"Extraordinary" modulation instability in optics and hydrodynamics
Guillaume Vanderhaegen, Corentin Naveau, Pascal Szriftgiser, Alexandre Kudlinski, Matteo Conforti, Arnaud Mussot, Miguel Onorato, Stefano Trillo, Amin Chabchoub, Nail Akhmediev
““Extraordinary” modulation instability in optics and hydrodynamics
Guillaume Vanderhaegen , Corentin Naveau , Pascal Szriftgiser , Alexandre Kudlinski , Matteo Conforti ,Arnaud Mussot , , Miguel Onorato , Stefano Trillo , Amin Chabchoub , and Nail Akhmediev University of Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Mol´ecules, Lille, France Institut Universitaire de France (IUF), France Dipartimento di Fisica, Universit degli Studi di Torino, 10125 Torino, Italy Department of Engineering, University of Ferrara, 44122 Ferrara, Italy Centre for Wind, Waves and Water, School of Civil Engineering,The University of Sydney, Sydney, NSW 2006, Australia Marine Science Institute, The University of Sydney, Sydney, NSW 2006, Australia Department of Theoretical Physics, Research School of Physics,The Australian National University, Canberra, ACT 2600, Australia
The classical theory of modulation instability (MI) attributed to Bespalov-Talanov in optics andBenjamin-Feir for water waves is just a linear approximation of nonlinear effects and has limitationsthat have been corrected using the exact weakly nonlinear theory of wave propagation. We reportresults of experiments in both, optics and hydrodynamics, which are in excellent agreement withnonlinear theory. These observations clearly demonstrate that MI has wider band of unstablefrequencies than predicted by the linear stability analysis. The range of areas where the nonlineartheory of MI can be applied is actually much larger than considered here.
INTRODUCTION
Well-known Bespalov-Talanov (BT) [1] and Benjamin-Feir (BF) [2, 3] instabilities discovered more than 60 yearsago (1966 and 1967, respectively) played a significant rolein understanding nonlinear phenomena in optics and hy-drodynamics [4–11]. The detailed description of thesefundamental results can be found in any common bookin nonlinear optics [12, 13], ocean waves [14, 15], andmore generally, in nonlinear dynamics book literature[16]. The theory tells us that a plane wave or a con-stant amplitude wave (CW) is unstable relative to smallamplitude perturbations with frequencies within certaindeterministic and finite range. These perturbations areunstable and can grow exponentially, thus, leading tomodulated waves with infinitely high amplitude. Clearly,such growth is unphysical and has to be reconsidered us-ing an approach beyond linear theory.Indeed, the accurate nonlinear theory [17] predicts sat-uration and the maximal amplitude of periodic waves ex-cited due to modulation instability (MI). This predictionis in accordance with conventional wisdom: “what goesup must come down”. In fact, this nonlinear stage of MIpredicted not only the exponential growth but the fol-lowing exponential decay back to the constant amplitudewave [17]. The latter was not obvious and required manyyears before this seemingly simple principle “must comedown” has been confirmed, first with the observation ofgrowth saturation in water waves [4], and then with thedemonstration of the full recursive behavior in opticalexperiments [18, 19]. If translated to the frequency do-main, this principle is, essentially, the Fermi-Pasta-Ulamrecurrence [20] (see also [4, 5, 9–11, 21]) .Despite these achievements, it has been recentlydemonstrated that not all secrets of modulation instabil- ity concealed by the linear approach have been revealedso far [22]. The results obtained in [22] demonstrate thatthe linear theory does not accurately predict the rangeof unstable frequencies. This fact is, once again, notobvious. An exact nonlinear theory is essential for re-vealing the full range of frequencies that are unstabledue to the modulation. Exact solutions of the nonlinearSchr¨odinger equation (NLSE) that describe the nonlin-ear stage of modulation instability are presently knownas Akhmediev breathers (AB) [23–29]. The latter form afamily of solutions with a free parameter that is directlyrelated to the whole interval of unstable frequencies in theBT and BF theories. However, even the AB solutions donot cover the whole range of unstable frequencies. Thefamily of ABs is actually a particular case of more gen-eral family of solutions that have been found in [30] andrefined recently in [22]. This extension expands the rangeof unstable frequencies predicted in the the BT and BFtheories. It has important ramifications for theory, ex-periment and