## Abstract

In this paper, we investigated the geometric parametric instability (GPI) in graded-index multimode fibers through the multimode generalized nonlinear Schrödinger equation. Our results clearly and intuitively indicate that the generations of GPI sidebands are nearly synchronous in the spectrums of all modes, and the shapes of these spectrums are nearly the same. The numerical results show that the energies of the GPI sidebands come from the pump sideband, and these sidebands are carried by similar spatial beam profiles due to the similar modal components. We also found that the large modal dispersion has an influence for the symmetry of these GPI sidebands.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In recent years, the potential applications of multimode fibers (MMFs) have attracted renewed interest due to the need for space division multiplexing (SDM) systems and imaging systems [1,2]. Compared with the single mode fibers (SMFs), MMFs have a larger mode field area and support more spatial modes, which can be used to improve the capacity of the communication system and provide the ultra-wideband tunable high spatial beam quality light source [3]. The complex coupling between multiple spatial modes means that nonlinear effects have spatiotemporal characteristics [4]. So, multimode fibers are also used as the platform to study complex nonlinear spatiotemporal dynamics such as spatial beam self-cleaning [5–10], intermodal four-wave mixing (IMFWM) [11,12], rogue waves [13], spatiotemporal mode-locking [14–16], multimode solitons [1,17], supercontinuum generation [18–20], and geometric parametric instability (GPI) [21–23].

Due to the interactions of nonlinearity and dispersion in the medium, parametric instability (PI) occurs in the wave propagation when a parameter of the medium is periodically modulated along the longitudinal direction [21]. PI is commonly known as the Faraday instability, which is a candidate mechanism to induce mode-locking in lasers [24] when the periodic modulation of the parameter of the medium is caused by an external forcing [22]. In some physical systems that naturally exhibit collective oscillations, PI is called geometric-type of parametric instability (GPI) [21]. Because of the periodic self-imaging [25,26] of the multimode beam in MMFs, the periodic oscillating field provides a periodic modulation for the refractive index along the longitudinal direction of the fiber, which leads to the GPI in the multimode fibers [21]. GPI in multimode fibers has attracted many researchers’ attention. In 2003, Longhi first theoretically predicted the existence of GPI in MMFs [27]. Krupa et al. then observed the GPI sidebands in MMFs for the first time and studied these sidebands’ spatial beam profiles [21]. They also carried out numerical simulations and obtained the GPI sidebands that agree well with the analytical predictions and experimental results. Eznaveh et al. proved in the experiment that the frequency positions of GPI sidebands could be adjusted by changing the core size. The sidebands could be redshifted by increasing the core sizes of the MMFs [28]. Matteo Conforti et al. proposed a 1 + 1D generalized nonlinear Schrodinger equation with a periodic nonlinear coefficient to describe the nonlinear pulse propagation in MMFs, which can rapidly reproduce GPI effectively [29]. C. Mas Arabi et al. proved that GPI sidebands with moiré-like patterns could be obtained by modulating the core diameter with a period close to the distance of self-imaging in MMFs [22]. H. E. Lopez Aviles et al. provided a rigorous analysis of GPI in GRIN MMFs. They presented the parametric problem in the Hill equation and solved it systematically with a Floquet approach [23].

Krupa et al. pointed out that the GPI sidebands are induced by the mechanism of QPM-FWM and observed the pump, and the GPI sidebands are all carried by bell-shaped spatial beam profiles [21]. Their results presented a solid impetus to the further researches of the GPI in MMFs. Other researchers also did some interesting work (such as Eznaveh et al. ‘s studies) to enrich the GPI research studies. However, their reports are all basically from the perspective of the beam. In this paper, unlike previous studies, we explore the GPI sidebands in graded-index multimode fibers from the viewpoint of spatial modes for the first time to the best of our knowledge. In this work, we selected the first five radially-symmetric modes for the numerical simulations and employed the multimode generalized nonlinear Schrödinger equation (MM-GNLSE) model [30–32] to describe the propagation of picosecond pulse in GRIN MMFs. We also observed the spectrum of each mode and found that the generations of all GPI sidebands are nearly synchronous, and the shapes of these spectrums are almost the same. Then, we further studied the energy source and the spatial distributions of the GPI sidebands. Finally, we implemented the numerical simulations that involve the modal dispersion term and found that the large modal dispersion influences the symmetry of these GPI sidebands.

