Mode attraction, rejection and control in nonlinear multimode optics

Novel fundamental notions helping in the interpretation of the complex dynamics of nonlinear systems are essential to our understanding and ability to exploit them. In this work we predict and demonstrate experimentally a fundamental property of Kerr-nonlinear media, which we name mode rejection and takes place when two intense counter-propagating beams interact in a multimode waveguide. In stark contrast to mode attraction phenomena, mode rejection leads to the selective suppression of a spatial mode in the forward beam, which is controlled via the counter-propagating backward beam. Starting from this observation we generalise the ideas of attraction and rejection in nonlinear multimode systems of arbitrary dimension, which paves the way towards a more general idea of all-optical mode control. These ideas represent universal tools to explore novel dynamics and applications in a variety of optical and non-optical nonlinear systems. Coherent beam combination in polarisation-maintaining multicore fibres is demonstrated as example.


INTRODUCTION
Multimode waveguides have become one of the major research topics of the last decade in optics.The initial interest in multimode optical fibres for next-generation optical communication systems 1,2 has rapidly spread to a diversity of applications, including astronomy 3 , high-speed datacentres 4 , imaging 5 and more recently integrated photonics 6 , which attests to the relevance of this field within the broad photonics community 7 .
The massive interest in multimode fibres has triggered the investigation of nonlinear effects in these systems.Whether it represents an issue to avoid or to exploit for high-power lasers 8 and optical communications 9 , nonlinear dynamics in multimode waveguides undoubtedly represents a rapidly growing research field.Indeed, nonlinear coupling among different spatial modes -mediated by cross-phase modulation, four-wave mixing and Raman scattering-and its interplay with temporal dispersion gives rise to complex, multi-faceted dynamics where much is still unknown [10][11][12][13][14][15] .
Over the last few years novel scenarios have been disclosed, including wideband supercontinuum mediated by intermodal nonlinear interactions 16,17 , nonlinear conversion via intermodal four-wavemixing 18,19 , spatio-temporal multimode solitons 20 and mode-locking 21 , pulse combining 22 and spatial beam self-cleaning 13,23 , to name a few.All the above-mentioned phenomena share a co-propagating geometry, whereas counter-propagating geometries in multimode waveguides have so far been left largely unexplored.
Nonlinear counter-propagating systems have been extensively studied in single-mode waveguides since the '80s, where new fundamental results have been demonstrated, along with relevant applications.This includes spatial/temporal chaos and bistability [24][25][26][27] as well as the existence of special polarisation states that play the role of robust attractor states.The notion of attraction plays a central role in mathematics and physics.The existence of system attractors has been observed in different kind of optical fibres, from isotropic 28 , to highly birefringent 29,30 up to randomly birefringent fibres 31 .While different types of fibre exhibit different kinds of attractors, the underlying attraction process is characterised by a similar dynamic 32 : independently of its input polarisation state, a forward signal (FS) is attracted towards an attractor state that is fixed by a counter-propagating backward control beam (BCB), launched at the opposite fibre end.The attraction process is driven by the Kerr-nonlinear interaction between the FS and BCB.As a result, for an increasing degree of nonlinearity, the attraction becomes more effective and robust to external perturbations (Fig. 1 a,b).
These two features -robustness and independence of the input conditions -are peculiarities of the counter-propagating setup, whereas they are not present in standard co-propagating systems 33,34 .Besides the fundamental interest, these outcomes have been exploited to implement a variety of novel optical devices, including lossless all-optical polarisers 35 , all-optical polarisation scramblers 36 and nonlinear focusing mirrors for temporal compression 37 .
Differently from single mode waveguides, counter-propagating setups in multimode waveguides have been little addressed so far.The existence of mode attractors in a fibre supporting two spatial modes has been envisaged 34,38,39 .However, this process has been measured and quantified for the first time only recently in our work Ref. [39].Moreover, there has not been any attempt to analyse the most general case of multimode waveguides supporting any number of modes.This is arguably due to the additional complexity intrinsic to the counter-propagating multimode dynamics, both in terms of fundamental understanding as well as of the experimental setup.
Nevertheless, the richness of recent results in nonlinear multimode waveguides, along with earlier outcomes in single-mode fibres with counter-propagating configurations as depicted above, let us to anticipate a variety of nonlinear phenomena in counter-propagating multimode systems.The scope of this paper is to initiate research in this direction.
Here, we generalize the idea of mode attraction in waveguides supporting an arbitrary number N of modes.Moreover, we predict and demonstrate experimentally a novel fundamental property that we name mode rejection.As illustrated in Fig. 1 c,d, this effect can be seen as the inverse of the attraction process.It is only by moving to waveguides supporting N>2 modes that mode rejection can be properly understood and distinguished from mode attraction.
In this work, the waveguides of choice are multimode and multicore optical fibres.It should be noted however that our results can be promptly adapted to different optical (e.g.integrated waveguides) or even non-optical systems exhibiting Kerr-like nonlinearity.The choice of multimode/multicore fibres is driven by two considerations.Firstly, these fibres benefit from a decade of improved manufacturing processes as well as a wide array of devices for efficient mode manipulation, decomposition and characterization, which are key steps in our investigation.Secondly, these fibres represent versatile platforms for exploring not only fundamental physics but also new applications.Here, we provide an example of coherent beam combination based on the new notion of mode rejection.
Our outcomes broaden our knowledge on the complex nonlinear multimode dynamics and open the path towards a more general idea of all-optical mode control.More generally, they shed new light into the fundamental ideas of attraction, rejection and control in the framework of nonlinear multimode systems, including but not limited to optical fibres.When the BCB is turned on and has a similar power to the FS (b), their nonlinear interaction modifies the mode content.If the input BCB is coupled to one circular polarization mode (left circularly polarised in this example), then irrespective of the mode content of the input FS, the output FS is attracted to the mode of the input BCB (100% of the power in the left circularly polarised mode).c, d, Mode rejection in multimode optical fibre.In this example the fibre supports 3 modes indicated with M1,M2,M3.When the BCB is turned off (c), the mode content of the output FS mirrors the input one (relative power P1 for mode M1, P2 for M2 and P3 for M3, where P1, P2 and P3 are arbitrary and P1+P2+P3 =100%).When the BCB is turned on and has a similar power to the FS (d), their nonlinear interaction modifies the mode content.If the input BCB is fully coupled to one spatial mode (M1 in this example), then irrespective of the mode content of the input FS, the output FS rejects the mode of the input BCB (0% power in mode M1).The specific amount of power on the remaining modes ( P2,out for M2 and P3,out for M3, where P2,out+P3,out =100%) depends on the system parameters.

