Optical polarization rogue waves and their identifications

Optical rogue waves are a class of pulses with extremely large amplitudes, whose probability of occurrence unexpectedly deviates from Gaussian-law statistics. To date, the mechanisms of rogue wave generation are still debated: investigations are under way, exploring the statistics of various pulse dimensions across different physical domains. Although polarization is one of the fundamental parameters of optical rogue waves, its statistics have received little attention until recently. Here, we review recent process of the polarization-dependent properties of optical rogue waves in ultrafast optics. Based on a two-dimensional statistical model, we introduce the concept of optical polarization rogue waves. Specifically, we consider the frequency of generation of waves with freak or rogue state of polarization, with a probability of occurrence deviating from a normal distribution. We demonstrate three nonlinear optical laser systems: a partially mode-locked laser, a dissipative soliton laser, and supercontinuum generation within a highly nonlinear fiber. Further, we identify optical polarization rogue waves in nonlinear laser systems, and discuss their generation mechanisms. Related results reveal that optical polarization rogue waves are embedded in optical systems with a deteriorated degree of coherence, which originates from vector four-wave-mixing processes. Polarization-dependent investigations will provide additional insight for our understanding of optical rogue waves.


Rogue waves
The concept of rogue waves (RWs) originates from oceanography. They are known as water waves with extremely large amplitude propagating in open oceans. Besides, these rare events emerge with unexpectedly large probabilities, which deviate from the power law wave amplitude statistics, and they leave without a trace [1][2][3][4][5][6][7][8][9]. These events lead to water waves as high as 30 m, which impose a great threat to ships and ocean liners, and even cause devastating damages. Motivated by the interest in their physics, and industrial applications, a huge amount of work has been carried out, to explore the origins and predictability of RWs. And some secrets behind RWs have been unveiled in recent groundbreaking research.
Experimentally, RWs can be identified by calculating probability density functions (PDFs): the presence of heavy-tailed statistics, typically L-shaped, which denotes the appearance of rogue events [10]. Several models have been developed, based on the weak nonlinear interactions between thousands of waves. Many of these models involve the nonlinear Schrödinger equation (NLSE): numerical simulations and analytical solutions such as coherent breathers or incoherent turbulence have been extensively compared. Various mechanisms generating RWs have been proposed so far: from the simple linear superpositions of random weak waves, to nonlinear effects such as modulation instability (MI) and breathers [11,12]. For a physical system with low coherence, RWs within diverse parameters may occur across different dimensions [13][14][15]. Advanced investigations have been carried out, involving, e.g. three-dimensional RWs [16][17][18], higher-order or super RWs [19][20][21], and the resonant interaction of three waves [22]. However, the steep wave profile of extreme events often appears too freak to be fully captured. Many efforts have been devoted to the experimental observation of RWs in oceans and in the laboratory. The debate about the origin of RWs continues, and, in our opinion, it will remain standing as long as there will be discrepancies between theory and experiments.

Optical rogue waves
Given the harsh conditions, studying RWs in their natural oceanic environment is problematic. The observation of RWs has been extended to other domains. More examples are found in the fluence profiles of multi-filaments, supercontinuum generation (SCG) in optical fibers, Bose-Einstein condensates, electrostatic surface plasma, atmospheric waves, liquid helium, capillary waves, gravity wave trains and even in financial markets [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. In 2007, due to phenomenological and physical analogies between extreme events in optics and hydrodynamics, the concept of optical RWs was introduced by D. R. Solli et al, based on long-tailed intensity histograms in the specifically filtered wavelength range of supercontinuum spectra [24]. As shown in figure 1.1, by using a real-time sepctrum detection based on dispersive Fourier transformation (DFT), they observed RWs in soliton-fission supercontinuum based on a micro-structured optical fiber. These results highlight the energy coupling between solitons and other wave packets, and the characteristic lifetimes during the formation of optical RWs.
Since then, great efforts have been devoted to finding examples of RWs in diverse optical systems, and different kinds of breathers or solitons have been proposed as the origin of optical RWs [25,[39][40][41]. Recently, highly developed real-time detection techniques, mainly based on the DFT and the time-lens, have been utilized to identify experimentally the presence of optical pulses with extremely high intensities. When a trough-to-crest height larger than 2 times the significant wave height (SWH) is found, the presence of RWs can be identified [42,43]. Moreover, the occurrence of optical RWs can be identified, on the basis of diverse parameters across multiple dimensions. Examples include: pulse intensity [24], spectral width [44], spatial intensity in two-dimensions [23,45], depth of dark pulses [46], and even rare pulses with right-skewed distributions [47]. The rogueness of a certain degree of freedom, rather than a ultra-high value, has been considered as a characteristic property of RWs. For a complete study of optical RWs, more parameters within other dimensions should be investigated further.

