Multi-scale polarisation phenomena

Multi-scale methods that separate different time or spatial scales are among the most powerful techniques in physics, especially in applications that study nonlinear systems with noise. When the time scales (noise and perturbation) are of the same order, the scales separation becomes impossible. Thus, the multi-scale approach has to be modified to characterise a variety of noise-induced phenomena. Here, based on stochastic modelling and analytical study, we demonstrate in terms of the fluctuation-induced phenomena and Hurst R/S analysis metrics that the matching scales of random birefringence and pump–signal states of polarisation interaction in a fibre Raman amplifier results in a new random birefringence-mediated phenomenon, which is similar to stochastic anti-resonance. The observed phenomenon, apart from the fundamental interest, provides a base for advancing multi-scale methods with application to different coupled nonlinear systems ranging from lasers (multimode, mode-locked, random, etc.) to nanostructures (light-mediated conformation of molecules and chemical reactions, Brownian motors, etc.).


INTRODUCTION
Many processes in nature have different temporal and spatial scales that lead to multi-scale complexity. To describe this complexity on different levels, multi-scale methods have been developed and explored for more than 100 years 1-3 . For example, in nonlinear fibre optics, three groups of scales have 13 orders of magnitude separation between the smallest length scale of 1.55 mm and the Southern Cross Cable Network, whose length is approximately 32 500 km. The shortest micron scale is related to the wavelength of light and the core diameter. Thus, Maxwell's equations have to be explored for characterisation of fibre in the dispersion relations context. The next metre scale corresponds to the fibre beat and correlation lengths, e.g., lengths at which a state of polarisation (SOP) reproduces itself and preserves deterministic evolution. The longest kilometre length scale is the fibre attenuation and gain scale, chromatic dispersion and the Kerr nonlinearity. At this length scale, the Manakov equation is obtained by averaging the randomly varying birefringence [4][5][6] .
The simplest manifestation of this effect, in the form of the resonance-like increase of the gain fluctuations as a function of the PMD parameter, has been first found theoretically by Lin and Agrawal 12 and experimentally by Popov and co-workers 24 .
Modern fibre Raman-based unrepeatered transmission systems use bidirectional pumping schemes 33 . The co-propagating pump and signal provide a major contribution to the pump-to-signal relative intensity noise (RIN) transfer, which also depends on the PMD value 34 . To study the statistical properties of forward pumped FRA is the key to unlocking the RIN characterisation, which is based on the vector models of FRA, and, thus, to developing efficient vector RIN suppression techniques.
To justify application of previously explored multi-scale techniques for studying statistical properties of co-propagating pump and signal SOPs, we, for the first time, use computer simulation of stochastic differential equations with application of the Klöden-Platen-Schurz algorithm, which provides the fastest convergence (see Supplementary  Information). We reveal both the polarisation pulling and resonancelike escape from polarisation pulling in terms of fluctuation-induced phenomena metrics, such as Kramers and intrawell relaxation lengths, gain, root mean square (RMS) gain fluctuations, as well as the spectrum, correlation function, the Hurst parameter and probability distribution function for projecting the signal SOP to the pump SOP. The FRA pump-signal SOPs interaction is defined by the rate of relative rotation of the signal SOP with respect to the pump SOP. According to the results of our analytical study, the stochastic modelling demonstrates that for some PMD parameters, which are typical for the currently used single mode fibres, this rate is on the same scale as the birefringence correlation length. Thus, the rate has to be included in the fast scale group with further averaging to obtain a correct description in the region of resonance-like escape from the polarisation pulling.

MATERIALS AND METHODS
Signal and pump states of polarisation evolution in terms of biased Brownian motion To provide an insight into the FIE phenomena for the fibre Raman amplification, we, first, outline similarities between the SOP evolutions in Raman and biased Brownian motion. The FIE from a metastable state of an excitable system with probability controlled by an external force is a fundamental phenomenon that is inherent in many physical processes, such as, diffusion in crystals, protein folding, activated chemical reactions and many others [28][29][30][31] . Figure 1a demonstrates the escape of the Brownian particle from the bottom of the potential well due to fluctuations and barrier height modulation. The potential well DU is a source of polarisation pulling (i.e., 'polarisation trapping' 10 ). For the adiabatic forcing case, specifically, when the period of barrier modulation T is much longer than the intrawell relaxation time t i , FIE takes the form of stochastic resonance (SR), such as, synchronisation between the activated escape events from the potential minimum with a periodic forcing, which results in the maximal signal-to-noise ratio at T 5 t k (t k is the Kramers time, which characterises the average residence time with respect to the FIE [28][29][30][31].
