Peregrine soliton emits dispersive waves within 1 graded-index multimode fibers without 2 higher-order dispersion

: 10 We investigate the propagation dynamics of the Peregrine soliton, a significant prototype 11 of rogue waves, within the graded-index multimode fibers, in the absence of higher-order 12 dispersion. The Peregrine soliton keeps the approximate evolution trend when propagating 13 within the graded-index multimode fibers to replace the single-mode fibers when preserving 14 the equivalent nonlinear effect. In addition, a series of dispersive waves (also called resonant 15 radiation) can be emitted by the Peregrine soliton, perturbated by the periodic beam oscillation 16 caused by the spatial self-imaging effect within the graded-index multimode fibers. To be more 17 exact, the location of the multiple resonant frequencies can be predicted using the modified 18 quasi-phase-matching conditions, which are verified by the numerically calculated results. We 19 can also manipulate the locations of spectral sidebands and the peak power of dispersive waves 20 by changing the self-imaging parameter of the graded-index multimode fibers. Our findings can 21 provide a deeper comprehension of the propagation characteristic of Peregrine soliton within 22 the graded-index multimode fibers and provide valuable instruction for further rich nonlinear 23 experiments. 24


INTRODUCTION
Currently, multimode fibers (MMFs) have attracted substantial attention in nonlinear optics, making potential contributions to theoretical scientific research and practical applications.It provides a complicated channel for supporting thousands of modes interacting with each other when propagating [1,2].Due to the spatial-division and mode-division multiplexing techniques [3][4][5], MMFs are regarded as a large bandwidth medium for optical fiber communication.
The MMFs can also provide a multitude of rich nonlinear features that cannot be achieved in single-mode fibers (SMFs) alone [6][7][8].In addition, MMFs have garnered increased interest in ultrafast photonics [9], Kerr self-cleaning effect [10][11][12], supercontinuum generation [13,14] and spatiotemporal mode-lock [7,15].One crucial feature of the graded-index (GRIN) MMFs is the spatial beam self-imaging effect [16,17], marked by the periodic compression and stretching of the input beam.Consequently, the spatial self-imaging effect induces a longitudinally varying effective beam area, leading to the formation of a periodic nonlinear refractive index grating [16][17][18].According to the previous research, the soliton pulse can emit linear dispersive waves (DWs) perturbed by the higher-order dispersion (HOD) within the SMFs [19].However, the optical beam undergoes a periodically evolving process when propagating through the GRIN MMFs, resulting in the appearance of multiple frequency peaks of the DWs as well without HOD [20,21].The resonant frequencies of the DWs also can be predicted by the quasi-phase matching (QPM) [22][23][24], giving rise to the phenomenon of the pairs of instability spectrum sidebands called geometric parametric instability (GPI) in MMFs which refractive index changing parabolically [6,[25][26][27].QPM also plays a significant role in elucidating the mechanism behind supercontinuum generation [28,29].
Peregrine soliton (PS) is considered a significant prototype of rogue waves (RWs).Its generation may be attributed to the strong modulation instability (MI) including linear random superposition effect and nonlinear random noise effect [30][31][32][33].Peregrine originally proposed the analytic expression of the PS in hydrodynamics [34], and the PS is also a solution of the nonlinear Schrödinger (NLS) equation.It represents the ultimate case of the Kuznetsov-Ma (KM) soliton and Akhmediev breather (AB) [35], exhibiting both spatial and temporal localization properties.However, it was not until 2010 that Kibler et al. observed the PS experimentally in optical fiber for the first time [36].Due to the characteristics of high peak power at the maximum compression point, the PS has attracted considerable interest in the generation of high-power pulses [37][38][39].The PS is also investigated in the MI [40][41][42], quadratic media [43], water waves [44], and plasmas [45].
Recently, there has been some numerical investigation into the propagation characteristics of PS pulses in the SMFs.Baronio et al. study the DWs radiated by fundamental PS pulse when perturbed by the TOD, whose radiation process is influenced by the intrinsic local longitudinally varying soliton wavenumber [46].In addition, Chowdury et al. investigate the fissive characteristic of a series of PS family, under the TOD and higher-order nonlinear effect [47][48][49].Furthermore, in 2017, Conforti et al. proposed a solved fast and efficient model, under weakly or moderate nonlinear conditions, to describe the nonlinear pulse propagation process exactly through the GRIN MMFs [23].The input field will be modulated by the periodic varying nonlinearity caused by the spatial self-imaging effects when propagating inside the GRIN MMFs.Therefore, we aim to investigate the evolution feasibility of PS within the GRIN MMFs and explore its underlying propagation characteristics.To the best of our knowledge, such an investigation has not been reported so far.
In this paper, we explore the propagation dynamics of the PS within the GRIN MMFs, in the absence of HOD.We show that the PS can radiate DWs, perturbated by the spatial self-imaging effect within the GRIN MMFs.Moreover, we propose a modified QPM condition to forecast the resonant frequencies of DWs analytically, and its accuracy can be validated through numerical results.Besides, we can also manipulate the generation of multiple DWs by changing the nonlinear parameter linked to the spatial self-imaging effect within GRIN MMFs.The structure of this paper is as follows: Section 2 introduces the theoretical model, including the derived (1 + 1) generalized NLS equation.Section 3 is dedicated to presenting the numerical results and engaging in discussions.Finally, in Sec. 4, we summarize our conclusions.