Interpretations for optical rogue waves
Recent years have witnessed a growing interest for optical RWs, while great debates on them are still ongoing in this community. In the pioneering works, the optical filtering technique has been taken seriously by several researchers, as a means to identify RWs [24,27,44]. Two different interpretations of optical RWs have been generally, although not completely, accepted. Both approaches involve two main phenomenological features of RWs. The first feature is the deviation of the statistics for the measured wave amplitude from the Gaussian behavior. The other one is the coherent build up of a RW in extended spatio-temporal systems. For optical fibers, the formation of RWs are determined mainly by the interplay between dispersion and nonlinearity, leading to the formation of time-frequency structures. While for spatially extended optical systems, two-dimensional structures are formed on the transverse wave front. Therefore, a reliable model for RWs should be described by partially differential equations, describing the action of dispersion and nonlinearity for the coherent build-up of giant waves [10]. On the other hand, we have to admit that the chaotic behavior or noise-induced intermittency, which can be described by ordinary differential equations, should to be considered for the classification of RWs [48]. The latter permit to describe the probability of RWs, which arises a fundamental question: the predictability of RWs. It was shown that RWs do not appear without a warning: they can preceded by a short phase of order [49][50][51]. Such a predictability can only be understood in terms of the language of turbulence [52][53][54][55][56].
A well-known theoretical model of RWs is provided by breather solutions of nonlinear optical systems. Such a model relies on the ocean-optics analogy. The dynamics of both ocean waves and pulses in optical fibers can be modeled by the same NLSE. For more complex systems, nonlinear wave dynamics involves an extended version of the Ginzburg-Landau equation. Figure 1.2 depicts wave evolution in the two physical systems, oceans and optical fibers. For optics, the NLSE describes the pulse envelope that modulates an electric field. While for hydrodynamics, the NLSE represents the envelope that modulates surface waves. Certain analogies do exist. Yet, such analogies should not be extended for different higher-order perturbations. The effect of perturbations on the solutions of the NLSE have been investigated [57]. It was found that RWs associated with first-order rational solution of the NLSE are robust against certain classes of perturbations.
An interesting phenomenon associated with RWs in optical fibers is the formation of breathers [58][59][60]. These highly coherent structures can be regarded as a kind of solitons on a finite background. These waves have certain analytical solutions. However it should be remembered that any real optical system is eventually dominated by linear loss. As a result of loss, a pure soliton is led out-of-balance by a mismatch between nonlinearity and dispersion. There are several kinds of breathers: the Akhmediev breathers, the Peregrine soliton, the Kuznetsov-Ma solitons. Higher-order strcutures are possible with stronger pulse localization and higher intensities. These breathers are frequently used for the interpretation of RWs, such as in MIs, in fiber resonators, or in SCG.
A great example is of RW generation mechanism is provided by MI. The nonlinear MI in optical fibers with anomalous dispersion is a simple and yet also complex nonlinear process [61,62]. It is simple because the solutions representing the balance between nonlinearity and dispersion are simple and beautiful. However, noise-seeded nonlinear processes lead to fluctuations of the MI gain spectra. Besides, perturbations arising from higher-order dispersion and Raman scattering make breathers formed during the nonlinear MI process relatively hard to capture. Thanks to single-shot detection based on DFT, real time optical spectra for any number of round trips can be detected, and a certain evidence has been found of the links between breathers and optical RWs [63].
In spite of the success in interpreting RWs as coherent wave structures, based on either solitons or breathers, a debate still exists. This originates from the experimental observation of incoherent solitons, which were firstly found in photorefractive crystals [64]. Such solitons result from the spatial self-trapping of incoherent light in a nonlinear medium with high noninstantaneous response. When the response time is much larger than the correlation time of a beam, the field fluctuations are averaged by the photorefractive nonlinearity. Additional achievements have been made, e.g. MI of incoherent waves [65,66], incoherent dark solitons [67], incoherent solitons in resonant interactions [68], and spectral incoherent solitons in optical fibers [69]. Statistically, the kinetic wave theory can provide a nonequilibrium description of soliton dynamics.
Recent experiments provide other insights on the emergence of RWs in optical turbulence [70]. Figure  1.3 depicts the average of the maximum intensity peak as a function of wave energy H. Depending on the degree of incoherence, three regimes have been identified: coherent rogue quasi-solitons, intermittent-like rogue quasi-solitons, and sporadic RWs. The PDFs of waves in the three regimes are totally different. In the first regime (i), due to the large localization of power, the statistics deviate from the normal distribution, and high power quasi-soliton structures emerge. After that, the system operates in the second regime (ii), and an intermittent dynamics could be found, where quasi-solitons erratically appear and disappear. Indeed, the trajectory of quasi-solitons is not invisible. Besides, the long tail of the corresponding PDF can be attributed to the emergence of significant fluctuations. Next comes the third regime (iii), which is a fully incoherent regime. During this regime, there are no quasi-soliton structures, but sporadic extreme events can be found, although with small probabilities.

