Ph.D. In Physics, XXIX Cycle, University of Rome “La Sapienza”

Nonlinear Optical Waves from

Nano-Disordered to Digital Systems

Ph.D. Thesis Project

PhD Candidate: Davide PierangeliSupervisor: Prof. Eugenio Del Re

General overview

Nonlinear optical waves are fascinating physical phenomena emerging when thelight-matter interaction cannot be treated in terms of harmonic oscillationsaround an equilibrium position. The concept of field mode then turns into com-plex objects without a linear equivalent. Spatial solitons [1], as localized andstationary propagating perturbations, or rogue waves [3, 2], as spatio-temporalpulses characterized by extremely elevated amplitudes, can emerge. Understan-ding the physical properties of highly nonlinear light propagation is attractinggreat interest from a fundamental and interdisciplinary perspective, fueled bythe promise of transferring these properties into innovative functionality. Infact, the spatio-temporal control of light represents a key step for the futuredevelopment of photonics. Experiments aimed at exploring nonlinear wave dy-namics are generally performed in systems that start from equilibrium, wherelong-range order and coherence form the basis for strong response. On theother hand non-equilibrium states have unique properties opening new physicalregimes and great possibilities for technological materials. For these reasons,combining the non-ergodic and critical properties of a complex media with non-linear optical propagation is an intriguing task in the contest of nonlinear opticsof disordered systems. A major thrust in this direction has been the introductionof compositional disorder in optical crystals. In particular, disorder at nanoscalein a perovskite ferroelectric can turn it into a relaxor, a medium where long-range order is frustrated but short-range order, in the form of correlated polar-nano-regions (PNRs), is anomalously enhanced [4]. These systems presents theexciting possibility of tapping into the complex-solid behavior to affect lightcontrol, and, in general, uncover new wave phenomena, such as the recently di-scovered scale-free optics and subwavelength anti-diffracting beam propagationin supercooled potassium-lithium-tantalate-niobate (KLTN) [5, 6, 7].

Part of my PhD research consists in the conception, realization and expla-nation of experiments that can allow us to understand how nonlinear opticalwaves can be affected by an underlying non-equilibrium dipolar response. Thisalso implies the study of the supporting dielectric and electro-optic propertiesof relaxor ferroelectrics in the critical region. On the other hand, giant di-sordered responses may be the key to observe, for the first time, spatial roguewaves driven by high nonlinearity in optical crystals. Another interesting aspectconsists of introducing microstructures in nanodisordered crystals, as nonlinearlattices, to explore the interplay in light propagation between periodic compo-nents, photorefractive nonlinearity and phase-transition properties. However,the interesting opportunities provided by nanodisordered ferroelectrics to ex-plore nonlinear waves are partially limited by the complexity of the systems,which affects both the experimental control and the theoretical analysis.

Another part of my PhD work is to overcome this limitation developing anew scheme based on digital media formed by Spatial Light Modulators (SLM),arrays of liquid crystals that can be electronically manipulated at the singleunit level to control the optical field phase and amplitude. Indeed, spatial lightmodulation has been shown to allow focusing through scattering media usingan adaptive-optics scheme in which information from the focal spot beyondthe scatterer is detected and fed back to the emitter [8, 9]. The idea is toimplement a feedback mechanism for using SLM as a nonlinear medium withdesired properties, opening in this way a new scenario with wide possibilities innonlinear optics.

In the next sections I will briefly describe in details some aspects forming myPhD research, presenting some results obtained and issues that I am addressing.In the first section nanodisordered ferroelectric are considered, and, in particu-lar, how PNRs result in the electro-optical response of a potassium-sodium-tantalate-niobate (KNTN) crystal near the dielectric anomaly. In the secondsection we study spatial photorefractive soliton propagation in a microstructu-red KLTN, exploring the effects of a lattice nonlinearity in the properties of thesolitary waves. Than, nonlinear beam propagation at the ferroelectric phase-transition is reported; preliminary results show the appearance of rare events,namely, spatial rogue waves, in which nonlinearity seems to play a key role.Finally, concerning the use of SLM as medium for nonlinear waves propagation,the basic idea and the first implemented steps are discussed.

