We report the cancellation of the soliton self-frequency shift in nonlinear optical fibers. A soliton which interacts with a group velocity matched low intensity dispersive pump pulse, experiences a continuous blue-shift in frequency, which counteracts the soliton selffrequency shift due to Raman scattering. The soliton self-frequency shift can be fully compensated by a suitably prepared dispersive wave.We quantify this kind of soliton-dispersive wave interaction by an adiabatic approach and demonstrate that the compensation is stable in agreement with numerical simulations

Amiranashvili, Shalva

Bandelow, Uwe

Pickartz, Sabrina

English

We report the cancellation of the soliton self-frequency shift in
nonlinear optical fibers. A soliton which interacts with a group velocity
matched low intensity dispersive pump pulse, experiences a continuous
blue-shift in frequency, which counteracts the soliton selffrequency shift
due to Raman scattering. The soliton self-frequency shift can be fully
compensated by a suitably prepared dispersive wave. We quantify this kind of
soliton-dispersive wave interaction by an adiabatic approach and demonstrate
that the compensation is stable in agreement with numerical simulations

Repositorium für Naturwissenschaften und Technik

Weierstraß-Institutfür Angewandte Analysis und StochastikLeibniz-Institut im Forschungsverbund Berlin e. V.Preprint ISSN 2198-5855Asymptotically stable compensationof soliton self-frequency shiftSabrina Pickartz, Uwe Bandelow, Shalva Amiranashvilisubmitted: November 30, 2016Weierstraß-InstitutMohrenstr. 3910117 BerlinGermanyE-Mail: sabrina.pickartz@wias-berlin.deuwe.bandelow@wias-berlin.deshalva.amiranashvili@wias-berlin.deNo. 2343Berlin 20162010 Mathematics Subject Classification. 78A60, 35Q60, 70H11.2010 Physics and Astronomy Classification Scheme. 42.65.-k, 42.81.Dp, 05.45.Yv, 03.65.Nk.Key words and phrases. Soliton self-frequency shift, Raman scattering, Soliton perturbation theory, Event hori-zons.U.B. and Sh.A. acknowledge support of Einstein Foundation Berlin and Research Center MATHEON under ProjectD-OT2.Edited byWeierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)Leibniz-Institut im Forschungsverbund Berlin e. V.Mohrenstraße 3910117 BerlinGermanyFax: +49 30 20372-303E-Mail: preprint@wias-berlin.deWorld Wide Web: http://www.wias-berlin.de/AbstractWe report the cancellation of the soliton self-frequency shift in nonlinear optical fibers.A soliton which interacts with a group velocity matched low intensity dispersive pumppulse, experiences a continuous blue-shift in frequency, which counteracts the soliton self-frequency shift due to Raman scattering. The soliton self-frequency shift can be fully com-pensated by a suitably prepared dispersive wave. We quantify this kind of soliton-dispersivewave interaction by an adiabatic approach and demonstrate that the compensation is sta-ble in agreement with numerical simulations.1 IntroductionA soliton propagating along a nonlinear optical fiber is subject to Raman scattering, the “red end”of the soliton spectrum is amplified at the expense of the “blue end” [1]. This results in what iscalled the soliton self-frequency shift (SSFS), a noticeable decrease of the carrier frequencyof subpicosecond solitons [2]. The Raman scattering plays an important role, e.g., in opticalsupercontinuum [3, 4], but in soliton communications systems it is desirable to cancel its effect.The SSFS depends strongly on pulse-width [5], so solitons with initially equal carrier frequencybut slightly different durations will gradually get different carrier frequencies (velocities) underRaman scattering, which results in jitter [6].SSFS compensation may be mediated by cross-phase modulation (XPM) between the solitonand an accompanying dispersive wave (DW) [7]. The latter appears naturally if a pulse whoseinitial power exceeds that of a soliton with the same duration, brakes down into just that solitonand a DW [2]. Initially both, soliton and DW, have the same carrier frequency. Other possibilitiesfor SSFS compensation include, e.g., bandwidth-limited amplification [8], pump by an additionalsoliton [9], interplay between Raman effect and Cherenkov radiation [10, 11]. In this Letter wedemonstrate how the SSFS can be counteracted by XPM interaction with a prearranged pumpDW, which (i) is much weaker than the soliton and (ii) has a distinctly different carrier frequency.Soliton collision with chirped DWs for SSFS compensation has been suggested in [12] basedon numerical results.A typical interaction of this kind is seen in Fig. 1. The electromagnetic power density [Fig. 1(a)]is plotted in space-time domain in a frame that co-moves with the unperturbed soliton. TheDW approaches the initially stationary soliton (zero delay) and, being almost perfectly reflected[13, 14], yields an interference picture seen to