Spatio-spectral light-by-light moulding in multimode fibre

Domain: chemistry
Published: 2026-03-07
Editorial outcome: major_revision

Abstract
Controlling complex light waves to achieve desired behaviours or characteristics on demand presents a significant challenge. This task becomes even more complicated when manipulating speckled light beams, owing to their inherently fuzzy intensity and phase structures. Here we demonstrate that a weak speckled second-harmonic signal in a multimode graded-index fibre can be manipulated via its conservative interaction with a high-power co-propagating fundamental pump wave. Specifically, the spatial quality of the signal can be either enhanced or degraded by varying the pump’s power or its modal distribution. The underlying physical mechanism is the optically induced mode conversion that can be controlled by the pump beam shape. This phenomenon enables new possibilities to manipulate complex light in nonlinear multimode waveguides. A striking example of this novel light-by-light control is the experimentally observed enhancement or partial suppression of the visible Raman Stokes cascade regulated by the second harmonic beam, while modulated by the mode power distribution of the fundamental beam.

Introduction
The manipulation of light-by-light through cross-phase modulation (XPM) via the Kerr effect was extensively studied in the 1980s and 1990s [1-7]. The Kerr-induced cross-action of an intense pump light beam can alter a weak copropagating probe beam's temporal, spectral and spatial profiles. This interaction occurs conservatively without any power exchange between the pump and the probe. For instance, the additional phase shift that is induced by the strong wave may enable phase-matching between a fundamental frequency (FF) beam and its second harmonic (SH), allowing the control of nonlinear frequency conversion processes [8]. A lesser-known method for conservative control between light beams of different frequencies is the optically induced mode coupling (OIMC). This parametric interaction permits frequency translation without exponential gain, thereby avoiding spontaneous noise emission. As a result, the frequency translation associated with OIMC preserves the quantum state of a signal [9-10]. OIMC can involve two pumps at different wavelengths: maximum conversion efficiency from a signal to an idler is obtained at phase matching, under various configurations of polarization states, spatial modes, and wavelengths for pumps, signals, and idlers [9,12-13].

Nonlinear effects in multimode optical fibres provide a leap in our capacity for all-optical light manipulation: recent experiments have revealed that controlling the input wavefront permits to shape the spatial, temporal, and spectral properties of highly multimode beams [14-15]. In this framework, the effect of Kerr beam self-cleaning (BSC) [16-27], which allows the transformation of a speckle into a quasi-single-mode beam at the output of a graded-index (GRIN) highly multimode optical fibre (MMF), is particularly intriguing. BSC was initially interpreted as a nonlinear interaction between linear beam self-imaging and Kerr nonlinearity. The resulting nonlinear refractive index long-period grating [16-17] facilitates the phase-matching of frequency degenerate inter-modal four-wave mixing (IM-FWM), thus enhancing the nonlinear power redistribution [17-19]. From the application point of view, the most important aspect of BSC is the enhancement of the beam spatial quality, that is, the quenching of the beam’s M2 factor. Although it is a purely conservative process, BSC has also been observed in lossy and gain fibres [25-26] and in active tapers [28-29]. This has opened a new avenue for developing powerful fibre lasers operating in Q-switched or mode-locked configurations [26-27, 30-33]. Experiments have shown that BSC maintains the spatial coherence of the input laser [34], which enables new applications, such as imaging through MMFs [35].

Nevertheless, explaining the underlying mechanism for BSC has led to an ongoing controversial debate. Currently, the most widely accepted theory of BSC is based on thermodynamic considerations [36-38]. Within such a theoretical framework, the concept of BSC has been extended to the nonlinear dynamics of two co-propagating, orthogonally polarized beams [39-42]. However, experimental demonstrations of BSC involving multiple beams have so far been limited to a single strong carrier wavelength, thus preventing applications of this effect to polychromatic beams. Our approach opens new opportunities for supercontinuum generation in multimode fibers by exploiting the spatial overlap between high- and low-power multicolor beams. New perspectives also include the improved spatial quality control of multimode laser sources, and the possibility of ultrafast routing and switching between different channels in the telecommunications domain.

In this work, we elevate the field of light manipulation in MMFs to a new dimension, involving the nonlinear interaction among multiple laser beams of different carrier wavelengths. We experimentally demonstrate that the evolution of a weak speckled beam in the visible range can be controlled using a combination of XPM and nondegenerate OIMC with an intense pump beam in the near infrared, which undergoes its own spatial reshaping through the Kerr BSC process, but also its own spectral broadening by means of the Raman scattering process and soliton generation dynamics for very high peak powers. In our experiments the FF infrared beam at 1030 nm is provided by a Yb laser, while the visible beam is obtained via a nonlinear conversion from FF to SH (515 nm). Both beams are simultaneously injected into a GRIN MMF. By varying the power and modal distribution of the FF beam, the M² quality factor of the visible SH beam at the MMF output can be either decreased (beam cross-cleaning, BXC) or increased (beam cross-spoiling, BXS). The associated modification of the mode power distribution of the SH beam results from phase-matched IM-FWM-induced energy exchanges between its guided modes, driven by the intense FF beam. This mechanism for controlling the power exchange between modes can be compared, by analogy, to an intra-fiber controllable active spatial diffuser, sending energy from mode to mode on demand. In a first series of experiments using short laser pulses, the effect is limited by dispersive temporal walk-off. In a second set of experiments, employing long laser pulses in stead, we show that beam cross-interactions can significantly alter the beam shape of Raman cascade of Stokes lines. In this case, our experiments reveal that it is possible to reshape the number and modal content of the Stokes lines generated by a visible light beam by controlling the spatial profile and power of the near-infrared beam. We corroborate our observations through numerical simulations. Additional elements can be found in the supplementary material. Nonlinear shaping of the spatial profile of a laser beam using another laser beam paves the way for applications such as beam switching or manipulation in ultrafast regimes, e.g., for deep biological imaging.

Results
Experimental results. In our experiments (Figure 1(a)), a Gaussian beam composed of 100 ps chirped pulses at 1030 nm is first split into its two orthogonal polarization components with adjustable powers (Figure 1(a)). The vertical component of the fundamental frequency (FF) pump beam is converted to its second harmonic (SH) by a type I BBO crystal positioned for strong phase mismatch conditions; the unconverted portion of the FF is filtered out. Next, the SH is temporally and spatially superimposed with the horizontal component of the FF beam by a beam splitter and a half-waveplate, before being coupled into the 50/125 µm GRIN fibre (see supplementary section 7). Under these conditions, both beams retain horizontal linear polarization, and their coupling efficiencies into the fibre can be adjusted. The green beam at 515 nm is weak enough, so that its own propagation in the fibre is purely linear. On the other hand, the pump beam may undergo spatial BSC thanks to its high peak power (see supplementary section 2). As a result, the transverse intensity profile of the pump beam at the fibre output evolves into a clean bell shape, which is observable in both near and far fields. By adjusting the delay line between the visible SH weak beam and the strong FF pump, so that the corresponding pulses no longer overlap in time, we verified that the weak SH beam remained unchanged, even in the presence of the FF beam. Conversely, when pulse synchronization among FF and SH pulses is well-tuned, we observed significant changes in the spatial distribution of the SH at 515 nm. These could be controlled by varying the power of the FF pump beam at 1030 nm, or its spatial coupling conditions. To further investigate this process, we analysed the near (Figures 1(c) and 1(d)) and far fields of the weak SH beam and performed numerical processing of both fields. This allowed the calculation of the beams’ spatial phase and M² coefficients, helping to identify their modal energy distribution (Figures 1(b,e,f), respectively).

For the first setting of the input coupling conditions of the FF beam into the fibre (i.e., straight on axis), we observed that increasing the FF power led to mode power redistribution for the SH, whereby lower-order modes (LOMs) of the SH are favoured with respect to higher-order modes (HOMs). This spatial beam reshaping, or BXC, is obtained by the OIMC process, which does not lead to energy exchange between the FF and the SH beams. Our analysis of near and far-field images at the fibre output confirms the beam cross-shaping. The M² coefficient of the SH beam, calculated by standard definitions [43], followed the same trend, decreasing along both the X and Y directions as the FF power grew larger (Figure 1(b), right). The maximum FF peak power in our experiment was approximately 40 kW, limited by input face fibre degradation. This power value remains far below levels typically associated with self-focusing effects.

We could reverse the beam interaction dynamics by slightly adjusting the FF beam’s input conditions (i.e., its focus point). Increasing the FF peak power led to a mode power redistribution for the SH, favouring HOMs. This BXS of the SH beam was accompanied by a corresponding increase of its M² coefficient (Figure 1(b), left). Near and far-field analyses confirmed this opposite trend. As we shall see in the subsequent theory section, the redistribution of energy between LOMs and HOMs of the SH beam is driven by the mode power distribution of the pump beam. That is, the FF beam modifies via OIMC the mode content of the weak SH beam. This may change the direction of energy flow among the modes of the SH beam, as testified by the corresponding decrease or increase of its quality factor. The occurrence of BXC or BXS in the SH beam is governed by nonlinear phase shifts, which either preserve or break phase-matching in the OIMC process, respectively.

In a second series of experiments (Figure 2), we excited additional modes in the FF to achieve a better spatial overlap with the SH. In this scenario, BSC efficiency for the FF beam was reduced, a bell-shaped beam is no longer formed at the fibre output (Figure 2 (a)). Nevertheless, by increasing the FF power we observed a gradual increase in the fundamental mode population of the SH beam (see Figure 2(b) and 2(c)). Maximum SH BXC was observed for input FF peak powers close to 23 kW, leading to a corresponding reduction of the M² coefficient for the SH beam. However, the process reversed at higher input FF powers (up to 100 kW), even without altering the input coupling conditions. Namely, the proportion of energy in the fundamental and other LOMs of the SH beam now decreases as the FF power grows larger (Figure 2(b)). In this regime, HOMs of the SH beam get progressively more populated, which can be interpreted as BXS, leading to an increased SH beam divergence at the fibre output. It is important to note that, to increase the peak power up to 100 kW, it was necessary to reduce the pulse duration from 100 ps down to 10 ps in order to avoid fibre damaging. This was achieved using the same initial 250 fs Fourier transform limited pump beam (see Methods) and changing the stretching passing from 100 ps to 10 ps.

As we have seen, the BXC efficiency can be limited either by the input FF peak power or by the temporal walk-off between FF and SH pulses. To further investigate the impact of BXC on multimode propagation, we conducted a third series of experiments where both individual SH and FF beams propagate in the fully nonlinear regime. We used a longer GRIN fibre (50 m) with a smaller core diameter (27 µm) (see also supplementary section 7); to minimize walk-off between the FF and the SH, we used another laser source that emits longer pulses, with a duration of 650 ps. By increasing the power of both SH and FF beams, while maintaining a peak power imbalance of 12:1 in favour of the infrared pump, we observed stimulated Raman scattering (SRS) cascades at both wavelengths, involving the generation of multiple Stokes lines (see Figure 3). The FF beam broadened spectrally beyond 1.3 µm, owing to Raman-shifted temporal soliton dynamics. On the other hand, SH propagation only produced Raman Stokes lines, because of the normal dispersion regime at the SH. Furthermore, both beams undergo nonlinear propagation leading to a spatial self-cleaning process and Raman beam clean-up, as already reported in the literature [18-20]. However, in the following paragraph, we present the impact of the cross-action between the two beams (FF+SH) on the SH beam, demonstrating BXC and BXS effects within the SH Raman cascade.

