Abstract
From optics to hydrodynamics, shock and rogue waves are widespread. Although they appear as distinct phenomena, transitions between extreme waves are allowed. However, these have never been experimentally observed because control strategies are still missing. We introduce the new concept of topological control based on the onetoone correspondence between the number of wave packet oscillating phases and the genus of toroidal surfaces associated with the nonlinear Schrödinger equation solutions through Riemann theta functions. We demonstrate the concept experimentally by reporting observations of supervised transitions between waves with different genera. Considering the box problem in a focusing photorefractive medium, we tailor the timedependent nonlinearity and dispersion to explore each region in the state diagram of the nonlinear wave propagation. Our result is the first realization of topological control of nonlinear waves. This new technique casts light on shock and rogue waves generation and can be extended to other nonlinear phenomena.
Introduction
In 1967 Gardner, Greene, Kruskal, and Miura developed a mathematical method—the inverse scattering transform (IST)^{1}—disclosing the inner features of nonlinear waves in hydrodynamics, plasma physics, nonlinear optics and many other physical systems^{2,3,4}. According to IST, one also predicts the periodical regeneration of the initial state, as in the FermiPastaUlamTsingou recurrence^{5,6}.
The nonlinear Schrödinger equation (NLSE)^{7} is a cornerstone of IST for detailing dispersive phenomena, such as dispersive shock waves (DSWs)^{8,9,10}, rogue waves (RWs)^{11,12,13,14}, and shape invariant solitons^{15,16,17}. DSWs regularize catastrophic discontinuities by means of rapid oscillations^{18,19,20,21,22}. RWs are giant disturbances appearing and disappearing abruptly in a nearly constant background^{23,24,25,26,27,28,29,30,31,32,33,34}. Solitons are particlelike dispersionfree wave packets that can form complex interacting assemblies, ranging from crystals to gases^{15,16,33,35,36,37}.
DSWs, RWs, and soliton gases (SGs) are related phenomena, and all appear in paradigmatic nonlinear evolutions, such as the box problem for the focusing NLSE^{38,39,40,41,42,43}. However, for the box problem in the smalldispersion NLSE, IST becomes unfeasible. In this extreme regime, the problem can be tackled by the socalled finitegap theory^{40,44}. It turns out that extreme waves are described in terms of one single mathematical entity, the Riemann theta function, and classified by a topological index, the genus \(g\) (see Fig. 1). In nonlinear wave theory, \(g\) represents the number of oscillating phases and evolves during light propagation: “single phase” DSWs have \(g=1\), RWs have \(g \sim 2\) and SGs have \(g\ > > \ 2\). This creates a fascinating connection between extreme waves and topology. Indeed, the same genus \(g\) allows a topological classification of surfaces, to distinguish, for examples, a torus and sphere (Fig. 1). The question lies open if this elegant mathematical classification of extreme waves can inspire new applications. Can it modify the basic paradigm by which the asymptotic evolution of a wave is encoded in its initial shape, opening the way to controlling extreme waves, from lasers to earthquakes?
Here, inspired by the topological classification, we propose and demonstrate the use of topological indices to control the generation of extreme waves with varying genera \(g\)^{41}. We consider the NLSE box problem where, according to recent theoretical results^{40}, light experiences various dynamic phases during propagation, distinguished by different genera. In particular, for high values of a nonlinearly scaled propagation distance \(\zeta\), one has \(g \sim \zeta\). By continuously varying \(\zeta\), we can change \(g\) and explore all the possible dynamic phases (see Fig. 1, where \(\zeta\) is given in terms of the observation time \(t\), detailed below). We experimentally test this approach in photorefractive materials, giving evidence of an unprecedented control of nonlinear waves, which allows the first observation of the transition from focusing DSWs to RWs.