the left of the soliton. The latter is deflected [redline in Fig. 1(a)] and compressed [red line in Fig. 1(b)]. The reflected part of the DW is frequencyshifted, as seen in the frequency domain [Fig. 1(d)]. The soliton frequency is also shifted dur-ing reflection [Fig. 1(c)]. Our numerical simulations use a generalized nonlinear Schrödingerequation (GNLSE) with Raman term [4]. Without the DW, both the soliton’s frequency and am-plitude degrade, whereas its trajectory is deflected towards larger delays (sole Raman effect,not shown). This degradation is completely compensated by the DW after an initial transientphase, moreover, we will see below that the compensation is stable.1550 60080 90 100 110 120010203040Frequency (THz)Propagation distance (cm)-1.5 -1.0 -0.5 0.0 0.5010203040Delay (ps)Propagation distance (cm)1.31.21.11.00.90 10 20-2.0-2.5 30 40Propagation distance (cm)130 500Power (arb. units)(c) (d)-30-20-100Power (dB)-50-40-30-20-100Spectral power (dB)(b)(a)GNLSEAdiabaticScatteredDWIncoming DWFigure 1: GNLSE calculation of a DW at 3.67 PHz interacting with a soliton at 0.67 PHz.(a) Power in space-time domain. The white dashed line is the soliton trajectory in the adiabaticapproximation. (b) Soliton peak power from GNLSE (red line) and adiabatic theory (dashed line).(c,d) spectral power for soliton and DW respectively. Dashed lines are from adiabatic equations(7–9).Dispersion properties of bulk silica meet all requirements for the interaction. Fig. 2 shows fre-quency dependencies of group velocity β′ and group velocity dispersion (GVD) β′′. The initialsoliton carrier frequency ωa and DW frequency ωb + Ω are chosen from opposite sides of thezero dispersion frequency (β′′(ωZDF) = 0) such that they provide almost equal group velocities.The reference frequency ωb corresponds to the equal group velocities, β′(ωa) = β′(ωb). In themoving frame we deal with the common delay τ = t− β′(ωa)z = t− β′(ωb)z. The propaga-tion distance z is measured along the fiber, t is the physical time variable. The frequency offsetΩ should lie in a small interval around ωb (shaded gray in Fig. 2) in order for the two pulses tointeract [14]. A nearly perfect DW reflection (Fig. 1a) can be understood as a fiber-optical analogof the event horizon [15] or with quantum mechanical scattering theory [16].In what follows, we quantify XPM interactions, like the one shown in Fig. 1. The standard solitonperturbation theory [17] was adapted to the problem at hand in [16]. It yields an adiabatic de-scription of how a soliton evolves when colliding with a DW. Here we extend the theory to includethe influence of Raman scattering, as outlined in the next section. Further we demonstrate howthis description suggests that a stable compensation of the SSFS can occur. We describe asimple procedure to specify DW parameters that will produce cancellation of the SSFS. The lastsection deals with the stability analysis, numerical tests, and discussion.24.890β΄ (fs/μm)β΄΄ (fs2 /μm)4.8864.8824.878-0.06-0.20.21.0 1.2 1.4 1.6 1.8 2.0 2.2Circular frequency (1015 rad/s)ωa ωZDF ωb(a)(b)Figure 2: A typical profile (bulk silica) of (a) the group delay β′(ω) and (b) GVD β′′(ω) thatleads to the collision phenomenon shown in Fig. 1. Collision can only be realized for initial DWfrequency offsets in a small interval (shaded gray) around the reference frequency of matchinggroup velocity. Initial frequencies of soliton and DW are indicated by bullets, frequency shifts areindicated by arrows.2 Model equationsAn adiabatic description of interactions, like in Fig. 1, is derived from two XPM-coupled GNLSEs,one centered at ωa for the soliton envelopeψ(z, τ), and one centered at ωb for the DW envelopeψb(z, τ). The DW equation is simplified such that it can be solved as a problem of plane wavescattering at a moving “solitonic” potential barrier. The soliton equation is reformulated so thatall higher-order terms in the GNLSE are treated as perturbationsi∂zψ + i [β′(ωa + ν)− β′(ωa)] ∂τψ− β′′(ωa + ν)2∂2τψ + n2aωa + νc|ψ|2 ψ = R (ψ, ψb) . (1)The most interesting point about this perturbation equation is that it explicitly takes into accounta varying soliton carrier frequency ωa + ν, where ν = ν(z), ν(0) = 0 is the yet unknowndetuning from the initial carrier frequency ωa. Especially the GVD β′′(ω) and the nonlinear termon the left-hand side of (1) follow the soliton frequency shift. Therefore also the deviation fromthe initial group delay β′(ωa) appears in the equation. All higher-order terms are contained inthe perturbation R (ψ, ψb)R(ψ, ψb) = −M∑m=3β(m)(ωa + ν)m![i∂τ ]m ψ − τ dνdzψ− 2n2ac[ωa + ν + i∂τ ][|ψb|2 ψ]− n2aci∂τ[|ψ|2 ψ]− fRn2ac[ωa + ν + i∂τ ] [I(ψ)ψ] , (2)I(ψ) =∫ τ−∞h(τ − τ ′) |ψ(z, τ ′)|2 dτ ′ − |ψ(z, τ)|2 .3The terms with derivatives of the wave vector β(ω) account for higher-order dispersion, the firstterm in the second line describes XPM, the