applications [31]. It means, that periodicperturbations of a plane wave (or CW) can grow in thesituations when we would not expect them to do so.Presenting simultaneously optical and hydrodynamicexperiments confirming this exceptional feature of mod-ulation instability in a single work has far reaching con-sequences. Observing the same effect at nearly oppo-site ends of spatial and time scales of MI in physics is aconvincing argument confirming the validity of the newfinding. It means that similar phenomena at other scalessuch as MI in plasma [32] or in Bose-Einstein condensate[33] must also be re-examined. In optics, the extension ofthe range of frequencies leading to MI might have mul-tiple applications for generating frequency combs [34],periodic pulse trains [35] and supercontinuum radiation[36]. In hydrodynamics, the new findings might result a r X i v : . [ n li n . PS ] S e p in reconsidering conditions leading to formation of roguewaves in the ocean [37]. THEORETICAL BACKGROUND
We start with the NLSE written in the normalisedform: iψ z + 12 ψ tt + | ψ | ψ = 0 (1)where ψ is the wave envelope function, z is the longitu-dinal co-ordinate, and t is the time in a frame movingwith group velocity. We are interested in doubly peri-odic waves, e.g., in solutions of Eq.(1) that are periodicboth in space and in time [22]. They comprise the three-parameter family of solutions with a single period alongeach axis, z and t . This family contains as particularcases other ‘elementary’ solutions and families [22]. Tobe specific, doubly periodic solutions of Eq. (1) can bepresented in general form: ψ ( t, z ) = [ Q ( t, z ) + iδ ( z )] e iφ ( z ) , (2)with the functions Q ( t, z ), δ ( z ) and φ ( z ) that can befound by a direct substitution of (2) into (1) [30]. Thereare two forms of such solutions, classified as A- and B-types depending on the parameters of the family. Eachtype contains MI as the limiting case. However, the lim-iting case of B-type solutions is the standard MI whilethe limiting case of A-type solutions is more general.This apparently puzzling asymmetry between the twofamilies finds its physical justification in the fact thatA-type solutions can be considered as the full nonlineardressing of solutions of the NLSE obtained in the lin-ear limit (when dispersion dominates over nonlinearity).As discussed in more details in [22], for very high mod-ulation frequencies, the deformation introduced by thenonlinearity is small and essentially the modulation expe-riences, upon evolution, only a periodic phase shift [38].However, when the frequency is reduced to sufficientlysmall values, the deformation due to the nonlinearity be-comes strong, thereby inducing a net amplification of theinput sidebands even outside the conventional MI band-width. Conversely, B-type solutions start to appear onlyat frequencies below the conventional band-edge of MI,as a result of the symmetry breaking nature of the onsetof conventional MI [10, 22]. Therefore, B-type solutionscannot be responsible for any unconventional MI.Thus, our point of interest in this work is the A-typesolutions. Then, the three functions in (2) are definedas follows. Namely, for the function δ ( z ), we have thefollowing expression: δ ( z ) = (cid:114) α − ν ) (cid:115) µz, k )1 + ν cn( µz, k ) sn( µz/ , k ) , (3) where m = k = 12 (cid:18) − η + ρ ( ρ − α ) AB (cid:19) , A = ( α − ρ ) + η , B = ρ + η , ν = A − BA + B , and µ = 4 √ AB . Thefunction δ varies within the interval 0 < δ < α .The phase φ ( z ) is given by: φ ( z ) = (cid:16) ρ + α ν (cid:17) z − α νµ (cid:20) Π(am( µz, k ) , n, k ) −− νσ tan − (cid:16) sd ( µz,k ) σ (cid:17) (cid:21) (4)where n = ν ν − , σ = (cid:113) − ν k +(1 − k ) ν , and sd( µz, k ) =sn ( µz,k ) dn ( µz,k ) , Π(am( µz, k ) , n, k ) is the incomplete elliptic in-tegral of the third kind with the argument am( u, k ) beingthe amplitude function.In contrast to δ and φ , the function Q depends on twovariables t and z . It is given by: Q ( t, z ) = sb − c + r + cn( pt, k q )1 + r cn( pt, k q ) , (5)where s ( z ) = sign [cn( µz/ , k )], r = M − NM + N , p = √ M N =2 (cid:112) ( α − ρ ) + η , k q = 12 + 2 ρ − α p , b = √ α − y,y ( z ) = δ ( z ) , c ± = (cid:114) (cid:104)(cid:112) ( y − ρ ) + η ± ( ρ − y ) (cid:105) ,M = (2 sb + c + ) + c − , and N = (2 sb − c + ) + c − . These functions and, consequently, the whole familyof solutions, depend on three arbitrary real parameters α , ρ, η [22, 30]. The periods in z and t also depend onthese parameters. They are given by: Z = 8 K ( k ) /µ, T = 4 K ( k q ) /p , respectively, where K ( k ) is the completeelliptic integral of the first kind. These free parametersprovide us with the possibility of accurately controllingthe wave evolution with periodic initial conditions andparticularly the development of modulation instability. INSTABILITY OUTSIDE THE CONVENTIONALMI BAND
