Topological Control of Extreme Waves

Giulia Marcucci∗ and Davide PierangeliDepartment of Physics, University Sapienza, Piazzale Aldo Moro 5, 00185 Rome, Italy and

Institute for Complex Systems, Via dei Taurini 19, 00185 Rome, Italy

Aharon J. AgranatApplied Physics Department, Hebrew University of Jerusalem, 91904 Israel

Ray-Kuang LeeInstitute of Photonics and Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan

Eugenio DelRe and Claudio ContiDepartment of Physics, University Sapienza, Piazzale Aldo Moro 5, 00185 Rome, Italy and

Institute for Complex Systems, Via dei Taurini 19, 00185 Rome, Italy(Dated: August 15, 2019)

From optics to hydrodynamics, shock and rogue waves are widespread. Although they appearas distinct phenomena, new theories state that transitions between extreme waves are allowed.However, these have never been experimentally observed because of the lack of control strategies.

We introduce a new concept of nonlinear wave topological control, based on the one-to-one corre-spondence between the number of wave packet oscillating phases and the genus of toroidal surfacesassociated with the nonlinear Schrodinger equation solutions by the Riemann theta function. Weprove it experimentally by reporting the first observation of supervised transitions between ex-treme waves with different genera, like the continuous transition from dispersive shock to roguewaves. Specifically, we use a parametric time-dependent nonlinearity to shape the asymptotic wavegenus. We consider the box problem in a focusing Kerr-like photorefractive medium and tailortime-dependent propagation coefficients, as nonlinearity and dispersion, to explore each region inthe state-diagram and include all the dynamic phases in the nonlinear wave propagation.

Our result is the first example of the topological control of integrable nonlinear waves. This newtechnique casts light on dispersive shock waves and rogue wave generation, and can be extended toother nonlinear phenomena, from classical to quantum ones. The outcome is not only important forfundamental studies and control of extreme nonlinear waves, but can be also applied to spatial beamshaping for microscopy, medicine and spectroscopy, and to the broadband coherent light generation.

In 1967 Gardner, Greene, Kruskal, and Miura de-veloped a mathematical method - the inverse scatter-ing transform (IST) [1] - disclosing the inner featuresof extreme nonlinear waves in hydrodynamics, plasmaphysics, nonlinear optics and many other physical sys-tems [2–4]. According to IST, one also predicts the peri-odical regeneration of the initial status, as in the Fermi-Pasta-Ulam-Tsingou recurrence [5, 6].

The nonlinear Schrodinger equation (NLSE) [7] is acornerstone of IST for detailing dispersive phenomena,as dispersive shock waves (DSWs) [8–10], rogue waves(RWs) [11–14] and shape invariant solitons [15]. DSWsregularize catastrophic discontinuities by mean of rapidlyoscillating undular bores [16–20]. RWs are giant distur-bances appearing and disappearing abruptly in a nearlyconstant background [21–28]. Solitons are particle-likedispersion-free wave packets that can form complex inter-acting assemblies, ranging from crystals to gases [15, 29].

DSWs, RWs, and soliton gases (SGs) are related phe-nomena, and all appear in paradigmatic nonlinear evolu-tions, as the box problem for the focusing NLSE [30–34].

∗ giulia.marcucci@uniroma1.it

However, the latter equation is solvable by IST only whenthe number of degrees of freedom in the IST descriptionis limited. Since this number grows as the inverse ofthe dispersion, when one considers the small-dispersionNLSE (SDNLSE), IST becomes unfeasible. In this ex-treme regime, the problem can be tackled by the so-calledfinite-gap theory [31]. It turns out that extreme wavesare described in terms of one single mathematical entity,the Riemann theta function, and classified by a topo-logical index, the genus g (see Fig. 1). In nonlinear wavetheory, g represents the number of oscillating phases, andevolves during light propagation: “single phase” DSWshave g = 1, RWs have g = 2 and SGs have g >> 2.This creates a fascinating connection between extremewaves and topology. Indeed, the same genus g allows atopological classification of surfaces, to distinguish, forexamples, a torus and sphere (Fig. 1). The questionlies open if this elegant mathematical classification of ex-treme waves can inspire new applications. Can it modifythe basic paradigm by which the asymptotic evolution ofa wave is encoded in its initial shape, opening the way tocontrolling extreme waves, from lasers to earthquakes?