Nondegenerate OIMC between FF and SH beams established a direct link between the visible and fundamental Raman cascades. This interaction allowed the visible Raman cascade to be either enhanced by the BXC (Figures 3(a) and 3(b)) or suppressed (Figures 3(c) and 3(d)) by the BXS effects, induced by the FF pump beam before its spectral broadening. Mutual beam interaction dynamics depends on the intensity, polarization, and input coupling conditions of the infrared FF beam. The observed nonlinear beam dynamics in the presence of the Kerr effect and SRS resembles the behaviour observed in the pure Kerr case. Now, BXC redistributes the energy among the transverse modes of the SH beam, which controls its Raman gain and the associated Stokes cascade. This allowed us to determine the mode content of the beam by separating its spectral components by means of a grating, as depicted in Figure 3(e). Interestingly, we observed that the oscillating transverse mode of the Raman cascade could change (Figures 3(f) and 3(g)) and that the number of Stokes lines could be decreased with a change in the transverse mode (Figures 3(h) and 3(i)). This light-by-light switching was observed across varying numbers of Stokes lines, demonstrating that spatial reshaping of the SH beam preceded the generation of its own Stokes lines. We also note that the modal cascade with the appearance of the LP02-like mode remains stable over all three Stokes lines, apparently blocking or stabilizing the Raman beam clean-up process (Figure 3(f) and 3(g)). We also measured that more than 80% of the energy contained in the three Stokes Raman lines oscillating in the fundamental mode is transferred into the LP02-like mode, clearly illustrating the highly energetic impact that this cross-process can introduce.

Numerical results. We conducted numerical simulations to analyse the spatial reshaping of the weak SH beam under BXC or BXS, influenced by the FF beam. We used two coupled equations, one for the FF and one for the SH beam, which interact via asymmetric OIMC and XPM, owing to their different wavelengths, powers, and temporal walk-off between their pulses (see Methods). We also included the linear random mode coupling terms. In the linear propagation regime, no evolution of the beams was observed; all excited modes retain their energy, and the M² coefficient remains constant for both beams (see Supplementary section 3a). Simulations show the presence of a spatial BSC process (Figure 4) in the pump beam whenever its power grows above a certain threshold value. As a result, the weak SH beam undergoes BXC, with spatial remodelling leading to power transfer into its fundamental mode at the expense of HOMs (Figure 4(f)). This BXC occurs without any energy exchange between the two beams: no second-order nonlinearity is included in the model. Moreover, when the input peak power of the FF beam is further increased, we observed that BXC starts to saturate, until it gradually reverses (Figure 5). In this regime, the LOMs of the SH beam are now depleted in favour of its HOMs (Figure 5(f)). This inversion of behaviour occurs without any modification of the spatial input coupling conditions of the FF or the SH beams, at variance with the experimental results that we illustrated in Figure 2. Here, the only parameter that varied in our simulations was the FF input peak power. It is important to note that the brightness improvement of the FF pump beam associated with BSC remains limited when compared with the brightness improvement of the weak SH beam under BXC.

We considered input Gaussian beams for both the FF and the SH, centred on the fibre axis in simulations. As a result, only even modes are excited (Figure 4(c)(h) and Figure 5(c)(h)). In the occurrence of both BXC and BXS of the weak SH beam, we observed that odd modes remain largely unpopulated; this phenomenon has also been highlighted in the simpler case of BSC, as noted in [44]. The authors identified that mode groups containing radially symmetric modes are more populated than odd modes, a natural consequence of injecting a radially symmetric coherent beam into the MMF. They also noted that nonlinear modal interactions have varying efficiencies, due to the significant differences in nonlinear overlap coefficients between modes, which significantly affect the rate of convergence toward a steady-state mode power distribution.

Temporal walk-off between the weak SH and the infrared FF beam plays a significant role in the mechanism of BXC or BXS. To evaluate this, we ran numerical simulations without temporal walk-off (see Supplementary section 3b). The results revealed a significant increase in modulation from the infrared to the green beam, along with a more pronounced change in the M² coefficient. Experimentally, temporal walk-off can be minimized by using two closely spaced wavelengths, or a longer pulse relative to the propagation time in the fibre, as realized in our third experiment in strong nonlinear propagation regime involving a remodelling of the Raman cascade.

Methods
Experiments. The detailed experimental setup is presented in Figure 1(a). It consists of a femtosecond-picosecond laser source delivering pulses between 250 fs and 100 ps (1030 nm), and a laser source with a pulse duration of 650 ps. The initial beam is separated into two parts using a polarizer and a half-wave plate. The first beam is converted into a BBO crystal cut for SH generation, and the remaining part of the infrared pump is removed using a BG18 filter. Thus, the SH is not generated inside the optical fiber, but it is coupled into it after conversion in a BBO crystal. The second part of the pump beam is attenuated by a passive filter and resynchronized with the SH of the first beam by means of two delay lines placed on each arm of the Mach-Zehnder interferometer. The synchronization accuracy is estimated to be 0.2 ps based on the optimization of the third harmonic generation correlation between SH and FF beams. A wideband dichroic plate is used. The two beams are temporally and spatially superimposed at the fibre input with linear polarization states by using a half-wave plate at the FF wavelength placed before the coupling lens. The fibre excitation process is carried out using a converging lens with a focal length of 50 mm. The output beams are analysed by two CCD cameras, allowing near- and far-field recordings for both beams. The first fibre is a 10 m long GRIN 50/125 µm multimode guide. A second GRIN fibre, 50 m long with a 27 µm core diameter, is instead used for Raman cascade generation. The images on Figures 3(f), 3(g), 3(h) and 3(i) are obtained by spatio-spectral analysis, using an optical grating to disperse the light in the horizontal dimension and reveal the far-field image of the beam in the vertical plane (Figure 3(e)). The results presented in Figures 2 and 3 are obtained by observing the green beam in the near-field domain and by readjusting only the coupling of the FF beam. The far-field associated with the green beam is then obtained in a second step, when collecting the final results.

Numerical simulations: For the spatial propagation in both FF beam and SH, we solved the GNLSE by considering an input Gaussian beam with a FWHM of 35 𝜇𝑚 for both the SH and the FF. These simulations consider diffraction, material dispersion, waveguide contribution and Kerr effect (𝑛 = 2.6⋅10−20𝑚2/𝑊). The scalar propagation for both field is solved via a split-step algorithm in the domain of spatial frequencies with a 2D fast Fourier transform for the diffraction term and in the spatial domain for waveguide contribution and Kerr effect. The two equations are incoherently coupled via the term of cross-phase modulation as follows:
𝜕𝐴 ∇2𝐴 (𝑘2(𝑥,𝑦)−𝑘2 ) 
𝐹𝐹 = 𝑖 ⊥ 𝐹𝐹 +𝑖 𝐹𝐹 𝐴 +𝑖𝛾 (|𝐴 |2+2|𝐴 |2)𝐴 ,
𝜕𝑧 2𝑘 2𝑘 𝐹𝐹 𝐹𝐹 𝐹𝐹 𝑆𝐻 𝐹𝐹 
𝐹𝐹 𝐹𝐹 (1)
𝜕𝐴 ∇2𝐴 (𝑘2(𝑥,𝑦)−𝑘2 ) 
{ 𝜕𝑧 𝑆𝐻 = 𝑖 2 ⊥ 𝑘 𝑆 𝑆 𝐻 𝐻 +𝑖 2𝑘 𝑆𝐻 𝑆𝐻 𝐴 𝑆𝐻 +𝑖𝛾 𝑆𝐻 (|𝐴 𝑆𝐻 |2 S+2|𝐴 𝐹𝐹 |2)𝐴 𝑆𝐻 

In equation 1 𝐴 and 𝐴 denote the complex field evelopes at SH and FF respectively and 𝛾 = 𝑖 𝑛 𝜔 ⁄𝑐 denote the nonlinear coefficients at the two carrier wavelengths. The first term in the right-hand side of the equation denotes the diffraction of the beam, the second one the waveguide condition and the third term accounts for the non-linear effects. Temporal effects are not included (to lighting the computational effort) and the consequences of the temporal walk-off effect are implemented by limiting the propagation length where the XPM interaction is active. In the simulations after 3 m the cross terms between SH and FF were eliminated. It is also worth noting that in the FF equation, the cross-phase modulation term proportional to |𝐴 𝑆𝐻 |2, could be omitted as the 𝐼 ≫ 𝐼 . While the power of the SH was kept constant at 𝑃 = 0.28𝑘𝑊 and the FF was increased until 𝑃 = 600𝑘𝑊. The wave-guide specifications were for a GRIN fibre 50/125. We assumed core and cladding refractive indices at FF of 1.4653 and 1.45 respectively. At SH the refractive indices are 1.4765 and 1.4615. The numerical simulation was performed with a longitudinal step of 2 𝜇𝑚, a transverse spatial domain of 90𝜇𝑚×90𝜇𝑚 with a resolution of 0.35 𝜇𝑚. The random mode coupling was simulated by weak random deformations of the refractive index with a coarse step applied at a random distance of average length of 5mm.