Polarization-dependent properties of RWs
The extensive experimental and theoretical investigations of optical RWs have been mostly focused on extreme events appearing in either the temporal or the spectral domains. Few studies have been carried out about extreme events in the polarization domain. Yet, as a fundamental parameter of light, the state of polarization (SOP) is of vital importance for any nonlinear optical process. In spite of that, most theoretical models of nonlinear optical process rely on a simple assumption of a fixed SOP. On the other hand, experimental results prove that oftentimes nonlinear optical processes are accompanied with an evolution of the SOP.
An experiment shown in figure 1.4 demonstrated polarization-dependent RWs dynamics in a fiber laser system without a saturable absorber (SA), based on an interaction mechanism involving a coupling between two polarization modes [71]. The cavity contains a 1480 nm laser diode, a pump polarization controller (PPoC), a 1480/1550 wavelength division multiplexer (WDM) before 1 m active fiber doped with Er 3+ , a polarization controller and 614 m of single-mode fiber (SMF). Figure 1.4 (b) depicts the PDFs of the amplitudes of pulses on a logarithmic scale. A characteristic L-shaped distribution can be observed for the PDFs associated with different regimes: lonely patterns, twin patterns, three sisters patterns, cross and accelerated patterns.
In this experiment, the mechanism of interactions among the RWs, and different patterns does not primarily result from the nonlinearity in a passive fiber, as the pulse amplitudes are relatively low, but more likely from interactions in the active part, likely from polarization hole burning due to polarization attraction [72]. The occurrence of vector behavior was attributed to population inversion in the active fiber. As the pulse interactions are mutually exclusive, all the RW patterns are different manifestations of the same underlying process.
As discussed before, incoherent solitons also provide a platform to observe optical RWs [73]. Figure 1.5 depicts schematically a fiber ring laser cavity mode-locked by a semiconductor saturable absorber mirror (SESAM). The laser cavity includes an erbium-doped fiber (EDF) pumped by a 980 nm laser diode (LD), a polarization-insensitive circulator, a 980/1550 WDM, a polarization controller (PC). A polarization beam splitter (PBS) is used to investigate vector dynamics. In this laser system, both polarization locking and polarization switching of incoherent solitons were observed. Figure 1.6 illustrates histograms for the x-polarization and the y-polarization of the detected single-shot spectra. As can be seen, both histograms exhibit long-tailed distributions. Waves with even four  times the SWH were found, which denotes the existence of extreme events. Besides, a great difference is shown in the two orthogonal polarization directions. The x-polarization is more likely to display explosive instabilities. Those results imply that optical RWs are polarization-dependent.
Despite the great achievements that have been made on polarization-resolved optical RWs, these do not represent RWs in the polarization domain, but rather temporal properties emerging in a specific SOP projection. At variance with the temporal or spatial RWs, we may introduce polarization rogue waves (PRWs) in the polarization domain, and be expressed give a new method for their idenfication. The SOP at a specific frequency (or wavelength) on the Poincaré sphere can asŜ = (s 1 , s 2 , s 3 ). Therefore, the relative distance, r, between two SOPs can be defined by Based on this new parameter, r, we may investigate the polarization dynamics by calculating its probability distribution, and identify PRWs when the PDF of r is deviating from a normal distribution [74].
In the following sections, we will examine the statistical polarization distributions in three nonlinear optical systems, specifically: a partially mode-locked laser (PML), a dissipative soliton (DS) laser, and SCG. We identify the emergence of PRWs in all of these three laser systems, and discuss their generating mechanisms. In section 2 we shall discuss PRWs in PML, which undergo a coherence transition. Next, section 3 is devoted to exploring polarization evolutions during the build-up of DSs. PRWs are identified when the coherence is deteriorated by higher gain. Section 4 contains the experimental results on the PRWs in supercontinuum, based on a DS propagating in a zero-dispersion fiber. Finally, a short summary is given in section 5.