Here, inspired by the topological classification, we pro-pose and demonstrate the use of topological indices tocontrol the generation of extreme waves with varying

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genera g [32]. We consider the SDNLSE box problemwhere, according to recent theoretical results [31], lightexperiences various dynamic phases during propagation,distinguished by diverse genera. In particular, for highvalues of a nonlinearly-scaled propagation distance ζ, onehas g ∼ ζ. By continuously varying ζ, we can change gand explore all the possible dynamic phases (see Fig. 1,where ζ is given in terms of the observation time t, de-tailed below). We experimentally test this approach inphotorefractive materials, giving evidence of an unprece-dent control of nonlinear waves, which allows the first ob-servation of the transition from focusing DSWs to RWs.

We consider the SDNLSE

ıε∂ζψ +ε2

2∂2ξψ + |ψ|2ψ = 0, (1)

where ψ = ψ(ξ, ζ) is the normalized complex field enve-lope, ζ is the propagation coordinate, ξ is the transversecoordinate and ε > 0 is the dispersion parameter. Wetake a rectangular barrier as initial condition

ψ(ξ, 0) =

{q for |ξ| < l0 elsewhere

, (2)

that is, a box of finite height q > 0, length 2l > 0 andgenus g = 0. In our work, we fix q = l = 1. Eq. (1)with (2) is known as the SDNLSE box problem, or thedam break problem, which exhibits some of the mostinteresting dynamic phases in nonlinear wave propaga-tion [31, 35]. The initial evolution presents the forma-tion of two wave trains counterpropagating that regular-ize the box discontinuities. These are single-phase DSWs(g = 1), able to generate undular bores. Their two wave-fronts superimpose in the central part of the box (seeFig. 1a) - occurring at ζ = ζ0 := l

2√2q

- and generate

a breather lattice of genus g = 2, a two-phase quasi-periodic wave resembling an ensemble of Akhmedievbreathers (ABs) [13, 25]. Since both ξ− and ζ−periodincrease with ζ, the oscillations at ξ ' 0 become locallyapproximated by Peregrine solitons (PSs) [13, 36]. Atlong propagation distance ζ >> ζ0, the wave train be-comes multi-phase and generates a SG with g ∼ ζ.

In figure 1a, we report the wave dynamics in physicalunits, as we make specific reference to our experimentalrealization of the SDNLSE box problem for spatial opti-cal propagation in photorefractive media (PR). In thesematerials, the optical nonlinearity is due to the time-dependent accumulation of free carriers that induces atime-varying low-frequency electric field. Through theelectro-optical effect, the charge accumulation resultsinto a time-varying nonlinearity. The correspondingtime-profile can be controlled by an external applied volt-age and the intensity level [37–39]. These features en-able to experimentally implement our topological con-trol technique. In PR, Eq. (1) describes an optical beamwith complex amplitude A(z, x, t) and intensity I = |A|2through the transformation (see Methods)

ζ = zεzD

, ξ = 2xW0, ψ = A√

I0, (3)

with W0 the initial beam waist along x-direction, zD =πn0W

20

2λ the diffraction length, n = n0 + 2δn0IIS

f(t) therefractive index, δn0 > 0 the nonlinear coefficient, IS thesaturation intensity, I0 the initial intensity. For PR

ε =λ

πW0

√IS

2n0δn0I0f(t), (4)

namely, the dispersion is modulated by the time-dependent crystal response function f(t) = 1 −exp (−t/τ), with the saturation time τ fixed by the inputpower and the applied voltage.