The modal decomposition performed both in the experimental and the simulations was done by means of a stochastic parallel gradient descent optimization algorithm [46]. The optimization was supervised by epochs of 100 times the number of modes considered, stopping criteria was either 100 epochs either an increase in the correlation experimental-simulation lower than 10-4. For the mode decomposition of the simulated data, the far field requirement was omitted as the phase of the beam was known.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript presents an approach utilizing optically induced mode coupling (OIMC) and cross-phase modulation (XPM) to realize spatial mode modulation between beams of different frequencies in multimode optical systems. The experimental results demonstrate two key effects: (i) In the absence of energy exchange, a strong fundamental frequency (FF) pump beam can bidirectionally modulate the spatial mode distribution of a weak second harmonic (SH) beam, leading to either beam cross-conversion (BXC), characterized by low-order mode enhancement, or beam cross-switching (BXS), characterized by high-order mode enhancement; (ii) This modulation mechanism significantly impacts the SH beam’s mode composition, beam quality, and nonlinear Raman cascade behavior.\nAdditionally, numerical simulations validate the influence of FF power on the SH mode distribution and highlight the crucial role of temporal walk-off in the modulation process. The simulated results seem to be in good agreement with experimental observations.\nOverall, the work provides a different perspective on optical control in multimode systems. However, I am unable to recommend the manuscript for publication in its current form. Below, I outline several questions and suggestions for the authors to address:\nGeneral comments:\nAs is well known, standard multimode fibers (MMFs), such as silica-based graded-index (GRIN) fibers, do not exhibit second-order nonlinear effects like second harmonic generation (SHG) under normal conditions due to the centrosymmetric nature of the material, which prohibits second-order susceptibility (χ(2)). Therefore, it is unclear how the authors achieve SHG, Raman Stokes cascades, and cross-phase modulation (XPM) via the Kerr effect simultaneously within the same GRIN MMF. The manuscript should clearly explain the physical mechanisms and nonlinear processes involved, especially how SHG is realized in such a system. If SHG occurs through special means—such as surface-induced symmetry breaking, quasi-phase matching, or use of non-silica dopants—those conditions and techniques must be explicitly detailed and justified.\nAs the authors are aware, beam self-cleaning (BSC) in multimode fibers is a well-established nonlinear phenomenon that has been extensively studied in the literature. It is generally understood within the framework of four-wave mixing (FWM), nonlinear mode coupling, and optical thermalization processes. In light of this, the manuscript should clarify how the proposed XPM-induced BSC fits within or deviates from this established theoretical framework. Specifically, the authors should elaborate on whether the observed effects are a novel manifestation of existing BSC mechanisms or represent a fundamentally new regime. A more rigorous theoretical discussion, possibly comparing the dynamics and physical conditions of XPM-induced BSC with traditional FWM- and thermalization-driven models, would be necessary and valuable for positioning this work within the broader context of nonlinear multimode optics.\nOther comments and suggestions:\n1. On the Energy Exchange Between FF and SH Beams: The authors should provide a more detailed evaluation, either experimental or theoretical, to confirm whether there is indeed no significant power exchange between the fundamental frequency (FF) and second harmonic (SH) beams. In high-power nonlinear regimes, energy transfer can critically affect the modal power balance of the SH beam, and this needs to be rigorously ruled out to support the core assumption of purely optically induced mode coupling (OIMC).\n2. On the Nature of the Mode Power Distribution in Figure 2(c): The power distribution of the SH modes shown in Figure 2(c) appears unusually smooth, which raises the question of whether it represents a single-shot measurement or an averaged statistical result. In practice, such idealized mode profiles are rarely observed, especially under strong nonlinear propagation. The authors are encouraged to clarify the data processing method and compare it to more realistic cases, such as Figure S5 in Nature Physics 2022, 18, 685–690, where fluctuations in power distribution are evident.\n3. On the Counterintuitive Modal Inversion at High FF Power: The authors report a reversal of modal energy flow at FF peak powers approaching 100 kW, where high-order modes (HOMs) of the SH beam become increasingly populated. This behavior is counterintuitive and requires further explanation. Specifically, the authors should assess whether the FF power in a 10 m fiber segment exceeds the thresholds for stimulated Raman scattering (SRS) or soliton formation. If such nonlinear effects are present, the observed behavior cannot be solely attributed to OIMC and may involve more complex nonlinear dynamics.\n4. On the Impact of Pulse Duration Compression: To achieve 100 kW peak power, the pulse duration was reduced from 100 ps to 10 ps, which implies a significant increase in spectral bandwidth due to the time-bandwidth product. The authors should clarify whether this spectral broadening introduces any additional dispersion or nonlinear effects that might influence OIMC or the Raman cascade dynamics. Such effects could affect the interpretation of mode control results.\n5. On the Feasibility of Temporal Overlap Between FF and SH Pulses: Given the group velocity mismatch between FF and SH wavelengths, achieving precise temporal and spatial overlap of low-coherence pulses is highly non-trivial. The authors are encouraged to provide a detailed and quantitative assessment of how this synchronization is achieved in the experiment, including potential experimental uncertainties. Without such justification, the validity of the claimed nonlinear interactions between temporally synchronized FF and SH pulses may be called into question.6. In the introduction section where the research background and methodology are presented, the motivation is currently summarized in only one sentence: “However, to our knowledge, experimental demonstrations of BSC involving multiple beams have been limited to a single carrier wavelength.” It would improve the manuscript if the authors could elaborate more on the research objectives and significance to better frame the study purpose.\n7. The clarity and presentation of the figures should be improved to enhance overall readability. In particular, important annotations or explanatory labels should be added to guide the reader. For example, in Figure 1, the two different mechanisms could be more clearly distinguished—either by explicitly labeling them or by using visual aids such as dividing lines, shading, or color coding. This would help readers better understand the conceptual differences and the structure of the figure at a glance.\n8. The manuscript frequently refers to supplementary materials for key data and conclusions, such as in the last paragraph of the first section. It would enhance the flow and comprehensiveness of the paper if some of this supplementary information were integrated into the main text. Also, the referencing of supplementary materials should be organized more systematically and judiciously throughout the article.\n9. Several statements in the manuscript are unclear and should be revised for better clarity. For example, in the key paragraph starting at line 65, the sentence “In our experiments, the visible green beam is the SH of a Yb laser, which is used as the FF infrared control beam” is confusing and needs clarification—particularly the identification of the control beam and the actual role of the fundamental frequency (FF) and second harmonic (SH) beams. Additionally, the phrase “This mechanism acts as an active diffuser…” is potentially misleading. In the context of speckled beams, a “diffuser” typically refers to a physical element (e.g., ground glass) that alters speckle size or spatial coherence. It would be helpful for the authors to clarify this analogy if any, or consider using a different phrase as it involves different physics.", "reviewer_2": "In this manuscript, the authors propose an approach to controlling the spatial quality of multi-mode fiber (MMF) guided light beams, through a light-by-light scheme. They simultaneously inject high-power fundamental frequency (FF) light and low-power second harmonic (SH) light into a graded-index (GRIN) MMF, and achieve spatial quality adjustment of the SH light by changing the power and modal power distribution of the FF light. Different from the light-by-light control in traditional optical switches, this manuscript explores the possibility of controlling the signal light using pump light at different wavelength. Two sets of experiments are carried out to demonstrate their concept: i) They co-propagate FF light and SH light in a 10 m 50/125 GRIN MMF, and then alter the intensity of FF light and characterize the output light fields to demonstrate their approach; ii) They choose a longer-pulse source and a 50 m MMF with a 27 µm core diameter to obtain stronger nonlinearity, leading to adjustment of the number and mode components of the Stokes lines.\nIn my opinion, the proposed approach even though interesting does not provide convincing and significant novelty or impact to guarantee its publication in such a high impact journal. Additional comments are listed below for the authors to take into account prior to resubmitting their manuscript to another journal.\n1.My major concern is, results in Fig. 1 and Fig. 2 are too weak to convince me. For instance, for the beam-cross cleaning (BXC) process in Fig. 1 (b), there are no obvious changes of M2. These discrete results are more like measured results that fluctuate within a limited range. I also don't see obvious beam-cleaning or beam-spoiling effect from the Fig. 1 (c) and (d). Similarly, results in Fig. 2 are also too weak. The mode distribution of SH light has only changed slightly, and there is no obvious beam-cleaning phenomenon.\n2.Furthermore, such a beam cleaning strength, which may only enable the M2 of SH beam optimize from 11 to 10, will greatly limit the application of the method in this manuscript. Since the spatial quality of beams with M2 greater than 10 is still very poor. Thus, I can’t judge that the proposed approach has significant impact.\n3.For the results with stronger nonlinearity (Fig. 3), is it possible to accomplish BXC process? The authors say that ‘Additionally, for both beams nonlinear propagation leads to both spatial BSC and Raman beam clean-up, which concentrate the beam energy into their LOMs (not shown here).’ It is confusing. A superior beam-cleaning result (e.g., optimized intensity profile or M2, etc.) in this case may be more convincing.\n4.Does the proposed approach only work for the same laser source? In this manuscript, the authors use the FF and SH beam of the same laser source to demonstrate their approach (for a better spatiotemporal overlap?). However, if one can spatial-split such a laser source into two parts (with the same wavelength), one part is able to clean-up the other part with high quality. Obviously, the process of spatial split may be simpler than using a BBO to obtain its SH beam. Thus, a demo that verifies another laser source (with wavelength different from the signal beam) can clean-up the signal beam may be more convincing, which may demonstrate the potentially broad application scenarios.\n5.Why did the authors choose to show the intensity profile of ‘40 kW’ in Fig. 1 (c) but display result of ‘26 kW’ in Fig. 1 (d)? To my point of view, for a fair comparison, the authors may provide intensity profiles for both ‘40 kW’ results.\n6.It seems that the enlarged spectrum images in Fig. 3 (a)-(d) do not completely correspond to their normal-sized ones. Are they the results of the same measurement?\n7.How do you control the nonlinear process in experiment (i.e., from BXC to BXS)? I suggest the authors to provide the detailed controlling method for results in Fig. 2 and Fig. 3.\n8.It would be better if the authors can provide their modal decomposition process (maybe in SI), since the spatial modal distribution is an important indicator in their manuscript.\nSome minor comments:\n9.Figure 2 (a), for the sub-image which showcases the near-field intensity profile of fundamental frequency beam, there is a strange line inside. The authors may check it.\n10.Figure 1 (b), the ‘15’ in the y-axis label is partially cut off.\n11.For a better illustration, the authors may provide color bars for the intensity profiles in this manuscript.", "reviewer_3": "Manuscript ID: 619254\nTitle: Spatio-spectral light-by-light moulding in multimode fibre\nAuthors: Yago Arosa, et. al.\nIn the field of beam transmission, multimode optical fibers (MMFs) have become the preferred medium for delivering high-power structured laser beams, attributed to their superior power-handling capabilities and modal diversity. Notably, the nonlinear effects intrinsic to MMFs present novel opportunities for achieving all-optical control. Nevertheless, the manipulation of speckled beams with complex spatial distributions remains a significant considerable challenge due to the inherently fuzzy intensity and phase structures. Diverging from prior research that predominantly concentrated on single-wavelength nonlinear control, this manuscript offers an innovative exploration of the nonlinear interaction mechanisms of dual-wavelength beams within MMFs. This approach broadens the scope of optical manipulation in MMFs, introducing a new dimension of multi-wavelength cooperative control. The findings illustrate that nonlinear interactions between fundamental-wavelength beam and its harmonic can be effectively regulated via optically induced mode coupling (OIMC) and cross-phase modulation (XPM), and thereby enhancing better spatial beam quality. Furthermore, by adjusting the modal power distribution of the second-harmonic signal, the visible Raman-Stokes cascade can be either enhanced or partially suppressed, highlighting the potential of multi-wavelength nonlinear effects for all-optical control. The idea is fresh,the proof-of-principle demonstration is sound, and the following are some suggestions that may help further improve the quality of the manuscript.