The evolution along the fibre length for signal and pump SOP is similar to the evolution of the Brownian particle in the potential well ( Figure 1a). As follows from Figure 1b, the pumpp p p~ðp 1 ,p 2 ,p 3 Þ and signalŝ s~ðŝ 1 ,ŝ 2 ,ŝ 3 Þ SOPs evolution comprises: (i) signal-to-pump SOP pulling (i.e., polarisation trapping caused by potential well buildup) [7][8][9][10][11][16][17][18][19][21][22][23][24][25][26][27] caused by the stimulated Raman scattering anisotropy; (ii) barrier modulation caused by the relative rotation of signal SOP with respect to the pump SOP at the rate b s 2 b p (b i 5 p/L bi , where L bi is the beat length, i 5 s, p) around the birefringence vector (BV) W i 5 (2b i cosh, 2b i sinh, 0) T , that randomly fluctuates in the equatorial plane. We assume that the birefringence strength 2b i is fixed and the orientation angle h is driven by a white-noise process (fixedmodulus model [4][5][6]. where ,…. represents averaging of the birefringence fluctuations along the fibre, d(z) is the Dirac delta function, and s 2 5 1/L c (L c is the birefringence correlation length). As a result of evolution, the signal wave is amplified and changes its direction as follows: S S(z)~s 0 (z,ŝ s(0))G aveŝ s(z), P P(z)~P 0 (z)p p p: Here, s 0 is part of the signal amplitude that is related to the pump and signal SOPs interaction, G ave~e xp is the averaged Raman gain, g is the Raman gain coefficient, P 0 is the pump power at distance z, P 0 (z)~P in exp ({a p z); P in is the input pump power, a s and a p are the signal and pump losses, respectively; L is the fibre length. The part of the Raman gain ,G., which is related to the pumpsignal SOPs coupling is We excluded the averaged Raman gain G ave in Equation (3). This allows us to concentrate on the vector nature of the processes under consideration. If the input pump and signal SOPs are parallel, the Raman gain adopts the maximum value, and, if the SOPs are orthogonal, then the Raman gain adopts the minimum value 7-27 . The gain difference is referred to as the polarisation-dependent gain (PDG) and is defined as follows  : To quantify de-correlation of the pump and signal SOPs and polarisation pulling in terms of the fluctuation-induced phenomena, we introduce the RMS gain fluctuations as follows 12,14 :

Random fluctuations
Pulling ∆U a 2 Figure 1 (a) Fluctuation-induced phenomena, where the escape probability is controlled by an external periodic force. DU -potential well, T -period of barrier modulation, r 5 1/t k -escape rate, t i -intrawell relaxation time, t k -residence time (the Kramers time); (b) Evolution of the pumpp and signalŝ states of polarisation (SOPs) and the local birefringence vector (BV) W i 5 (2b i cosh, 2b i sinh,0) T on the Poincaré sphere. Vectorsŝ andp rotate around the local axis W at rates b p and b s , vector W rotates randomly in the equatorial plane at the rate s 5 L c 21/2 (L c is the correlation length). Anisotropy of fibre Raman amplification, which results in the signal-to-pump SOP polarisation pulling, i.e., builds up a potential well, while relative rotation of the signal SOP, with the rate b p -b s , plays a barrier modulation role, and the random fluctuation of BV defines the noise.

Multi-scale polarisation phenomena
V Kalashnikov et al 2 Thus, we introduced the ,G., PDG, and s 2 G metrics to further justify the different multi-scale techniques 7-20 using stochastic modelling.