For a given propagation distance L (the length of thephotorefractive crystal), the genus of the final state isdetermined by the detection time t, which determinesε, ζ = L

εzD, and g, correspondingly. The genus time-

dependence is sketched in Fig. 1a. The output wave pro-file depends on its genus content, which varies with t.

Following the theoretical approach in [31], the two sep-aratrix equations divide the evolution diagram in Fig. 1ain three different areas: the flat box plateau with genusg = 0, the lateral counterpropagating DSWs with genusg = 1, and the RWs after the DSW-collision point (cor-responding to the separatrices intersection) with genusg = 2. The two separatrices (dashed lines in Fig. 1a)have equations

x = x0 ±W0

2t0(t− t0) = x0 ± v(t− t0), (5)

with (t0, x0) the DSW-collision point, with t0 ' τIsn0W20

64I0δn0L2

and x0 given by the central position of the box. It turnsout that the shock velocity is

v =W0

2t0=

32δn0L2

Isn0W 20U0τ

P, (6)

proportional to the input power, as experimentallydemonstrated below (Fig. 3b, other parameters are de-tailed below).

Eqs. (5) express the genus time-dependence for its firstthree values g = 0, 1, 2. It allows designing the wave-shape, before the experiment, by associating a specificcombination of the topological indices, and to predictthe detection time corresponding to the target topology.In other words, by properly choosing the experimentalconditions, we can manage to predict the occurrence of agiven extreme wave by using the expected genus g. Ac-cording to Eq. (4), we use time t and initial waist W0 tovary ε. The accessible states are outlined in the phasediagram in Fig. 1b, in terms of ε and W0. ChoosingW0 = 100µm as in Fig. 1a, by varying t one switchesfrom DSWs to RWs, and then to SGs.

The caseW0 = 140µm is illustrated in Fig. 2 by numer-ical simulations (the corresponding experimental resultsare in Fig. 3). The two focusing DSWs and the SG arevisible at the beginning and at the end of temporal evolu-tion, respectively (see phase-diagram in Fig. 1b). As soon

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as an initial super-Gaussian wave (Fig. 2b, see Methods)starts to propagate, two DSWs appear on the beam bor-ders (Fig. 2c) and propagate towards the beam centralpart (Fig. 2d). When the DSWs superimpose, ABs aregenerated (Fig. 2e). From the analytical SDNLSE so-lutions for the focusing dam break problem [31], we seethat ABs have ξ-period increasing with ζ. Moreover, onefinds that ∂tζ > 0, therefore the period in the x-directionmust increase with time, and central peaks appear uponevolution. These peaks are well approximated by PSs,for large t, as confirmed by Figs. 2f,g.

Figure 3 shows the experimental observation of thecontrolled dynamics simulated in Fig. 2. Figure 3asketches the experimental setup, detailed in Methods. Aquasi-one-dimensional box-shaped beam propagates in aphotorefractive crystal, and the optical intensity distri-bution is detected at different times. The observation forW0 = 140µm is reported in Fig. 3c. We observe an ini-tial DSW phase that undergoes into a train of large am-plitude waves. In this regime, we identify breather-likestructure (ABs, inset in Fig. 3c) that evolves into a SG atlarge propagation time. The DSW phase is investigatedvarying the input power. We find a linear increasing be-havior of the shock velocity when increasing the power(Fig. 3b), as predicted by Eq (6).

Figure 4 illustrates the numerically determined dynam-ics at smaller values of the beam waist (W0 = 40µm),a regime in which the generation of single PSs is evi-dent. The intensity profile is reported in Fig. 4a. Asshown in Fig. 1b, one needs to carefully choose W0 forobserving a specific transition. In particular, we are in-terested in the PS/SG transition, which has never beenobserved before. For W0 = 40µm, the super-Gaussianwave (Fig. 4b) generates a PS (Fig. 4c), formed in a fewseconds (Fig. 4c). The following dynamics shows the SGgeneration (Figs. 4e-g): the PS is alternately destroyedand reformed, and ABs occur on the beam lateral sides.