\n(1) The crucial issue: The method for controlling beam quality presented in the manuscript appears to build upon previous all-optical control work research involving beams of the same wavelength (Journal of Lightwave Technology. 2023, 41, 3164). In comparison to the interaction between beams of identical wavelengths, what benefits are associated with employing two beams of different wavelengths to modulate beam quality in an MMF? Is there potential for further enhancement of beam quality?\n(2) This manuscript primarily examines the impact of the pump beam power on the signal beam, yet it lacks comprehensive details regarding the spatial mode of the pump beam itself. Does the spatial mode of the pump beam affect its interaction with the signal beam, particularly considering the nonlinear overlap coefficients between modes? Could the authors provide further in-depth analysis of this aspect?\n(3) The manuscript states that the propagation of the signal beam within the MMF is linear, but it does not provide detailed information about its polarization state. Could the polarization state of the signal beam influence its subsequent evolution? For example, might different polarization states affect the mode coupling efficiency or mediate the phase-matching conditions?"}, "author_response": "Answer to Reviewers\nSubmission of the revised version of manuscript NCOMMS-25-42447-T\nWe would like to thank the Reviewers for their valuable time and constructive comments on\nour manuscript submitted to Nature Communications and titled: “Spatio-spectral light-by-light\nmoulding in multimode fibre”.\nWe have amended our manuscript to address the Reviewers’ concerns carefully. Our point-by-point response and a detailed list of changes made in the revised version are enclosed in the following Response Letter. We have also attached a version in which all major content changes are highlighted in red. We believe that we have successfully addressed all issues raised by the three Reviews.\n----------------------------------------------------------------------------------------------------------------\nResponse to Reviewer 1:\nReviewer 1 Comment:\nThe manuscript presents an approach utilizing optically induced mode coupling (OIMC) and\ncross-phase modulation (XPM) to realize spatial mode modulation between beams of different\nfrequencies in multimode optical systems. The experimental results demonstrate two key\neffects: (i) In the absence of energy exchange, a strong fundamental frequency (FF) pump beam\ncan bidirectionally modulate the spatial mode distribution of a weak second harmonic (SH)\nbeam, leading to either beam cross-conversion (BXC), characterized by low-order mode\nenhancement, or beam cross-switching (BXS), characterized by high-order mode\nenhancement; (ii) This modulation mechanism significantly impacts the SH beam’s mode\ncomposition, beam quality, and nonlinear Raman cascade behavior.\nAdditionally, numerical simulations validate the influence of FF power on the SH mode\ndistribution and highlight the crucial role of temporal walk-off in the modulation process. The\nsimulated results seem to be in good agreement with experimental observations.\nOverall, the work provides a different perspective on optical control in multimode systems.\nHowever, I am unable to recommend the manuscript for publication in its current form. Below,\nI outline several questions and suggestions for the authors to address:\nOur Answer:\nWe appreciate the concise summary provided by Reviewer 1, and have carefully considered all of the\ncomments and suggestions. In the following, we provide detailed responses and revisions that directly\naddress each of the Reviewer’s concerns.\nReviewer 1 Comment:\nGeneral comments:\nAs is well known, standard multimode fibers (MMFs), such as silica-based graded-index\n(GRIN) fibers, do not exhibit second-order nonlinear effects like second harmonic generation\n(SHG) under normal conditions due to the centrosymmetric nature of the material, which\nprohibits second-order susceptibility (χ(2)). Therefore, it is unclear how the authors achieve\nSHG, Raman Stokes cascades, and cross-phase modulation (XPM) via the Kerr effect\nsimultaneously within the same GRIN MMF. The manuscript should clearly explain the\nphysical mechanisms and nonlinear processes involved, especially how SHG is realized in such\na system. If SHG occurs through special means—such as surface-induced symmetry breaking,\nquasi-phase matching, or use of non-silica dopants—those conditions and techniques must be\nexplicitly detailed and justified.\nOur Answer:\nWe agree with the Reviewer that SH generation is not possible in silica, and thus cannot occur\nin multimode silica fibers without modification of the material electronic structure. In our\nexperiment, SHG is not produced within the fiber itself, but instead in a type I BBO crystal\nplaced in one arm of the setup, as shown in Figure 1(a). To clarify this point, we have added\nthe following sentence to the \"Methods\" section: “Thus, the SH is not generated inside the\noptical fiber but it is coupled into it after conversion in a BBO crystal”.\nReviewer 1 Comment:\nAs the authors are aware, beam self-cleaning (BSC) in multimode fibers is a well-established\nnonlinear phenomenon that has been extensively studied in the literature. It is generally\nunderstood within the framework of four-wave mixing (FWM), nonlinear mode coupling, and\noptical thermalization processes. In light of this, the manuscript should clarify how the\nproposed XPM-induced BSC fits within or deviates from this established theoretical\nframework. Specifically, the authors should elaborate on whether the observed effects are a\nnovel manifestation of existing BSC mechanisms or represent a fundamentally new regime. A\nmore rigorous theoretical discussion, possibly comparing the dynamics and physical conditions\nof XPM-induced BSC with traditional FWM- and thermalization-driven models, would be\nnecessary and valuable for positioning this work within the broader context of nonlinear\nmultimode optics.\nOur Answer:\nIndeed, the BSC arises from the coherent interaction between transverse modes in a multimode\nfiber, and directly affects the power distribution among the modes. The combined effects of\nlinear (random) mode coupling, nonlinear Kerr process, and intermodal beating (self-imaging)\ninduce a periodic index grating that enables four-wave mixing and drives energy exchange\nbetween modes. Under these conditions, the coupling strength is determined by self- and cross-\nphase modulation between the modes of a single monochromatic coupled beam.\nThis interaction, governed by the initial power and its initial modal energy distribution, induces\na self-organized process that converges towards the limited number of stable states, most\nnotably the beam cleaning state.\nIn the present study, we replace the self-phase modulation with cross-phase modulation, which\ncan be introduced by an additional beam either coupled into or generated within the fiber core\nat different initial wavelengths. While this does not change the principle of energy transfer\namong modes, it significantly enhances its impact and enables new states to emerge at the fiber\noutput. Among them, the main and interesting effect is the ability to control the direction of\nenergy transfer between modes by means of a control beam, thereby modifying the dynamics\nof energy propagation inside the fiber.\nThis new regime not only enables control of a weak beam at any different wavelength within\nthe fiber, but can also either improve or degrade its brightness. Furthermore, it allows one to\naccelerate or slow down the intermodal energy transfer driven by random coupling. In this\nwork, the use of another copropagating beam at a different wavelength introduces additional\nopportunities for achieving advanced control of light in multimode fibers.\nAs far as the comparison of our system and experimental results with “thermalization-driven\nmodels”, in our Supplementary sections SI3 and SI4, we have presented numerical\ncomputations of the entropy evolutions of the interacting fundamental and second-harmonic\nbeams. These intriguing results will certainly stimulate further theoretical studies for extending\ncurrent thermodynamic treatments to the case of multiple interacting beams. However, the\ndevelopment of such theory is beyond the scope of our present work.\nTo follow the Reviewer's recommendation and to better illustrate the difference with the\nstandard BSC mechanism, we have added the following paragraph to the discussion section.\n“The cross-beam interaction can be regarded as an extension of the BSC mechanism, in which\nthe self-phase modulation driving the cleaning process is replaced by a nonlinear phase shift\n(or cross-phase modulation) induced through interaction with an additional beam. In other\nwords, the longitudinal grating formed in the BSC regime, arising from the combined action\nof the Kerr effect and the self-imaging process in the parabolic fiber, can strongly impact other\nbeams, whether weak or strong, that propagate simultaneously within the fiber. This constitutes\na new nonlinear propagation regime that is more versatile and readily controllable.”\nReviewer 1 Comment:\nOther comments and suggestions:\nOn the Energy Exchange Between FF and SH Beams: The authors should provide a more\ndetailed evaluation, either experimental or theoretical, to confirm whether there is indeed no\nsignificant power exchange between the fundamental frequency (FF) and second harmonic\n(SH) beams. In high-power nonlinear regimes, energy transfer can critically affect the modal\npower balance of the SH beam, and this needs to be rigorously ruled out to support the core\nassumption of purely optically induced mode coupling.\nOur Answer:\nWe agree with the Reviewer on the importance of verifying the absence of energy transfer\nbetween the pump wave and the signal wave at the second harmonic (SH) in the multimode\nfiber. We confirmed this by examining the output spectrum across the full range of interest\n(from 400 nm to 1500 nm) while injecting the pump wave only. The absence of SH clearly\ndemonstrates the lack of phase-matching between the two wavelengths, which rules out any\nenergy transfer. Additionally, a simple filter at the SH wavelength combined with a power\nmeter can be used to measure the constant energy of SH beam when coupling the FF beam,\nfurther confirming the absence of energy exchange.\nIt is also well known that a poling process can be realized in multimode fibers to achieve weak\nSHG. In our earlier work published in 2017 (D. Ceoldo et al., “ Second harmonic generation\nin multimode graded-index fibers: spatial beam cleaning and multiple harmonic sideband\ngeneration,” Optics Letters, Vol. 42, Issue 5, pp. 971-974 (2017)), we demonstrated that this\nrequires the simultaneous injection of both beams with sufficiently high peak power to induce\na coherent nonlinear beating, forcing the electronic charge displacement in a periodic\nconfiguration, thus establishing a longitudinal phase-matching. Although the SHG has also\nbeen observed without such preparation (Eftekhar, M. A. et al, “Instant and efficient second-\nharmonic generation and downconversion in unprepared graded-index multimode fibers,”\nOptics Letters 42(17) 3478-3481 (2017)), the poling process typically requires several hours\nof exposure and grows slowly with time. In the present study, we have never applied such high\npeak powers over extended times, and therefore no SHG occurs in the fiber itself. This confirms\nthe absence of phase-matching and the absence of any energy exchange between the SH and\nfundamental beams. For this reason, we have not modified the manuscript and have retained\nthe statement mentioning the absence of energy transfer between wavelengths.\nReviewer 1 Comment:\n(OIMC).\n2. On the Nature of the Mode Power Distribution in Figure 2(c): The power distribution of the\nSH modes shown in Figure 2(c) appears unusually smooth, which raises the question of\nwhether it represents a single-shot measurement or an averaged statistical result. In practice,\nsuch idealized mode profiles are rarely observed, especially under strong nonlinear\npropagation. The authors are encouraged to clarify the data processing method and compare\nit to more realistic cases, such as Figure S5 in Nature Physics 2022, 18, 685–690, where\nfluctuations in power distribution are evident.\nOur Answer:\nWe agree with the Reviewer that the power distribution of the SH modes appears to be\nsignificantly smoother than the modal distribution reported in the reference cited by the\nReviewer. However, the number of modes supported by our fiber at 532 nm is considerably\nlarger than in the referenced work. In our case, nearly 2000 modes could be excited under the\ninitial coupling conditions. Moreover, optical coupling of the SH wavelength is achieved with\na large mode diameter, due to the chromatic aberration between the SH and the optimized\nfundamental waveform. This is clearly illustrated in Figure 2(a), where the speckle pattern at\nthe SH is visible, along with the small (< 1 µm) spatial elementary lobes observed in the SH\nnear field. It is also worth noting that the energy distribution at the fiber output covers the entire\nfiber core area, confirming the excitation of a large number of high-order modes. In addition,\nthe fiber used in our experiment is also significantly longer (10 m) than that used in the\nreferenced work (50 cm). As a result, random mode coupling plays a much greater role in our\ncase, leading to a smoother redistribution of energy among modes and promoting an evolution\ntowards a Rayleigh-Jeans type energy distribution.