second is the self-steepening term. The perturbationterm containing dν/dz results from the necessary reformulations [16] of the standard solitonperturbation equation (as found for example in [17]) in order to include ν in the GVD. The termin the third line describes Raman scattering. The included Raman response function for fusedsilica readsh(τ) =ν21 + ν22ν1e−ν2τ sin ν1τ, (3)and we used parameters fR = 0.18, 1/ν1 = 12.2 fs, and 1/ν2 = 32 fs [18].The fundamental soliton solution of the unperturbed (1) reads in a general formulationψ =1σ√|β(2)(ωa + ν)|c[ωa + ν]naeΘcosh 1σ[τ − T ], (4)with frequency offset ν = ν(z) from the initial carrier frequency ωa, duration σ = σ(z), tem-poral delay T = T (z), and a phase Θ = Θ(z). It is assumed that all soliton parameters arez-dependent and slowly evolve under the influence of the DW. The soliton perturbation theoryprovides ODEs for these parameters after |ψb|2 is specified. The initial parameter values aredenoted by σ(0) = σa, etc.The DW equation was derived by a series of simplifications applied to a full GNLSE. It was lin-earized with respect to the DW envelopeψb, also higher-order effects (dispersion, self-steepening,and Raman scattering) have no strong influences on a plane DW, and are neglected accordingly.The resulting equation readsi∂zψb −β′′(ωb)2∂2τψb + n2b2ωbc|ψ|2ψb = 0. (5)It is mathematically equivalent to the quantum mechanical Schrödinger equation for a scatteringproblem: a plane wave with the power Pb is reflected at a moving barrier with a hyperbolic secantshape. It can be solved analytically [19]:|ψb|2 = PbT∣∣∣∣∣F(a, b, c,1− tanh τ−Tσ2)∣∣∣∣∣2, (6)where F is a Gaussian hypergeometric function, the quantityT =sinh2(πΩ̄σ)cosh2(πs) + sinh2(πΩ̄σ),is the transmission coefficient of the scattering problem, both using parametersa, b =12− iΩ̄σ ± is, c = 12− iΩ̄σ, Ω̄ = Ω− Bβ′′(ωb),B = β′(ωa + ν)− β′(ωa)−β′′(ωa + ν)[ωa + ν]σ2+β′′′(ωa + ν)6σ2+ · · · ,s =12√16|β′′(ωa + ν)|β′′bωbωa + νn2bn2a− 1.4Now, |ψb|2 from (6) is inserted into (2) and then (1) is used to derive ODEs for the solitonparameters σ(z), ν(z), and T (z). The phase Θ(z) can safely be ignored. The soliton durationσ(z) happens to be explicitly expressed in terms of ν(z)σ(ν) =β′′(ωa + ν)β′′(ωa)σa[1 + νωa]2 . (7)The ODE for the soliton frequency offset readsdνdz= −πfR4σβ′′(ωa + ν)R1(ν) (8)+4µTσLa[1 +νωa] ∫ 10dζ |F (a, b, c, ζ)|2 [2ζ − 1] ,and for the soliton delay we getdTdz= B − 3πfR8σβ(2)(ωa + ν)ωa + νR2(ν) (9)+4µTωaLa∫ 10dζ |F (a, b, c, ζ)|2 [1− [2ζ − 1] arctanh (2ζ − 1)] ,withR1(ν) =∫ ∞−∞dω iω3ĥ(ω)− 1sinh2 π2σ(ν)ω,R2(ν) =∫ ∞−∞dω ω2ĥ(ω)− 1 + 13ω∂ωĥ(ω)sinh2 π2σ(ν)ω.Here, ĥ(ω) is the Fourier transform of Raman response function from (3) with h(τ < 0) ≡0. Note that only the real parts of both integrands contribute to R1,2(ν). The dimensionlessparameter µ is the input DW power Pb divided by the initial peak power of the soliton Paµ =PbPawith n2aPa =|β(2)(ωa)|cωaσ2a=cωaLa,where La = σ2a/|β′′(ωa)| is the dispersion length of the initial soliton. An adiabatic (slow)evolution of the soliton parameters requires µ 1.Equations (7–9) provide self-consistent quantitative description of optical solitons controlled bycollisions with DWs. They might look complicated, but are more easily accessible than the orig-inal full GNLSE. One can, e.g., easily quantify a DW that succeeds in cancellation of the SSFS,and the stability of the cancelation can be investigated, as we will see in the next section.3 Controlled SSFS cancellationTo begin with, we switch off the DW and consider a single soliton (ωa = 0.67 PHz, σa = 40 fs)traveling along an optical fiber subject to Raman scattering, higher-order dispersion (Fig. 2), and580 90 100 110010203040Frequency (THz)Propagation distance (cm)1.00.90.80.70.6Power (arb. units)0 10 20 30 40Propagation distance (cm)4.00.0 1.0 2.0 3.0Delay (ps)010203040Propagation distance (cm)(b)120 130(a)(c)-50-40-30-20-100Spectral power (dB)-30-20-100Power (dB)Figure 3: Space-time (a) and spectral (c) representation of a single soliton propagating along thefiber under influence of Raman effect from numerical simulation using full GNLSE. Additionaldashed lines result from the adiabatic equations (7–9). (b) Soliton’s peak power from the GNLSE(red) and from the adiabatic equations (dashed). See Sec. 3 for parameters.self-steepening. We compare numerical solution of the full GNLSE (Fig. 3) to the predictions ofthe adiabatic equations (7–9) (dashed lines in Fig. 3). Note especially that the evolution of solitonpower is accurately described, in contrast to predictions by standard soliton perturbation theorywhich state that the soliton amplitude is not affected by the Raman effect [1, 17].Now we return to the scattering of the DW, as shown in Fig. 1(a) for the full GNLSE solution.Despite of the complex interaction character and approximations made when deriving (5), solu-tions of the above