Optical polarization rogue waves in a PML
The coherence transition from high to low is common in many physical systems, where both stable coherent and low coherence structures can be observed [25, 70, 74 80]. Whereas, for a PML, spectral and temporal outputs are unstable. These outputs have also been called 'noise-like pulses' , due to their appearance as noise-like bursts of radiation. Abundant works have been performed, revealing the existence of RWs in various laser systems [81,82]. Here, we use the term of partially mode-locked laser with the emphasis of marking the difference with respect to the highly mode-locked lasers in section 3. Besides, the PML which is discussed here originates from a specific coherence-evolving mechanism: the generation of high coherent sidebands, and also to the presence of low-coherence nonlinear coupling processes. Primarily, energy transfers from a continuous wave (CW) central mode into parametric instability (PI) gain sidebands. PI is a special case of MI, where two pump photons at frequency of ω 0 are annihilated, and two new photons with Stokes or anti-Stokes frequencies are generated. As this degenerate four-wave-mixing (DFWM) process occurs within a cavity configuration, PI sidebands are highly coherent under the combined action of nonlinearity and periodic dispersion [64,[83][84][85][86][87][88][89][90][91][92][93][94][95][96]. The loss of coherence of PML is associated with cascaded vector mutual wave mixing of the PI sidebands within non-polarization-maintaining fibers, where bunches of pulses with irregular varying time duration and intensity are observed [97][98][99][100].

Experimental setup
The fiber cavity shown in figure 2.1(a) contains an EDF pumped by two CW lasers through two WDMs, SMF, dispersion compensating fiber (DCF), a polarization independent optical isolator (ISO), a PC, an optical coupler (OC), and a SA fabricated by filling reduced graphene oxide (rGO) flakes into cladding holes of a photonic crystal fiber (PCF), to provide enough nonlinear phase for mode-locking [101,102]. Specifically, a high-speed polarization state analyzer (PSA) is utilized to detect the laser after filtering by a tunable filter (TF). In figure 2.1(b), the 100 single-shot spectra detected by DFT are consistent with the time-averaged optical spectrum. The full width at half maximum (FWHM) of the optical spectrum is 16 nm.

