay [fs]

(b)

0 200 400 600 800 1000 1200 14000

500

1000

∆t [fs]

∆Tso

l [fs]

(c)

Fig. 9. Experiment; (a) representative PCF output spectrum, (b) in-tensity autocorrelation of the laser, (c) min½ΔTsol� (dots) andmax½ΔTsol� (crosses) versusΔt, and the noninteracting case (dashed).

Andresen et al. Vol. 28, No. 7 / July 2011 / J. Opt. Soc. Am. B 1721

onset of propagation, a departure from the simplifiedanalytical expression of [25] is observed due to the SSFS,which leads to a ΔT sol-decreasing interaction at small Δt.When Epair is not initially composed of solitons, the constitu-ent pulses first evolve into solitons in shedding a dispersiveresidue; the interaction through SRS between the leading so-liton and the trailing residue leads to a net decrease of ΔTsol.If, in addition, the constituent pulses initially overlap, thisgives rise to an interaction that can increase or decreaseΔT sol, depending on Δϕ. In our experiments, all of the men-tioned effects were found to be significant. It is thus apparentfrom our treatment that, in order to access the most simpledomain treated in Section 3.A, Epair must be preshaped witha phase and amplitude shaper.

As an experimental example, we demonstrated how the co-herent addition of solitons in a PCF can simultaneously red-shift an input pulse pair and through interference increase thespectral brightness by a factor of 2 compared to the incoher-ent sum. The same example demonstrated how a small differ-ence in the relative delay at the PCF input can translate into alarge change at the PCF output. Or, quoting the simulations, achange in the relative phase without any change in the tem-poral envelope could produce similar results, thus facilitatingnonmechanical scanning of a delay over hundreds of femto-seconds.

We note the fundamental feature of the present approach tocontrolling the nonlinear interaction in optical fibers: in add-ing solitons, which are stable under propagation regardless ofthe magnitude of the GVD, this is a universal approach, whichis not dependent on the exact fiber parameters as long as thefiber has anomalous dispersion and is sufficiently nonlinear.

Besides the obvious prospects of extending the describedmethod to more general input pulse trains containing morethan two pulses with unbalanced intensities, an interesting fu-ture potential for nonlinear pulse shaping by superposition offundamental solitons could be to experimentally synthesizethe soliton trains predicted in [37,39] to propagate as a singleentity.

We believe that nonlinear pulse shaping has potentialapplications in nonlinear microscopy where high spectralbrightness is required, such as CARS microscopy, SRS micro-scopy, and two-photon fluorescence microscopy.

ACKNOWLEDGMENTSWe thank Sebastien Michel for stimulating discussions. Finan-cial support was provided by the French National ResearchAgency (grant SOFICARS RIB 2007), the Région Provence-Alpes-Côte-d’Azur, and the CNRS-Weizmann NaBi EuropeanAssociated Laboratory. We are grateful to Amplitude Systemsfor making the t-pulse laser available for the experiments.

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