adiabatic equations (dashed lines in Fig. 1) provide reasonable quantitativeestimates of the soliton trajectory and power, and an impressivly accurate prediction for thesoliton carrier frequency. We can now derive initial parameter ranges such that a stable SSFScompensation arises.For this we inspect the ODE (8) for soliton frequency shift, which is self-consistent due to (7).Its first summand describes the influence of Raman scattering, the second describes the influ-ence of the imposed DW. We consider an exemplary initial soliton with ωa = 0.67 PHz andσa = 40 fs and look for pairs of DW frequency offset Ω and normalized intensity µ, such thatsoliton frequency evolves stable with ν(z) = ν(0) = 0 for all z. Fig. 4(a) shows DW parame-ters (Ω, µ) such that ν(z) ≡ 0 is a stationary solution of (8), i.e., (dν/dz)ν=0 vanishes. TheSSFS cancellation might be expected for 0.1 ≤ Ω/(PHz) ≤ 0.3, otherwise the required DWpower quickly increases with µ and soliton evolution is no longer adiabatic. Furthermore, wecan identify a parameter range for which the compensation is asymptotically stable. To this end60.00.20.40.60.81.01.2μ-505Ω (THz )κ (cm-1 )0 10 20 30 40 50(a)(b)μ=0.0225Figure 4: (a) Pairs of Ω and relative DW amplitude√µ for which solitons frequency does notchange. Amplitude is plotted instead of power to facilitate readability. (b) Stability analysis. Theregion of stable SSFS compensation is shaded gray.the derivative with respect to ν of the right-hand side of (8) should be negative for ν = 0. Therelevant quantityκ = ∂ν(dνdz)∣∣∣∣ν=0is shown in Fig. 4(b). For κ > 0 the SSFS reappears if fiber length exceeds 1/κ. Withinthe small region of asymptotically stable (κ < 0) compensation, we must choose an initialparameter pair (Ω, µ) such that the relative DW power µ 1, in order for the adiabatic theoryto be a good approximation. As threshold we generously choose µ < 0.0225. The accordingparameter region is shaded gray in Fig. 4.Fig. 5 shows the results of a simulation with parameters produced by the above procedure(Ω = 0.286 PHz, µ = 0.0081). The transient phase is very short. The soliton stabilizes at about0.666 PHz (cf. with the initial ωa = 0.67 PHz). Fig. 5 (b) depicts soliton power evolution. We seethat the soliton lost only 4% of its peak power after propagating a distance of 40 cm. Travelingalone, the soliton lost about 30% of its peak power due to the effects of Raman scattering, seeFig. 3(b).In conclusion, we explained how the SSFS can be cancelled in an asymptotically stable mannerby XPM interaction with a properly prepared small-amplitude wave, and provided a simple wayto calculate the required wave parameters. U.B. and Sh.A. acknowledge support of EinsteinFoundation Berlin and Research Center MATHEON under Project D-OT2.References[1] G.P. Agrawal, Nonlinear Fiber Optics, 4th edn. (Academic, New York, 2007)[2] F.M. Mitschke, L.F. Mollenauer, Opt. Lett. 11(10), 569 (1986)[3] D.V. Skryabin, A.V. Gorbach, Rev. Mod. Phys. 82(2), 1287 (2010)780 90 100 110 120 130010203040500 550Propagation distance (cm)Frequency (THz)1.000.980.960.94Power (arb. units)0 10 20 30 40Propagation distance (cm)1.01010203040Propagation distance (cm)-0.2 0.0 0.2 0.4 0.6Delay (ps)0.8 1.0600(d)-50-40-30-20-100Spectral power (dB)-30-20-100Power (dB)(c)(a)(b)Incoming DWScattered DWFigure 5: Compensation of SSFS by collision with a suitably prepared DW. (a) Evolution ofsoliton together with a DW in time domain as results from GNLSE. (b) Soliton amplitude isalmost unchanged. (c) Soliton frequency is locked. (d) the scattered DW frequency is now welldefined, cf., Fig 1(d). See Sec. 3 for parameters.[4] J.M. Dudley, G. Genty, S. Coen, Rev. Mod. Phys. 78(4), 1135 (2006)[5] J.P. Gordon, Opt. Lett. 11(10), 662 (1986)[6] J.K. Lucek, K.J. Blow, Phys. Rev. A 45(9), 6666 (1992)[7] D. Schadt, B. Jaskorzynska, J. Opt. Soc. Am. B 5(11), 2374 (1988)[8] K.J. Blow, N.J. Doran, D. Wood, JOSA B 5(6), 1301 (1988)[9] M. Manousakis, N. Moschonas, P. Papagiannis, H. K., J. Mod. Opt. 52(17), 2549 (2005)[10] D.V. Skryabin, F. Luan, J.C. Knight, P.S.J. Russell, Science 301(5640), 1705 (2003)[11] F. Biancalana, D.V. Skryabin, A.V. Yulin, Phys. Rev. E 70(1), 016615 (2004)[12] I. Babushkin, S. Amiranashvili, C. Brée, U. Morgner, G. Steinmeyer, A. Demircan, IEEEPhotonics Journal 8(3), 7803113 (2016)[13] A. Efimov, A.V. Yulin, D.V. Skryabin, J.C. Knight, N. Joly, F.G. Omenetto, A.J. Taylor, P. Rus-sell, Phys. Rev. Lett. 95(21), 213902 (2005)[14] A. Demircan, S. Amiranashvili, G. Steinmeyer, Phys. Rev. Lett. 106(16), 163901 (2011)[15] T.G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, U. Leonhardt, Science319(5868), 1367 (2008)8[16] S. Pickartz, U. Bandelow, S. Amiranashvili, Phys. Rev. A 94(3), 033811 (2016)[17] A. Hasegawa, M. Matsumoto, Optical Solitons in Fibers (Springer, 2003)[18] K.J. Blow, D. Wood, IEEE J. Quantum Electron. 25(15), 2665 (1989)[19] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, 2nd edn. (Pergamon, 1965)9