Equations above provide an exact wave dynamics withtwo frequencies. Thus, the MI which is periodic alongthe t -axis is a particular case of these solutions. Indeed,there is a range of parameters ρ and η when the solutionrepresents the growth of a periodic perturbation on topof a continuous wave. This happens when 0 < ρ < η →
0. This range corresponds to exact conditionsof modulation instability with the exponential growth ofperiodic perturbation with a frequency defined by ρ . Onthe other hand, for parameters ρ and η beyond this range,the evolution has all features of modulation instabilitybut the growth of the perturbation takes different form.This more general evolution is periodic in z . The solu-tion is closest to the continuous wave when the evolutionvariable z = ±Z /
4. Starting from one of these values of z leads to the growth of modulations on the backgroundCW. One example is given in Fig. 1(a) that shows waveintensity profiles at z = −Z / z = 0 when the modulation ismaximal (blue curve). Pulses within this periodic pulsetrain are maximally compressed. The wave intensity pro-file returns back to the initial condition at z = + Z / Time t | | ² G a i n ( d B ) Frequency ( b )( a ) FIG. 1. (a) Transformation of a periodic perturbation on topof the CW (red curve) into a train of pulses (blue curve).Parameters of the solution here are: ρ = 0, η = 1, α =1. Modulation frequency ω = 2 . > ω depends on theparameter ρ that changes in the interval [ − , ω > ω ≤ The amplification of the periodic component of the so-lution calculated numerically from the exact solution isshown in Fig. 1(b). Here, the frequency range [0 ,
2] isthe standard band of modulation instability. Amplifica-tion within this range is not surprising. However, theamplification is not zero when the frequency ω >
2. Theamplification here might seem smaller than within theband [0,2]. However, the amplitudes of the pulse trainsreached due to the growth are of the same order of mag-nitude as within the band. Thus, the effect is easily mea-surable in experiments. Moreover, the frequency rangeshown in Fig. 1(b) is nearly 1.5 times the conventional MIbandwidth ω ∈ [0 , η . Forlarger values of η , the amplification within the standardMI band is smaller. However, the amplification outsideof this band does not depend on η . Thus, at larger valuesof η , the MI effect is nearly the same order of magnitudewithin and outside of the standard band.Another remarkable feature of the MI visible inFig.1(a) is the period of the pulse train which is twice theperiod of the initial modulation. Every second maximaof the periodic perturbation grows while the juxtaposingmaxima decay. This feature adds flexibility to potentialapplications of the effect. The red curve in Fig. 1(a) andanalogous curves calculated for other values of parame-ters have been used as initial conditions in the optical and water wave experiments as well as in numerical sim-ulations presented below. OPTICAL EXPERIMENTS
For optical experiment, we used a setup similar to theone used in [10, 39] and devoted to investigate nonlin-ear stage of MI within its conventional bandwidth. Itsschematic is shown in Fig. 2. The input in the form of O S A S M F - O s c ill o . P u l s e , s i d e b a nd s , ph a s e p r o c e ss o r s i d e b a nd s p r o c e ss o r C W ff f f f f ff f f f - f m f + f m f + f m f - f m ∆ f = f -f P h a s e l o c k e d P u m p S i g n a l f B a c k r e fl e c t e d li g h t P S ( z ) P P ( z ) ϕ S ( z ) ϕ P ( z ) R a m a n pu m p i n S p e c t r a l F il t e r C o up l e r R a m a n pu m p o u t L a s e r C W L a s e r S ee d l o c a l o s c ill a t o r FIG. 2. Experimental setup: f , are the frequencies of themain laser and the local oscillator laser, respectively. Here f m is the input modulation frequency (pump frequency at f ,input sideband frequencies at f ± f m ). The backscatteredsignal from the SMF-28 fiber goes through a circulator to beanalysed via heterodyning (beating with the local oscillator)and then filtered (waveshaper) to isolate the power and phaseevolutions of the pump and the first-order side-band pair inthe MI spectral comb; OSA - optical spectrum analyzer. continuous wave with periodic perturbation is created byCW laser 1. The intensity and the phase of the pump andthe sidebands are precisely controlled. The resulting 3-wave input is injected into a L = 18 .