## 2. Theoretical model

In GRIN MMFs, due to the equal spacing of the modal propagation constants [33], the multimode beam produces a periodic self-imaging effect during the propagation process [21]. Because of the Kerr effect, the field conducts a longitudinal periodic modulation on the refractive index [5,34] in a way that is similar to an equivalent long-period fiber Bragg grating. So that, the quasi-phase-matching (QPM) condition for generating GPI sidebands can be satisfied [21], $\textrm{2}{\textrm{k}_P} - {k_S} - {k_A} ={-} 2\pi h/\xi$, where ${\textrm{k}_P},{k_S},{k_A}$ represent the pump wave vector, GPI Stokes wave vector, and GPI anti-Stokes wave vector respectively, and $h = 1,2,3\ldots .$ It’s possible to calculate the frequency offsets of GPI sidebands by the formula ${f_h} ={\pm} \sqrt h {f_m}$, where $2\pi {f_m} = \sqrt {2\pi /(\xi {K^{^{\prime\prime}}})}$, ${f_m}$ is the first-order frequency offset of GPI sidebands, $\xi = \pi \rho /\sqrt {2\Delta }$ is the self-imaging period, ${K^{^{\prime\prime}}}$ is the group velocity dispersion, $\rho$ is the fiber core radius, and $\Delta $ is the relative index difference [21].

We considered the first five-orders radially-symmetric spatial modes and used a parabolic GRIN MMF with $\rho = 25um,\Delta = 8.78 \times {10^{ - 3}}$. The value of group velocity dispersion for the fundamental mode is $16.35 \times {10^{ - 27}}{s^2}/m$ at the pump wavelength of 1064 nm. Because of different group velocity dispersion, there are some differences among the locations of GPI sidebands for the five modes. Therefore, we calculated the frequency offsets of the first four-orders sidebands for each mode, respectively. As shown in Table 1, the sideband frequency shifts of the higher-order modes are larger. With the increase of the sideband order, the frequency shift differences between the higher-order modes and the fundamental mode become larger. It means that the positions of GPI sidebands can be adjusted by selecting different spatial modes.

The description of pulse propagation in multimode fibers often involves complex spatiotemporal dynamics. There are two models for the numerical description of pulse propagation in multimode fibers [29]: the (3 + 1) D generalized nonlinear Schrödinger equation (GNLSE) [1,25,33,35], and the MM-GNLSE. In the experiment, beam propagation usually involves abundant guided modes. However, it is almost impossible to consider so many modes in the numerical simulation because of an expensive computation. The (3 + 1) D GNLSE was used more to study the GPI in previous studies. To study the generation process and the spatial distribution characters of the GPI sidebands more clearly, we employed the MM-GNLSE to describe the propagation of pulse in GRIN MMFs. This model decomposes the electromagnetic field into a series of spatial modes, which is helpful to study the GPI from a more intuitive angle. The evolution of the electric field temporal envelope for the spatial mode p can be described as:

## 3. Simulation results and discussions

In the simulation, we neglected the Raman scattering effect, material absorption, and higher-order dispersion. The second term in the Eq. (1) represents the modal dispersion. To simplify the analysis, we also ignored this term here. We employed a pulse with a pulse width of 100 Ps and adjusted the peak power to 270 kW. Initial pulse energy is distributed to the first five radially-symmetric modes (40% in $L{P_{01}}$, 20% in $L{P_{02}}$, 17% in $L{P_{03}}$, 15% in $L{P_{04}}$, and 8% in $L{P_{05}}$).