\nThe calculation of modal decomposition is based on near and far fields obtained experimentally\nwith a CCD camera. One image is a result of a large number of pulses (>1000 pulses)\npropagating in the fiber without modification of the initial coupling conditions.\nIn order to further clarify the calculation conditions, in the Supplementary Material, we have\nintroduced the following text containing a more detailed explanation of the used method,\ntogether with images comparing the beam pattern measured with CCD camera and numerically\nreconstructed.\n“SI6 Numerical modal decomposition method\nTo perform the modal decomposition, we followed a procedure similar to the one reported in\nKim et al. [19] based on a stochastic parallel gradient descent (SPGD) method. The output field\nfrom the fiber is expressed as a superposition of guided LG modes. The near (and far) field\nintensity distribution can then be reconstructed as a weighted sum of these LG modes (or their\nFourier transform), with the unknown modal amplitudes and phases determined through an\niterative optimization as shown in the flow chart in SI6a.\nFigure SI6a: Flow chart showing the optimization procedure for the modal decomposition\nFirst, we addressed possible mismatches in sampling and alignment by interpolating the\nmeasured intensity distributions. Specifically, both near and far field intensity profiles were\nresampled onto a common grid with identical spatial resolution. This direct interpolation\nensured that differences between the two datasets arose only from modal content, and not from\npixel resolution or coordinate offsets. Once the experimental and simulated profiles were\nmatched in resolution, a first correlation between the experimental NF and the modes was done\nto obtain an initial set of relative power for each mode (𝑎 ), we set all modes to be in phase\n𝑖\ninitially. This algorithm iteratively perturbs the modal coefficients: relative powers (𝑎 ) and\n𝑖\nphases(𝜑 ) with random updates and retains those changes that reduce the cost function,\n𝑖\ndefined as the normalized mismatch between experimental and reconstructed intensity profiles.\nBy dynamically adjusting the perturbation size during the iterations, the SPGD approach\nenhances convergence speed while avoiding local minima. The optimization was structured in\nepochs; each defined as 100 iterations per mode considered. The algorithm terminated either\nafter 100 epochs or once the improvement in correlation between the experimental and\nreconstructed profiles fell below 10−4. In the case of modal decomposition applied to simulated\ndata, the far-field condition was omitted since the full phase information of the beam was\navailable. By enforcing agreement at both distances, the uniqueness of the modal\ndecomposition was guaranteed, avoiding the conjugate-pair ambiguity that can appear when\nusing a single measurement plane. In Figure SI6b some examples of the reconstructed data are\nshown.\nFigure SI6b: Examples of reconstructed images obtained experimentally by using CCD\ncamera, by using our numerical method to obtain modal decomposition at the fiber end.\nReviewer 1 Comment:\n3. On the Counterintuitive Modal Inversion at High FF Power: The authors report a reversal\nof modal energy flow at FF peak powers approaching 100 kW, where high-order modes\n(HOMs) of the SH beam become increasingly populated. This behaviour is counterintuitive\nand requires further explanation. Specifically, the authors should assess whether the FF power\nin a 10 m fiber segment exceeds the thresholds for stimulated Raman scattering (SRS) or soliton\nformation. If such nonlinear effects are present, the observed behaviour cannot be solely\nattributed to OIMC and may involve more complex nonlinear dynamics\nOur Answer:\nWe agree with the Reviewer that the high peak powers coupled into multimode optical fibers\ncan induce additional nonlinear conversions, especially above 25 kW for 10 meters of fiber,\nwhere the infrared part of the spectrum starts to extend significantly beyond 1.2 µm (see\nFigures 3(b) and 3(d)). In our study, we consider that optically induced mode coupling (OIMC)\nis not limited to the wavelength coupled into the fiber, but can also affect modes generated at\nother wavelengths through nonlinear conversion. As the reviewer noted, the resulting nonlinear\ndynamics are therefore very complex. To clarify this point, we have expanded the following\nparagraph by adding the sentences highlighted in red. It is also important to emphasize that our\nnumerical simulations account for spectral broadening without including Raman contribution\nin the equations.\n“In this paper, we elevate the field of light manipulation in MMFs to a new dimension,\ninvolving the nonlinear interaction among multiple laser beams of different carrier\nwavelengths. We experimentally demonstrate that the evolution of a weak speckled beam in the\nvisible range can be controlled using a combination of XPM and nondegenerate OIMC with\nan intense pump beam in the near infrared, which undergoes its own spatial reshaping through\nthe Kerr BSC process but also its own spectral broadening by means of the Raman scattering\nprocess and soliton generation dynamics for very high peak powers”.\nReviewer 1 Comment:\n4. On the Impact of Pulse Duration Compression: To achieve 100 kW peak power, the pulse\nduration was reduced from 100 ps to 10 ps, which implies a significant increase in spectral\nbandwidth due to the time-bandwidth product. The authors should clarify whether this spectral\nbroadening introduces any additional dispersion or nonlinear effects that might influence\nOIMC or the Raman cascade dynamics. Such effects could affect the interpretation of mode\ncontrol results.\nOur Answer:\nIn our work, we used a Fourier-transform-limited laser pulses with a duration of 250 fs, which\nwe could stretch up to 100 ps (see the beginning of the paragraph titled: “Experiment” in the\n“Method” section). Thus, for both the 100 ps and 10 ps pulses, the same spectral bandwidth is\nintroduced into the 10-m-long optical fiber, with only the pulse duration being varied. In this\nconfiguration, we estimated that the temporal walk-off between the SH and the pump beam at\n1030 nm was close to 0.8 ns, while the dispersion term remained unchanged due to the same\nspectral bandwidth of the pulse. We also note that the dispersion impact on the pulse used in\nthe experiment is 5 ps for 10 meters of fiber. Therefore, the primary influence on our\nexperiment arises from the temporal walk-off, which we investigated numerically (see Results).\nWe clearly demonstrated that this parameter is responsible for the observed decrease in\nefficiency of the BXC and the BXS. Using a 650 ps pulse duration over 50 m of fibers, we\nobserved a much more efficient process, capable of significantly modifying the Raman cascade\nprocess associated with the SH beam (see Figure 3).\nWe clarified this point in the text by adding the following sentence: “This was achieved using\nthe same initial 250 fs Fourier transform limited pump beam (see Methods) and changing the\nstretching passing from 100 ps to 10 ps.”\nReviewer 1 Comment:\n5. On the Feasibility of Temporal Overlap Between FF and SH Pulses: Given the group\nvelocity mismatch between FF and SH wavelengths, achieving precise temporal and spatial\noverlap of low-coherence pulses is highly non-trivial. The authors are encouraged to provide\na detailed and quantitative assessment of how this synchronization is achieved in the\nexperiment, including potential experimental uncertainties. Without such justification, the\nvalidity of the claimed nonlinear interactions between temporally synchronized FF and SH\npulses may be called into question\nOur Answer:\nThe synchronisation between the FF and the SH beam is reached by using a Mach-Zehnder\ninterferometer, where one arm of the FF beam is converted into the SH by using a BBO crystal\nof 2 cm long (see Figure 1(a)). Thus, the main path difference between the FF and SH waves\nis introduced by the BBO crystal, placed in only one path of the interferometer, and by the\nchromaticity of other elements such as focusing lens, dichroic plate, waveplates, and green\nfilter. The difference in the optical path is then compensated for by the delay lines placed in\nthe interferometer’s arms. The uncertainty of the synchronisation is lower than 0.2 ps compared\nwith the 10 ps, 100 ps and 650 ps of pulse durations used. The synchronisation at the\ninterferometer output is verified through nonlinear cross-correlation by means of the process\nof third-harmonic generation in a BBO crystal cut for sum frequency generation involving the\nFF and the SH waves.\nWe introduced the following sentence in the “Method” section in order to clarify that\nparameter: “The synchronization accuracy is estimated to be 0.2 ps based on the optimization\nof the third-harmonic generation correlation between SH and FF beams.”\nReviewer 1 Comment:\n6. In the introduction section where the research background and methodology are presented,\nthe motivation is currently summarized in only one sentence: “However, to our knowledge,\nexperimental demonstrations of BSC involving multiple beams have been limited to a single\ncarrier wavelength.” It would improve the manuscript if the authors could elaborate more on\nthe research objectives and significance to better frame the study purpose.\nOur Answer:\nFollowing the Reviewer’s suggestion, we further elaborated the introduction providing the\nmotivations and significance of our work by adding the following text: “However,\nexperimental demonstrations of BSC involving multiple beams have so far been limited to a\nsingle strong carrier wavelength, thus preventing applications of this effect to polychromatic\nbeams. Our approach opens new opportunities for supercontinuum generation in multimode\nfibers by exploiting the spatial overlap between high- and low-power multicolor beams. New\nperspectives also include the improved spatial quality control of multimode laser sources, and\nthe possibility of ultrafast routing and switching between different channels in\ntelecommunications the domain.”\nReviewer 1 Comment:\n7. The clarity and presentation of the figures should be improved to enhance overall readability.\nIn particular, important annotations or explanatory labels should be added to guide the reader.\nFor example, in Figure 1, the two different mechanisms could be more clearly distinguished—\neither by explicitly labeling them or by using visual aids such as dividing lines, shading, or\ncolor coding. This would help readers better understand the conceptual differences and the\nstructure of the figure at a glance.\nOur Answer:\nFollowing the reviewer's suggestion, we modified Figure 1 to improve clarity and readability.\nThe results illustrating beam cross-cleaning and beam cross-spoiling are now better separated\nand clearly identified.\nReviewer 1 Comment:\n8. The manuscript frequently refers to supplementary materials for key data and conclusions,\nsuch as in the last paragraph of the first section. It would enhance the flow and\ncomprehensiveness of the paper if some of this supplementary information were integrated into\nthe main text. Also, the referencing of supplementary materials should be organized more\nsystematically and judiciously throughout the article.\nOur Answer:\nDue to the main text length constraints, many results are presented in the Supplementary\nMaterial. However, following the Reviewers' recommendation, we have reorganized the\nsupplementary section. Each section now includes a brief summary (inserted below) giving the\nreader a quick overview of its contents.\nSupplementary material summary\nSI1 - A simple model:\nSection SI1 of the Supplementary Material introduces a simple model illustrating the concept\nof beam cross-cleaning (BXC) and the beam cross-spoiling (BXS). This model is based on\ncross-phase modulation between two modes at the SH and two modes at the FF.\nSI2 - Fundamental beam self-cleaning:\nSection SI2 shows the evolution of the self-cleaning process of the fundamental beam for a\ngiven initial coupling. This corresponds to the first case described in the main paper.\nSI3 - Numerical simulations:\nSection SI3 presents results of numerical simulations, illustrating the linear case of cross-\nprocess propagation and its negligible impact on the spatial evolution. In addition, the role of\ntemporal walk-off between SH and FF waves is highlighted, showing its strong influence on\nthe energy transfer efficiency between modes at the second harmonic.\nSI4 - M² and entropy evolutions:\nSection SI4 shows the numerical evolution of the beam entropy at the SH as a function of the\nFF power. The speed of evolution of the entropy depends on the power of the fundamental\nwave, thus allowing its growth speed to be decreased or increased\nSI5 - Speckle by speckle cross-cleaning\nSection SI5 shows that a speckle at the SH wavelength can be strongly modified by a speckle\nat the fundamental wavelength by means of the cross-cleaning process and give rise to a\nconvergence of the energy at the SH on a single high-order mode. This calculation shows the\npotential efficiency of the cross-cleaning process, as well as the possible reduction of the\nentropy during propagation in a transient regime, i.e., before reaching a thermalization state.\nSI6 - Numerical approach description of the modal decomposition method\nSection SI6 details the numerical approach used to perform the modal decomposition of the SH beam.\nThis uses near-field and far-field images recorded at the output fiber.\nReviewer 1 Comment:\n9. Several statements in the manuscript are unclear and should be revised for better clarity.\nFor example, in the key paragraph starting at line 65, the sentence “In our experiments, the\nvisible green beam is the SH of a Yb laser, which is used as the FF infrared control beam” is\nconfusing and needs clarification—particularly the identification of the control beam and the\nactual role of the fundamental frequency (FF) and second harmonic (SH) beams. Additionally,\nthe phrase “This mechanism acts as an active diffuser…” is potentially misleading. In the\ncontext of speckled beams, a “diffuser” typically refers to a physical element (e.g., ground\nglass) that alters speckle size or spatial coherence. It would be helpful for the authors to clarify\nthis analogy if any, or consider using a different phrase as it involves different physics.