Vector models of the fibre Raman amplifier and multi-scale techniques Here, we present two analytical models (where different averaging techniques have been used) and stochastic equations to validate these models. In the first model, the generic multi-scale technique has been applied, where only the randomly varying birefringence scale has been considered as the fastest scale 4-6 . Next, we average the fast birefringence fluctuations (details are found in the Supplementary Information) and neglect the pump depletion, cross-phase and selfphase modulations (XPM and SPM) and time dependence, i.e. group velocity dispersion (GVD). This approximation is valid for the pump powers P in ,1 W, signal powers s 0 ,10 mW 12,21 , D p . 0.01 ps km 21/2 12 . It has been estimated 12 that the GVD can be neglected when the fibre length L is much smaller than the dispersion length L D~T For pulse duration T p 5 2.5 ps, jb 2 j 5 5 ps 2 km 22 , we have L D . 100 km. Thus, GVD can be neglected for L ,20 km 12 .
Taking into consideration Equation (2), we obtain the following equations, which describe the evolution of pump and signal SOPs, which are related to the pump-signal SOPs coupling: S~sG ave , s~s 0ŝ ,ŝ j j~1, s 0~s j j~ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s 2 1 zs 2 2 zs 2 3 q , P~P 0 (z)p,p j j~1: Thus, the multi-scale method includes averaging of the fast birefringence fluctuations and results in the averaged gain value and in the absence of pump and signal SOPs correlation. The method neglects gain fluctuations. Thus, one condition for the validity of the method is the low gain fluctuations. Equation (6) has been developed using unitary transformation to exclude the pump SOP fluctuations due to the random birefringence. The applied transformation preserves the length of the pump and signal SOP vectors as well as the scalar and vector products. As a result, evolution of the signal SOP includes a term (the second one), which accounts for the relative rotation of the signal SOP with respect to the pump SOP. However, Kozlov and coworkers 7 have applied unitary transformations to the pump and signal SOPs to exclude both the pump and signal SOP fluctuations due to random birefringence and, as a result, have obtained equations that differ from Equation (6) and those derived by Sergeyev and co-workers 13 . The stimulated Raman scattering and XPM introduce a coupling between the pump and signal SOPs. Thus, the adopted transformations 7 do not preserve either the vector and scalar lengths or the vector products.
To justify the multi-scale method that results in Equation (6), we use stochastic equations derived from the coupled Manakov-PMD equations to calculate the part of the gain, which is related to the pump-signal SOPs coupling, gain fluctuations and correlation properties of signal and pump SOPs (details are found in Supplementary Information) 12 : Here g h is defined in Equation (1). Direct averaging of the randomly varying birefringence, including the scale of signal and pump waves interaction, results in the following equations [13][14][15][16][17][18][19][20] : Here We also find the RMS gain fluctuations from Equation (5) using the following equations 17 : To justify validity of the multi-scale technique, we calculate gain AEGae, RMS, PDG and correlation between the pump and signal SOPs by solving Equations (6)- (9).
To quantify polarisation pulling and escape in terms of the FIE phenomena, we use the previously suggested approach 17 to calculate parameters, which are equivalent to parameters used in the excitable systems 29 models (the Kramers length ,L K . and intrawell relaxation length ,L R .). First, Equation (8) is simplified as follows: Multi-scale polarisation phenomena V Kalashnikov et al 3 Using the linear stability analysis of Equation (10) nearx 0 h i,ŷ 0 , we find eigenvalues: We introduce the intrawell relaxation length L R~1 =jRe L 1,2 ð Þj. If Im L 1,2 ð Þ=0, the system escapes by oscillating around the stateŝ x 0 h i,ŷ 0 . Thus, we define the Kramers length as L K~2 pL=jIm L 1,2 ð Þj. To study the long-range memory effects for the Raman-induced polarisation pulling and escape, we provide the Hurst rescale range R/S analysis to obtain the Hurst parameter 3,35-46 . First, for time series X i (i 5 1,2…N) the mean value m N and the cumulative deviate series C N,k are calculated as follows 3 : Next, the range R N and the standard deviation S N are calculated as The rescale range is found as R N /S N . Then, the time series of N points are divided into two N/2-points time series, and the rescale range R N/2 /S N/2 is calculated for both time series and is, then, averaged. This process is repeated for partial series which comprise n 5 N/4, N/8,… points. The Hurst parameter H is estimated by fitting the power law of averaged R n /S n for nR', e.g.  (1,0,0). The gain ,G. is normalised to G ave in agreement with Equation (3).