THEORETICAL ANALYSIS
The refraction index of the GRIN MMFs exhibits a parabolic variation form within the core as where  0 is the core index,  is the core radius, Δ = ( 0 −  1 )/ 1 , and  1 represents the cladding index.The Kerr self-focusing effect of the fibers is linked to the Kerr coefficient  2 and the local intensity .We assume that the pulse beam is consistently confined to the fiber core due to the Kerr self-focusing effect and neglects all polarization effects (the electric field is polarized along the -axis and this polarization state remains fixed with the propagation direction).In this manner, the spatial-temporal nonlinear propagation of the Gaussian shape beam can be simplified as the following Gross-Pitaevskii equation under the paraxial and slowly varying envelope approximations [2,23,50]: where  is the slowly varying electric field envelope,  0 =  0  0 / represents the wavenumber at the frequency  0 and  2 links to the group-velocity dispersion (GVD).We could numerically solve this equation using the split-step Fourier-transform method [51], but it would demand substantial computing resources owing to the full four-dimensional Gaussian beam propagating inside GRIN MMFs experiences the spatial self-imaging effect in the linear regime or the weakly nonlinear regime, where the transverse beam profile reproduces along the propagation direction, as reported in previous studies [18,23,52].The amplitude of the well-known Gaussian beam can be expressed in the following form: [23,53]: where  0 is the spot size of the input beam in  = 0,  =  2  2 0  2 0  0 /2 0 is a dimensionless parameter linked to the beam collapse,  0 corresponds to the peak power of the pulse, and 2Δ represents the self-imaging period of the GRIN MMFs.The oscillated beam width  will recover to its initial value  0 at each self-imaging period length   .We assume that the self-imaging portion of the beam keeps steady [18], to drive the solution of Eq. ( 2) as  (, , , ) = (, ) •  (, , ).The complicated (3 + 1) equation can be simplified to a (1 + 1) form that is computationally efficient, revealing that (, ) satisfies the following quasi-one-dimension NLS equation: where (, ) is the pulse envelope, the nonlinear parameter  =  0  2 /(     ) is related to the effective area     , and the periodically varying function  () =  2 ()/ 2 0 , which indicates the spatial beam width recurrence and oscillation of the initial Gaussian beam due to the spatial self-imaging effect.We can automatically normalize the NLS equation to indicate the beam propagation in fibers using the following dimensionless variables [51]: where , , and  correspond to the normalized propagation distance, time, and pulse envelope corresponding to the dispersion length   =  2 0 /| 2 |, initial pulse duration  0 , and peak power  0 , respectively.The normalized NLS equation is as follows: where,  = √  0   is the soliton numbers, and the periodic varying function can be rewritten as: where the dimensionless parameter  represents the extent of the beam width compression during each self-imaging period.For instance, at the distance  = (  /2), the width of the beam  will become √  times the initial value  0 .When the value  varies from 0.1 to 10, there are no qualitative differences in the optical pulse propagation dynamics [53].It is worth noting that  = 1 represents an extraordinary case in which the beam width remains constant with propagation direction, equivalent to the condition in an SMF without any periodic oscillation effect.Here, we choose  = 0.25, indicating that the beam width will reduce to half of the original size during each cycle.For a typical GRIN MMF with  = 25 and Δ = 0.01, the corresponding self-imaging period   is 0.555 less than 1.In contrast, the dispersion length   in the typical GRIN MMFs more than 10 for  0 > 0.1 ps at the center wavelengths near 1550 [54].As a result, the value of  is then greater than 100, indicating that the beam width can oscillate hundreds of times within one dispersion length.In 2018, Ahsan et al. numerically investigated the stability of the optical soliton within the GRIN MMFs under different average-nonlinearity N = / 4 √ , which was defined as the nonlinear term average over the self-imaging period   within the GRIN MMFs [16,53,55].Here, we choose N = 1 to ensure that the PS pulse inside the GRIN MMFs sustains an equivalent nonlinear term, according to the NLS equation [51].
The dispersion length is relevant to the evolution of soliton, which can not occur under the condition that the beam width changes on a scale of 1 or less [56].It is worth noting that  = 1 virtually does not exist because the temporal variation can influence the spatial width periodic varying in this scenario.Based on this, we select two  values (100 and 10) for the numerical simulation, corresponding to the situations where the dispersion length is larger than or ten times the self-imaging period.These different scenarios allow for a better understanding of the complex spatial-temporal dynamics within the GRIN MMFs.