\nOur Answer:\nWe reformulated the first sentence as follows: \"In our experiment, the FF infrared beam at\n1030 nm is provided by a Yb laser, while the visible beam is obtained via nonlinear conversion\nfrom FF to SH (515 nm).”\nThe second sentence has been reformulated as: “This mechanism for controlling the power\nexchange between transverse modes can be compared, by analogy, to an intra-fiber\ncontrollable active spatial diffuser, sending energy from mode to mode on demand.”\nSince the reference to a diffuser appears in multiple locations in the manuscript, we have also\nrephrased it in the following way:” It should also be emphasized that the FF beam disturbs the\nSH beam, by increasing or decreasing the energy transfer dynamics between its modes, similar\nto the effect of a local diffuser placed within the fiber core.\n----------------------------------------------------------------------------------------------------------------\nResponse to Reviewer 2:\nReviewer 2 Comment:\nIn this manuscript, the authors propose an approach to controlling the spatial quality of multi-\nmode fiber (MMF) guided light beams, through a light-by-light scheme. They simultaneously\ninject high-power fundamental frequency (FF) light and low-power second harmonic (SH) light\ninto a graded-index (GRIN) MMF, and achieve spatial quality adjustment of the SH light by\nchanging the power and modal power distribution of the FF light. Different from the light-by-\nlight control in traditional optical switches, this manuscript explores the possibility of\ncontrolling the signal light using pump light at different wavelength. Two sets of experiments\nare carried out to demonstrate their concept: i) They co-propagate FF light and SH light in a 10\nm 50/125 GRIN MMF, and then alter the intensity of FF light and characterize the output light\nfields to demonstrate their approach; ii) They choose a longer-pulse source and a 50 m MMF\nwith a 27 µm core diameter to obtain stronger nonlinearity, leading to adjustment of the number\nand mode components of the Stokes lines.\nIn my opinion, the proposed approach even though interesting does not provide convincing and\nsignificant novelty or impact to guarantee its publication in such a high impact journal.\nAdditional comments are listed below for the authors to take into account prior to resubmitting\ntheir manuscript to another journal.\nOur Answer:\nWe are sorry about the unfavorable assessment of the novelty and impact of our paper by\nReviewer 2. However, we respectfully disagree with the Reviewer on this point: we firmly\nbelieve that our study provides numerous new findings that will significantly contribute to the\nscience and technology of light-by-light control in optical fibers or fiber laser sources.\nFor instance, the key highlights of the novelty and impact of our paper include:\nWe have introduced and demonstrated experimentally and numerically the ability of complex\nmedia such as multimode fibers to control the brightness of a weak or strong beam via its\ninteraction with an intense light of different wavelengths. This control can force spatial\nfocusing or defocusing, thanks to the manipulation of an energetic pump beam whose\npropagation and coupling are fully controlled. Thus, this pump beam can act as an ultrafast\nexternal optical control device, able to change the propagation dynamics and nonlinear\nconversion of other beams. Our results will be extremely useful for many applications, such as\nsupercontinuum generation, where the pump beam can strongly influence the spatial and\nspectral distribution of the converted wavelength, and also in the field of lasers, where the laser\nwavelength could interact with the spatial distribution of the pump beam. Telecommunications\ncould also exploit such a modal connection to implement spatial switching for all-optical data\nrouting.\nFrom a fundamental perspective, our results offer important insights for a thermodynamic\ndescription of the light propagation, raising questions about cross-condensation phenomenon\npreviously observed and reported in other domains (see for example: the Claisen cross-\ncondensation - L. Claisen and A. Claparède, “Condensationen von Ketonen mit Aldehyden”,\nBer. Dtsch. Chem. Ges., vol. 14, no. 2, 1881, pp. 2460-2468, ISSN 0365-9496, DOI\n10.1002/cber.188101402192, or the cross-aldolic-condensation in chemistry - L. G. Wade,\nOrganic Chemistry, Upper Saddle River (New Jersey), Prentice Hall, 2005 (6th ed.), ISBN\n978-0-13-169957-1 and 0-13-169957-1, p. 1056-1066).\nAnother very interesting result, presented in this paper, is the cross-beam alteration of the low\nintensity beam, which cannot be achieved through the spatial self-cleaning process. This\nmechanism, which is essentially the inverse of the spatial self-cleaning process, expands our\nability to regulate the energy transfer to higher-order modes, thereby increasing the beam\nexpansion rate during propagation, that is, it can accelerate or slow down beam thermalization\nin a multimode fiber. Consequently, the flow of beam entropy can be regulated by the power\nof a copropagating beam. Beyond this demonstration, we also clearly show that the temporal\nwalk-off can significantly limit or enhance energy transfer, making it a key parameter to\nconsider. As a result, wavelengths close to or far from the pump wavelength can be affected\ndifferently, introducing an additional interesting mechanism for spatial shaping in the\nstimulated Raman process.\nWe have also demonstrated that a limited transformation effect at a given wavelength (e.g. the\nSH of the pump wave) can drive the stimulated Raman effect from this SH wave to strongly\nmodify its spatial gain distribution and generate Stokes waves in a high-order mode in a stable\nstate. This was observed for the first three Stokes lines, which transferred nearly all the energy\nfrom the fundamental mode to an LP02-like mode, thereby highlighting the high efficiency of\nthis spatial transformation. The almost total energy transfer achieved within the stimulated\nRaman process represents a major novel and impactful result, showing the full potential of this\ncross-modulation effect.\nMoreover, we show, for the first time to our knowledge, that the Raman beam clean-up in\nmultimode fiber may be affected by copropagating beams, enabling the transformation of a\nLP01 mode into a higher-order mode.\nFinally, we have also numerically demonstrated that a low-intensity speckle beam can be\nefficiently converted into a quasi-single mode by interaction with another pump beam speckle\nat a different wavelength, achieving a total power transfer of more than 52% to the main mode\ncompared to 6% when cross-cleaning is not operating.\nFor all these reasons, we respectfully disagree with the Reviewer's opinion on the significance\nof our work, thus precluding our manuscript from being considered for publication in Nature\nCommunication.\nReviewer 2 Comment:\n1.My major concern is, results in Fig. 1 and Fig. 2 are too weak to convince me. For instance,\nfor the beam-cross cleaning (BXC) process in Fig. 1 (b), there are no obvious changes of M2.\nThese discrete results are more like measured results that fluctuate within a limited range.\nI also don't see obvious beam-cleaning or beam-spoiling effect from the Fig. 1 (c) and (d).\nSimilarly, results in Fig. 2 are also too weak. The mode distribution of SH light has only\nchanged slightly, and there is no obvious beam-cleaning phenomenon.\nOur Answer:\nWe have redrawn Fig. 1 (b) to clearly show that the M2 changes are obvious, and that there is\na significant decrease or increase of the observed M2 values, which could not be replaced by a\nhorizontal line as it would happen if the results were the outcome of random fluctuations, as\nthe Reviewer has supposed, without violating the error bars of our measurements. Moreover,\nthe Reviewer should not focus solely on the first part of the article but rather consider the\noverall content of our manuscript. In our first set of experiments, we measured the evolution\nof the spatial distribution of the SH beam energy as a function of the pump beam power using\nshort pulses. The observed effect was limited, which we attributed to the significant temporal\nwalk-off between the two waves. This hypothesis was numerically confirmed, and we\nsubsequently modified our experiment to overcome this limitation.\nIt should be noted that this limited impact due to walk-off is clearly stated in the manuscript\nwith the following sentence: “In our first series of experiments, the effect was limited by\ndispersive temporal walk-off because of the relatively short pulses involved.”\nWe further emphasized the limited impact of cross-action when using pulse durations from 10\nps to 100 ps with the following sentence already included in the text: “As we have seen, the\nBXC efficiency can be limited either by the input FF peak power or by the temporal walk-off\nbetween FF and SH pulses”.\nThus, using the same beams with a longer pulse duration, we demonstrated in Figure 3 that the\nSH beam can be sufficiently affected to modify its entire Raman cascade, producing the first\nthree Stokes lines in the LP02-like mode instead of the initial LP01 mode. As a result, the\nenergy observed in the initial fundamental mode without cross-cleaning is almost entirely\ntransferred to the LP02-like mode, with an efficiency exceeding 80%.\nFinally, to highlight the strong potential of our approach, we simulated the complete\ntransformation of a low-power speckle beam into a high-order quasi-single mode beam with\nan efficiency greater than 52% (see Figure SI5 in the Supplementary Material). This\ndemonstrates that the beam-cleaning or beam-degradation effect can be sufficiently efficient to\nsignificantly change the nonlinear beam evolutions.\nTo further emphasize this point, we added the following sentences to the text: “We also\nmeasured that more than 80% of the energy contained in the three Stokes Raman lines\noscillating in the fundamental mode is transferred into the LP -like mode, clearly illustrating\n02\nthe highly energetic impact that this cross-process can introduce.”\nReviewer 2 Comment:\n2.Furthermore, such a beam cleaning strength, which may only enable the M2 of SH beam\noptimize from 11 to 10, will greatly limit the application of the method in this manuscript. Since\nthe spatial quality of beams with M2 greater than 10 is still very poor. Thus, I can’t judge that\nthe proposed approach has significant impact.\nOur Answer:\nHere again, the Reviewers' comments seem restricted to the beginning of our article, and\noverlook the significant changes observed in the Raman cascade, where the first three Stokes\nRaman lines initially generated in the LP01 mode are completely reshaped into the LP02-type\nmode with an efficiency exceeding 80%. In addition, our numerical results presented in the\nSupplementary Material show the transformation of a speckle beam by another speckle into a\nsingle high-order mode (HG2.1). Thus, the total energy cumulated on the HG2.1 mode reaches\n52% of the total power. These two examples clearly illustrate the great potential of our cross-\nreshaping process.\nReviewer 2 Comment:\n3.For the results with stronger nonlinearity (Fig. 3), is it possible to accomplish BXC process?\nThe authors say that ‘Additionally, for both beams nonlinear propagation leads to both spatial\nBSC and Raman beam clean-up, which concentrate the beam energy into their LOMs (not\nshown here).’ It is confusing. A superior beam-cleaning result (e.g., optimized intensity profile\nor M2, etc.) in this case may be more convincing.\nOur Answer:\nYes, the results shown in Figure 3(a,b) clearly demonstrate that the beam cleaning of the green\nbeam (already measured and presented on Figures 1 and 2) can indeed be obtained, thereby\nmodifying its power density and generating additional Raman Stokes lines.\nThe Reviewer stated that the following sentence was confusing: \"Additionally, for both beams\nnonlinear propagation leads to both spatial BSC and Raman beam clean-up, which concentrate\nthe beam energy into their LOMs (not shown here)\" and suggested that “A superior beam-\ncleaning result (e.g., optimized intensity profile or M2, etc.) in this case may be more\nconvincing”.\nWe agree with the Reviewer that the original sentence is unclear, as we do not present results\nfor the infrared beam, but instead show new, unpublished results for the SH beam. We have\ntherefore replaced the sentence with the following: “Furthermore, both beams undergo\nnonlinear propagation leading to a spatial self-cleaning process and Raman beam clean-up,\nas already reported in the literature [18-20]. However, in the following paragraph, we present\nthe impact of the cross-action between the two beams (FF+SH) on the SH beam, demonstrating\nBXC and BXS effects within the SH Raman cascade.”\nWe also agree with the Reviewer’s remark that “A superior beam-cleaning result would be\nmore convincing”. This is precisely what we illustrate in Figure 3, where the SH beam exhibits\nan increase in Raman Stokes lines under BXC and a reduction in Raman Stokes lines under\nBXS.\nReviewer 2 Comment:\n4.Does the proposed approach only work for the same laser source? In this manuscript, the\nauthors use the FF and SH beam of the same laser source to demonstrate their approach (for\na better spatiotemporal overlap?). However, if one can spatial-split such a laser source into\ntwo parts (with the same wavelength), one part is able to clean-up the other part with high\nquality. Obviously, the process of spatial split may be simpler than using a BBO to obtain its\nSH beam. Thus, a demo that verifies another laser source (with wavelength different from the\nsignal beam) can clean-up the signal beam may be more convincing, which may demonstrate\nthe potentially broad application scenarios.