Elsevier

Chaos, Solitons & Fractals

Volume 84, March 2016, Pages 73-80
Chaos, Solitons & Fractals

Extreme events in complex linear and nonlinear photonic media

https://doi.org/10.1016/j.chaos.2016.01.008Get rights and content

Ocean rogue waves (RW) are huge solitary waves that have for long triggered the interest of scientists. The RWs emerge in a complex environment and it is still under investigation if they are due to linear or nonlinear processes. Recent works have demonstrated that RWs appear in various other physical systems such as microwaves, nonlinear crystals, cold atoms, etc. In this work we investigate optical wave propagation in strongly scattering random lattices embedded in the bulk of transparent glasses. In the linear regime we observe the appearance of extreme waves, RW-type, that depend solely on the scattering properties of the medium. Interestingly, the addition of nonlinearity does not modify the RW statistics, while as the nonlinearities are increased multiple-filamentation and intensity clamping destroy the RW statistics. Numerical simulations agree nicely with the experimental findings and altogether prove that optical rogue waves are generated through the linear strong scattering in such complex environments.

Introduction

Ocean rogue or freak waves are huge waves that appear in relatively calm seas in a very unpredictable way. Numerous naval disasters leading to ship disappearance under uncertain conditions have been attributed to these waves. Since sailors are well known story makers these monster, destructive waves that were in naval folklore perhaps for thousands of years penetrated the realm of science only recently and after quantitative observations [1], [2]. Since then, they seem to spring up in many other fields including optics [3], [4], [5], [6], [7], BEC and matter waves, finance, etc [8], [9], [10], [11], [12]. Unique features of rogue waves, contrary to other solitary waves, are both their extreme magnitude and also their sudden appearance and disappearance. In this regard they are more similar to transient breather events than solitons. Since the onset of both necessitates the presence of some form of nonlinearity in the equation of motion describing wave propagation, it has been tacitly assumed that extreme waves are due to nonlinearity. Intuitively, one may link the onset of a rogue wave to a resonant interaction of two or three solitary waves that may appear in the medium. However, large amplitude events may also appear in a purely linear regime [1], [2], [4], [6]; a typical example is the generation of caustic surfaces in wave propagation [13], [14].

Propagation of electrons or light in a weakly scattering medium is a well-studied classical problem related to Anderson localization and caustic formation. Recent experiments in the optical regime [15] have shown clearly both the theoretically predicted light localization features as well as the localizing role of (focusing) nonlinearity in the propagation [15], [16], [17], [18], [19], [20]. In these experiments a small (of the order of 103) random variation of the index of refraction in the propagation leads to eventual localization while at higher powers, where nonlinearity is significant, localization is even stronger. Thus, destructive wave interference due to disorder leads to Anderson localization that may be enhanced by self-focusing nonlinearity. In the purely linear regime propagation in two dimensions in a weakly random medium has shown that branching effects appear through the generation of caustic surfaces [13], [14], while linear rogue waves have been observed with microwaves [4].

In this work we focus on an entirely different regime of wave propagation, in strongly scattering optical media that consist of Luneburg-type lenses randomly embedded in the bulk of glasses. Spherical or cylindrical Luneburg lenses (LLs) have very strong focusing properties directing all parallel rays impinging on them to a single spot on the opposite side surface. The index variation is very large, viz. of the order of 40% and thus a medium with a random distribution of Luneburg-type lenses departs strongly from the Anderson regime investigated in [15], [16], [17], [18], [19], [20]. In the experimental configuration used in this work we used “Luneburg Holes (LH)” or anti-Luneburg lenses instead of LLs; the LHs have a purely defocusing property. In the methods section we demonstrate that our observations discussed in the following are generic and independent of the type of scatterers.

Experimental and numerical observation of rogue waves. Focusing tightly a femtosecond IR beam into the bulk of fused silica substrates induces nonlinear absorption allowing the selective modification of the material [21]. Under appropriate irradiation conditions one may create LH-type structures and by placing those in a controlled way in space to create three dimensional LH lattices like the ones shown in Fig. 1(a).