Nonlinear electro-optical response of relaxor ferroelectrics

Nanodisordered ferroelectrics are perovskites in which the introduction of com-positional disorder, through isovalent elements, induces the so called relaxorbehavior. It is characterized by a high temperature paraelectric phase and alow temperature ferroelectric dipolar glass phase, connected through a diffusephase transition around a peak room-temperature Tm. Low-frequency dielectricspectroscopy measures shows interesting properties in this temperature range,as giant response (εr = 104 − 105), non-ergodicity, breaking of the mean-fieldbehavior and also frequency dispersion of the peak. This means that the sy-stems respond differently on different spatial scales, implying the presence ofcorrelated PNRs that microscopically rule all those properties. In particular,frequency dispersion presents freezing of some relaxation times for a tempe-rature just below the glass transition (Vogel-Fulcher relaxation) signaling ablocked disordered state. Non-ergodic features have been shown to influencenonlinear light dynamics propagating into the medium; effects of local non-equilibrium, as aging or enhanced photorefractive response, result in the alte-ration of the stability and symmetry properties of spatial soliton [10, 11]. Toinvestigate if PRNs and their microscopic freezing may affect also the electro-optical macroscopic response we have performed cross-polarizers experiment ina K1−xNaxTaxNb1−yBbyO3 : Cu crystal, in which the field induced birefrin-gence is measured. Dielectric and optical results are reported in Fig. 1. Theelectro-optical refractive index modulation as a function of temperature and ap-plied electric field shows a giant electro-optic effect [12] undergoing an ultra-fasttemperature dynamics in moving towards the transition. The anomalous scalingmodifies the functional form of the electro-optical response, that is, as expec-ted, quadratic dependent by the bias field deep into the paraelectric phase and

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Dielectric Optical

KNTN

FE/DG phase

PEphase

(a)

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(a)

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Figura 1: Dielectric and optical results in nanodisordered KNTN. (left) Staticdielectric constant showing the features of a relaxor: (a) diffuse phase transitionat Tm with thermal hysteresis and peak frequency dispersion (inset). (b) Tran-smission microscopy images revealing the paraelectric state and the field inducedferroelectric order. (right) Cross-polarizers setup (a) and measured electro-opticeffect at different temperature in the paraelectric phase, showing fast tempera-ture scaling and linear dependence (inset) at low fields near Tm. (c) Polarizationresponse fitted through Eq.(1) and corresponding analysis providing an effectivefreezing temperature T0 for the PNRs reorientational fluctuations.

for high fields, but surprisingly became linear for the lower temperature at lowfields. A linear electro-optic effect is characteristic of non-centrosimmetric me-dia and suggest a broken symmetry. In fact, under the inversion of the externalfield we have a globally symmetric response in which the PNRs non-symmetricresponse on nanometric scales emerge as a linear contribution, i.e., a symmetrypreserving electro-optic effect. We have explained this observation adding inthe macroscopic polarization a dipolar reorientational term (Langevin-like) dueto PNRs,

P = pPNR + pχp= ρp0 tanh

[p0|E|

kB(T − T0)

]u + ε0χpE . (1)

This model fits well the experimental data and provides the low-fields linearcontribution observed in the induced birefringence when p0|E|/kB(T −T0)� 1,having

∆n ' −(1/2)n3gε20(εr − 1)2E2 − n3gρp0 tanh

[p0|E|

k(T − T0)

]ε0(εr − 1)|E|. (2)

Interestingly, it provides also a freezing temperature (Fig. 1(c)), demonstratingthat the anomalous scaling is related to reduced effective thermal fluctuationsof the dipolar component [13]. The coincidence between freezing temperatu-re revealed macroscopically (optics) and microscopically (dielectric) is underinvestigation, but there are strong indications in this direction.

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Microstructuring nanodisordered crystals: lattice nonlinea-rity and continuous solitons

Nanodisorderd ferroelectric crystals can also be microstructured. The intro-duction of a built-in oscillating low-frequency dielectric constant in a KLTNvolume, obtained through the spatial variation of the ferroelectric transitiontemperature, results in a periodically modulated electro-optic nonlinearity, ac-tivatable through an external bias field [14]. This is interesting because inperiodic systems the coupling between spectral components of the optical fieldand nonlinearity has been extensively investigated allowing diffraction control[15] and giving rise to self-localized states, non-spreading during propagation,such as discrete and gap solitons [16]. Experiments on discrete trapping are ge-nerally based on photonic lattices made from etched waveguide arrays or createdthrough optical induction in photorefractive media, so that the nonlinear wavehas always evolved in a fixed linear/nonlinear pattern, in conditions in whichthe lattice is not appreciably affected by the wave (Fig. 2(a)). Differently, inour case the microstructure leads to a lattice nonlinearity, a periodic variationin the nonlinear response that is in turn negligible in the linear response. Thismeans the lattice itself depends on the propagating soliton, and both latticeand soliton are strongly interacting during propagation (Fig. 2(b)). We haveexperimentally investigated spatial solitons propagation in this new systems andthe typical phenomena are shown in Fig. 2(c-d). In the linear case, when thelattice is not activated via the external field, the input (1+1)D waves diffractat the output of the KLTN crystal. The activation of the lattice nonlinearityleads first to a delocalized discrete pattern, while a continuous soliton formsat the steady state. This unexpected continuous behavior, even though strong