Data and discussion
We measured the SOP of each filtered wavelength. Figures 2.2(a)-(c) show that SOPs for wavelengths far away from the center line bifurcate into a cross-like shape on the Poincaré sphere. Multiple wave mixing processes generate new frequencies with separately evolving azimuth and ellipticity angles, resulting in perpendicular lines aligning with either a meridian or a parallel curve. SOP turbulence is accompanied by the loss of temporal coherence. Irregular SOPs located outside of the main polarization directions emerge when the pump power P exceeds 250 mW. Four-wave-mixing (FWM) cascades with nonlinear chaotic phase-matching conditions originate from nonlinear net phase shifts in the resonator. Whenever the pump power is larger than 600 mW, cascade FWM leads to a fully developed turbulent evolution with random SOPs. As depicted in figure 2.3, irregular points with distance far away from each other are observed, with abnormal probabilities. This results in a distorted Gaussian PDF. L-shaped PDFs denote the formation of PRWs, whose generation is attributed to a stochastic mixing of vector laser modes.

DS based on carbon-nanotubes
DSs have been investigated in diverse laser systems, operating with different rare earth-doped gain media [103][104][105]. Figure 3.1(a) schematically depicts a typical ring cavity containing 15 m EDF and 2.7 m SMF. The SA is based on single-wall carbon nanotubes. Here, we detect the single-shot spectra by a DFT method, and also record their SOP evolution dynamics with high speed PSA.

Data and discussion
In figure 3.1(b), for pump powers smaller than 53 mW, the laser operates from random amplified spontaneous emission (ASE) into an unstable pulse regime, until it enters into a stable regime. For the unstable regime, we observe a giant CW component, located on the broadened edge of the pulsed component. By further increasing the pump power, one gradually broadens the optical spectrum. Too large gain deteriorates the mode-locking state. This deterioration can be identified from the noisy frequencies appearing in the two fronts of the optical spectrum for the pump power of 70 mW, which originate from   the presence of multiple wave couplings. We can see that, for a pump power below 55 mW, optical clusters are formed, while for a pump power of 70 mW, we identify DS explosions. During DS explosions, the balance of nonlinearity, dispersion, and loss/gain is perturbed by the surplus gain at large pump powers [23,106,107]. Part of the DS energy dissipates into CWs via the explosion, and the DS maintains its property of being highly coherent. The corresponding SOP evolutions give more information about the formation of DSs [24,74,[108][109][110][111]. We measured the SOPs of different filtered wavelengths. Figure 3.3 shows that, when a DS is formed, its SOPs are evolving from a random cloud into a narrow range. Such a stable SOP structure is preserved even when its absolute value is changed, by biasing the fiber between the output OC and the PSA. During a DS  explosion, SOPs exhibit fluctuations and tend to be spreading further on the Poincaré sphere, especially for frequencies located on the two spectral edges.
To reveal the SOP dynamics more clearly, we consider the phase diagram for the ellipticity angle χ versus the spherical orientation angle ψ: .
Here, s 1 , s 2 , s 3 are the Stokes parameters. As shown in figure 3.4(a), the SOPs follow a random distribution, where ASE dominates. Figure 3.4(b) shows that all SOPs are located within a well-defined region, although they still behave erratically. For a stable DS, as shown in figure 3.4(c), a rectangle-shaped optical spectrum results into a fixed distribution along a folding trajectory. Whereas deteriorated distributions are observed during DS explosions, and additional frequencies with distant SOPs are detected.