RESULTS
Theory of mode attraction and rejection.Our theoretical investigation focuses on two distinct scenarios.Firstly, we consider a polarization maintaining (PM) fibre of length L, supporting the propagation of an arbitrary number N of spatial modes linearly polarised along one of its birefringence axes.The FS (BCB) enters the fibre at z=0 (z=L) and excites a combination of spatial modes, whose amplitude is indicated by f n (z,t) ( b n (z,t) ), n={1,2,…N}.We find that their spatio-temporal evolution is described by the set of coupled nonlinear Schrödinger equations Eqs.(1) (see Methods).
Secondly, we consider the case of a single-mode isotropic fibre.Differently from the previous case, it can only support the propagation of N=2 modes.These are quasi-degenerate modes with orthogonal circular polarization, whose dynamics is well-known and is described by Eqs.(2) with N=2 (see Methods).One of the key-points of our analysis is that we extend the study of Eqs.(2) to an arbitrary dimension N>2.Despite the existence of fibre systems where Eqs.(2) with N>2 could apply is yet unknown, however this extension allows for a direct comparison between Eqs.(1) and Eqs.(2) in the general case N>2, which ultimately leads to a new understanding of the nonlinear mode dynamics in counter-propagating systems.
We note that Eqs.(1) and Eqs.(2) are similar but differ in the last term on the right-hand side, which describes the power exchange among forward and backward modes.The main outcome of our theoretical investigation is that, under certain conditions on the Kerr coefficients, both Eqs.(1) and Eqs.(2) are integrable: therefore, they can be solved analytically (see Methods and Supplementary information note 1).The analytical solution discloses a new fundamental property.
In the case of Eqs.
and therefore |f m (L)| 2 ∼ P f , which means that all the power of the output FS gets coupled into the same mode-m of the input BCB.This generalises the idea of mode attraction in the case of N-dimensional systems, with N arbitrary.
In conclusion, Eqs.( 1) and ( 2) describe the two opposite processes of mode rejection and attraction in the case of arbitrary dimension N. Their difference is intimately related to the nature of the energyexchange term that couples the forward and the backward beams.The schematic in the Supplementary information note 5 summarises these results.We highlight that it is only in systems supporting more than 2 modes (N>2) that one can fully appreciate the idea of mode rejection.Indeed, in bimodal systems the rejection of one mode could be misleadingly interpreted as the attraction towards the other mode 34,39 .
Our theoretical analysis is supported by a simulation tool that numerically solves Eqs.(1) and (2).Besides confirming the theoretical predictions, the simulations reveal that mode rejection and attraction take place in the most general case where the Kerr coefficients are fully arbitrary (See Supplementary information note 2).Therefore, these phenomena represent universal features of nonlinear counter-propagating systems.commercial polarization maintaining fibre (PM-6MF) from Thorlabs supporting 6 distinct modes at λ=1040 nm (LP01,11a,11b, 21a,21b,02).In our experiments each spatial mode is linearly polarised along one of the birefringence axes of the fibres under test (see Supplementary information note 3).The substantial birefringence of the fibres ensures that the input polarization is maintained (>10 dB polarization extinction ratio).The spatial mode profiles in the near-field (NF) and the far-field (FF) of the DCF and TCF are reported at the bottom and can be compared against the camera images shown in Figs.3-5 to appreciate the mode rejection dynamics.For the Thorlabs fibre just the NF modes are reported.
Experimental results on mode rejection.As anticipated, the existence of optical fibres where Eqs.(2) with N>2 may apply is yet unknown.On the other hand, the case of Eqs.(1) with N>2 is relevant to PM multimode fibres.Therefore, in our experiments we have conducted a systematic investigation in a variety of PM multimode and multicore fibres, in order to provide the first experimental evidence of the mode rejection principle.Details on the fibres and experimental parameters are provided in Fig. 2b and Supplementary information note 3, including the choice of the optical power to achieve the high levels of system nonlinearity required.In our setup (Fig. 2a), an inhouse built linearly polarized source (central wavelength λ=1040 nm, 500 ps pulses) is split into two beams that form the input FS and BCB.Cameras are placed in the near-and far-field to measure the spatial intensity profiles of the output beams, and these images are then used to estimate the mode content of the output FS via a mode decomposition technique covering all guided modes of the fibre (see Methods and Supplementary information note 4 for details).
The first fibres under test were a homemade dual-core fibre (DCF) and tri-core fibre (TCF).Multicore fibres represent a novel platform for exploring nonlinear effects in counter-propagating systems, which allow generalisation of our theoretical predictions beyond standard single-core multimode fibres.Furthermore, these fibres allow us to characterize the robustness of mode rejection against a variety of simultaneous fabrication imperfections, which include random variations of core radii, shape and core-to-core distance.Spatial light modulators and phase-plates were used to control the intensity and phase of light coupled into each core, and hence to excite an arbitrary combination of modes.Fig. 3 displays the experimental results obtained in a 1-m long DCF that supports the propagation of an even and an odd supermode, here indicated with SM e and SM o .The FS and BCB are co-polarised.On one side, the input FS is launched with fixed power, coupled to different arbitrary combinations of modes.At the opposite fibre end, the input BCB is launched with variable power, and it is coupled to the odd (Fig. 3a) or the even (Fig. 3b,c) supermode.In line with our theoretical predictions, experiments confirm that, for increasing values of BCB power, the output FS gradually rejects the BCB mode.The numerical simulation of Eqs.(1) shows a good match with the experiments when using comparable parameters (pulse width, peak powers and Kerr coefficients, see also Supplementary information note 3).power of input FS on the top of each panel) and launched with fixed power of 3.75 kW,6.5 kW and 6.2 kW in a, b, c respectively.If the input BCB is coupled to SMo (a), then the output FS rejects SMo and is therefore mainly coupled to SMe.When the BCB power =5.1kW, ~90% of the output FS power is coupled to SMe.Consequently, the output FS far-field exhibits one single lobe and resembles the mode SMe-FF of Fig. 2b, corresponding to in-phase combination of the cores.On the contrary, if the input BCB is coupled to SMe (b, c), then the output FS undergoes rejection of SMe and is therefore mainly coupled to SMo .When the BCB power = 5.6kW, ~81% and 98% of the output FS power is coupled to SMo in cases b and c, respectively.Consequently, the output FS far-field exhibits two distinct symmetric lobes and resembles the mode SMo-FF of Fig. 2b, corresponding to out-of-phase combination of the cores.Error bars of ±3% are added to the measured relative power of the rejected mode, which represents the estimated uncertainty of our mode decomposition algorithm.
Note that, irrespective of the mode distribution of the input FS, according to our theoretical model and our numerical simulations full rejection (i.e., no output FS power coupled into the input BCB mode) could be in principle achieved by appropriately increasing the forward and backward powers P f and P b .However, in our experiments the maximum coupled peak power P f +P b is limited to 12 kW by the optical source in use, which explains why only partial rejection was observed.It is worth noting that the power requirement could be substantially mitigated by resorting to highly nonlinear fibres, which includes fibres with non-silica host.
As previously mentioned, when a fibre supports just two guided modes, as is the case for the DCF, then rejection of one mode corresponds to attraction towards the other mode.This opens up the possibility to lock the output FS to either the even or the odd supermode in an all-optical way.In the case under analysis, when the BCB is coupled to the odd (even) supermode, then the output FS undergoes rejection of that mode, and consequently it is attracted towards the even (odd) supermode, which corresponds to a condition of in-phase (out-of-phase) coherent combination of the two cores.This is confirmed by the observation of the far-field intensity profile of the output FS on the camera, which shows a transition towards an in-phase (out-of-phase) coherent combination when increasing the power of the BCB coupled to the odd (even) supermode, as reported in Fig. 3 and in the Supplementary movies (see Supplementary information note 4).
A striking property of mode rejection is its robustness against external perturbations, which is illustrated in Fig. 4a.Moving and/or bending the fibre modifies the launching conditions of the FS and may lead to random coupling among the fibre modes.When the BCB is turned off, this results in random variations of the output FS.However, when the BCB is turned on and is coupled to one mode (SM e in Fig. 4a), then mode rejection takes place, irrespective of the launching conditions.As a result, we observe the effective rejection of that mode in the output FS (relative power of SM e < 10% in Fig. 4a).Consequently, we can robustly lock the output FS to either one or the other supermode.