Results
Timedependent spatial box problem
We consider the NLSE
where \(\psi =\psi (\xi ,\zeta )\) is the normalized complex field envelope, \(\zeta\) is the propagation coordinate, \(\xi\) is the transverse coordinate and \(\epsilon \ > \ 0\) is the dispersion parameter. We take a rectangular barrier as initial condition
that is, a box of finite height \(q\ > \ 0\), length \(2l\ > \ 0\), and genus \(g=0\). In our work, we fix \(q=l=1\). Equation (1) with (2) is known as the NLSE box problem, or the dam break problem, which exhibits some of the most interesting dynamic phases in nonlinear wave propagation^{40,45}. The initial evolution presents the formation of two wave trains counterpropagating that regularize the box discontinuities. These wave trains are singlephase DSWs (\(g=1\)). Their two wavefronts superimpose in the central part of the box (see Fig. 1a)–occurring at \(\zeta ={\zeta }_{0}:= \frac{l}{2\sqrt{2}q}\) – and generate a breather lattice of genus \(g=2\), a twophase quasiperiodic wave resembling an ensemble of Akhmediev breathers (ABs)^{13,28}. Since both the \(\xi \) and \(\zeta \) periods increase with \(\zeta\), the oscillations at \(\xi \simeq 0\) become locally approximated by Peregrine solitons (PSs)^{13,46,47,48}. At long propagation distances \(\zeta \ > > \ {\zeta }_{0}\), the wave train becomes multiphase and generates a SG with \(g \sim \zeta\).
In Fig. 1a, we report the wave dynamics in physical units, as we make specific reference to our experimental realization of the NLSE box problem for spatial optical propagation in photorefractive media (PR). In these materials, the optical nonlinearity is due to the timedependent accumulation of free carriers that induces a timevarying lowfrequency electric field. Through the electrooptic effect, the charge accumulation results into a timevarying nonlinearity. The corresponding timeprofile can be controlled by an external applied voltage and the intensity level^{49,50,51}. These features enable to experimentally implement our topological control technique. In PR, Eq. (1) describes an optical beam with complex amplitude \(A(z,x,t)\) and intensity \(I= A{ }^{2}\) through the transformation (see Methods)
with \({W}_{0}\) the initial beam waist along \(x\)direction, \({z}_{{\rm{D}}}=\frac{\pi {n}_{0}{W}_{0}^{2}}{2\lambda }\) the diffraction length, \(n={n}_{0}+\frac{2\delta {n}_{0}I}{{I}_{{\rm{S}}}}f(t)\) the refractive index, \(\delta {n}_{0}\ > \ 0\) the nonlinear coefficient, \({I}_{{\rm{S}}}\) the saturation intensity, \({I}_{0}\) the initial intensity. For PR
namely, the dispersion is modulated by the timedependent crystal response function \(f(t)=1\exp \left(t/\tau \right)\), with the saturation time \(\tau\) fixed by the input power and the applied voltage^{5}.
Genus control
For a given propagation distance \(L\) (the length of the photorefractive crystal), the genus of the final state is determined by the detection time \(t\), which determines \(\epsilon\), \(\zeta =\frac{L}{\epsilon {z}_{{\rm{D}}}}\), and \(g\), correspondingly. The genus timedependence is sketched in Fig. 1a. The output wave profile depends on its genus content, which varies with \(t\).
Following the theoretical approach in ref. ^{40}, the two separatrix equations divide the evolution diagram in Fig. 1a into three different areas: the flat box plateau with genus \(g=0\), the lateral counterpropagating DSWs with genus \(g=1\), and the RWs after the DSWcollision point (corresponding to the separatrices intersection) with genus \(g=2\). The two separatrices (dashed lines in Fig. 1a) have equations
with \(({t}_{0},{x}_{0})\) the DSWcollision point, \({t}_{0}\simeq \frac{\tau {I}_{{\rm{S}}}{n}_{0}{W}_{0}^{2}}{64{I}_{0}\delta {n}_{0}{L}^{2}}\), and \({x}_{0}\) given by the central position of the box. It turns out that the shock velocity is
proportional to the input power, as experimentally demonstrated and detailed below.
Equation (5) expresses the genus timedependence for its first three values \(g=0,1,2\). It allows designing the waveshape, before the experiment, by associating a specific combination of the topological indices, and to predict the detection time corresponding to the target topology. In other words, by properly choosing the experimental conditions, we can predict the occurrence of a given extreme wave by using the expected genus \(g\). According to Eq. (4), we use time \(t\) and initial waist \({W}_{0}\) to vary \(\epsilon\). The accessible states are outlined in the phase diagram in Fig. 1b, in terms of \(\epsilon\) and \({W}_{0}\). Choosing \({W}_{0}=100\,\upmu\)m as in Fig. 1a, by varying \(t\) one switches from DSWs to RWs, and then to SGs.