Asymptotically stable compensation of soliton self-frequency shift

We report on the stable cancellation of the soliton self-frequency shift (SSFS) in nonlinear optical fibers. A soliton, which scatters a group velocity matched pump wave in a so-called optical event horizon regime, may experience a blueshift in frequency, which counteracts the SSFS induced by Raman scattering. The SSFS can easily be compensated by a suitably prepared pump wave, but usually the compensation is unstable and is destroyed after a certain propagation length. We study this kind of soliton-pump wave interaction by an adiabatic approach and quantify the parameter range in which the stable SSFS compensation is possible. The theory enjoys agreement with numerical simulations of soliton propagation

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Asymptotically stable compensation of the soliton self-frequency shift

Weierstraß-Institutfür Angewandte Analysis und StochastikLeibniz-Institut im Forschungsverbund Berlin e. V.Preprint ISSN 2198-5855Asymptotically stable compensationof soliton self-frequency shiftSabrina Pickartz, Uwe Bandelow, Shalva Amiranashvilisubmitted: November 30, 2016Weierstraß-InstitutMohrenstr. 3910117 BerlinGermanyE-Mail: sabrina.pickartz@wias-berlin.deuwe.bandelow@wias-berlin.deshalva.amiranashvili@wias-berlin.deNo. 2343Berlin 20162010 Mathematics Subject Classification. 78A60, 35Q60, 70H11.2010 Physics and Astronomy Classification Scheme. 42.65.-k, 42.81.Dp, 05.45.Yv, 03.65.Nk.Key words and phrases. Soliton self-frequency shift, Raman scattering, Soliton perturbation theory, Event hori-zons.U.B. and Sh.A. acknowledge support of Einstein Foundation Berlin and Research Center MATHEON under ProjectD-OT2.Edited byWeierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)Leibniz-Institut im Forschungsverbund Berlin e. V.Mohrenstraße 3910117 BerlinGermanyFax: +49 30 20372-303E-Mail: preprint@wias-berlin.deWorld Wide Web: http://www.wias-berlin.de/AbstractWe report the cancellation of the soliton self-frequency shift in nonlinear optical fibers.A soliton which interacts with a group velocity matched low intensity dispersive pumppulse, experiences a continuous blue-shift in frequency, which counteracts the soliton self-frequency shift due to Raman scattering. The soliton self-frequency shift can be fully com-pensated by a suitably prepared dispersive wave. We quantify this kind of soliton-dispersivewave interaction by an adiabatic approach and demonstrate that the compensation is sta-ble in agreement with numerical simulations.1 IntroductionA soliton propagating along a nonlinear optical fiber is subject to Raman scattering, the “red end”of the soliton spectrum is amplified at the expense of the “blue end” [1]. This results in what iscalled the soliton self-frequency shift (SSFS), a noticeable decrease of the carrier frequencyof subpicosecond solitons [2]. The Raman scattering plays an important role, e.g., in opticalsupercontinuum [3, 4], but in soliton communications systems it is desirable to cancel its effect.The SSFS depends strongly on pulse-width [5], so solitons with initially equal carrier frequencybut slightly different durations will gradually get different carrier frequencies (velocities) underRaman scattering, which results in jitter [6].SSFS compensation may be mediated by cross-phase modulation (XPM) between the solitonand an accompanying dispersive wave (DW) [7]. The latter appears naturally if a pulse whoseinitial power exceeds that of a soliton with the same duration, brakes down into just that solitonand a DW [2]. Initially both, soliton and DW, have the same carrier frequency. Other possibilitiesfor SSFS compensation include, e.g., bandwidth-limited amplification [8], pump by an additionalsoliton [9], interplay between Raman effect and Cherenkov radiation [10, 11]. In this Letter wedemonstrate how the SSFS can be counteracted by XPM interaction with a prearranged pumpDW, which (i) is much weaker than the soliton and (ii) has a distinctly different carrier frequency.Soliton collision with chirped DWs for SSFS compensation has been suggested in [12] basedon numerical results.A typical interaction of this kind is seen in Fig. 1. The electromagnetic power density [Fig. 1(a)]is plotted in space-time domain in a frame that co-moves with the unperturbed soliton. TheDW approaches the initially stationary soliton (zero delay) and, being almost perfectly reflected[13, 14], yields an interference picture seen to the left of the soliton. The latter is deflected [redline in Fig. 1(a)] and compressed [red line in Fig. 1(b)]. The reflected part of the DW is frequencyshifted, as seen in the frequency domain [Fig. 1(d)]. The soliton frequency is also shifted dur-ing reflection [Fig. 