Multi-scale polarisation phenomena V Kalashnikov et al 4 works in reverse for the anti-persistent statistics. Specifically, a positive increment in the past will result in a more probable negative trend in the future. Hurst first suggested an application of the R/S analysis to study water storage in the Nile River 36,37 . Since then, the Hurts parameter has been proven as a feasible metric for analysing long-range dependence in network traffic 39,40 , turbulence 41 , heartbeat 42 , coalition of neurons dynamics 43 , detection of low observable targets within sea clutter 44 , identification and prediction of epileptic seizures, earthquakes, and crashes in financial market 3,45,46 .

RESULTS AND DISCUSSION
Equation (7) have been solved using the Wolfram Mathematica 9.0 computer algebra system using the built-in Klöden-Platen-Schurz method, which provided the fastest convergence compared with the Runge-Kutta and Milstein algorithms (see the Supplementary Information). The averaging procedure was performed for an ensemble of N 5 100 stochastic trajectories. This provides the precision of 1/N 1/2 , 10%, which is sufficient to justify the analytical results obtained by Sergeyev and co-workers [13][14][15][16][17][18][19][20] . We used the following parameters: the Raman gain coefficient g was 0.8 W 21 km 21 ; the input signal s 0 and the pump P in powers were 10 mW and 1 W, respectively; the fibre length L was 5 km; the correlation length of birefringence vector L c was 100 m. The Stokes parameters for the pump and the input signal fields, which correspond to the maximum and minimum PDG values, were:p~(1,0,0),ŝ~(1,0,0) for the maximum gain and maximum PDG;p~(1,0,0),ŝ~({1,0,0) for the minimum gain and maximum PDG;p~(0,0,1),ŝ~(0,0,1) for the maximum gain and minimum PDG; andp~(0,0,1),ŝ~(0,0,{1) for the minimum gain and minimum PDG. Based on our previous publications and on Equations (7), the Poincaré sphere reference frame is chosen to have a local birefringence asW i,un~( 2b i ,0,0) for all stochastic realisations. Thus, all trajectories for the Stokes parameter s 0 have the same starting point with respect to the chosen reference frame with further divergence being caused by the random birefringence fluctuations, as shown in the Figure 2 inset.
The dependences of gain AEGae, which, according to Equation (3), is part of the Raman gain, are related to the pump-signal SOPs coupling (black solid and dashed curves), and RMS gain fluctuations s G (red solid and dashed curves) on the PMD parameter are shown in Figure 2. The solid and dashed curves correspond to the numerical solution of Equations (7)- (9). Based on Figure 2, the stochastic calculations of the gain and RMS gain fluctuations, which are derived using Equation (7), perfectly fit in the whole range of the PMD parameters using results from our previously developed model, which is based on Equations (8) and (9) [13][14][15][16][17][18][19][20] . This is a significant result because it provides analysis tools for long fibre communication systems without using time-consuming calculations, which are based on the solution of the underlying stochastic equations.
Based on Equation (6), we conclude that in view of the exponential decay of the term related to escape from polarisation pulling, an application of the multi-scale technique with averaging, excluding scale of the pump-signal SOPs interactions, results in polarisation pulling in all ranges of the PMD parameters. As a result, the gain values coincide with gain values obtained from Equations (7)- (9) in the limit of D p R0 and are close to the values for an ideal Raman polarizer 11 , by taking into account normalisation of ,G. to G ave in Equation (3). However, the averaging technique, which accounts for the signal and pump SOPs interaction [13][14][15][16][17][18][19][20] scale, better agrees with the stochastic modelling results (Figure 2). In addition, these analytical techniques, which resulted in Equations (8) and (9), predict a resonant enhancement of the RMS gain fluctuations within the range of PMD parameters of 10 22 to 10 21 ps km 21/2 , which are typical for the modern single mode fibres (the red dashed curves in Figure 2 in comparison with the red solid curves, which are obtained numerically from Equation (7)).
Though the analytic theory predicts a constant asymptotic of 0.34 dB for the PDG parameter (red curve in Figure 3), the numerical PDG disappears approximately monotonically with PMD (solid curves in Figure 3), which is in agreement with the Ref. 10 results. Nevertheless, the averaging of the N 5 100 trajectories provides a precision D,1/N 1/2 of 0.4 dB. Thus, the asymptotic cannot be validated for the parameters used here.