28 km long SMF-28fiber (group velocity dispersion β = − × − s m − ,nonlinear coefficient γ = 1 . × − W − m − ). The lossis actively compensated by using a counter-propagatingRaman pump emulating an almost fully transparent opti-cal fiber [10]. Power and phase distributions of the pumpand the first order sideband (signal) are obtained using amulti-heterodyning technique between the backscatteredsignal and the local oscillator [10].In order to apply the theory in the previous section tooptical fibers, the variables must be renormalized. Tothis end, the dimensional distance Z , time T (in theframe traveling at light group-velocity), and field Ψ (with | Ψ | giving directly the power in Watts) are obtained bythe following rescaling Z = ( z − z ) L NL , T = t T s , Ψ = ψ (cid:112) P P , (6) L NL = ( γP P ) − , T s = (cid:112) | β | L NL , (7)where L NL is the characteristic nonlinear length scale as-sociated with CW power P p , and T s is the relative tempo-ral scale associated with dispersion. Here z is a suitableshift that accounts for the fact the input Z = 0 corre-sponds to a point of weak modulation in the solution(whereas z = 0 is the point of maximum amplificationin the solution). For practical purposes, we can approxi-mate z ≈ Z /
4, valid for weak enough input modulation.In this scaling, the MI cutoff frequency is f C =2 / (2 πT s ) = 1 / ( π (cid:112) | β | L NL ). The pump power P P inexperiments is set to 180 mW leading t.o a cutoff fre-quency of the conventional MI gain band at f C = 33 . f m is located outside the MI gain band i.e. f m > f C .The intensity of the sidebands is set at 5.3 dB below thepump power. The experimental spectra of the 3-waveinput and the spectrum of spontaneous MI, i.e. conven-tional MI gain band profile, are plotted in Fig. 3. Theinitial relative phase between the pump and the signal is − π in order to excite the A-type waves. -60 -40 -20 0 20 40 60 Frequency detuning f (GHz) R e l a � v e p o w e r ( d B / d i v ) Spontaneous MI3-wave input
FIG. 3. Experimental 3-wave input spectrum (purple solidline) compared with the spontaneous MI spectrum (yellowdotted line). The two grey areas beyond dashed black linesmark the range of frequencies f > f C , located outside of theconventional MI gain band ( f ≤ f C ). Experimental data for f m = 40 GHz and analytical so-lution for the same set of parameters are shown in Fig. 4.Fig. 4(a) displays the experimental power evolution ofthe pump (blue) and the signal (red). The correspond-ing theoretical prediction is shown by dashed curves. Asexpected, first, we observe amplification of the signal anddepletion of the pump. The process reverses at around2 . . Z = 0 and its first maximum(2 . η in experiments are higher (for ω = ω m and η = 0 . . Analy � cal solu � onsExperimental data Δ Φ ( r a d ) ( a ) D i s t a n c e Z ( k m ) Time T (ps) D i s t a n c e Z ( k m ) Time T (ps) D i s t a n c e Z ( k m ) Time T (ps)Distance Z (km)
Distance Z (km) P o w e r ( W ) (W) (W) E x p . E x p . T h e o . T h e o . μ s i n ( Δ Φ ) μ cos( ΔΦ ) PumpSignal ( b ) ( c )( d ) ( e )( f ) ( g ) D i s t a n c e Z ( k m ) Time T (ps) (rad) (rad)
FIG. 4. Three waves evolution along the fiber when the side-band detuning f m = 40 GHz ( ω m = 2 . η = 1 . ρ = 0 . α = 1 . shows the nearly linear evolution of the pump-signal rel-ative phase (∆Φ) over the fiber length. The experimentalcurve fits the theory almost perfectly. Importantly, theinitial phase is recovered after two cycles of power evolu-tion (around Z = 10 km), whereas successive maximumamplification stages turn out to be mutually out of phase(sidebands shifted by π ), which is a unique feature of A-type solutions [22].It is also very convenient to illustrate the dynamics ofthe process in the phase space ( µ cos(∆Φ), µ sin(∆Φ))where µ is the signal power normalised to its maximumvalue. Such trajectories are shown in Fig. 