The investigation for the presence of a rogue wave is performed by probing a laser beam through the volume of the lattice and imaging the output. This approach is advantageous because it allows the study of both linear and nonlinear phenomena, depending only on the probe beam intensity.

For the linear observations a low power continuous wave 633 nm laser beam was used as probe. A large number of different lattices were studied until “rogue” events were observed as seen in Fig. 1(b). The corresponding “rogue” event intensity profile is shown in Fig. 1(c) and the distribution of the intensities, in semilog scaling, in Fig. 1(d) and permit to conclude that this signal cannot be anything else than an optical rogue wave, contiguous to the definition of the phenomenon, that is, RWs deviate from the Rayleigh distributions showing long tail distributions [1,2,4].

In Fig. 2 we present the light propagation in a random LH lattice (Fig. 2a) under steady state conditions. We observe that the presence of scatterers with strong defocusing properties forces light to form propagation channels (Fig. 2b) that can lead in the generation of very large amplitude rogue wave events (Fig. 2c). Such events have amplitudes larger than twice the significant wave height (SWH) in the medium and are directly attributed to wave coalescence induced by the strong scattering of light by the LHs. Although the medium is purely linear, the induction of caustic surfaces leads to resonant events that have clear rogue wave signatures. In Fig. 2(d) is shown the intensity profile where a rogue wave occurs. Obviously, the highest peak is larger than twice the SWH resulting in a rogue wave event.

In Fig. 2(e) we plot the distribution of intensities (in semilog axis). By the central limit theorem and the simple random wave prediction for the probability distribution of wave intensities I, the intensities have to follow the Rayleigh law, meaning a distribution type P(I)=eI where I=|E|2 (E is the electric field) is normalized to one [1], [2], [4], [14], [18]. However, when extreme events appear, the intensities distribution deviates from simple exponential and long tails appear [1], [2], [4], [8], clearly seen in both our experimental Fig. 1(d) and numerical results Fig. 2(e).

Experimental and numerical parametric studies: In order to study the dependence of the phenomenon on the scattering strength of the lattice we vary the LH lattice randomness as well as the refractive index profile amplitude. This is done by fabricating various system configurations with different distribution of LHs as well as different maximal Δn differences in the index of refraction between the host medium (glass) and the center of the LH. Interestingly changing the disorder level did not alter the general RW statistics picture. On the other hand, the refractive index variation of the sample influenced the phenomenon strongly. Specifically we found that there is a threshold in Δn below which no rogue waves were observed. The results for small values (Δn < 1%) Fig. 3(a), to intermediate values (Δn ≈ few%) Fig. 3(b) and high values (Δn ≈ 30%) Fig. 3(c), clearly shows the dependence of the rogue wave generation on the scattering strength of the medium.

In our numerical analysis we investigate random lattices of the type shown in Fig. 2(a) while changing the maximal index variation. In Fig. 4 we present the distribution of intensities for three different index variations, viz. Δn=10% Fig. 4(a), Δn=20% Fig. 4(b) and Δn=30% Fig. 4(c). We found that the long tails at high intensities disappear as the index variation decreases, with rogue events appearing for index variations roughly above 20%. The qualitative as well as quantitative agreement of experimental and theoretical results in the linear regime demonstrates that in the present context the onset of RW extreme events is due to strong scattering in the complex LH lattice.

The role of nonlinearity: An obvious question arises as of the role of nonlinearity in the same processes. For answering this question experimentally we increased the intensity of the probing radiation (using high power femtosecond pulses) exciting thus nonlinear modes through Kerr nonlinearity. In Fig. 5 one can observe the total beam, Kerr-induced, self-focusing in the bulk of a glass without any lattice inscribed in it as the input beam power is increased from (a) to (d).