RESULT AND DISCUSSION
PS is a spatial-temporal doubly localized solitary wave with an additional phase compared to the plane wave.The expression of the PS is as follows: where the exponential term represents the nonlinear Kerr shift of the background [34,57].Here, we take Eq. ( 9) with  = −3 as the injected field, simulating the evolution of the PS pulse through 6  distance from  = −3 to 3.
As shown in Figs.1(a) and 1(c), the overall trend of the temporal evolution of the PS within the GRIN MMFs are much the same when compared to it in SMF, shown in Ref. [46].
It shows that the power of the PS at the maximum compression point position is nearly nine times higher than the background, even in the worst-case scenario perturbated by the strong spatial self-imaging effect  = 10.Moreover, the phase distribution of the PS within the GRIN MMFs [see Figs.1(b) and 1(d)], is also consistent with the theoretical case.In addition, we can observe the alternation of strength and weakness power distribution of the PS near the maximum compression point within the GRIN MMFs at the case  = 10 shown in Fig. 1(c).It demonstrates that the strong spatial self-imaging effect can perturb the evolution of the PS but does not disrupt its propagation trend.We can find that the phenomenon of the alternation of strength and weakness power of PS becomes more pronounced at the center area.This can be explained by the fact that the central area is more susceptible to being modulated by the spatial self-imaging effect caused by the periodically varying effective beam area within the GRIN MMFs.As a result of periodic modulation, the linear DWs are emitted by the PS at both wings of the maximum compression point which is the peak power location of the PS pulse [see Fig.As the solution of the NLS equation in SMFs, the PS exhibits a local nonlinear deviation (along with ) of the longitudinal phase   (, ), which is induced by the background phase shift [44].In previous research, the overall phase of the PS was deduced as the sum of the background and local contribution, which can be expressed explicitly as follows [46]: The maximum phase shift can be reached at the pulse center  = 0, and the longitudinal profile is shown in Ref. [46].The derivative of such a phase in Eq. ( 10) represents the longitudinally varying nonlinear wavenumber of the PS, which is expressed as the superposition of the background and local wavenumber: The local contribution reaches a maximum around the maximum compression point  = 0, resulting in   ( = 0) = 11/3.Such a local variation of the PS can play an important role in exploring the characteristics of the DWs emitted by the PS within the GRIN MMFs.The GRIN MMFs processing the spatial self-imaging effect create an infinite number of sidebands, whose frequency shifts from the center frequency can be estimated by the QPM condition [16,20,23,50].Here we briefly deduce the analytic expression of resonant frequencies of the DWs within the GRIN MMFs: where  is an integer and   represents the wavevector of the DW.In the general NLS equation, the wavevector of the DW can be written as   = ≥2   ! Ω  [51], Ω = 2( −  0 ) 0 represents the resonant frequencies of the DWs.The wavenumber of the PS can be obtained by the following normalized equation [46]: where   is the position of the pulse peak power within the GRIN MMFs.The normalized QPM equation in the absence of HOD can be written in the following form: By solving the roots of Eq. ( 14), we can analytically obtain the resonant frequencies at different values of wavenumber  in Fig. 5   confirms the evolution feasibility of the PS within the GRIN MMFs by successfully predicting the resonant frequencies modulated by the local phase deviation, which was previously observed only in SMFs [44,46].