Supervised transition from shock to rogue waves
The case \({W}_{0}=140\,\upmu\)m is illustrated in Fig. 2a by numerical simulations. The two focusing DSWs and the SG are visible at the beginning and at the end of temporal evolution, respectively (see phase diagram in Fig. 1b). As soon as an initial superGaussian wave (Fig. 2b, see Methods) starts to propagate, two DSWs appear on the beam borders (Fig. 2c) and propagate towards the beam central part (Fig. 2d). Experimental proof of the genuine nonlinear nature of the beam evolution at this regime, not due to modulation instability arising from noise in the central part of the box, is shown in Supplementary Information (Suppl. Fig. 1). When the DSWs superimpose, ABs are generated (Fig. 2e). From the analytical NLSE solutions for the focusing dam break problem^{40}, we see that ABs have \(\xi\)period increasing with \(\zeta\). Moreover, one finds that \({\partial }_{t}\zeta \ > \ 0\), therefore the period in the \(x\)direction must increase with time, and central peaks appear upon evolution. These peaks are well approximated by PSs, for large \(t\), as confirmed by Fig. 2f, g.
The occurrence of RWs in the large box regime is proved also by statistical analysis, illustrated in Supplementary Information (Suppl. Fig. 2h, i).
Figure 3 shows the experimental observation of the controlled dynamics simulated in Fig. 2. Figure 3a sketches the experimental setup, detailed in Methods. A quasionedimensional boxshaped beam propagates in a photorefractive crystal, and the optical intensity distribution is detected at different times. The observations of shock velocities and beam propagation for \({W}_{0}=140\,\upmu\)m are reported in Fig. 3b, c, respectively. In Fig. 3c, we see an initial DSW phase that evolves into a train of large amplitude waves. In this regime, we identify a breatherlike structure (ABs, inset in Fig. 3c) that evolves into a SG at large propagation time. The DSW phase is investigated varying the input power. We find a linear increasing behavior of the shock velocity when increasing the power (Fig. 3b), as predicted by Eq. (6). The shock velocity is proportional to the distance between the two counterpropagating DSWs at a fixed time. We measured the width \(\Delta x\) of the plateau at time \(\bar{t} \sim 30\) s. Referring to Eq. (6), we obtain the normalized velocity \(\bar{v}=v/{v}_{0}\), with \({v}_{0}=L/\bar{t}\).
Peregrine solitons emergence
Figure 4 illustrates the numerically determined dynamics at smaller values of the beam waist (\({W}_{0}=10\,\upmu\)m), a regime in which the generation of single PSs is evident. The intensity profile is reported in Fig. 4a. As shown in Fig. 1b, one needs to carefully choose \({W}_{0}\) for observing a RWs generation without the DSWs occurrence. For \({W}_{0}=10\,\upmu\)m, the superGaussian wave (Fig. 4b) generates a PS (Fig. 4c–e). The following dynamics shows the higherorder PS emergence (Fig. 4f, g), each order with a higher genus.
Figure 5a–g report the experimental results for the case \({W}_{0}=30\,\upmu\)m. Observations of the Peregrinelike soliton generation are shown, both in intensity (Fig. 5a–d) and in phase (Fig. 5e–g). For a small initial waist, a localized wave, well described by the PS (Fig. 5b, d), forms and recurs without a visible wave breaking. This dynamics is in close agreement with simulations in Fig. 4d–g, where the PS is repeatedly destroyed and generated, each time at a higher order. Phase measurements are illustrated in Fig. 5e–g. Each PS has twophase signatures: a longitudinal smooth phase shift of \(2\pi\) and a transversal rectangular phase shift profile, with height \(\pi\) and basis as wide as the PS width^{47,48}. Such signatures are here both experimentally demonstrated. From Fig. 5e, which shows the interference pattern during the first PS occurrence, we obtain the longitudinal phase shift behavior in Fig. 5g, by a cosinusoidal fitting along the central propagation outline. Figure 5f reports the experimental transversal phase shift profile along \(x\). A comparison with the measured interference fringes is also illustrated in the inset, which directly shows the phase jump (topological defect). Stressing the significance of these results is very important, because they are a proof of the topological control: the genus is determined by the input waist and time of detection. Indeed, the longitudinal phase shift represents the transition from genus \(0\) to \(2\), whereas the transverse PS phase shift outline unveils the value \(g=2\), equal to the number of phase jumps (first from \(0\) to \(\pi\), then again from \(\pi\) to \(0\)). This is summarized in Fig. 5h, which sketches numerical simulations of phase behavior at \({W}_{0}=10\,\upmu\)m, normalized in \([\pi ,\pi ]\). Figure 5h gives a picture of genera changes, PS occurrence and phase discontinuities. The genus is zero and the phase profile is flat until the first PS occurrences. After that, the phase value changes and the phase transverse profile presents two jumps of \(\pi\).
