CIVIL ENGINEERING 365 ALL ABOUT CIVIL ENGINEERING


Formally speaking, a Rogue Wave has been defined, in the oceanographical context, as an extreme wave appearing suddenly and having an amplitude larger than the rest of a given amplitude ensemble. This gives a very low probability for the excitation of such an extremely large wave, been therefore a rather rare event. Accordingly, it becomes necessary to give a technical definition in order to study this phenomena on different contexts. In this way, we require to compute a given quantity that could be compared for different system parameters in order to clearly define if the large amplitude events found in the dynamics are extreme or not. Along this work, we will define that an intensity (I_nequiv |u_n|^2) corresponds to an extreme event if (I_n>2I_s), where (I_s) corresponds to a threshold intensity. (I_s) is defined as the average value of the highest intensity tertile of the corresponding probability density function (PDF) distribution12. Therefore, for a given realization, we analyze all sites intensities (I_n) and obtain a PDF distribution where we define the intensity (I_s) and count the number of EEs over the entire intensity ensemble.

Figure 2

(a) Semi-log PDF distribution for a single realization of diagonal disorder ((W_{beta }=0.1)), and for a given propagation distance (z=95). Red and blue vertical lines indicate (I_s) and (2I_s), respectively. (b) Percentage of EEs versus propagation distance z for same realization described in (a).

As the intensity distribution and, therefore, the number of EEs change at different propagation distances, we develop a full dynamical characterization: For a given realization of disorder, we numerically integrate model (1) in the interval (zin {0,z_{max}}). Then, for every step in z, we analyze the corresponding spatial profile ( |u_n(z)|^2) and count the lattice intensities to create a PDF distribution, where we identify the percentage of EEs. Figure 2a shows an example of a PDF distribution for diagonal disorder and for a given distance z. We observe a heavy-tailed intensity distribution, what is an indicator of the existence of EE29. For every step in z, we obtain the percentage of EEs versus propagation distance, as shown in Fig. 2b. Here, we observe an increasing number of EEs versus distance, which is associated with light spreading across the lattice and reflections occurring for (zgtrsim 10). As a reflected wave superposes to slower propagating fronts, this causes a spurious increment of local intensities (I_n), what adds extra counts to the overall statistic. Afterwards, light spreading is more homogeneous and the number of EEs reduces up to a rather constant value ((sim 0.38) in the example), observing some kind of thermalization or saturation phenomena. This means that the spatial pattern, although still fluctuating, behaves similarly for an increasing value of z. We, therefore, observe an EE saturation after some propagation distance, what strongly depends on the degree of disorder.

Taking this information into account, we define an EE dynamically averaged value (({overline{EE}})), obtained by averaging the number of EEs in the interval (zin {85,95}) for a given realization and given disorder, and then averaging again over 100 disorder realizations. In the chosen z-interval, dynamics has already relaxed, without the strong effect of first reflections at surfaces. So, we can correctly characterize the appearance of large amplitude events for a given degree of disorder. We collect the information of ({overline{EE}}) versus disorder strength in Fig. 3a, b, for diagonal and non-diagonal disorder, respectively. Blue curves show the averaged data as described above, which consider all lattice amplitudes, without any filter, when counting intensity peaks to form the corresponding PDF distribution. We observe a rather strange behavior in both blue curves. First, ({overline{EE}}) decreases to a minimum around ({W_{beta },W_{V}}sim {0.15,0.05}), from zero to weak disorder, and then ({overline{EE}}) increases to a maximum located close to ({W_{beta },W_{V}}sim {0.6,0.3}). Afterwards, in both cases, the number of ({overline{EE}}) decays slowly. We observe that the typical discrete diffraction pattern, having two characteristic main lobes [as shown in Fig. 1c], increases the statistic of large amplitude events. Without filtering, all intensity are counted and the intensities at lobes (that include several sites) are been considered as EEs. Of course, this is out of the definition of a low statistic large amplitude and rare event, which is necessary to define a RW. In other words, every time we would propagate light on a homogenous lattice we would observe a RW. Clearly, a RW is a rare event, therefore the described observation (discrete diffraction) can not be categorized as such. This also affects the dynamics at weak disorder, because in that regime propagation is a mixture between discrete diffraction and trapping at disordered regions, as shown in Fig. 1d. On the other side, we know that for larger disorder we will end up with Anderson localization20 [see Fig. 1e]. Obviously, this regime can not be counted as a RW, because, as Anderson taught us, this observation will always occur for a disordered system; therefore, it would not be a rare event at all. For a large disorder strength, profiles are completely localized with (Rrightarrow 1/N), having a single excited site. Therefore, if not filter is included in data, the number of EE must go to the limit of 1/N, with only one site having a large amplitude. By filtering the data, this peak will be exactly (I_s) and, therefore, no RWs will be defined, as it should be. Clearly at this regime, the overall statistic increases when counting all amplitude sites and the localized profile generates spurious large amplitude events.

Figure 3
figure3

Averaged extreme events ({overline{EE}}) versus disorder strength for (a) diagonal and (b) non-diagonal disorder. Different colors corresponds to different applied filters, as indicated in figure.

As our work is focused on identifying the effect of disorder on the appearance of RWs on 1D lattices, we implement a filtering process. We first notice that low amplitude peaks increase the amount of data and, therefore, decrease the threshold value (2I_s). So, we decide to simply avoid counting low amplitude data by applying an intensity filter, which is defined as a given percentage of the largest peak ((I_{max})) at a given realization. To wit, (X%) means that we will only count sites with intensities larger than (Xcdot I_{max}/100). The result of applying this filtering process is presented in Fig. 3, using different colors depending on the indicated filter value. First of all, for both cases, we notice that the inclusion of a data filter produces a similar tendency. We observe how the number of averaged ({overline{EE}}) decreases for an ordered system, what indicates a correct effect of our suggested filtering process. The previously described minimum decreases and is shifted to a smaller value of disorder for an increasing applied filter. In both cases, we observe that a filter of around (20%) completely eliminates the ({overline{EE}}) count at zero disorder. So, with this level of filtering we are indeed avoiding to count a non rare effect as discrete diffraction. In addition, our filter is also eliminating Anderson localization as a RW, reducing to zero the averaged ({overline{EE}}) for larger disorder (in this regime, a lower filter of (sim 15%) is, in fact, enough). Our filter is indeed shifting the threshold intensity (I_s) allowing to classify only very rare large amplitude events as RWs. Nicely, considering the dynamics at weak disorder, we have been able to limit the ({overline{EE}}) region as a result of the propagation of linear extended waves plus weak random localization on distorted regions across the lattice. That means that, on a 1D disordered photonic lattice, we would observe an extreme event only when a good balance between transport and weak localization effects is achieved and large events become statistically possible. As we observe in Fig. 3, the maximum number of ({overline{EE}}), after filtering (20%) of peaks, is quite low, with a value lower than (1%) of total counted peaks.



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