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Figura 2: Spatial soliton propagation in microstructured KLTN. Sketch of nonli-near waves propagation in (a) a photonic lattice and (b) in a lattice nonlinearity.(c-d) Experimental results showing the beam dynamics from the starting delo-calized discrete pattern to the continuous soliton. (c1) Cylindrical input beam,(c2) diffracting output, (c3) delocalized discrete propagation in the initial stageof the nonlinear dynamics and (c4) continuous soliton at the steady state. (d1)Input, (d2) Output and (d3) diffractionless wave for a 2D beam.

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discrete features are present in the underlying media, has roots in the couplingbetween periodic and non-periodic terms in the supporting nonlinearity. In fact,the build-up of the light-induced space-charge field leads progressively the un-derlying lattice into a latent state independently both of the lattice amplitudeand of the wave width and size. Discrete effects are shown to appear when theelectro-optical grating is partially decoupled from the photorefractive response;on the other hand, the coupling allow the transition from discrete to continuouslight propagation. The whole picture has been confirmed through numericalsimulation based on the generalized nonlinear Shroedinger equation with thelattice nonlinearity. This study point out how the periodic properties of a me-dia can be made to not emerge in the propagating waveform if they are filteredout by a strong interplay between the nonlinear waves and the nonlinear lattice[17]. This new periodic optical medium opens up perspectives for exploring thephysical correlation between nonlinear waves and nonlinear lattices.

Spatial rogue waves in photorefractive ferroelectrics under-going the phase-transition

We have experimentally investigated optical waves propagation in the highlynonlinear regime provided by a photorefractive ferroelectric crystal undergoingthe phase-transition. In this condition, fluctuations, on spatial scales of the orderof λ, and scattering are expected to strongly affect the nonlinear beam propaga-tion via a non-equilibrium disordered index of refraction pattern. This regimehas been studied, with the experimental setup peculiar of soliton propagationshown in Fig.3, focusing cylindrical Gaussian beams (λ = 532nm, P = 0.1mW ,FWHM = 8µm) on a sample of KLTN, K1−xNaxTaxNb1−yBbyO3 : Cu, withx = 0.04 and y = 0.38, illuminated through uniform background intensity. Thecrystal is kept at T = Tm + 1K and then an external electric field is appliedalong the x-direction, the same along which the propagating beam is polari-zed. In proximity of Tm a bias field larger than the actual coercive field areable to induce the ferroelectric ordering. The thermal gradient select differentlocal phases that strongly affect the beam propagation and the transmitted in-tensity distribution present features strictly dependent on the medium phase(Fig. 3). In particular, we are able to identify a region corresponding exactlyto the critical point; here, a disordered “speckle-like” output distribution withmicrometric spot of various intensity is observed (Fig. 3(b)). In this regionlarge intensity fluctuations appear, with bright spot of extreme intensity. InFig. 4(a) a rare events is reported; the detected intensity is about twenty timeslarger than the averaged intensity and the measured transverse dimension is3µm, implying super-resolution too. Further analysis about the statistical pro-perties of the phenomena indicates this event as a rogue waves. In fact, takingseveral uncorrelated images, a long-tail statistics has been found in this highlynonlinear regime (Fig. 4(b)). The observed behavior strongly deviates fromthe Gaussian statistics (standard speckle distribution) and it is well fitted bya stretched exponential distribution. Several events exceed the hydrodynamicsthreshold defining a rogue event. Nonlinearity seems to play a key role in theemergence of spatially localized extreme amplitudes in our system; in fact, whena weaker nonlinear regime is considered, the long-tail disappear and Gaussiandistribution govern the dynamics (Fig. 4(c)).

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Figura 3: Experimental setup to explore light propagation in a critical media.Cylindrical Gaussian beams co-propagating with background plane-waves intoa KLTN crystal kept in biased condition in proximity of the ferroelectric phase-transition. (left) Sketch showing the effect of the thermal gradient via coercivefields on the nonlinear beam propagation. (a) Modulation instability in a localparaelectric state, (c) critical opalescence associated to ferroelectric order and(b) speckle-like distribution occurring at the transition point.