The statistical properties of the PS intensity are illustrated in Supplementary Information (Suppl. Fig. 2f, g), and they confirm the occurrence of RWs in the small box regime.
Discussion
The topological classification of nonlinear beam propagation by the genera of the Riemann theta functions opens a new route to control the generation of extreme waves. We demonstrated the topological control for the focusing box problem in optical propagation in photorefractive media. By using the timedependent photorefractive nonlinearity, we could design the final state of the wave evolution in a predetermined way and explore all the possible dynamic phases in the nonlinear propagation.
Such a novel control strategy enabled the first observation of the continuous transition from dispersive shock to rogue waves and soliton gases, demonstrating that different extreme wave phenomena are deeply linked, and also that a proper tuning of their topological content in their nonlinear evolution allows transformations from one state to another. The further numerical and experimental analysis reported in Supplementary Information proves that this new control paradigm in thirdorder media has a broad range of validity, where it is not affected by linear effects, like modulation instability or loss, but its nature is genuinely nonlinear.
In conclusion, our result is the first example of the topological control of integrable nonlinear waves. This new technique casts light on dispersive shock waves and rogue wave generation. It is general, not limited to the photorefractive media, and can be extended to other nonlinear phenomena, from classical to quantum ones. These outcomes are not only important for fundamental studies and control of extreme nonlinear waves, but further developments in the use of topological concepts in nonlinear physics can allow innovative applications for engineering strongly nonlinear phenomena, as in spatial beam shaping for microscopy, medicine and spectroscopy, and coherent supercontinuum light sources for telecommunication.
Methods
Photorefractive media
Starting from Maxwell’s equations in a medium with a thirdordernonlinear polarization, in paraxial and slowly varying envelope approximations, one can derive the propagation equation of the complex optical field envelope \(A(x,y,z)\):
with \(z\) the longitudinal coordinate, \(x,y\) the transverse coordinates and \(n={n}_{0}+\delta n(I)\) the refractive index, weakly depending on the intensity \(I= A{ }^{2}\)\(\left(\delta n(I)\ <<\ {n}_{0}\right)\).
Equation (7) is the nonlinear Schrödinger equation (NLSE) and rules laser beam propagation in centrosymmetric Kerr media. For PR, the refractive index perturbation depends also parametrically on time, i.e., \(\delta n=\delta n(I,t)\). In fact, the amplitude of the nonlinear selfinteraction increases, on average, with the exposure time up to a saturation value, on a slow timescale, typically seconds for peak intensities of a few kWcm^{−2 }^{51}.
In our centrosymmetric photorefractive crystal, at first approximation \(\delta n=\frac{\delta {n}_{0}}{{\left(1\,+\,\frac{I}{{I}_{{\rm{S}}}}\right)}^{2}}f(t)\), with \(f(t)\) the response function. \(\delta {n}_{0}\) includes the electrooptic effect coefficient^{49,50,51}. For weak intensities \(I\ <<\ {I}_{{\rm{S}}}\), we obtain a Kerrlike regime with \(\delta n=2\delta {n}_{0}\frac{I}{{I}_{{\rm{S}}}}f(t)\), apart from a constant term. We consider the case \({\partial }_{y}A \sim 0\) (strong beam anisotropy), thus we look for solutions of the \((1+1)\)dimensional NLSE for the envelope \(A \sim A(x,z)\):
with \(\rho (t)=\frac{2\pi \delta {n}_{0}}{\lambda {I}_{{\rm{S}}}}f(t)\) and the field envelope initial profile
One obtains Eq. (8) from Eq. (1) through the transformation (3). We stress that, in this case, the dispersion parameter depends on time, as follows from Eq. (4).