Figure 5 reports the experimental results for the casesW0 = 50µm (Fig. 5a) and W0 = 30µm (Figs. 5b,c).Fig. 5a presents the observation of the controlled transi-tion from focusing DSWs to Peregrine-like solitons. Fora smaller initial waist (Fig. 5b), a localized wave, welldescribed by the PS (Figs. 5c-e), forms and recurs with-out a visible wave breaking. This dynamics is in closeagreement with simulations in Figs. 4d-g, where the PSis repeatedly destroyed and generated.

In conclusion, the topological classification by thegenus of the Riemann theta function opens a new route toexperimentally control the generation of extreme waves.We demonstrated the topological control for the focus-ing box problem in optical propagation in photorefrac-tive media. By using the time-dependent photorefrac-tive nonlinearity, we were able to change the final stateof the wave evolution in a predetermined way and ex-plore all the possible dynamic phases. This enables thefirst observation of the transitions from shock to roguewaves. This also demonstrates that different extremewave phenomena are deeply linked and that proper tun-

ing of their topological content in the nonlinear evolutionenables transformations from one state to another.

These results are general, and not limited to the pho-torefractive media. Further developments in the use oftopological concepts in nonlinear physics can allow in-novative applications for engineering strongly nonlinearphenomena, as in spatial beam shaping for microscopy,medicine and spectroscopy, and coherent supercontin-uum light sources for telecommunication.

I. METHODS

A. Photorefractive Media

Starting from Maxwell’s equations in a medium with athird-order-nonlinear polarization, in paraxial and slowlyvarying envelope approximations, one can derive thepropagation equation of the complex optical field enve-lope A(x, y, z):

ı∂zA+1

2k∇2A+

k

n0δn(I)A = 0, (7)

with z the longitudinal coordinate, x, y the transverse co-ordinates and n = n0+δn(I) the refractive index, weaklydepending on the intensity I = |A|2 (δn(I) << n0).

Eq. (7) is the nonlinear Schrodinger equation (NLSE)and rules laser beam propagation in centrosymmetricKerr media. For PR, the refractive index perturbationdepends also parametrically on time, i.e., δn = δn(I, t).In fact, the amplitude of the nonlinear self-interactionincreases, on average, with the exposure time up to asaturation value, on a slow timescale, typically secondsfor peak intensities of a few kWcm2 [39].

In our centrosymmetric photorefractive crystal, at firstapproximation δn = −δn0(

1+ IIS

)2 f(t), with f(t) the response

function. δn0 includes the electro-optic effect coeffi-cient [37–39]. For weak intensities I << IS , we obtain aKerr-like regime with δn = 2δn0

IISf(t), apart from a con-

stant term. We consider the case ∂yA ∼ 0 (strong beamanysotropy), thus we look for solutions of the (1 + 1)-dimensional NLSE for the envelope A ∼ A(x, z):

ı∂zA+1

2k∂2xA+ 2ρ(t)|A|2A = 0, (8)

with ρ(t) = 2πδn0

λISf(t) and the field envelope initial profile

A(x, 0) =

{ √I0 for |x| ≤ 1

2W0

0 elsewhere. (9)

One obtains Eq. (8) from Eq. (1) through the transfor-mation (3). We stress that, in this case, the dispersionparameter depends on time, as follows from Eq. (4).