On the contrary when the same intense beam goes through a glass with a lattice inscribed in it things are considerably different. At the limit of small nonlinearity, around the critical power Pcr, although an amplification of the waves already existing in the linear regime is observed, the linear RW statistics are not modified, namely the intensities still deviate from Rayleigh distribution showing long tails. This is shown in Fig. 6(a) where a linear RW is further amplified maintaining though its intensity aspect ratio compared to the neighboring lower level waves. As the input power is increased gradually, the lower height waves are amplified as well resulting to a small amplitude multi-filamentation image, Fig 6(b). Further increase in the input beam power, and thus higher nonlinearity, results to the saturation of the intensity of all modes, starting from the higher to the lower ones, since higher order –defocusing– nonlinearities lead to intensity clamping [22]. This is shown in Fig. 6(c) where a higher input laser power pushes many small waves up to the clamping intensity. From the above it is clear that the generation of RWs in the strongly scattering system is a result of linear interference mechanisms while nonlinearity will either accentuate the phenomenon, when it is relatively small, or completely destroy the RW statistics when it is high. It is interesting to refer here at a recent report on laser filamentation merging and RW events [23]. Actually, these observations can be nicely explained in the frame of our present findings, since the merging of the filaments (although a nonlinear effect) leads to RWs not because of the nonlinearity but because of linear thermal effects and turbulence induced in the medium by the accumulated heat from the high repetition rate and power of the employed laser system.

Further, our experimental findings on the nonlinearity role are nicely reproduced by numerical simulations (Fig. 7). We introduce a focusing nonlinearity (Kerr effect) in the dielectric constant reading ɛ=n2=ɛL+χ|E|2, where E is the electric field, ɛL the linear part of the dielectric constant and χ the nonlinear parameter varying from 0 up to 105 (depending on the strength of the nonlinearity; in normalized values). As in the experiments we can see that the linear observed RW statistics (Fig. 7a,e) are not affected in the presence of a relatively small nonlinearity (Fig. 7b,f). In this case most waves are simply amplified without destroying the RW statistics but slightly increasing the queue of the intensity distribution (Fig. 7f) as expected from the higher amplitudes. This situation dramatically changes at higher nonlinearities (Fig. 7c,g and Fig. 7d,h) where more and more waves are amplified, completely destroying the rogue wave statistics, that is, the long tails have been disappeared and the intensities follow again the Rayleigh distribution, in full agreement with our experimental observations.

Section snippets

Conclusion

Rogue waves are extreme waves that appear in diverse systems; we focused on complex media where randomly placed elements introduce strong light scattering and interference patterns. In the purely linear regime the coalescence of these light channels and the resulting complexity leads to the appearance of extreme, transient waves. There is a clear departure from the Rayleigh law at large intensities where RWs are produced. Most importantly we have shown both experimentally and numerically that

Experimental

The inscription of the scatterers is performed via laser induced refractive index modification. A pulsed IR laser beam (pulse duration 30 fs, central wavelength 800 nm) is focused tightly with an objective lens (x20, NA 0.45). The intensity at the focal volume of the objective lens is high enough to excite nonlinear phenomena such as, nonlinear absorption and avalanche ionization, which in turn can alter permanently the refractive index of optically transparent solid materials, like silica

Author contributions

All authors have contributed to the development and/or implementation of the concept. G.P.T. proposed the original theoretical system with the LLs, while S.T. suggested the experimental realisation with the LHs. S.T. designed and supervised the experimental work and I.J.P. performed the experiments and analysis. M.M. preformed the simulations and analysis, while G.P.T. supervised the theoretical work. All authors contributed to the discussion of the results and to the writing of the manuscript.

Additional information

Competing financial interests: The authors declare no competing financial interests.

Acknowledgments

We gratefully acknowledge the assistance of M. Thévenet and D. Gray at the early experimental stages of this study. This work was supported by the THALES projects “ANEMOS” and “MACOMSYS”, and the Aristeia project “FTERA” (grant no 2570), co-financed by the European Union and Greek National Funds. We also acknowledge partial support through the European Union program FP7-REGPOT-2012-2013-1 under grant agreement 316165 and partial support of the Ministry of Education and Science of the Russian

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