To characterise the transition from polarisation pulling to escape from pulling, we determined Kramers and the intrawell relaxation lengths from Equations (11) and (12) and the Hurst parameter for the pump-to-signal SOP projection x h i:p . s h i= s j j h i (insets 1 and 2 in Figure 3). As follows from Figure 3 (inset 1), the transition from polarisation pulling to escape has the threshold at D p < 0.02 ps km 21/2 , which, according to Equation (12), corresponds to the escape rate Im(L 1,2 ) j j §0 for ,x. evolving along the fibre length. In contrast to our previous results on SR in the fibre Raman amplification 17 , here the escape from polarisation pulling happens in many uncorrelated steps rather than in one step, as for SR. Therefore, we have increased gain fluctuations instead of increased the signal-tonoise ratio, e.g., the stochastic anti-resonance 17 .
To gain insight into statistical properties of the pump and signal SOPs interaction, we studied the stochastic evolution of the signalto-pump SOP projection x h i:p . s h i= s j j h i along the fibre instead of the s 0 evolution (Figures 4-6). By comparing  1,0,0),ŝ~(1,0,0). evolution of ,x. reflects the statistics of SOP interactions by including small scales, whereas for s 0 the small-scale statistics disappears due to the propagation distance averaging. The asymptotic behaviour (D p R0) of ,x. demonstrates the Raman-induced polarisation pulling effect [7][8][9][10][11][16][17][18][19][21][22][23][24][25][26][27] , where the Raman amplification plays an effective fibre polariser role because x h i:p . s h i= s j j h i?1, i.e., the signal SOP is attracted to the pump SOP (see the top row of Figure 4 and the black solid curve in Figure 2a). For D p R0, the fibre becomes effectively 'isotropic' 10 . Thus, the Raman amplification anisotropy results in the strong amplification of the co-polarised to pump signal SOP and the attenuation of the cross-polarised signal SOP.
For the initially cross-polarised pump and signal SOPs, this attraction occurs (the top row of Figure 5) with a lower rate and is initiated by the birefringence fluctuations due to the escape from the metastable state with x h i:p.s h i= s j j h i?{1. As a result, the average gain AEGae remains minimal for the considered fibre length (Figure 2b and 2d). An important polarisation pulling property for both considered initial signal SOPs is minimisation of RMS fluctuations of the average gain (points A on the red curves in Figure 2a and 2b) and a regular structure of the spectral energy density (Figure 6a and 6b). This is due to 'fine graining' of the birefringence fluctuations, which play a role of white noise perturbations around a stable polarisation state (Figure 2a and 2c and the top row in Figure 4), or perturbations pulling out a metastable polarisation state (Figure 2b and 2d and the top row in Figure 5). The corresponding correlation functions demonstrate a damped oscillation behaviour (insets in Figure 6a and 6b).
The opposite extreme case is L c ..L b (the large PMD parameters, i.e., the case of a 'standard Raman amplifier' 10,11 ), when the deterministic evolution, which is induced by the fibre birefringence, prevails over the stochastics. In this case, a single mode fibre is similar to the polarization-maintaining (PM) fibre, which has comparatively rare stochastic switches of the birefringence axis fluctuations. Thus, the RMS gain fluctuations decrease (e.g., the points C on the red curves in Figure 2a and 2b), and ,G. approaches a constant small but nonzero value (black curves in Figure 2a and 2c). Because the evolution is driven by the fast pump-signal decorrelation, the average gain is minimal (but non-zero) for the initially co-polarised pump and signal. This means that there is a weak correlation between the pump and signal SOPs (bottom row in Figure 4).
The decrease of SOP correlation manifests itself in the Hurst parameter reduction H ,1 (inset 2 in Figure 3). For the initially crosspolarised pump and signal SOPs, the residual correlation (bottom row in Figure 5) maximises the gain ,G. (black curves in Figure 2b and 2d and the bottom row in Figure 5). An oscillatory evolution, which underlies localisation, along the fibre reveals itself in the modulated power spectrum densities and the corresponding correlation functions (Figure 6e and 6f).