A further relevant feature of mode rejection is that it can be controlled via the relative angle of polarisation orientation between the FS and the BCB.When the FS and BCB are orthogonally polarised, the power-exchange term of Eqs.( 1) is reduced by a factor of 3 (see Methods), which leads to a substantial suppression of the power exchange between the FS and BCB, ultimately compromising the mode rejection process.This is confirmed by the results in Fig. 4b, which shows the evolution of mode rejection as a function of the relative angle α between the input FS and the BCB in the DCF.When α=0 deg (i.e., the FS and the BCB are co-polarised), a strong rejection of the BCB mode is observed (relative power of SM e =10% in Fig. 4b), which gradually reduces for increasing values of α.When α=90 deg (i.e. the FS and BCB are orthogonally-polarised), the far-field intensity of the output FS resembles the case where the BCB is turned off, indicating that mode rejection is substantially suppressed.The input FS is coupled to a combination of the two supermodes, whereas the input BCB is coupled to SMe.FS and BCB are co-polarised.The fibre is perturbed 5 different times.Each time, the perturbation consists in compressing or bending the fibre with different levels of intensity and in different points.For each perturbation, the mode decomposition of the output FS is computed, and the output FS far-field intensity is imaged when the BCB is either turned off or on.When BCB is on, robust rejection of SMe occurs in the output FS (<10% power in all 5 cases).b, Mode decomposition and far-field intensity of the output FS as a function of the relative angle α between the polarisation orientations of the input FS and the BCB.The far-field with BCB off is reported for comparison.Error bars of ±3% are added to the measured relative power, which represents the estimated uncertainty of our mode decomposition algorithm.
Our experiments with the TCF, reported in Fig. 5, show similar outcomes, and again suggest that mode rejection is robust against standard fibre fabrication imperfections (see Supplementary information note 3).Indeed, the TCF supports 3 guided supermodes (SM 1 ,SM 2 , SM 3 ), and in good agreement with the numerical simulations mode rejection is observed when the BCB is coupled to any of these.Further experiments were carried out in a single-core few-mode PM fibre supporting up to 6 modes at λ=1040 nm.Although the specific evolution of the power carried by each mode is dependent on the input FS mode content, rejection of the input BCB mode was consistently achieved, irrespective of the input FS.Fig. 6 reports a few illustrative examples (see also Supplementary information note 4 and related movies).Dynamic of the backward control beam.Eqs. ( 1) are invariant with respect to the exchange between FS and BCB modes.However, different boundary conditions apply to the input FS and to the input BCB, which reflects their different roles.
Because the input BCB plays the role of the control beam, its mode content is fixed and coupled to one single mode.For this reason, the output FS systematically undergoes rejection irrespectively of the input FS mode content.On the other hand, the input FS plays the role of a probe beam with arbitrary mode content.Therefore, the output BCB does not undergo any rejection dynamics, because different input FSs lead to different output BCB mode contents.
It is interesting to note that, in the special case of a two-mode-fibre having the Kerr coefficients all identical, the mode content of the output BCB is attracted towards the orthogonal modal state of the input FS 34 when they have the same power.This is confirmed by our experiments in the home-made DCF, where the above-mentioned condition on the Kerr coefficients is met with excellent approximation (γ 11 ~ γ 12 ~γ22 , see Supplementary information note 3).In the example illustrated in Fig. 7 the input BCB is coupled to supermode SM o , whereas the input FS to a combination of 60% SM e and 40% SM o .As expected, the output FS undergoes rejection of the input BCB mode.On the other hand, the output BCB achieves a mode content of ~ 40% SM e and ~ 60% SM o when the two beams have the same power (see green dashed vertical line).
These considerations have an important consequence when the input FS has a random mode content in time.In that case the amount of disorder of the input FS is transferred to the output BCB, whose mode content becomes therefore randomly distributed.This dynamic is similar to that of polarization attraction phenomena in single-mode fibres 34 , where a mutual exchange in the degree of polarization is achieved between FS and BCB, that is to say, the polarization attraction undergone by the output FS takes place at the expense of a depolarization of the output BCB.Fig. 7: Comparison between the dynamics of the output FS and the output BCB in the home-made DCF.In the case of panel a, the input FS is launched with fixed power of 5.05 kW, whereas the BCB power is variable (see horizontal axis).In the case of panel c, the input BCB is launched with fixed power of 5.05 kW, whereas the FS power is variable (see horizontal axis).Panel b shows the far-field intensities.The input FS is a combination of ~60% SMe and ~40% SMo , whereas the input BCB is coupled to SMo.Consequently, the output FS undergoes rejection of SMo as illustrated in panel a.On the other hand, the output BCB does not undergo rejection, as illustrated in panel c.Indeed, the mode content of the output BCB in stationary regime strictly depends on the fibre parameters and the input FS, which is in general arbitrary.In this example, because the fibre is bimodal and the Kerr coefficients are almost identical, then the output BC reaches a state orthogonal to the input FS (~40% SMe and ~60% SMo) when the 2 beams have the same power (5.05 kW, see green dashed line in panel c).
Towards the idea of mode control.Mode attraction and rejection take place when the input BCB is coupled to one individual mode.We envisage further novel scenarios in the most general case where the input BCB is coupled to an arbitrary set of modes.In this framework, one may wonder whether for a given input FS it is possible to achieve a specific output FS mode distribution by opportunely setting the input BCB.Once again the theoretical framework developed for the special case γ nn =γ and γ nm =(1/2)γ (n≠m) gives a clue on some possible scenario.For example, following the steps introduced in the Supplementary information note 1 , we find that for a given input FS it is possible to focus all the output FS power into one single mode, say mode-k, provided that the input BCB has the following configuration: | ̂()| ∼ 0,  ̂() ∼ −i | ̂(0)| e i  / if n ≠ k, where  = and   is the relative phase between the input FS mode-k and mode-n.
Note that this outcome should not be confused with the notion of mode attraction, which is fundamentally different.Indeed, mode attraction implies that for a fixed input BCB coupled to one mode, then the output FS is attracted towards that mode for any arbitrary input FS.On the contrary, in the case mentioned above, the input BCB is not fixed, but depends on the specific input FS mode power distribution and relative phase.
We have performed some preliminary experiments in our TCF where the input BCB is coupled to a combination of modes, rather than just one mode.In Fig. 8 we observe that, for the same input FS, different input BCBs give rise to substantially different output FS mode distributions.Fig. 8: Preliminary experiments of all-optical mode control.Differently from Figs. 3-7, where the input BCB is coupled to one single mode leading to mode rejection, now the input BCB is coupled to a combination of modes (see relative power of input FS and BCB on the top of panels a and b).The bottom images show the far-field intensities of the output FS when the BCB power is either 0 W or 6.6 kW.The input FS is almost identical in panels a and b, and consequently the corresponding output FS is almost identical in the two cases when BCB is turned off (see the far-field with BCB=0 W in panels a and b).On the contrary, the input BCB is different in panels a and b.We see that different mode distributions of the input BCB give rise to different mode distributions of the output FS (see the far-field with BCB=6.6 KW in panels a and b).These results seem to suggest that, by properly setting the input BCB, the output FS could be shaped on demand in time and space.Error bars of ±3% are added to the measured relative powers, which represents the estimated uncertainty of our mode decomposition algorithm.
These preliminary results pave the way towards a more general idea of mode control, that is to say, the ability to control on demand and all-optically (and then potentially at ultrafast rate) the mode content of the output FS via the BCB.The latter may be even generated via back-reflection of the FS, thus without the need for an independent and additional optical source 31 .A new idea of light-self organization then emerges, where the output FS is shaped in time and space depending on its input state, leading to new opportunities like all-optical switching or highly scalable coherent combination in multimode and multicore waveguides, to name a few.These ideas are of course speculative at the moment and additional extensive investigation will be required to put them into practice.However, the theoretical and experimental results reported in this work represent ideal tools to address the above-mentioned opportunities and related challenges.