1(c)]. Our numerical simulations use a generalized nonlinear Schrödingerequation (GNLSE) with Raman term [4]. Without the DW, both the soliton’s frequency and am-plitude degrade, whereas its trajectory is deflected towards larger delays (sole Raman effect,not shown). This degradation is completely compensated by the DW after an initial transientphase, moreover, we will see below that the compensation is stable.1550 60080 90 100 110 120010203040Frequency (THz)Propagation distance (cm)-1.5 -1.0 -0.5 0.0 0.5010203040Delay (ps)Propagation distance (cm)1.31.21.11.00.90 10 20-2.0-2.5 30 40Propagation distance (cm)130 500Power (arb. units)(c) (d)-30-20-100Power (dB)-50-40-30-20-100Spectral power (dB)(b)(a)GNLSEAdiabaticScatteredDWIncoming DWFigure 1: GNLSE calculation of a DW at 3.67 PHz interacting with a soliton at 0.67 PHz.(a) Power in space-time domain. The white dashed line is the soliton trajectory in the adiabaticapproximation. (b) Soliton peak power from GNLSE (red line) and adiabatic theory (dashed line).(c,d) spectral power for soliton and DW respectively. Dashed lines are from adiabatic equations(7–9).Dispersion properties of bulk silica meet all requirements for the interaction. Fig. 2 shows fre-quency dependencies of group velocity β′ and group velocity dispersion (GVD) β′′. The initialsoliton carrier frequency ωa and DW frequency ωb + Ω are chosen from opposite sides of thezero dispersion frequency (β′′(ωZDF) = 0) such that they provide almost equal group velocities.The reference frequency ωb corresponds to the equal group velocities, β′(ωa) = β′(ωb). In themoving frame we deal with the common delay τ = t− β′(ωa)z = t− β′(ωb)z. The propaga-tion distance z is measured along the fiber, t is the physical time variable. The frequency offsetΩ should lie in a small interval around ωb (shaded gray in Fig. 2) in order for the two pulses tointeract [14]. A nearly perfect DW reflection (Fig. 1a) can be understood as a fiber-optical analogof the event horizon [15] or with quantum mechanical scattering theory [16].In what follows, we quantify XPM interactions, like the one shown in Fig. 1. The standard solitonperturbation theory [17] was adapted to the problem at hand in [16]. It yields an adiabatic de-scription of how a soliton evolves when colliding with a DW. Here we extend the theory to includethe influence of Raman scattering, as outlined in the next section. Further we demonstrate howthis description suggests that a stable compensation of the SSFS can occur. We describe asimple procedure to specify DW parameters that will produce cancellation of the SSFS. The lastsection deals with the stability analysis, numerical tests, and discussion.24.890β΄ (fs/μm)β΄΄ (fs2 /μm)4.8864.8824.878-0.06-0.20.21.0 1.2 1.4 1.6 1.8 2.0 2.2Circular frequency (1015 rad/s)ωa ωZDF ωb(a)(b)Figure 2: A typical profile (bulk silica) of (a) the group delay β′(ω) and (b) GVD β′′(ω) thatleads to the collision phenomenon shown in Fig. 1. Collision can only be realized for initial DWfrequency offsets in a small interval (shaded gray) around the reference frequency of matchinggroup velocity. Initial frequencies of soliton and DW are indicated by bullets, frequency shifts areindicated by arrows.2 Model equationsAn adiabatic description of interactions, like in Fig. 1, is derived from two XPM-coupled GNLSEs,one centered at ωa for the soliton envelopeψ(z, τ), and one centered at ωb for the DW envelopeψb(z, τ). The DW equation is simplified such that it can be solved as a problem of plane wavescattering at a moving “solitonic” potential barrier. The soliton equation is reformulated so thatall higher-order terms in the GNLSE are treated as perturbationsi∂zψ + i [β′(ωa + ν)− β′(ωa)] ∂τψ− β′′(ωa + ν)2∂2τψ + n2aωa + νc|ψ|2 ψ = R (ψ, ψb) . (1)The most interesting point about this perturbation equation is that it explicitly takes into accounta varying soliton carrier frequency ωa + ν, where ν = ν(z), ν(0) = 0 is the yet unknowndetuning from the initial carrier frequency ωa. Especially the GVD β′′(ω) and the nonlinear termon the left-hand side of (1) follow the soliton frequency shift. Therefore also the deviation fromthe initial group delay β′(ωa) appears in the equation. All higher-order terms are contained inthe perturbation R (ψ, ψb)R(ψ, ψb) = −M∑m=3β(m)(ωa + ν)m![i∂τ ]m ψ − τ dνdzψ− 2n2ac[ωa + ν + i∂τ ][|ψb|2 ψ]− n2aci∂τ[|ψ|2 ψ]− fRn2ac[ωa + ν + i∂τ ] [I(ψ)ψ] , (2)I(ψ) =∫ τ−∞h(τ − τ ′) |ψ(z, τ ′)|2 dτ ′ − |ψ(z, τ)|2 .3The terms with derivatives of the wave vector β(ω) account for higher-order dispersion, the firstterm in the second line describes XPM, the second is the self-steepening term. The perturbationterm containing dν/dz results from the necessary reformulations [16] of the standard solitonperturbation equation (as found for example in [17]) in order to include ν in the GVD. The termin the third line describes Raman