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B. Numerical Simulations

We solve numerically Eq. (1) by a one-parameter-depending beam propagation method (BPM) with a sym-metrized split-step in the code core [40]. We use a high-order super-Gaussian initial condition

ψ(ξ, ζ = 0) = q exp

{−1

2

(ξ

l

)24}. (10)

For each temporal value, Eq. (1) solutions have differentdispersion parameter ε and final value of ζ, because fromEqs. (3) it reads ζfin = 4L

ε(t)kW02 , where L is the crys-

tal length. In Figs. 2,4, we show the numerical results.The propagation in time considers ψ(ξ, ζfin), which cor-responds to detections at end of the crystal.

C. Experimental Set-up

A y-polarized optical beam at wavelength λ = 532nmfrom a continuous 80 − mW NdYAG laser source isfocused by a cylindrical lens down to a quasi-one-dimensional beam with waist U0 = 15µm along the y-direction. The initial box shape is obtained by a maskof tunable width, placed in proximity of the input faceof the photorefractive crystal. A sketch of the opticalsystem is shown in Fig. 3a. The beam is launched intoan optical quality specimen of 2.1(x) × 1.9(y) × 2.5(z)mm

K0.964Li0.036Ta0.60Nb0.40O3 (KLTN) with Cu and V im-purities (n0 = 2.3). The crystal exhibits a ferroelectricphase transition at the Curie temperature TC = 284K.Nonlinear light dynamics are studied in the paraelectricphase at T = TC + 8K, a condition ensuring a largenonlinear response and a negligible effect of small-scaledisorder [41]. The time-dependent photorefractive re-sponse sets in when an external bias field E is appliedalong y (voltage V = 500V). To have a so-called Kerr-like (cubic) nonlinearity from the photorefractive effect,the crystal is continuously pumped with an x-polarized15mW laser at λ = 633nm. The pump does not interactwith the principal beam propagating along the z axis andonly constitutes the saturation intensity IS . The spatialintensity distribution is measured at the crystal outputas a function of the exposure time t by means of a high-resolution imaging system composed of an objective lens(NA = 0.5) and a CCD camera at 15Hz.

In the present case, evolution is studied at a fixed valueof z (the crystal output) by varying the exposure time t.In fact, the average index change grows and saturatesaccording to a time dependence well defined by the sat-uration time τ ∼ 100s once the input beam intensity,applied voltage, and temperature have been fixed.

II. DATA AVAILABILITY

All data are available in this submission.

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ACKNOWLEDGEMENTS

We acknowledge P. M. Santini, P. G. Grinevich, andV. E. Zakharov for fruitful conversations on related top-ics. The present research was supported by PRIN2015NEMO project (2015KEZNYM grant), H2020 Quan-tERA QUOMPLEX (grant number 731473), H2020 Pho-Qus (grant number 820392), and Sapienza Ateneo (2016and 2017 programs).

AUTHOR CONTRIBUTION

G.M. and D.P. equally contributed to this work. G.M.,R.K.L, and C.C. conceived the idea and the theoreticalframework; D.P., E.D., and C.C. conceived its experi-mental realization. G.M. and C.C. developed the theo-retical background. G.M. performed the numerical sim-ulations. D.P. carried out experiments and data analy-sis. A.J.A. designed and fabricated the photorefractivecrystal. All authors discussed the results and wrote thepaper.

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COMPETING INTEREST

The authors declare no competing interests.

ADDITIONAL INFORMATION

Correspondence and requests for materials should beaddressed to G.M.

7

a

b

FIG. 1. Topological classification of extreme waves. a Final states of the wave for a fixed initial waist W0 = 100µmshowing the generation of focusing dispersive shock waves (g = 1), rogue waves (g = 2) and a soliton gas (g > 2) after differenttime intervals in a photorefractive material (see text). b Phase diagram reporting the final states in terms of the parameter εand the initial beam waist. Transitions occur by fixing waist and varying ε or, equivalently, the observation time t. Differentsurfaces displayed in proximity of the various wave profiles, corresponding to the different regions in the phase diagram, outlinethe link between the topological classification of extreme waves in terms of the genus g and the topological classification oftoroidal Riemann surfaces (for a sphere, g = 0, for a torus, g = 1, etc.).