4(c). The theo-retical curve shown by the dashed line is strictly periodicand corresponds to the A-type solution. The quantita-tive agreement is also pretty good if we focus on thelocations of the curve maxima. Figure 4(c) also givesa pictorial view of the fact that the sidebands amplifi-cation is connected to the nonlinear deformation of theorbit with respect to circular orbits (characteristic of thepurely linear limit ω → ∞ , not shown). The net gain in-deed arises, in the figure-of-eight-shaped orbit, from theratio of the signal at the maximum elongation (horizontalaxis, ∆Φ = 0 , π ) and at the maximum orbital squeezing(input, ∆Φ = − π/ (W) (W) E x p . E x p . T h e o . T h e o . ( a ) ( b )( c ) ( d ) D i s t a n c e Z ( k m ) Time T (ps) D i s t a n c e Z ( k m ) Time T (ps) D i s t a n c e Z ( k m ) Time T (ps) D i s t a n c e Z ( k m ) Time T (ps) (W) (W)
FIG. 5. Spatio-temporal power evolution for two other val-ues of the pump-signal frequency shift. (a),(b) f m = 34 GHz( ω m = 2 . f m = 37 GHz ( ω m = 2 . η = 1 . ρ = 0 . α = 1 .
0; and (c),(d) η = 1 . ρ = 0 . α = 1 . two signal frequency shifts ( f m = 34 and 37 GHz respec-tively) are still outside of the MI band but located closerto the cutt-off frequency. Again, the chess-board like pat-tern of these plots confirms the A-type nature of thesesolutions. We can also notice that when approaching thecut-off frequency, the spatial periods (along z ) increase,as can be seen from Figs.4 (d) and 5 (a) and (c). Impor- tantly, maximal wave amplitudes reached at the points ofmaximal compression are of the same order of magnitudefor all cases shown in these figures.As mentioned, the spatial (longitudinal) period de-pends on the shift between the modulation and the pumpfrequencies. When the modulation frequency is outsideof the MI band, this period decreases with the modu-lation frequency moving away from the pump. Experi-mental verification of this behaviour is shown in Fig. 6(a).Fig. 6(b) shows the corresponding theoretical plot. While E x p . N u m . ( a ) ( b ) D i s t a n c e Z ( k m ) Frequency detuning (GHz)
34 36 38 40 D i s t a n c e Z ( k m ) Frequency detuning (GHz)
34 36 38 400.110.100.090.080.070.06 0.110.100.090.080.070.06 (W) (W)
FIG. 6. 2-D plot of the signal power as a function of distance Z (vertical axis) and pump-signal frequency shift (horizontalaxis). (a) experiment and (b) numerics. the frequency shift increases from 34 GHz to 41 GHz, thenumber of longitudinal periods along the same distance ≈
18 km changes from 3 to 3.8. This means that eachlongitudinal period decreases from ≈ ≈ .
73 km.Agreement between the experimental data and the the-ory is also good as the two plots in Fig. 6 demonstrate.Thus, our optical experiments confirm, clearly, the fact ofexistence of modulation instability outside of the conven-tional instability band. The measurements are in goodagreement with the theoretical predictions based on theexact solutions of the NLSE.
WATER WAVE EXPERIMENTS
The hydrodynamic experiments have been performedin a uni-directional wave tank, installed at the Universityof Sydney and shown in Fig. 7. Its dimensions are 1 m
FIG. 7. Sketch of the L=30 meters long water tank at theUniversity of Sydney. In red the locations of the gauges. × ×
30 m. The tank was filled with fresh waterto the height of 0 . . < f < z for plotting the experimentalpatterns, as shown below. The wave envelopes have beencomputed using the Hilbert transform while the spectraldata have been calculated using the fast Fourier trans-form of the water surface elevation data.Although the water wave envelope obeys the samedimensionless focusing NLSE as the normalized opticalfield in optical fiber, the spatial and time scales turn outto be extremely different. We start from dimensionaldeep-water time-NLSE [15] characterized by the second-order dispersion coefficient β = − /g ( g = 9 .