DISCUSSION
In this work, we investigated the nonlinear dynamics of multimode optical waveguides with a counter-propagating geometry.We identified two classes of multimode systems that exhibit the two opposite dynamics of mode rejection and attraction.
The first class is characterised by the occurrence of mode rejection, and is described by Eqs.(1).It is only when moving to systems supporting more than two modes that the peculiar dynamics of mode rejection can be properly understood and differentiated from mode attraction.We provide systematic evidence of mode rejection in a variety of home-made multicore fibres and commercial multimode fibres.Our experimental outcomes highlight the robustness of mode rejection against standard fabrication imperfections and external perturbations, as well as the ability to control this process via the relative state of polarization among the FS and the BCB.
The second class of multimode systems is characterised by effective mode attraction towards the BCB mode, and is described by Eqs.(2).Isotropic single-mode fibres exhibiting polarization mode attraction represent the lowest-dimensional example (N=2) belonging to this class.Although polarization mode attraction has been predicted and demonstrated almost two decades ago, our theoretical analysis generalises the idea of mode attraction to systems of arbitrary dimension N. The scope of this generalisation is two-fold.Firstly, it allows a direct comparison between Eqs.( 1) and ( 2), and therefore between the fundamental notions of rejection and attraction, in systems of arbitrary dimension N. Second, it sheds light on the universality of these processes, which goes beyond bidimensional systems.Although currently it is yet unknown the existence of multimode systems where Eqs.(2) with N>2 applies, however these may be found in the future, either in optical fibres or different physical systems.
We note that the ideas of mode attraction and rejection are intrinsically related to an input BCB that is coupled to one single mode.However, an even more rich scenario emerges when the input BCB is coupled to a combination of modes, which brings forward the more general idea of modal control.This may inspire a plethora of new applications, including all-optical switching for space-divisionmultiplexing, or tuneable coherent combination for high-power lasers, of which the experimental outcomes illustrated in Figs. 3 and 4 are indeed a basic example in the case of a dual-core fibre.
Our theoretical analysis may be adapted to the case where the BCB (FS) is generated as a backreflection of the FS (BCB) through a set of mirrors 31 or a cavity.This could lead to new understanding and concepts in multimode optical parametric oscillators and lasers, and related applications.
In conclusion, by predicting the existence of mode attraction and rejection in multimode waveguides of arbitrary dimension N, and by providing experimental evidence of the latter, this work sets the basis for a novel understanding of the complex dynamics in nonlinear multimode waveguides and suggests new scenarios and applications that can be explored by means of the theoretical, numerical and experimental tools introduced in this paper.The schematic in the Supplementary information note 5 reports a conceptual map of the above-mentioned outcomes.
It is worth noting that the ideas and theoretical results introduced in this paper encompass but are not limited to optical waveguides.The fundamental notions of attraction and rejection may be used to investigate any nonlinear system described by formally similar sets of equations, which includes classical wave thermalization and condensation phenomena [40][41][42] or non-optical systems like hydrodynamics 43,44 and matter-wave Bose-Einstein condensates 45 .