8

a

b c d

e f g

FIG. 2. Controlling the extreme wave genus. a Numerical simulation of the control of the final state after a propagationdistance L = 2.5mm for an initial beam waist W0 = 140µm (I0 = P

U0W0= 0.38W/m2). Axis x represents the beam transverse

direction, axis t the time of output detection. b Initial beam intensity: a super-Gaussian wave centered at x = 150µm of heightI0 and width W0. c-d Focusing dispersive shock waves occurrence: (c) represents the beam intensity at t = 5s, when the wavebreaking has just occurred, so two lateral intense wave trains regularize the box discontinuity and start to travel towards thebeam central part; (d) the beam intensity at t = 11s, which exhibits the two counterpropagating DSWs reaching the centerx = 150µm. e-g Akhmediev breathers and Peregrine solitons generation: beam intensity at (e) t = 49s, (f) t = 98s, and (g)t = 120s, after the two dispersive shock waves superposition and the formation of Akhmediev breathers with period increasingwith t. Since a Peregrine soliton is an Akhmediev breather with an infinite period, increasing t is tantamount to generatingcentral intensity peaks, locally described by Peregrine solitons.

9

a b

c

FIG. 3. Experimental demonstration of the extreme wave genus control. a Experimental setup. A CW laser is madea quasi-one-dimensional wave by a cylindrical lens (CL), then a tunable mask shapes it as a box. Light propagates in a pumpedphotorefractive KLTN crystal, it is collected by a microscope objective and the optical intensity is detected by a CCD camera.The inset shows an example of the detected input intensity distribution (scale bar is 50µm). b Normalized shock velocity,measured through the width of the oscillation tail at fixed time, versus input power. The blue squares are the experimentaldata, while the dashed pink line is the linear fit. c Experimental observation of optical intensity I/I0 for an initial beam waistW0 = 140µm. Axis x represents the beam profile, transverse to propagation, collected by the CCD camera, while axis t is timeof CCD camera detection. Output presents a first dispersive-shock-wave phase, a transition to a phase presenting Akhmedievbreather structures and, at long times, a generation of a soliton gas. The inset is an exemplary wave intensity profile detectedat t = 63s (dotted blue line), along with the theoretical Akhmediev breather profile.

10

a

b c d

e f g

FIG. 4. Simulation of the topological control for a small waist. a Numerical simulation of the control of the final stateafter a propagation distance L = 2.5mm for an initial beam waist W0 = 40µm (I0 = P

U0W0= 1.33W/m2). Axis t expresses time

of detection, while x is the beam transverse coordinate. b Initial beam intensity: a super-Gaussian wave centered at x = 150µm.c-d Peregrine soliton generation: (c) represents the beam intensity at t = 5s, before the formation of the Peregrine soliton,(d) the beam intensity at t = 15s, which exhibits the Peregrine soliton profile. e-g Soliton gas generation: beam intensity at(e) t = 27s, (f) t = 42s, and (g) t = 51s, where the Peregrine soliton is alternately destroyed and reformed, and Akhmedievbreathers occur on the beam lateral sides.

11

a

b

c d e

FIG. 5. Experimental topological control for a small waist. a-b Experimental observation of optical intensity I/I0 forinitial beam waists (a) W0 = 50µm and (b) W0 = 30µm. Axis t is time of output detection, whereas x represents the directiontransverse to propagation. In (a) the transition from focusing dispersive shock waves to Peregrine-soliton-like structures isshown, while in (b) we cannot appreciate the wave breaking and only observe Peregrine soliton formation and recurrence.c-e Intensity outlines corresponding to numbered dashed line in (b), where the Peregrine soliton is alternately destroyed andreformed. The dotted blue lines represent the experimental waveforms, while the pink continuous line is a fitting functionaccording to the analytical Peregrine soliton profile.