81 m/s is the gravitational acceleration) and the nonlinear coef-ficient γ = − κ ( γβ >
0, focusing regime), where κ isthe wavenumber of the carrier, with the carrier frequencyfixed through the dispersion relation ω = √ gκ . In orderto introduce a normalization akin to Eqs. (6-7), the di-mensional distance along the tank Z , the dimensionaltime T , and the envelope of water wave elevation Ψ, canbe expressed in terms of nonlinear length L NL and tem-poral scale T s , fixed by the input envelope elevation a , asfollows (see also [41]) Z = ( z − z ) L NL , T = t T s , Ψ = ψ a, (8)where L NL = κ a , T s = (cid:113) gκ a , and z is a suitableshift for which considerations analogous to those made inoptics are still valid. It is worth mentioning that in thiscase, the time T is also measured in the frame movingwith the group velocity c g = ω κ . Obviously, this scalingis not unique. An equivalent choice often employed inthe case of water waves can be written in terms of thewavenumber κ and the wave steepness ε = aκ : Z = z/ ( κε ), T = √ t/ ( ωε ), Ψ = ψε/κ .Operating with the scaling in Eq. (8), we obtained thetheoretical spatio-temporal patterns shown in Figs. 8-10[see right panels (b)], which we compare with experi-mental data [left panels (a) in the same figures]. Thechoice of the parameters of the NLSE solution used forgenerating these patterns is given in the figure captions.In our water waves experiment, typically, L NL ≈
10 m and T s ≈ . L NL ≈ T s ≈
10 ps of the optical experiment. Accordingly,also the MI cut-off frequency, which reads, in thiscase, f C = π (cid:113) gκ a , turns out to be several orders ofmagnitude lower than the one in optics. For instance,with κ = 10 m − and a = 0 .
01 m (case of Fig. 8),we obtain f C = 0 .
22 Hz. The envelope evolution aspredicted by theory takes into account the second-orderStokes correction to the water surface elevation [31]. D i s t a n c e Z ( m ) Time T (s) T h e o . D i s t a n c e Z ( m ) (m)Time T (s) (m) E x p . ( a ) ( b ) FIG. 8. (a) Experimental and (b) theoretical plots of spatio-temporal wave evolution that start with extraordinary mod-ulation instability. The values of parameters in the NLSEsolution used to prepare the initial conditions in the exper-iment are: η = 1 . ρ = − . α = 1 .
0. Wave ampli-tude a = 0 .
01 m, the carrier wavenumber κ = 10 m − , thecorresponding wave steepness ε = 0 .
1, and the modulationfrequency f m = 0 .
37 Hz is well above the cut-off f C = 0 . These spatio-temporal patterns are very similar tothose obtained in optical experiments. Remarkably, ourmaxima (two periods) have been achieved within thelength of the tank as can be seen from Fig. 8. The re-sulting chessboard structure of this pattern correspond-ing to the A-type doubly periodic wave is also clearlyseen. Three recurrences to a nearly constant amplitudewave are clearly visible despite unavoidable dissipationelements, always present when performing laboratory ex-periments. Note that for the given carrier wave param-eters, it would not be possible to observe more than onecycle of MI-growth-decay or AB within the given effectivepropagation distance of 25 m.In order to reaffirm the observation, two more exam-ples of the spatio-temporal pattern are shown in Fig. 9.These plots contain less then one period of evolution thatincludes one full recurrence to initial conditions at around15 meters mark in (a) and around 19 meters mark in (c).In each case, the carrier steepness ε has been adjusted tobe just below the threshold of wave breaking. The latterhappens due to the excessive wave amplitude amplifica-tion.One essential difference of experimental patterns inFigs. 8(a) and 9(a) from the optical ones is slightly tiltedvertical stripes. The reason is the asymmetry of the waterwave profiles, which is the result of significant breatheramplification of a factor of three and above. The con- D i s t a n c e Z ( m ) Time T (s) D i s t a n c e Z ( m ) Time T (s) E x p . T h e o . T h e o . D i s t a n c e Z ( m ) Time T (s) E x p . ( a ) ( b )( c ) ( d ) D i s t a n c e Z ( m ) Time T (s) (m) (m)(m) (m)
FIG. 9. (a,c) Experimental and (b,d) theoretical plots ofspatio-temporal wave evolution that start with modulationinstability. The values of parameters in the NLSE solutionused to prepare the initial conditions in the experiment are:(a,b) η = 1 . ρ = 0 . α = 1 .
0; and (c,d) η = 1 . ρ = 2 . α = 1 .
0. Wave amplitude a = 0 .
01 m in each case.The wavenumber of the carrier is (a) κ = 10 m − and (c) κ = 8 m − . The corresponding wave steepness ε = 0 . f C = 0 .