METHODS
Theoretical framework and numerical simulations.The following theoretical analysis applies to multimode and multicore fibres that, as previously mentioned, represent the waveguides of choice in our experiments.Nevertheless, this analysis could be readily adapted to different kind of Kerr-nonlinear waveguides by taking into account the related specific nonlinear cubic response.Here we consider two beams centred at the same carrier wavelength that are counter-propagating in a multimode optical fibre and coupled to a combination of N guided spatial modes.We focus on a polarization-maintaining (PM) step-index geometry as it represents the ideal scenario to investigate the complex multimode dynamics, without the burden induced by random polarisation variations.We follow a standard procedure starting from the Maxwell equations and then we ignore the fast-oscillating terms that average out to 0 34,46,47 .We also ignore Raman scattering, which indeed turns out to be negligible for the power levels used in our experiments.We finally derive the following set of coupled Schrödinger equations for the forward and backward modal amplitudes: Where: =2 for linearly co-polarised counter-propagating beams, whereas =2/3 when these are orthogonally polarized; fn(z,t) and bn(z,t) are respectively the amplitude of the forward and backward mode-n; vn is the related group velocity; γnm are the nonlinear Kerr coefficients computed from the intramodal and intermodal effective areas 48 (see Supplementary information note 3 for details on the parameters).The latter require the calculation of the spatial mode profiles, which are computed with a finite-element-method software (Comsol Multiphysics 5.6) and take into account the actual refractive index profile of the fibres under test as measured by means of an Optical Fiber Analyzer (Rayphotonics IFA-100).While the first and second term on the right-hand-side of Eqs.(1) describe respectively self and cross-phase modulation, the last term represents the intermodal power exchange among forward and backward modes.In order to simplify our analysis, the group velocity dispersion (GVD) and the higher-order dispersion terms are not included in Eqs.(1) 34,49 .Indeed, the largest GVD coefficient at the wavelength of interest (λ=1040 nm) is β2max ∼ 25 ps 2 /km, whereas the pulses used in our experiments are t0=0.5 ns wide.This corresponds to a minimum dispersion length t0 2 /β2max of several km, therefore order of magnitudes longer than the fibres under test.Similarly, propagation losses are negligible due to the short length of the fibres in use.Note that in the counter-propagating setup the input forward and backward fields are injected respectively at z=0 and z=L.Therefore, the boundary conditions for Eqs.(1) that fix the initial state of the forward and backward modes are fn(z=0,t) and bn(z=L,t), respectively.A further boundary condition defines the fields at t=0 inside the fiber ( 0<z<L).We assume the fibre is empty, that is to say, fn(z,t=0)=bn(z,t=0)=0.
Fig. S2 reports the results obtained from the numerical solution of Eqs.(1).Note that, differently from Eqs.(S1) that describes the stationary problem, Eqs.(1) account for the full space-time dynamics and then allow investigating the temporal evolution of the FS.When the BCB is off, there is no power exchange among the modes, therefore the mode power distribution of the output FS mirrors the input one.However, when the BCB is on, after a short transient time the output FS relaxes towards a stationary state.As predicted, rejection of mode-1 takes place in the output FS: the relative power of mode-1 goes from 0.40 when the BCB is off (Fig. S2a) down to ~0.018 when the BCB is on (Fig. S2b), in excellent agreement with our estimation of 0.019.A striking feature of Eqs.(1,2) is that they exhibit the same rejection and attraction dynamics even in the general case where the Kerr coefficients   are arbitrary.Indeed, we still observe a relaxation towards a stationary state and again we identify the nonlinear interaction among forward and signal modes as the underpinning "driving force" leading to rejection or attraction.
To prove this, we have run 500 simulations of Eqs.( 1) with 4 modes and by using an arbitrary set of Kerr-coefficients (see the caption of Fig. S4 for details).In each simulation, the fiber length is 1 m, the total FS and BCB power is 5kW and the BCB is coupled to mode-1.However, the relative power and phase of the input FS modes is random, which results in dispersed power distribution functions as reported in Fig. S4, top row.The bottom row of Fig. S4 shows instead the corresponding output power distribution function, from which we clearly observe an effective rejection of mode-1.Indeed, in all the 500 instances the power of the output FS coupled to mode-1 is lower than 10% (and <5% in 430 out of 500 instances).It is worth noting that these outcomes hold true even in the case where the input FS fluctuates in time, provided that the time scale of the fluctuations is longer than the characteristic response time of the system 1 , which here can be roughly approximated with c/( avg •P f ),  avg being the average Kerr coefficient.
We have performed the same numerical analysis in the case of Eqs.( 2), and as shown in Fig. S5 we observe now a clear attraction towards mode-1.Indeed, in all the 500 instances most of the output FS power is carried by mode-1, and consequently the relative power of the others mode is almost null (seeFig.S5, bottom row).These results seem therefore to indicate that mode rejection and attraction are universal features of multimode nonlinear systems ruled by Eqs.( 1) and ( 2), respectively.