\nOur Answer:\nYes, the Reviewer is correct: we used a single laser source to ensure perfect temporal\nsynchronization of the two beams. The Reviewer also noted: “However, if one can spatial-split\nsuch a laser source into two parts (with the same wavelength), one part is able to clean-up the\nother part with high quality.”\nThis is exactly what we performed in the spatial domain. We used a Mach-Zehnder\ninterferometer to spatially separate the FF beam and its SH. The residual FF beam generating\nthe SH in one arm was suppressed with a filter, ensuring complete independence between the\nSH and the FF waves.\nThe Reviewer further suggested: “Thus, a demo that verifies another laser source (with\nwavelength different from the signal beam) can clean-up the signal beam may be more\nconvincing, which may demonstrate the potentially broad application scenarios”.\nThe main difficulty with this approach lies in synchronizing two independent laser sources,\nparticularly when short pulses are used. However, the use of another wavelength closer to the\nFF beam is feasible with a different experimental setup. For instance, the BBO crystal could\nbe replaced with a simple single-mode silica fiber capable of producing a single Stokes Raman\nradiation only 13 THz away from the FF wave. In this case, the temporal walk-off would be\nconsiderably reduced, as we have already proven in the article by using a 650 ps pulses with\nboth the FF and SH beams.\nReviewer 2 Comment:\n5.Why did the authors choose to show the intensity profile of ‘40 kW’ in Fig. 1 (c) but display\nresult of ‘26 kW’ in Fig. 1 (d)? To my point of view, for a fair comparison, the authors may\nprovide intensity profiles for both ‘40 kW’ results.\nOur Answer:\nWe agree with the Reviewer and have replaced the 26 kW image with one obtained at 40 kW.\nReviewer 2 Comment:\n6.It seems that the enlarged spectrum images in Fig. 3 (a)-(d) do not completely correspond to\ntheir normal-sized ones. Are they the results of the same measurement?\nOur Answer:\nFigures 3(a) and 3(b) were obtained by alternately adding and removing only the infrared beam,\nfollowed by re-measuring the total output spectrum across the entire spectral range. The\nspectroscope automatically optimizes the measurement level.\nFigures 3(c) and 3(d) were obtained for a different coupling of the FF pump wave compared to\nFigures 3(a) and 3(b). Similarly, Figure 3(d) was obtained under the same conditions as Figure\n3(c), but by adding only the FF pump wave.\nReviewer 2 Comment:\n7.How do you control the nonlinear process in experiment (i.e., from BXC to BXS)? I suggest\nthe authors to provide the detailed controlling method for results in Fig. 2 and Fig. 3.\nOur Answer:\nThe nonlinear process in Figure 2 is controlled by monitoring the near-field pattern of the SH\nbeam and readjusting only the coupling of the FF wave. This adjustment is performed using\nthe Rmax@1030 nm deflecting mirror in the interferometer arm of the FF beam. The same\nprocedure was applied for Figure 3, where we varied the FF coupling to obtain the maximum\neffect on the SH beam, thereby generating the Raman cascade.\nWe clarified this point in the Methods section by adding the following sentences:\n“The results presented in Figures 2 and 3 are obtained by observing the green beam in the\nnear-field domain and by readjusting only the coupling of the FF beam. The far-field\nassociated with the green beam is then obtained in a second step, when collecting the final\nresults.”\nReviewer 2 Comment:\n8.It would be better if the authors can provide their modal decomposition process (maybe in\nSI), since the spatial modal distribution is an important indicator in their manuscript.\nOur Answer:\nThe modal decomposition of the SH beam is presented in Figures 2 and 3 to illustrate the impact\nof cross-cleaning on the modal composition. For Figure 3, the analysis is performed in the\nspectro-spatial domain by using a diffraction grating along the horizontal axis (wavelength),\nwhile the spatial domain is displayed along the vertical axis. Consequently, a 2D spatial\ndecomposition cannot be performed because spectral dispersion modifies the horizontal\ndimension. Changes in the spatial content of the second harmonic for the Stokes waves are\nestimated by analyzing the profile size and measuring the divergence associated with the SH\nbeam.\nReviewer 2 Comment:\nSome minor comments:\n9.Figure 2 (a), for the sub-image which showcases the near-field intensity profile of\nfundamental frequency beam, there is a strange line inside. The authors may check it.\nOur Answer:\nWe removed the parasitic line.\nReviewer 2 Comment:\n10.Figure 1 (b), the ‘15’ in the y-axis label is partially cut off.\nOur Answer:\nWe modified the image.\nReviewer 2 Comment:\n11.For a better illustration, the authors may provide color bars for the intensity profiles in this\nmanuscript.\nOur Answer:\nWe added the color bar in Figures 1 and 2. The colors in Figure 3 are the correct colors, and\nwavelengths appear on the spectra.\n----------------------------------------------------------------------------------------------------------------\nReviewer 3 Comment:\nIn the field of beam transmission, multimode optical fibers (MMFs) have become the preferred\nmedium for delivering high-power structured laser beams, attributed to their superior power-\nhandling capabilities and modal diversity. Notably, the nonlinear effects intrinsic to MMFs\npresent novel opportunities for achieving all-optical control. Nevertheless, the manipulation\nof speckled beams with complex spatial distributions remains a significant considerable\nchallenge due to the inherently fuzzy intensity and phase structures. Diverging from prior\nresearch that predominantly concentrated on single-wavelength nonlinear control, this\nmanuscript offers an innovative exploration of the nonlinear interaction mechanisms of dual-\nwavelength beams within MMFs. This approach broadens the scope of optical manipulation in\nMMFs, introducing a new dimension of multi-wavelength cooperative control. The findings\nillustrate that nonlinear interactions between fundamental-wavelength beam and its harmonic\ncan be effectively regulated via optically induced mode coupling (OIMC) and cross-phase\nmodulation (XPM), and thereby enhancing better spatial beam quality. Furthermore, by\nadjusting the modal power distribution of the second-harmonic signal, the visible Raman-\nStokes cascade can be either enhanced or partially suppressed, highlighting the potential of\nmulti-wavelength nonlinear effects for all-optical control. The idea is fresh, the proof-of-\nprinciple demonstration is sound, and the following are some suggestions that may help further\nimprove the quality of the manuscript.\nOur Answer:\nWe thank the Reviewer for the very positive comments, which emphasize that our approach\nbroadens the scope of optical manipulation in MMFs by introducing a new dimension of multi-\nwavelength cooperative control.\nReviewer 3 Comment:\n(1) The crucial issue: The method for controlling beam quality presented in the manuscript\nappears to build upon previous all-optical control work research involving beams of the same\nwavelength (Journal of Lightwave Technology. 2023, 41, 3164). In comparison to the\ninteraction between beams of identical wavelengths, what benefits are associated with\nemploying two beams of different wavelengths to modulate beam quality in an MMF? Is there\npotential for further enhancement of beam quality?\nOur Answer\nThe additional results presented in this paper, compared to previously reported monochromatic\ncross-cleaning processes in a multimode fiber (Journal of Lightwave Technology, 2023, 41,\n3164), show that spatial cleaning can also influence other wavelengths of both low and high\nintensities, even in the presence of strong or weak temporal walk-off.\nOur results further introduce a reverse process capable of degrading the spatial profile of a\nbeam. The beam manipulation demonstrated here is sufficiently effective to modify the\nnonlinear dynamics of a Raman cascade within the same multimode fiber.\nThese findings also open the door to new applications, such as space-time supercontinuum\nshaping, enhancing their potential for multispectral imaging systems. In telecommunications,\nthis type of modal control could be exploited for space-time encoding as well as for all-optical\nswitching and routing. From a more fundamental perspective, within a thermodynamic\ndescription of light propagation, our results may pave the way for the multicolor cross-\ncondensation effect in multimode optical fibers.\nReviewer 3 Comment:\n(2) This manuscript primarily examines the impact of the pump beam power on the signal beam,\nyet it lacks comprehensive details regarding the spatial mode of the pump beam itself. Does\nthe spatial mode of the pump beam affect its interaction with the signal beam, particularly\nconsidering the nonlinear overlap coefficients between modes? Could the authors provide\nfurther in-depth analysis of this aspect?\nOur Answer:\nYes, it is clear that the initial spatial distribution of the FF pump power directly affects the\nnonlinear phase imprinted on the SH beam modes. This additional phase controls the four-\nwave mixing process and the energy exchange between the SH beam modes. This aspect is\nclearly analyzed in the Supplementary Material, where we develop a simple model involving\ntwo second-harmonic modes and two FF modes. The model illustrates the fundamental\nphysical mechanisms of energy exchange between modes of the same wavelength, mediated\nby the contribution of the fundamental wave power (see Supplement SI1). By coupling the FF\nwave into the fiber, we can enforce energy exchange between the SH modes while controlling\nthe energy transfer direction. The initial distribution of the fundamental modes is clearly\nillustrated in Supplement SI2 for a given initial condition, where the coupled energy is\nprimarily concentrated in the lower-order modes. Consequently, by adjusting the FF beam\ncoupling, we were able to obtain either the BXC or the BXS processes of the SH beam.\nReviewer 3 Comment:\n(3) The manuscript states that the propagation of the signal beam within the MMF is linear,\nbut it does not provide detailed information about its polarization state. Could the polarization\nstate of the signal beam influence its subsequent evolution? For example, might different\npolarization states affect the mode coupling efficiency or mediate the phase-matching\nconditions?\nOur answer:\nYes, the Reviewer is correct. The polarization state of the weak SH beam can modify the\ncoupling process by exciting different modes depending on its polarization orientation.\nMoreover, the cross-phase modulation between the two beams is also polarization dependent.\nInitially, both beams are linearly polarized when coupled into the optical fibers. A second-\nharmonic half-wave plate, placed in front of the fiber, allows the SH beam to be controlled and\nsteered without modifying the polarization state of the fundamental beam. Given the\nconvincing results already obtained, we have chosen not to study the impact of the polarization\nstates of the SH and FF beams on modal evolution in this study. This is a complex topic that\nwill be addressed in future work. It is also worth noting that the impact of the polarization state\non the monochromatic cross-cleaning between two FF beams in multimode fibers has already\nbeen published (see Journal of Lightwave Technology, 2023, 41, 3164) (Reference 40)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "As stated in my previous report, the work presented in this manuscript is interesting and pertains to the timely and active area of optical thermalization. My initial concern is that some aspects of the underlying physics are not clearly presented. However, the authors addressed all my comments satisfactory, so I’m happy to accept this revised manuscript for publication.\nIndeed, the authors provided detailed explanation to the issues pointed out in my previous review. For instance, they explained well the absence of phase-matching and the absence of any energy exchange between the SH and fundamental beams; They explained better the temporal overlap between FF and SH pulses; They also explained well that random mode coupling plays a much greater role in their fiber system (as compared to a previous work), leading to a smoother redistribution of energy among modes and promoting an evolution towards a Rayleigh-Jeans type energy distribution. The authors also provided detailed discussion and additional results in Suppl Mater (see, e.g., SI6 Numerical modal decomposition method). The authors have also reorganized the supplementary sections, with a brief summary of the contents.\nAgain, I think the authors answered our questions very seriously, provided very detailed and well-founded answers to the questions I raised, supplemented relevant experimental results, and made corresponding modifications in the article. I thus highly recommend the acceptance of this manuscipt.", "reviewer_2": "I thank the authors for their detailed and thoughtful responses to my initial comments and for the revisions made to the manuscript and figures. The authors have addressed most of my major concerns: the inclusion of error bars for the M² measurements in Fig. 1 is convincing, and the expanded discussion on the novelty of the work is appreciated. In particular, the additional data on Raman cascade control—demonstrating a dramatic energy transfer (>80%) to the LP02-like mode—effectively alleviates my earlier concerns regarding the significance of the observed effects. Before this manuscript can be accepted for publication in Nature Communications, I recommend the following revisions for the authors’ consideration:\n1. The authors could provide more detailed discussions on how their light-by-light moulding effects might be applied in real-world scenarios.\n2. It would be helpful to include more details on the GRIN fiber used in the experiments, such as its refractive index profile and a cross-sectional image.\n3. In response to my previous comment, I would appreciate further clarification on the modal decomposition method used to obtain the modal spectra (e.g., in Fig. 1f and Fig. 2b,c), as this is an important characterization in this work.