22 Hz in (a) and aκ = 0 .
08 withcut-off frequency f C = 0 .
15 in (c). The modulation frequencyis f m = 0 .
25 Hz in (a) and f m = 0 .
16 Hz in (c). sequence is the nonlinear Stokes contributions that arealways present in water waves at these scales [42, 43].Despite these deviations, the patterns in Figs. 8(a) and9(a) clearly confirm the presence of the modulation in-stability and its nonlinear evolution beyond the standardunstable frequencies of MI in the BF theory.Generally, when increasing the amplification factor ofthe breather, the steepness has to be decreased in order toavoid wave breaking. The latter violates the condition ofthe flow to be irrotational and thus, prohibits applicabil-ity of the Euler equations and subsequently, the validityof the NLSE [14]. Indeed, when this threshold of wavebreaking is exceeded, spilling as well as recurrent break-ing occurs and the pattern changes significantly and doesnot follow the theoretical NLSE predictions. One exam-ple is shown in Fig. 10. Here, the value of the breatherparameter ρ is increased in comparison to the previouscases. Modulation instability still develops but there isno obvious recurrence back to initial conditions as can beseen from Fig. 10(a). CONCLUSIONS
In conclusion, we have experimentally confirmed thatmodulation instability is more complex phenomenonthan the one predicted by the simplified linear stabilityanalysis. The most striking difference that the more ac- D i s t a n c e Z ( m ) Time T (s) T h e o . E x p . ( a ) ( b ) D i s t a n c e Z ( m ) Time T (s) (m) (m)
FIG. 10. (a) Experimental and (b) theoretical plots of spatio-temporal wave evolution that start with modulation instabil-ity. The values of parameters in the NLSE solution used toprepare the initial conditions in the experiment are: η = 1 . ρ = 2 . α = 1 .
0. Wave amplitude a = 0 .
01 m. Thewavenumber of the carrier in (a) is κ = 8 m − , consequentlythe wave steepness ε = 0 .
08. The modulation frequencyhere f m = 0 . f C = 0 .
15 Hz. Modulation instability develops but wavebreaking prevents the recurrence back to initial conditions. curate nonlinear analysis using exact breather frameworkreveals is the fact that periodically perturbed continuouswaves develop the growth of perturbation not only withinthe standard modulation instability band described bythe BF and BT theories but also outside of it. To be moreaccurate, the frequency range of unstable growth of theperturbation extends beyond the standard MI threshold.Another dramatic difference from the standard theorycan be seen when observing the nonlinear stage of MI.The subsequent evolution beyond the initial growth cre-ates a specific chess-board like periodic spatio-temporalpattern of wave propagation. Temporal maxima of thegenerated pulse trains change position every half periodof spatial evolution. The effect tightly related to thisphenomenon is the fact that the frequency of the pulsetrain at the point of maximum compression is half of thefrequency of initial modulation. Such phenomenon mayfind applications in frequency comb devices facilitatingthe atomic clock synchronisation when the frequenciesdiffer by an octave [44].Having these unusual features revealed in nonlinearanalysis, we can call the effect ‘extended’ or ‘extraordi-nary’ modulation instability. Importantly, we were ableto track and confirm this ‘extraordinary’ modulation in-stability in two different physical media, namely, in opticsand in hydrodynamics, proving the interdisciplinary sig-nificance of the extended MI. In fact, these are the twoareas of physics where the wavelength differs by four or-ders of magnitude, and the modulation frequencies by tenorders of magnitude. This twofold confirmation of the ef-fect shows that it is ubiquitous and does not depend onthe scale of the physical system that we operate with.The effects should be also observable in other areas ofphysics such as astrophysics [45], plasma [46, 47], meta-materials [48] and in Bose-Einstein condensate [49, 50].We envisage that the new phenomenon can be usefulin applications such as generation of optical frequencycombs and pulse trains with prescribed parameters: pe-riods, amplitudes and duty cycles. Moreover, we antic-ipate novel modelling approaches for extreme events innonlinear dispersive media.
ACKNOWLEDGMENTS
Agence Nationale de la Recherche (Programme In-vestissements dAvenir); Ministry of Higher Educationand Research; Hauts de France Council; European Re-gional Development Fund (Photonics for Society P4S,FUHNKC, EXAT). [1] V. I. Bespalov and V. I. Talanov,
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