Note 3. Fibre characterization and experimental parameters
Tables S1 and S2 show the microscope measurements of the fibre diameter, core size and pitch for the dual-core-fibre (DCF) and three-core-fibre (TCF) fabricated in our cleanrooms.The measurements are taken over two fibre sections, at the start and then at the end of the pulled fibre, whose total length is about 1 km.Note that in the experiments we use however short fibre segments (1m for the DCF, 40 cm for the TCF).While ideally the cores size Dc1, Dc2 and Dc3 should be identical (5 um), in practice due to fabrication errors the difference is up to ~4% in the same section.Similarly, there is a ~2% variation from the core-to-core pitch P12 to P13 in the TCF.36 OD: outer fibre diameter, Dc1,Dc2,Dc3: diameter of core1,core2,core3; P12: distance between the centres of core 1 and 2.; P13: distance between the centres of core 1 and 3; α= 30 deg.
The microscope measurements reported above, along with a standard measurement of the refractive index profile, are then used to model the DCF and the TCF with a finite-element-method software (Comsol Multyphisics 5.6), and then to compute the spatial mode profiles (reported in Fig. 2b), the group velocity and the dispersion of each spatial mode, as well as the effective areas from which we derive the nonlinear Kerr coefficients, reported in Table S3 and S4.Note that both the DCF and the TCF are polarization-maintaining and modes are linearly polarized along the horizontal or vertical direction.This is confirmed by our experiments, where we find an extinction-ratio >10 dB.The data reported in Tables S3 and S4 are almost independent of the mode polarization.
For each one of the experiments illustrated in Fig. 3 (DCF) and Fig. 5 (TCF), we report in Table S5 the corresponding peak power of the input FS and input BCB, along with their mode decomposition.Note that the power P f of the input FS is fixed, whereas the power P b of the BCB is gradually increased from 0 to its maximum value.The values of peak power have been chosen to achieve a high nonlinear regime, which is necessary condition to observe the mode rejection dynamics.As reported in the section "Methods-Theory of mode rejection", the parameter T=(T f T b ) 1/2 = Lγ(P f P b ) 1/2 represents the overall system nonlinearity.As a rule of thumb, our simulations indicate that we need T>2 to observe effective mode rejection.In the fibres under test, where the Kerr coefficients are generally different each other, γ is replaced by their arithmetic average γ avg .Moreover, because we use pulsed beams, the fibre length L is replaced by the actual interaction length L in =2•t 0 •c (c=speed of light in the fibre, t 0 =pulse width, 0.5 ns).In conclusion, the parameter T eff = L in γ avg (P f P b ) 1/2 , reported in the last row of Table S5, is used to evaluate the system nonlinearity.The peak powers P f and P b are properly chosen so as to meet the condition T eff >2.The mode decomposition provides an estimation of both the relative power and relative phase for each mode.In the case of the DCF (Fig. 3a,b,c) the relative phase reported in Table S5 is the one between SM e and SM o .In the case of the TCF (Fig. 5a,b,c) the first value indicates the relative phase between SM 1 and SM 2 , the second value between SM 1 and SM 3 .When running our numerical simulations of Eqs.(1), which are reported in Fig. 3 and Fig. 5 along with the experimental results, we have used the fibre and experimental parameters reported in Tables S3-S5.The temporal pulses are modelled with a super-gaussian shape of 6 th order.The pulse width (500 ps in the experiments) is the only free-parameter in our simulations.It is adjusted between 500ps and 700ps to get the good quantitative agreement with experiments illustrated in Figs. 3 and Fig. 5.Note that in the case of the PM-6MF, we were unable to fit quantitatively the experimental results via numerical simulations.Indeed, our simulations indicates that when the number of modes becomes significant (typically >4), then the specific dynamic of each individual mode is extremely sensitive to small variations of the input conditions and fibre parameters.This however does not change the fundamental outcome, namely mode rejection of the BCB mode is always achieved.Finally, note that the estimated beat-length among the pairs of modes (LP 11a , LP 11b ) and (LP 21a , LP 21b ) is < 1mm, therefore much shorter than the fibre employed in the experiments.This makes the fibre to support in effect 6 distinct modes rather than just 4. As in the case of the DCF and TCF, the PM-6MF is polarization-maintaining, with a polarization extinction-ratio >18 dB.In Fig. S6 we represent the birefringent axes of the DCF, TCF and PM-6MF.We have recorded some Supplementary movies illustrating the rejection dynamics in the fibres under tests.The movies are named after the corresponding panel in Figs.3,5 and 6 (e.g.Supplementary Movie for Figure3a.movreports the results illustrated in Fig. 3a).In each movie, the input FS is launched with a fixed power (see Note 3 for experimental parameters).On the contrary, the input BCB power is gradually increased from 0 to its maximum value.For a given BCB power, each movie frame reports the following information (see Fig. S7):