Figure 1 shows the first fourth-orders GPI sidebands where the propagation distance z = 0.3 m. The red and black dotted lines respectively represent the positions of the pump sideband and the first four-orders GPI sidebands. The fundamental mode occupies the most energy, and the intensity of the pump sideband is the largest. Therefore, we used the group velocity dispersion value of the fundamental mode to calculate the sideband frequency shifts. This figure indicates that these sidebands’ peaks basically appear on the dotted lines correspond to the theoretical values. Note that numerical simulation's sideband peaks move slightly towards the pump peak compared with theoretical predictions, which is caused by the high initial peak power [23].

To explore the generations of the GPI sidebands more clearly and intuitively, we further investigated the spectra of the first five radially-symmetric modes and the total spectrum, respectively, shown in Fig. 2. The red dotted line in each spectrum diagram represents the pump sideband location, and the black dotted lines indicate the locations of GPI sidebands by the analytical predictions. It is interesting that the spectrum shapes of these modes are nearly the same as the total spectrum, and the generations of GPI sidebands are almost synchronous during the evolution (see Visualization 1). Note that we uniformly selected the intensity of the fundamental mode's pump peak as the reference for the intuitive presentation of the sideband intensity.

To further studied the energy which flows into the GPI sidebands, we selected the fundamental mode as an example (choosing any of these spatial modes is right) and calculated the mean increased pump energy ${E_{MIPE}}$ and the mean increased mode energy ${E_{MIME}}$ of the fundamental mode, which are written as:

Next, we selected the spectrum with a bandwidth of 2-4 THz (adjusted according to the spectrum width of the sideband) near each sideband's peak value and study the spatial beam distributions of these spectral components. As shown in Fig. 4, the first four GPI anti-Stokes, Stokes sidebands, and the pump peak sideband are all carried by a similar spatial beam profile, where the propagation distance z = 0.282 m. The reason for this result is that the modal components of all sidebands are nearly the same.

Finally, we implemented the numerical simulations which involve the modal dispersion term. The first five subgraphs in Fig. 5 represent the spectra of the first five radially- symmetric modes and the total spectrum of the five spatial modes, where the propagation distance z = 0.252 m. Different from Fig. 2, Fig. 5 shows a slight asymmetry in each subgraph caused by the modal dispersion. However, the spectrum shapes of these spatial modes are still very similar, which indicates that it’s reasonable to ignore the modal dispersion term for a more intuitive and simpler analysis when we studied the generation of GPI in GRIN MMFs.

In the studies by Krupa et al., the GPI sidebands and the pump sideband are carried by the same spatial beam profile during the generation of the GPI sidebands [21]. We speculated the reason is that there are a significant number of spatial modes in their studies [21], and the pump underwent the process of Kerr self-cleaning before a substantial spectral broadening [18]. So, the main energy is distributed among the low-order modes (most of the energy flows into the fundamental mode). These spatial modes come from the same or adjacent modal groups, which means the small modal dispersions. Therefore, the modal components and the spatial beam distributions of the pump sideband and the GPI sidebands are the same.

## 4. Conclusion

In conclusion, we explored the generation and the spatial distributions of GPI sidebands in GRIN MMFs from the viewpoint of spatial modes. In this paper, we selected the first five radially-symmetric modes and employed the MM-GNLSE model to reproduce the GPI sidebands numerically. As expected, the numerical simulations agree well with the analytical predictions. Our studies indicate that the energy of the pump sideband flows into the GPI sidebands and all modes have a similar spectrum. Our results also show that the pump sideband and the GPI sidebands have a similar spatial distribution due to the nearly same modal components, and the large modal dispersion influences the symmetry of the spectrum of each mode. Our work shows the generation of the GPI sidebands from a clear and visualized view and provides a supplement for Krupa et al. ‘s studies on GPI.

It is well known that the GPI in GRIN MMFs can be used to induce the generation of supercontinuum from the visible light to the mid-infrared region [18,21], which offers an application in generating new applied light sources, especially in the biomedical imaging domain. Our studies provide some theoretical references for the further applications of the GPI in GRIN MMFs.

## Funding

National Natural Science Foundation of China (11874019, 61875058, 92050101); Guangzhou Key Laboratory for Special Fiber Photonic Devices and Applications, South China Normal University.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

No data were generated or analyzed in the presented research.

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