scattering. The included Raman response function for fusedsilica readsh(τ) =ν21 + ν22ν1e−ν2τ sin ν1τ, (3)and we used parameters fR = 0.18, 1/ν1 = 12.2 fs, and 1/ν2 = 32 fs [18].The fundamental soliton solution of the unperturbed (1) reads in a general formulationψ =1σ√|β(2)(ωa + ν)|c[ωa + ν]naeΘcosh 1σ[τ − T ] , (4)with frequency offset ν = ν(z) from the initial carrier frequency ωa, duration σ = σ(z), tem-poral delay T = T (z), and a phase Θ = Θ(z). It is assumed that all soliton parameters arez-dependent and slowly evolve under the influence of the DW. The soliton perturbation theoryprovides ODEs for these parameters after |ψb|2 is specified. The initial parameter values aredenoted by σ(0) = σa, etc.The DW equation was derived by a series of simplifications applied to a full GNLSE. It was lin-earized with respect to the DW envelopeψb, also higher-order effects (dispersion, self-steepening,and Raman scattering) have no strong influences on a plane DW, and are neglected accordingly.The resulting equation readsi∂zψb − β′′(ωb)2∂2τψb + n2b2ωbc|ψ|2ψb = 0. (5)It is mathematically equivalent to the quantum mechanical Schrödinger equation for a scatteringproblem: a plane wave with the power Pb is reflected at a moving barrier with a hyperbolic secantshape. It can be solved analytically [19]:|ψb|2 = PbT∣∣∣∣∣F(a, b, c,1− tanh τ−Tσ2)∣∣∣∣∣2, (6)where F is a Gaussian hypergeometric function, the quantityT =sinh2(piΩ¯σ)cosh2(pis) + sinh2(piΩ¯σ),is the transmission coefficient of the scattering problem, both using parametersa, b =12− iΩ¯σ ± is, c = 12− iΩ¯σ, Ω¯ = Ω− Bβ′′(ωb),B = β′(ωa + ν)− β′(ωa)− β′′(ωa + ν)[ωa + ν]σ2+β′′′(ωa + ν)6σ2+ · · · ,s =12√16|β′′(ωa + ν)|β′′bωbωa + νn2bn2a− 1.4Now, |ψb|2 from (6) is inserted into (2) and then (1) is used to derive ODEs for the solitonparameters σ(z), ν(z), and T (z). The phase Θ(z) can safely be ignored. The soliton durationσ(z) happens to be explicitly expressed in terms of ν(z)σ(ν) =β′′(ωa + ν)β′′(ωa)σa[1 + νωa]2 . (7)The ODE for the soliton frequency offset readsdνdz= −pifR4σβ′′(ωa + ν)R1(ν) (8)+4µTσLa[1 +νωa] ∫ 10dζ |F (a, b, c, ζ)|2 [2ζ − 1] ,and for the soliton delay we getdTdz= B − 3pifR8σβ(2)(ωa + ν)ωa + νR2(ν) (9)+4µTωaLa∫ 10dζ |F (a, b, c, ζ)|2 [1− [2ζ − 1] arctanh (2ζ − 1)] ,withR1(ν) =∫ ∞−∞dω iω3hˆ(ω)− 1sinh2 pi2σ(ν)ω,R2(ν) =∫ ∞−∞dω ω2hˆ(ω)− 1 + 13ω∂ωhˆ(ω)sinh2 pi2σ(ν)ω.Here, hˆ(ω) is the Fourier transform of Raman response function from (3) with h(τ < 0) ≡0. Note that only the real parts of both integrands contribute to R1,2(ν). The dimensionlessparameter µ is the input DW power Pb divided by the initial peak power of the soliton Paµ =PbPawith n2aPa =|β(2)(ωa)|cωaσ2a=cωaLa,where La = σ2a/|β′′(ωa)| is the dispersion length of the initial soliton. An adiabatic (slow)evolution of the soliton parameters requires µ 1.Equations (7–9) provide self-consistent quantitative description of optical solitons controlled bycollisions with DWs. They might look complicated, but are more easily accessible than the orig-inal full GNLSE. One can, e.g., easily quantify a DW that succeeds in cancellation of the SSFS,and the stability of the cancelation can be investigated, as we will see in the next section.3 Controlled SSFS cancellationTo begin with, we switch off the DW and consider a single soliton (ωa = 0.67 PHz, σa = 40 fs)traveling along an optical fiber subject to Raman scattering, higher-order dispersion (Fig. 2), and580 90 100 110010203040Frequency (THz)Propagation distance (cm)1.00.90.80.70.6Power (arb. units)0 10 20 30 40Propagation distance (cm)4.00.0 1.0 2.0 3.0Delay (ps)010203040Propagation distance (cm)(b)120 130(a)(c)-50-40-30-20-100Spectral power (dB)-30-20-100Power (dB)Figure 3: Space-time (a) and spectral (c) representation of a single soliton propagating along thefiber under influence of Raman effect from numerical simulation using full GNLSE. Additionaldashed lines result from the adiabatic equations (7–9). (b) Soliton’s peak power from the GNLSE(red) and from the adiabatic equations (dashed). See Sec. 3 for parameters.self-steepening. We compare numerical solution of the full GNLSE (Fig. 3) to the predictions ofthe adiabatic equations (7–9) (dashed lines in Fig. 3). Note especially that the evolution of solitonpower is accurately described, in contrast to predictions by standard soliton perturbation theorywhich state that the soliton amplitude is not affected by the Raman effect [1, 17].Now we return to the scattering of the DW, as shown in Fig. 1(a) for the full GNLSE solution.Despite of the complex interaction character and approximations made when deriving (5), solu-tions of the above adiabatic equations (dashed lines in Fig. 1) provide reasonable quantitativeestimates of the soliton trajectory and power, and an impressivly accurate prediction for thesoliton carrier frequency. We can now derive initial parameter ranges such that a stable