Table S5: Experimental parameters
(1) the current BCB power; (2) the corresponding relative power of the rejected mode in the output FS (obtained from mode decomposition); (3) the measured far-field intensity profile of the output FS; (4) the reconstructed far-field intensity profile obtained when using the mode weights and phase from mode decomposition; (5) the 2D spatial correlation coefficient (Corr) between the measured and the reconstructed farfield 2 .It should be noted that Corr is typically >98%, which attests the accuracy of our mode decomposition algorithm.

Fig. 1 :
Fig. 1: Illustration of mode attraction and rejection in multimode waveguides.The input FS (FSin) and input BCB (BCBin) are launched at the two opposite ends of an optical fibre.a, b, Mode attraction in single-mode isotropic optical fibres.In these fibres, the left and right circular polarisation states play the role of robust attractors.When the BCB is turned off (a), the mode content of the output FS (FSout) mirrors the input one (right and left circular polarizations have respectively relative powers Pr and Pl , where Pr and Pl are arbitrary and Pr+Pl =100%).When the BCB is turned on and has a similar power to the FS (b), their nonlinear interaction modifies the mode content.If the input BCB is coupled to one circular polarization mode (left circularly polarised in this example), then irrespective of the mode content of the input FS, the output FS is attracted to the mode of the input BCB (100% of the power in the left circularly polarised mode).