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Figura 4: Rogue waves observation in KLTN. (a) Transmitted intensity distribu-tion characterized by large fluctuations and with the appearance of a bright spotof extreme intensity. (b) Long-tail statistics associated to the highly nonlinearregime. The distribution function strongly deviates from the Gaussian behavior,with several rare events, appearing preferentially around the beam averagedmidpoint (inset), exceeding the rogue waves threshold. (c) The standard be-havior is experimentally verified weakening the strength of the photorefractivenonlinearity.

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We are explaining the appearance of rogue waves in KLTN [20], or, in gene-ral, in an optical crystals, where rare events have never been observed before,as the result of the interplay between a stochastic component and a nonlinearresponse. Indeed, from previous study, a key rule to the rogue waves emergenceseems to be played by the onset of a chaotic, turbulent or instable dynamics;in the spatial domain this implies the presence of blocked or induced disorderproviding interaction and coupling of different spatial region [18]. Recent fin-dings have further highlighted the role of randomness showing that, even in theabsence of a nonlinearity, random fields synchronization can trigger the onsetof extreme spatio-temporal events, opening even more intriguing questions onthe part played by the nonlinear dynamics [19]. Here, differently from otheroptical crystals, a disordered component is involved in light propagation and itis provided by the ferroelectric transition; the nonlinear mechanism, the pho-torefractive response, seems to be crucial in the amplification of instabilities.The fundamental role played by the nonlinear response has been further under-lined through numerical simulation of the generalized nonlinear Schroedingerequation in highly nonlinear conditions; results show that a microscopic drivermechanism to the emergence of extreme events may be related to the collisionsand mergers of generated solitons.

Nonlinear SLM: towards digital optical materials

The observation of nonlinear optical waves in crystals implies complex response,often hard to model. If on one hand this is interesting, on the other generatein a controlled fashion these waves would open up great possibilities. In thisregard, we are developing a new scheme to generate spatial nonlinear self-effectson a propagating beam. It is based on a SLM operating on the wave as a phasemodualation, as the nonlinear index modulation operates in the numerical beampropagation method (BPM), usually used to simulate nonlinear spatial light dy-namics. In details, we are implementing an adaptive optics scheme [21] in whichthe SLM is put in loop with a CCD cameras and is fed with nonlinear phaseholograms as a function of the revealed field (Fig. 5). In this way at every loopthere is an half step of physical free propagation and an half step of nonlinearresponse via the SLM, so that the feedback configuration allows us to affect thebeam spatial properties inside the loop itself. In fact, the intensity distributionis elaborated according to an algorithm that accounts for the renormalization ofoverall transmitted power, rescaling and repositioning of the beam, and, mostimportantly, carries out the desired nonlinear operation. As a whole, this sy-stems can be thought as an artificial nonlinear materials which properties canbe varied to support systematically nonlinear optical waves. Especially, thesewaves can be made to interact with any other system put in the loop.

The first considerations that have arisen concern the automation of the loop,the fine-tuning with the CCD cameras doing the feedback and in particular,the use of the SLM in a real space configuration. In fact, even though someencouraging signs has been obtained using input plane-waves (as usually donefor other purposes, beam shaping for example), only in this case an hologramcan act correctly as a nonlinear response. However, being the single unit ofthe liquid crystals matrix 8µm sized, is not yet entirely clear if micrometricbeams can be manipulated in this manner, eventually using a parallel telescopiccavity. To date, we are working with millimetric laser beam and we are trying

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liquid crystal matrix

Pixel Pitch: 8.0 μm1920 X 1080 resolution

CCD Cam

SLM

Figura 5: Adaptive optics experimental scheme for digital nonlinearity. A phase-only SLM is put in loop with a CCD cameras revealing the emitted optical fieldand fed it back as an hologram in which proper nonlinear operation can beincluded. The beam inside the loop can thus experiences nonlinear effect infree-space and with suitable feedback algorithm nonlinear optical waves can bein principle generated.

to achieve in free-space the fundamental effect of nonlinear light interaction ina medium: self-focusing.

The research is in collaboration with A.J. Agranat and his group at the AppliedPhysics Dep. of the Hebrew University of Jerusalem, who provides optical quali-ty nanodisordered ferroelectric crystals, and with G.B. Parravicini at the PhysicsDep. of University of Pavia, who performs dielectric spectroscopy analysis onsuch systems.

References

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[21] M.Leonetti, S.Karbasi, A.Mafi and C.Conti, Nat. Commun. 5, 1 (2014).

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