The intermediate case of L b <L c =4 (D p < 10 22 410 21 ps km 21/2 ) demonstrates a resonant enhancement of polarisation stochastic evolution, where the RMS gain fluctuations have a set of spikes (in the vicinity of points B in Figure 2). Such spikes correspond to enhanced 'wandering' of the trajectories for the signal-to-pump SOP projections ,x. (Figures 4 and 5). In view of the increased rather than decreased gain fluctuations for point B, this phenomenon is contrary to the SR [28][29][30][31] and, thus, is referred to as stochastic antiresonance 29 . The stochasticity intensification is demonstrated by the threshold-like dropping of the Hurst parameter to H ,0.7 and the corresponding collapse of the Kramers length (insets in Figure 3). This switching between the statistical scenarios is the distinguishing characteristic of the 'stochastic anti-resonance' under consideration. The average polarisation state remains 'localised' (the middle rows of   [4][5] but its sensitivity to the input SOP disappears with the PMD parameter growth (Figure 3). This means that the PDG decreases with D p (solid lines in Figure 3). Therefore, the Raman gain in the vicinity of the standard deviation peak behaves as an 'effective depolariser', which diminishes the PDG.

CONCLUSIONS
In summary, using stochastic modelling, we provided insights into multi-scale polarisation phenomena for the FRA as a function of its birefringence properties (PMD parameter). We demonstrated that for the low PMD values the fibre become almost isotropic. Thus, the Raman amplification anisotropy leads to polarisation pulling when the signal SOP is attracted to the pump SOP [7][8][9][10][11][16][17][18][19][21][22][23][24][25][26][27] . Therefore, the length of the pump-to-signal SOP interaction (beat length of the relative rotation of the pump SOP with respect to the signal SOP) is much longer than the birefringence correlation length. Thus, averaging over the correlation length scale using the generic multi-scale technique [4][5][6] leads to the results that are close to those obtained using stochastic modelling (Figures 2 and 3). Due to the decreased interaction length (increase in PMD), deterministic rotation of the signal SOP with respect to the pump SOP is intensified and results in escape phenomena, which is similar to stochastic anti-resonance, in view of the increased RMS gain fluctuations ( Figure 2). When the rotation rate approaches the correlation length, the scale averaging of the correlation length is no longer valid and cannot provide correct results for the gain (Figure 2). Only by including the scale of the signal-to-pump SOPs interactions, we demonstrate that it is possible to obtain the correct results (Figures 2 and 3). A further decrease of the interaction length corresponds to an almost deterministic birefringence case, where the pump and signal SOPs rotate without interaction ( Figure 2). Detailed statistical analysis of the pump-to-signal SOP projection evolution along the fibre unveiled different types of fractional Brownian motions as a function of PMD values in terms of the Hurst parameter H. For the low PMD values, the polarisation pulling leads to HR1, which corresponds to the persistent statistics. For the PMD values that correspond to the gain fluctuations maximum, the Hurst parameter decreases to H 5 0.7 and, therefore, approaches the Brownian motion with H 5 0.5. Further increase in the PMD parameter corresponds to the almost deterministic SOPs evolution and, thus, the persistent statistics with HR0.8.
The obtained results are further generalised by accounting for the pump depletion, XPM and SPM, and time dependence (GVD and  Figure 6 The power spectral densities of x h i:p . s h i= s j j h iat L 5 5 km and the z-dependences of the corresponding correlation function (insets) forp~(1,0,0), s~(1,0,0) (a "maximum gain"; (a), (c) and (e) graphs) andp~(1,0,0),ŝ~({1,0,0) (a "minimum gain"; (b), (d) and (f) graphs). The PMD parameters correspond to points A (upper row of the graphs in Figure 6), B (middle row of the graphs in Figure 6) and C (bottom row of the graphs in Figure 6) in Figure 2.
Multi-scale polarisation phenomena V Kalashnikov et al 8 walk-off between the pump and signal waves). This manipulation provides an opportunity to gain insight into the RIN 33 and extreme statistics in the FRAs 47 as a function of the PMD parameters and to adapt the developed methods to characterise the parametric 48 and Brillouin 49 amplifiers. Additionally, these results can be applied, in the context of new multi-scale methods development, to study the complex nonlinear coupled systems, such as lasers (multimode, mode-locked, random) 50-52 , nanostructures (light-mediated conformation of molecules and chemical reactions, Brownian motors 53 ), and other systems [39][40][41][42][43][44][45][46] .