Numerical simulations
We solve numerically Eq. (1) by a oneparameterdepending beam propagation method (BPM) with a symmetrized splitstep in the code core^{52}. We use a highorder superGaussian initial condition
For each temporal value, Eq. (1) solutions have different dispersion parameter \(\epsilon\) and final value of \(\zeta\), because from Eq. (3) it reads \({\zeta }_{{\rm{fin}}}=\frac{4L}{\epsilon (t)k{W}_{0}^{2}}\), where \(L\) is the crystal length. In Fig. 2 and 4, we show the numerical results. The propagation in time considers \(\psi (\xi ,{\zeta }_{{\rm{fin}}})\), which corresponds to detections at end of the crystal.
Experimental setup
A \(y\)polarized optical beam at wavelength \(\lambda =532\) nm from a continuous \(80\) mW Nd:YAG laser source is focused by a cylindrical lens down to a quasionedimensional beam with waist \({U}_{0}=15\,\upmu\)m along the \(y\)direction. The initial box shape is obtained by a mask of tunable width, placed in proximity of the input face of the photorefractive crystal. A sketch of the optical system is shown in Fig. 3a. The beam is launched into an optical quality specimen of \(2.{1}^{(x)}\times 1.{9}^{(y)}\times 2.{5}^{(z)}\) mm \({{\mathrm{K}}}_{0.964}{\mathrm{Li}}_{0.036}{\mathrm{Ta}}_{0.60}{\mathrm{Nb}}_{0.40}{{\mathrm{O}}}_{3}\) (KLTN) with Cu and V impurities (\({n}_{0}=2.3\)). The crystal exhibits a ferroelectric phase transition at the Curie temperature \({T}_{{\rm{C}}}=284\) K. Nonlinear light dynamics are studied in the paraelectric phase at \(T={T}_{{\rm{C}}}+8\) K, a condition ensuring a large nonlinear response and a negligible effect of smallscale disorder^{53}. The timedependent photorefractive response sets in when an external bias field \(E\) is applied along \(y\) (voltage \(V=500{\,\mathrm{V}}\)). To have a socalled Kerrlike (cubic) nonlinearity from the photorefractive effect, the crystal is continuously pumped with an \(x\)polarized \(15\) mW laser at \(\lambda =633\) nm. The pump does not interact with the principal beam propagating along the \(z\) axis and only constitutes the saturation intensity \({I}_{{\mathrm{S}}}\). The spatial intensity distribution is measured at the crystal output as a function of the exposure time \(t\) by means of a highresolution imaging system composed of an objective lens (\({\mathrm{NA}}=0.5\)) and a CCD camera at \(15\) Hz.
In the present case, evolution is studied at a fixed value of \(z\) (the crystal output) by varying the exposure time \(t\). In fact, the average index change grows and saturates according to a time dependence well defined by the saturation time \(\tau \sim 100\) s once the input beam intensity, applied voltage, and temperature have been fixed.
Data availability
All data are available in this submission.
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Acknowledgments
We acknowledge M. Conforti, S. Gentilini, P.G. Grinevich, A. Mussot, P.M. Santini, S. Trillo, and V.E. Zakharov for fruitful conversations on related topics. We thank MD Deen Islam for technical support in the laboratory. The present research was supported by PRIN 2015 NEMO project (grant number 2015KEZNYM), H2020 QuantERA QUOMPLEX (grant number 731473), H2020 PhoQus (grant number 820392), PRIN 2017 PELM (grant number 20177PSCKT), Sapienza Ateneo (2016 and 2017 programs), and Ministry of Science and Technology of Taiwan (1052628M007003MY4).
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G.M. and D.P. equally contributed to this work. G.M., R.K.L., and C.C. conceived the idea and the theoretical framework; D.P., E.D., and C.C. conceived its experimental realization. G.M. and C.C. developed the theoretical background. G.M. performed the numerical simulations. D.P. carried out experiments and data analysis. A.J.A. designed and fabricated the photorefractive crystal. All authors discussed the results and wrote the paper.
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Marcucci, G., Pierangeli, D., Agranat, A.J. et al. Topological control of extreme waves. Nat Commun 10, 5090 (2019). https://doi.org/10.1038/s41467019128150
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