SSFScompensation arises.For this we inspect the ODE (8) for soliton frequency shift, which is self-consistent due to (7).Its first summand describes the influence of Raman scattering, the second describes the influ-ence of the imposed DW. We consider an exemplary initial soliton with ωa = 0.67 PHz andσa = 40 fs and look for pairs of DW frequency offset Ω and normalized intensity µ, such thatsoliton frequency evolves stable with ν(z) = ν(0) = 0 for all z. Fig. 4(a) shows DW parame-ters (Ω, µ) such that ν(z) ≡ 0 is a stationary solution of (8), i.e., (dν/dz)ν=0 vanishes. TheSSFS cancellation might be expected for 0.1 ≤ Ω/(PHz) ≤ 0.3, otherwise the required DWpower quickly increases with µ and soliton evolution is no longer adiabatic. Furthermore, wecan identify a parameter range for which the compensation is asymptotically stable. To this end60.00.20.40.60.81.01.2μ-505Ω (THz )κ (cm-1 )0 10 20 30 40 50(a)(b)μ=0.0225Figure 4: (a) Pairs of Ω and relative DW amplitude√µ for which solitons frequency does notchange. Amplitude is plotted instead of power to facilitate readability. (b) Stability analysis. Theregion of stable SSFS compensation is shaded gray.the derivative with respect to ν of the right-hand side of (8) should be negative for ν = 0. Therelevant quantityκ = ∂ν(dνdz)∣∣∣∣ν=0is shown in Fig. 4(b). For κ > 0 the SSFS reappears if fiber length exceeds 1/κ. Withinthe small region of asymptotically stable (κ < 0) compensation, we must choose an initialparameter pair (Ω, µ) such that the relative DW power µ 1, in order for the adiabatic theoryto be a good approximation. As threshold we generously choose µ < 0.0225. The accordingparameter region is shaded gray in Fig. 4.Fig. 5 shows the results of a simulation with parameters produced by the above procedure(Ω = 0.286 PHz, µ = 0.0081). The transient phase is very short. The soliton stabilizes at about0.666 PHz (cf. with the initial ωa = 0.67 PHz). Fig. 5 (b) depicts soliton power evolution. We seethat the soliton lost only 4% of its peak power after propagating a distance of 40 cm. Travelingalone, the soliton lost about 30% of its peak power due to the effects of Raman scattering, seeFig. 3(b).In conclusion, we explained how the SSFS can be cancelled in an asymptotically stable mannerby XPM interaction with a properly prepared small-amplitude wave, and provided a simple wayto calculate the required wave parameters. U.B. and Sh.A. acknowledge support of EinsteinFoundation Berlin and Research Center MATHEON under Project D-OT2.References[1] G.P. Agrawal, Nonlinear Fiber Optics, 4th edn. (Academic, New York, 2007)[2] F.M. Mitschke, L.F. Mollenauer, Opt. Lett. 11(10), 569 (1986)[3] D.V. Skryabin, A.V. Gorbach, Rev. Mod. Phys. 82(2), 1287 (2010)780 90 100 110 120 130010203040500 550Propagation distance (cm)Frequency (THz)1.000.980.960.94Power (arb. units)0 10 20 30 40Propagation distance (cm)1.01010203040Propagation distance (cm)-0.2 0.0 0.2 0.4 0.6Delay (ps)0.8 1.0600(d)-50-40-30-20-100Spectral power (dB)-30-20-100Power (dB)(c)(a)(b)Incoming DWScattered DWFigure 5: Compensation of SSFS by collision with a suitably prepared DW. (a) Evolution ofsoliton together with a DW in time domain as results from GNLSE. (b) Soliton amplitude isalmost unchanged. (c) Soliton frequency is locked. (d) the scattered DW frequency is now welldefined, cf., Fig 1(d). See Sec. 3 for parameters.[4] J.M. Dudley, G. Genty, S. Coen, Rev. Mod. Phys. 78(4), 1135 (2006)[5] J.P. Gordon, Opt. Lett. 11(10), 662 (1986)[6] J.K. Lucek, K.J. Blow, Phys. Rev. A 45(9), 6666 (1992)[7] D. Schadt, B. Jaskorzynska, J. Opt. Soc. Am. B 5(11), 2374 (1988)[8] K.J. Blow, N.J. Doran, D. Wood, JOSA B 5(6), 1301 (1988)[9] M. Manousakis, N. Moschonas, P. Papagiannis, H. K., J. Mod. Opt. 52(17), 2549 (2005)[10] D.V. Skryabin, F. Luan, J.C. Knight, P.S.J. Russell, Science 301(5640), 1705 (2003)[11] F. Biancalana, D.V. Skryabin, A.V. Yulin, Phys. Rev. E 70(1), 016615 (2004)[12] I. Babushkin, S. Amiranashvili, C. Brée, U. Morgner, G. Steinmeyer, A. Demircan, IEEEPhotonics Journal 8(3), 7803113 (2016)[13] A. Efimov, A.V. Yulin, D.V. Skryabin, J.C. Knight, N. Joly, F.G. Omenetto, A.J. Taylor, P. Rus-sell, Phys. Rev. Lett. 95(21), 213902 (2005)[14] A. Demircan, S. Amiranashvili, G. Steinmeyer, Phys. Rev. Lett. 106(16), 163901 (2011)[15] T.G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, U. Leonhardt, Science319(5868), 1367 (2008)8[16] S. Pickartz, U. Bandelow, S. Amiranashvili, Phys. Rev. A 94(3), 033811 (2016)[17] A. Hasegawa, M. Matsumoto, Optical Solitons in Fibers (Springer, 2003)[18] K.J. Blow, D. Wood, IEEE J. Quantum Electron. 25(15), 2665 (1989)[19] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, 2nd edn. 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Asymptotically stable compensation of soliton self-frequency shift

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