Experimental setup
We further identify the presence of optical PRWs in supercontinuum generation, pumped by an amplified DS, generated as described in section 3. Figure 4.1(a) depicts the experimental setup, consisting of a DS laser source, a commercial erbium-doped fiber amplifier (EDFA), and 20 m of highly nonlinear fiber (ZDF). After SCG, signal is filtered by a TF. The output SOP is detected by a high-speed PSA. After the EDFA, the average power is amplified from 0.5 mW up to 100 mW, and the pulse duration is compressed from 29 ps down to 1.5 ps.  Figure 4.1(b) shows the SCG optical spectrum as a function of pump power, that increases with a step of 1 mW. The supercontinuum is rather flat after the EDFA: the 3 dB flatness in the blue side can be larger than 350 nm. We classified the SCG process in terms of two different stages: (1) the scalar stage, (2) the vector stage. For 15 mW, we observe a scalar stage, where cascaded symmetric sidebands are generated around the DS spectrum. Different orders of Stokes and anti-Stokes sidebands appear. The gain g(Ω) of the primary MI sidebands reads as

Results and discussion
where, β 2 , γ, and P are the group velocity dispersion (GVD), the nonlinear coefficient, and the peak power, respectively. Owing to the nominally zero GVD, sideband shifts are unusually large, and they increase with the power. After the scalar stage, a slight interference pattern appears. For 40 mW, vector FWMs contributes to spectrum broadening, and new frequencies are generated in both forward and backward directions (that is, away from and back into the pump frequency). Eventually, a broad and flat spectrum is formed via the merging of all FWMs processes.
SOPs tends to be scattered around in the vector stage, as shown in figures 4.2(a)-(f). For pump powers larger than 30 mW, the SOPs scatter over a broad area. Meanwhile, the degree of SOP scattering increases with pump power, because of stronger FWMs. Eventually, the phases of newly generated frequencies are loosely fixed and become chaotic. This process deteriorates the coherence of SCG. Figure 4.3 shows the PDF of r at 1570 nm, which is representative of the behavior within the spectral range of the SC. For 40 mW, the PDF is quasi-Gaussian. Whereas, when the pump power grows above 70 mW, a heavy tail appears. We identify the PRWs whenever r is larger than twice of the SWH. However, the probability to find PRWs for pump powers of 100 mW decreases unexpectedly, and the PDF returns to a Gaussian shape. The reason could be that frequencies undergo intense cascaded FWM in both forward and backward directions, and the energy coupling among different wavelengths and SOPs is rapidly varying. New frequencies with larger r are generated, while they are also annihilated.  Specifically, we may calculate the PDFs for wavelengths ranging from 1541 nm to 1557 nm in figure 4.4. SOPs emerging at 1557 nm are scattered within a single domain. For wavelengths situated farther away, the SOPs scatter within different domains. For 100 mW pumped supercontinuum, the histograms are highly sensitive to the specific filtered wavelength. For example, at 1543 nm and 1547 nm, SOPs with rogue positions are observed on the Poincaré sphere, as denoted by the heavy tails at large r values. However, double and triple PDFs may appear, depending on the wavelength that we selected.
When compared with intensity distributions for conventional RWs, PRWs in SCG are slightly different. Quantitatively, PRWs are identified on the basis of the PDF of r, while conventional RWs denote the presence of abnormally large intensities within a wide frequency range. The temporal intensities for PRWs may not be large when considering their specific generation procedure. However, we believe that there are intrinsic connections between PRWs and conventional RWs, since both waves are only formed by highly nonlinear processes.

Conclusion
We review recent process of studies on the polarization-dependent characteristics of optical RWs. The associated experiments show that the generation of RWs in nonlinear optical systems is generally polarization-dependent. A new statistical model based on the relative distance, r, between the SOPs of different wavelength components on the Poincaré sphere was proposed, in order to quantitatively identify the presence of PRWs. This measure permits to identify frequencies with a freak state of polarization, that is, with abnormal probabilities that deviate from a normal distribution. Next, we gave a short summary of the identification of PRWs in three nonlinear laser systems. Specifically: a partially mode-locked fiber laser, a dissipative soliton laser, and supercontinuum generation pumped by an amplified dissipative soliton. We identified the presence and properties of PRWs in those nonlinear laser systems. All of these laser systems undergo a transition from a high coherence to a low coherence regime, due to vector four-wave-mixing processes. Investigations of PRWs provide additional insight for our understanding of the nonlinear optical processes associated with the generation of RWs.