CONCLUSION
In conclusion, we have reported a numerical study on the propagation dynamics of PS pulse through the GRIN MMFs.Firstly, we have found that the propagation characteristics of the PS within the GRIN MMFs are the same in SMFs, both in terms of temporal evolution and phase variation.Under the periodic modulation effect induced by the spatial beam self-imaging effect, the PS can emit multiple DWs even in the absence of HOD.The resonant frequency locations of the DWs can be predicted using the modified QPM condition.Finally, we can manipulate the peak power and the resonant frequencies of the DWs by changing the  values.This study is helpful in the understanding of the rich nonlinear phenomena associated with the generation of DWs and GPI sidebands within the GRIN MMFs.Disclosures.The authors declare no conflicts of interest.

Fig. 1 .
Fig. 1.(a) and (c) Temporal evolutions of the PS pulse over a distance  = 6  when it is perturbed by the spatial self-imaging effect ( = 100 and  = 10) within the GRIN MMFs.(b) and (d) Phase distribution of the PS corresponding to (a) and (c).
1(c)].Different from the typical DWs perturbed by HOD, the oscillation caused by the spatial self-imaging effect can also induce the generation of the DWs within the GRIN MMFs.

Figure 3
Figure 3 shows three X-FROG spectrograms calculated numerically at three locations ( = −3, 0, and 3) within the GRIN MMFs.These spectrograms graphically reveal the generation of DWs when the PS pulses propagate through the fiber.At the initial position of  = −3, the entire spectrum concentrates on the zero-dispersion wavelength regime of the GRIN MMFs.The spectrum broadens with further propagation, and at near  = 0, a series of frequency peaks appeared symmetrically at the normal-GVD and anomalous-GVD regimes, which correspond to the resonant frequencies of the DWs emitted by the PS [see Fig. 3(b)].It indicated that the PS pulse emits DWs when modulated by the periodic varying nonlinear parameter within the GRIN MMFs.With further propagation, one can see in Fig. 3(c) with  = 3, the DWs evolve completely with a broad spectrum and higher delay time.As shown in Figs.4(a) and 4(c), a series of multiple symmetrical spectral sidebands occurred at  = 10 and  = 100, respectively.Different from the theoretical Fourier spectrum of the PS in SMFs, the additional sidebands represent the frequencies of the DWs caused by the spatial self-imaging effect within the GRIN MMFs.Corresponding to Fig 3(b), where the DWs formed at the position  = 0, we can observe these spectral sidebands appearing at the peak power point.It is worth noting that different  values can generate different spacing of a series of sidebands, and the frequency intervals are proportional to .These spectrum sidebands can be filtered out

Fig. 4 .
Fig. 4. (a) and (c) The spectrum evolution of the PS within the GRIN MMF over 6  distance at  = 100 and  = 10 respectively.(b) and (d) The DWs filtered from these frequencies sidebands of the corresponding spectrum via the inverse Fourier transform.
(a).The intersection point between the dotted black line (with different ) and the solid red parabola, signifies the spectral frequencies ( −  0 ) 0 .Correspondingly, one can observe multiple spectrum peaks symmetrically distributed in the output spectrum (solid red) at  = 3 [see Fig.5(b)].The blue arrow connects these analytic roots to a series of simulated spectrum peaks, revealing a high consistency between analytic and simulated resonant frequencies.In addition, it also substantiates the possibility of the local phase deviation of the PS within the GRIN MMFs via the modified QPM equation.It further

Fig. 5 .
Fig. 5. (a) and (b) The predicted spectral locations of the DWs derived from the QPM conditions in contrast to the simulated output spectrum obtained at the 6  distance within the GRIN MMF when  = 10.

Figure 6
Figure 6 visualizes the variation of spectral intensity distribution at  = 3 with .When increasing , the interval of resonant frequencies becomes wider, consistent with the analytical varying trend.The power of the DWs strengthens with the decrease of  caused by the strong self-imaging effect.When the value of  is between 5 and 10, there is a similar continuous spectrum lying near the center wavelength area as shown in Fig. 6.This can be interpreted as a small self-imaging parameter inducing the interval of frequency sidebands gradually diminishing and overlapping.The results show that we can manipulate the distribution of the resonant frequencies and the power of the DWs by changing  within the GRIN MMFs.

Fig. 6 .
Fig. 6.Output spectra at  = 3 as a function of  values from 5 to 50.

Funding.
National Natural Science Foundation of China (61975130); Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515010084).