Fig. 2 :
Fig. 2: Experimental setup and test fibres.a, Schematic of the experimental setup and beam path for the FS (in yellow) and the BCB (in red).PBS=polarization beam splitter; SLM=spatial light modulator; PM=power meter; PP=phase plate; FF=farfield; NF=near-field.The fibre ends z=0 (input FS, output BCB) and z=L (output FS, input BCB) are indicated with black dashed lines.b, Microscope image of the 3 test fibres and related spatial modes used in the experiments: a homemade dualcore fibre (DCF) supporting 2 modes (SMe,o) , a homemade tri-core fibre (TCF) supporting 3 modes (SM1,2,3) and a

Fig. 3 :
Fig. 3: Mode rejection in the DCF.Mode decomposition of output FS in the DCF fibre (1 m long) for different launching conditions (experiments(exp): dots; simulations(simu): lines).The bottom images show the far-field intensities of the output FS for 3 distinct values of BCB power.The input FS is coupled to different combinations of SMe and SMo (see relative

Fig. 4 :
Fig.4: Robustness and control of mode rejection.Input FS and BCB are linearly polarized and launched with ~ 5kW peak power in the DCF.a, Mode decomposition of the output FS for off/on BCB.The input FS is coupled to a combination of the two supermodes, whereas the input BCB is coupled to SMe.FS and BCB are co-polarised.The fibre is perturbed 5 different times.Each time, the perturbation consists in compressing or bending the fibre with different levels of intensity and in different points.For each perturbation, the mode decomposition of the output FS is computed, and the output FS far-field intensity is imaged when the BCB is either turned off or on.When BCB is on, robust rejection of SMe occurs in the output FS (<10% power in all 5 cases).b, Mode decomposition and far-field intensity of the output FS as a function of the relative angle α between the polarisation orientations of the input FS and the BCB.The far-field with BCB off is reported for comparison.Error bars of ±3% are added to the measured relative power, which represents the estimated uncertainty of our mode decomposition algorithm.

Fig. 5 :
Fig. 5: Mode rejection in the TCF.Mode decomposition of the output FS in the homemade TCF (40 cm long) for 3 different launching conditions, and comparison with numerical simulations.The input FS is coupled to different combinations of guided supermodes SM1,2,3 (see relative power of input FS on the top of each panel) and launched with fixed power of 4.23 kW, 4.23 kW and 4.33 kW in panels a, b, c respectively.The input BCB is mainly coupled to SM1(a), SM2(b) or SM3(c).We observe almost full rejection of SM1 and SM2 in a and b, respectively, when the BCB has ~ 6.2kW(a) or 4.5kW(b) peak power.Rejection of SM3 reported in c is instead less effective.Nevertheless, we observe a clear trend of rejection in line with numerical simulations, indicating that full rejection maybe achieved by increasing the total power of FS and/or BCB.Error bars of ±3% are added to the measured relative power of the rejected mode, which represents the estimated uncertainty of our mode decomposition algorithm.

Fig. 6 :
Fig. 6: Mode rejection in the 6-mode fibre (PM-2000 from Thorlabs, 1 m long).The input FS is coupled to different combinations of modes and launched with fixed power of 4.00 kW, 4.10 kW and 3.20 kW in panels a, b, c respectively.The input BCB is mainly coupled either to LP01(a, b) or LP11b(c).Error bars of ±3% are added to the measured relative power of the rejected mode, which represents the estimated uncertainty of our mode decomposition algorithm.

Fig. S2 : 2 /
Fig.S2: Temporal evolution of the output FS relative power |f n (t,L)| 2 /P f of mode-n (n={1,2,3,4}) from the simulation of Eqs.(1).When the BCB is off (a), the mode content of the output FS is unchanged with respect to the input one.The time delay TOF indicates the time-of-flight.When the BCB is on (b), then after a short transient time the system relaxes towards a stationary state.The input BCB is coupled to mode-1, therefore the output FS rejects mode-1, whose output relative power |f 1 (t,L)| 2 /P f is as low as ~0.018 when the stationary state is achieved.One may perform a similar analysis in the case of mode attraction.We simulate Eqs.(2) (again in the ideal case   =  and   = (1/2)) and we consider the same input conditions and parameters used in the previous example.In analogy with the previous case, |  () | 2 = | 1 (0)| 2 /  and |  ()| 2 = | 1 ()| 2 /  describes the input and output relative power carried by the forward mode-1, respectively.The results are illustrated in Fig.S3.Now mode attraction takes place in the output FS.The relative power of mode-1 goes from 0.40 when the BCB is off (Fig.S3a) up to ~0.97 when the BCB is on (Fig.S2b),once again in excellent agreement with our estimate ( |  ()| 2 =| 1 ()| 2 /   ~ 1 −  (|  () |) 2 / 2 =0.968.

Fig. S3 :
Fig.S3: Temporal evolution of the output FS relative power |f n (t,L)| 2 /P f of mode-n (n={1,2,3,4}) from the simulation of Eqs.(2).When the BCB is off (a), the mode content of the output FS is unchanged with respect to the input one.Mode attraction towards mode-1 takes place when the BCB is on (b): the output relative power |f 1 (t,L)| 2 /P f grows up to ~0.97.

Fig. S7 .
Fig. S7.Example of video frame from Figure5a.movand related information . The condition |D R (L)|→0 implies therefore that f m (L) ∼0: namely, irrespective of the mode distribution of the input FS in z=0, the output FS in z=L carries no energy in the mode-m, that is to say the mode-m is rejected at the output.

Table S2 :
Microscope measurements of the TCF intramodal nonlinear Kerr coefficient of the SMe(o) mode, γ12: intermodal Kerr coefficient between the SMe and the SMo mode; β1, β2: inverse group velocity, and group velocity dispersion coefficient of the modes.

Table S4 :
estimation of the TCF parameters