\n4. In the revised Fig. 1b, please replace “13,0” with “13.0”.", "reviewer_3": "This revision have well responded and solved issues in the previous review."}, "author_response": "Second answer to Reviewers\nSubmission of the revised version 2 of manuscript NCOMMS-25-42447-T\nWe would like to thank the Reviewers for their valuable time and constructive comments on\nour manuscript submitted to Nature Communications and titled: “Spatio-spectral light-by-light\nmoulding in multimode fibre”.\nWe have amended our manuscript to address carefully the remaining Reviewers’ concerns. Our\npoint-by-point response and a detailed list of changes made in the revised version are enclosed\nin the following Response Letter. We have also attached a version in which all major content\nchanges are highlighted in red. We believe that we have successfully addressed all issues raised\nby the Reviewers.\n----------------------------------------------------------------------------------------------------------------\nResponse to Reviewer 1:\nReviewer 1 Comment:\nAs stated in my previous report, the work presented in this manuscript is interesting and\npertains to the timely and active area of optical thermalization. My initial concern is that some\naspects of the underlying physics are not clearly presented. However, the authors addressed\nall my comments satisfactory, so I’m happy to accept this revised manuscript for publication.\nIndeed, the authors provided detailed explanation to the issues pointed out in my previous\nreview. For instance, they explained well the absence of phase-matching and the absence of\nany energy exchange between the SH and fundamental beams; They explained better the\ntemporal overlap between FF and SH pulses; They also explained well that random mode\ncoupling plays a much greater role in their fiber system (as compared to a previous work),\nleading to a smoother redistribution of energy among modes and promoting an evolution\ntowards a Rayleigh-Jeans type energy distribution. The authors also provided detailed\ndiscussion and additional results in Suppl Mater (see, e.g., SI6 Numerical modal\ndecomposition method). The authors have also reorganized the supplementary sections, with a\nbrief summary of the contents.\nAgain, I think the authors answered our questions very seriously, provided very detailed and\nwell-founded answers to the questions I raised, supplemented relevant experimental results,\nand made corresponding modifications in the article. I thus highly recommend the acceptance\nof this manuscript.\nOur Answer:\nWe appreciate the positive evaluation of our revised paper provided by Reviewer 1, and her/his\nrecommendation for acceptance.\n----------------------------------------------------------------------------------------------------------------\nReviewer 2 Comment:\nI thank the authors for their detailed and thoughtful responses to my initial comments and for\nthe revisions made to the manuscript and figures. The authors have addressed most of my major\nconcerns: the inclusion of error bars for the M² measurements in Fig. 1 is convincing, and the\nexpanded discussion on the novelty of the work is appreciated. In particular, the additional\ndata on Raman cascade control—demonstrating a dramatic energy transfer (>80%) to the\nLP02-like mode—effectively alleviates my earlier concerns regarding the significance of the\nobserved effects. Before this manuscript can be accepted for publication in Nature\nCommunications, I recommend the following revisions for the authors’ consideration:\nOur Answer:\nWe thank the Reviewer 2 for her/his positive comments on our revised manuscript, and her/his\nrecommendation for acceptance.\nReviewer 2 Comment:\n1. The authors could provide more detailed discussions on how their light-by-light moulding\neffects might be applied in real-world scenarios.\nOur Answer:\nTo address this recommendation, we have identified two potential applications. The first relates\nto the development of a new all-fiber nonlinear component capable of generating a single-mode\nsecond harmonic signal within a multimode fiber. The goal of this component is to enhance the\nfrequency conversion capabilities of fiber lasers, and to enable ultrafast signal processing by\nexploiting second-order nonlinearity in fibers. Indeed, it has already been demonstrated that\nthe poling efficiency can be significantly improved in a graded-index multimode fiber after\nself-cleaning process of the pump beam [1-2]. As a result, the induced χ(2) nonlinearity can be\neffectively exploited and controlled to produce a second-harmonic beam that propagates in the\nsingle fundamental mode (see Fig. 8 in Ref. [1]). This behavior is primarily enabled by the self-\ncleaning of the pump wave and its spatiotemporal cross-interaction with its second harmonic\nbeam along the co-propagation. Therefore, a novel all-fiber multimode component producing\na second-harmonic signal in a single guided mode can be obtained. It is worth noting that mode-\nlocking, routing, addressing, ultrafast filtering may be implemented by means of the second-\nharmonic generation process [3,5].\n[1] D. Ceoldo et al., “Second harmonic generation in multimode graded-index fibers: spatial\nbeam cleaning and multiple harmonic sideband generation”, Optics Letters, 42 (5), 971-974\n(2017).\n[2] M. A. Eftekhar et al., “Instant and efficient second-harmonic generation and down-\nconversion in unprepared graded-index multimode fibers”, Optics Letters, 42 (17), 3478-3481\n(2017).\n[3] V. Couderc et al., “Low repetition rate of a mode locked Nd: YAG laser using quadratic\npolarization switching,” Optics communications 220 (4-6), 413-417 (2003).\n[4] C. Simos et al., “Temporal reshaping of optical pulses using quadratic spatial soliton\ngeneration and spatial filtering,”IEEE Photonics Technology Letters 14 (5), 636-638 (2002)\n[5] PH Pioger et al., “Ultra-fast saturable absorber through spatial self-trapping and\nfiltering in Ti: PPLN film waveguides,” Applied physics B 77 (6), 571-576 (2003).\nThe second application involves the simultaneous propagation of multiple wavelengths in a\nmultimode fiber alongside an intense fundamental beam to perform multiplex coherent anti-\nStokes Raman scattering (M-CARS) spectroscopy and nonlinear fluorescence imaging. In such\nimaging systems, the pump pulse first generates a broad supercontinuum in the infrared region,\nproducing a wide range of new frequencies. As the pump wave undergoes spatial beam self-\ncleaning, it can further modify the infrared radiation (via cross-cleaning interaction), forcing it\nto propagate in low-order modes. As a result, the transverse spatial resolution of the nonlinear\nimage is significantly improved, reaching sub-micrometer resolution [1,2].\n[1] N. Ould-Moussa et al., “Spatiotemporal beam self-cleaning for high-resolution nonlinear\nfluorescence imaging with multimode fibres,” Scientific Reports 11, 18240 (2021).\n[2] T. Mansuryan et al., “Spatial Division Multiplexing for Multiplex Coherent Anti-Stokes\nRaman Scattering,” Journal of Lightwave Technology 41 (22), 6875-6883 (2023).\nBeyond the two examples mentioned above, it is also important to emphasize that a fiber laser\nsource, which typically integrates a pump beam and the laser signal, can also utilize a cross-\ninteraction process to reshape its beam. Indeed, using a pulsed pump beam in a multimode fiber\nmay strongly influence the final spatial distribution of the laser radiation, thus forcing single-\nmode laser emission in multimode fibers. To illustrate all these potential applications, we have\ninserted the following sentences, together with two relevant references, at the end of the\ndiscussion section.\n“Beyond this demonstration of the cross-action between two beams propagating\nsimultaneously in a multimode fiber, we would also like to highlight that these multiwave\nnonlinear interaction processes may find numerous potential applications. Let us mention, for\ninstance, multiplexed coherent anti-Stokes Raman spectroscopy and nonlinear fluorescence\nspectroscopy in multimode fibers, where multiple wavelengths propagate simultaneously,\nenabling high-resolution transverse imaging of biological samples over a wide spectral range\n[35, 47]. Another example involves the process of second-harmonic generation in multimode\nfibers, which gives rise to single-mode emission owing to the cross action between the\nfundamental pump and its harmonic [48].\n[47] T Mansuryan, A Tonello, K Krupa, A Desmouliere, G Ndong Ntoutoume, V Sol, C Lefort,\nM Zitelli, M Ferraro, F Mangini, Y Sun, Y Arosa Lobato, B Wetzel, S Wabnitz, V Couderc,\n“Spatial Division Multiplexing for Multiplex Coherent Anti-Stokes Raman Scattering,”\nJournal of Lightwave Technology 41 (22), 6875-6883 (2023).\n[48] D. Ceoldo, K Krupa, A Tonello, V Couderc, Daniele Modotto, Umberto Minoni, G Millot,\nStefan WabnitzD. Ceoldo et al., “Second harmonic generation in multimode graded-index\nfibers: spatial beam cleaning and multiple harmonic sideband generation”, Optics Letters, 42\n(5), 971-974 (2017).”\nReviewer 2 Comment:\n2. It would be helpful to include more details on the GRIN fiber used in the experiments, such\nas its refractive index profile and a cross-sectional image.\nOur Answer:\nWe included all the features of the used optical fibers in the SI7 section of Supplementary\nMaterial, as follow:\nSI7 - Features of multimode fibers\nThe characteristics of the GRIN multimode optical fibers used in the experiments are presented\nbelow. The transverse refractive index profile was measured using the ARDEN nPA600\nrefractive index profiler. The first germanium-doped fiber has a diameter of 50 µm and a\nrefractive index difference Δn of 15 × 10⁻³. The second fiber has a diameter of 27 µm and a\nrefractive index difference of 30 × 10⁻³. The refractive index profile shows a parabolic\nevolution for both fibers due to germanium doping.\nFigure SI7: (a) 2D image of the two fibers used and measured with the ARDEN nPA600\nprofiler; (b) Their corresponding 1D profiles. The first profile (on the right) has a diameter of\n50 µm, while the second profile (on the left) has a diameter of 27 µm. The red curves on the\n1D profiles are parabolic fittings.\nReviewer 2 Comment:\n3. In response to my previous comment, I would appreciate further clarification on the modal\ndecomposition method used to obtain the modal spectra (e.g., in Fig. 1f and Fig. 2b,c), as this\nis an important characterization in this work.\nOur Answer:\nThe detailed methodology employed for modal decomposition is clearly described in section\nSI6 of Supplementary Material, where we also cite the reference that inspired our approach.\nWe would like to note that both the explanation and the schematic illustrating this data-\nprocessing procedure, based on near-field and far-field measurements, were already provided\nin our previous response. For completeness and clarity, this explanation has been included\nagain in current revision.\nSI6 - Numerical modal decomposition method\nTo perform the modal decomposition, we followed a procedure similar to the one reported in\nKim et al. [19] based on a stochastic parallel gradient descent (SPGD) method. The output field\nfrom the fiber is expressed as a superposition of guided LG modes. The near (and far) field\nintensity distribution can then be reconstructed as a weighted sum of these LG modes (or their\nFourier transform), with the unknown modal amplitudes and phases determined through an\niterative optimization as shown in the flow chart in SI6a.\nFigure SI6a: Flow chart showing the optimization procedure for the modal decomposition\nFirst, we addressed possible mismatches in sampling and alignment by interpolating the\nmeasured intensity distributions. Specifically, both near and far field intensity profiles were\nresampled onto a common grid with identical spatial resolution. This direct interpolation\nensured that differences between the two datasets arose only from modal content, and not from\npixel resolution or coordinate offsets. Once the experimental and simulated profiles were\nmatched in resolution, a first correlation between the experimental NF and the modes is done\nto obtain an initial set of relative power for each mode (𝑎 ), we set all modes to be in phase\n𝑖\ninitially. This algorithm iteratively perturbs the modal coefficients: relative powers (𝑎 ) and\n𝑖\nphases(𝜑 ) with random updates and retains those changes that reduce the cost function,\n𝑖\ndefined as the normalized mismatch between experimental and reconstructed intensity profiles.\nBy dynamically adjusting the perturbation size during the iterations, the SPGD approach\nenhances convergence speed while avoiding local minima. The optimization was structured in\nepochs; each defined as 100 iterations per mode considered. The algorithm terminated either\nafter 100 epochs or once the improvement in correlation between the experimental and\nreconstructed profiles fell below 10−4. In the case of modal decomposition applied to simulated\ndata, the far-field condition was omitted since the full phase information of the beam was\navailable. By enforcing agreement at both distances, the uniqueness of the modal\ndecomposition was guaranteed, avoiding the conjugate-pair ambiguity that can appear when\nusing a single measurement plane. In figure SI6b, some examples of the reconstructed data are\nshown.\nFigure SI6b: examples of reconstructed images obtained experimentally by using CCD\ncamera, by using our numerical method to obtain modal decomposition at the fiber end.\nReviewer 2 Comment:\n4. In the revised Fig. 1b, please replace “13,0” with “13.0”.\nOur Answer:\nWe replaced it.\n-----------------------------------------------------------------------------------------------------------\nReviewer 3 Comment:\nThis revision has well responded and solved issues in the previous review.\nOur Answer:\nWe thank reviewer 3 for the positive feedback on our revised manuscript, and for the\nrecommendation of its acceptance."}, {"round": 3, "reviewer_comments": {"reviewer_2": "The authors have addressed all my comments, and now I can recommend it for publication in Nature Communications."}, "author_response": ""}]

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