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Abstract
We investigate the modal energy flow of the femtosecond-pulsed beam self-cleaning on LP11 mode with the influence of different factors such as the initial fraction of LP11 mode, initial peak power, distribution of high-order modes and the numerical aperture of the fiber. It is interesting that there is a critical value of the initial peak power, Pcr, which is the watershed, not only in the quantitatively dominant transverse mode converting from LP11 mode to LP01 mode, but also in the behavior of HOMs of the transition from Attractor to chaos. Our simulation results may provide a novel perspective to understanding the beam self-cleaning on LP11 mode.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Over the last decade, the multi-mode fibers (MMFs) are extensively investigated owing to the potential application in spatial division multiplexing system [1,2]. Besides, due to supporting a large number of spatial modes and allowing of high power transmission, MMFs provide an excellent platform to observe the complex spatiotemporal dynamics phenomenon, such as multi-mode solitons [2–4], spatiotemporal mode-locking [5–8], supercontinuum generation [9–11], and spatial beam self-cleaning [12–17]. Particularly, the Kerr beam self-cleaning (KBSC) has attracted much research attention, because it is the nonlinear phenomenon in which the initial speckled beam can evolve into a bell-shaped beam of good quality after the propagation in MMF [12,15]. So far, the physical mechanism of the KBSC is still an open issue, and there are several concepts are introduced to offer an explanation: self-organized instability [14], nonlinear irreversible mode-coupling process [13], wave condensation which is well known in hydrodynamic 2D turbulence [16,17], and the thermalization of modal distribution in multimode fiber [18–20]. The research of Fan O. Wu et al. showed that the beam self-cleaning (BSC) phenomenon can be regarded as a thermalization phenomenon where most of power flows into the lowest group of modes, and the output modal distribution is in accord with the Rayleigh-Jeans (RJ) distribution which results from the thermal equilibrium [18]. These explanation mentioned above, which are related to the BSC on LP$_{01}$ mode, have provided some interesting methods and inspiration for someone to understand and explore the nonlinear phenomenon in MMFs.
More recently, the spatial beam self-cleaning on LP$_{11}$ mode is achieved experimentally [21–26]. It has been observed with diverse experimental conditions as follows: different kinds of fibers including GRIN MMFs [22–26], step-index MMF [21], and tapered fiber [26]; normal [21–23,25,26] to anomalous dispersion regimes [24]; and initial pulse with different duration of sub-nanosecond and picosecond [21–26]. These previous research have reported the threshold power [21–24], the output beam profiles evolution upon the power [21–24], and the spatiotemporal dynamics [24]. Also, the effect of initial conditions such as pulse duration, wavelength, and input beam position with respect to the core center have been investigated [21,24]. However, the BSC on LP$_{11}$ mode is not investigated as extensively as the BSC on LP$_{01}$ mode, and the relative physical mechanism is still not clear.
In this work, we systematically study the modal energy flow of the beam self-cleaning on LP$_{11}$ mode with the femtosecond input pulse. The energy flow of this process is influenced by different factors including the initial fraction of LP$_{11}$ mode, the initial peak power, and the numerical aperture (NA) of the fiber. Interestingly, the system exists a critical value of the initial peak power P$_{cr}$, which is the watershed of the transition of quantitatively dominant mode and the behavior of HOMs converting from Attractor into chaos. The existence of the P$_{cr}$ implies that the beam self-cleaning on LP$_{11}$ is observed inevitably for a certain range of initial peak power with the proper initial modal distribution and suitable parameter of the fiber.
2. Results and analysis
The pulse propagation in MMF relates to the evolution of spatial distribution and temporal domain simultaneously. With the modal decomposition, the spatiotemporal dynamics and the modal energy flow in MMF can be exhibited clearly. To explore the modal energy flow in the coming part, we solve the generalized multimode nonlinear Schrödinger equation (GMMNLSE) [27,28]. The evolution of the electric field temporal envelope for the transverse mode p upon the propagation distance z can read as follows:
2.1 Modal energy flow
We adopt a 12m-long MMF with parabolic refractive index profile, 0.2 NA and core diameter of 50 $\mu$m. The nonlinear index of refraction $n_2$ is 3.2$\times$10$^{-20}$$m^2W^{-1}$. A 90-femtosecond Gaussian pulse centered at 1064 nm with initial peak power of 200 kW was launched into this fiber. The beam self-cleaning of the pulse propagation containing different number of transverse modes are observed, as shown in the Fig. 1. In the propagation containing the first 10 modes of the fiber, the equal fractions (19$\%$) of the total energy are initially set to the first five modes severally, and 5$\%$ of the total energy is set to the last five modes considered as the background noise. When only the first 5 modes are propagated, the total energy is initially divided equally between all modes. The spatial beam self-cleaning upon the propagation distance z of two cases are shown in the Fig. 1(a) and (b) respectively. Within the short distance, the initially centrifugal beam quickly evolves into the double-peaked intensity pattern which is similar to the LP$_{11}$ mode. However, the output beam profiles of two cases are not absolutely alike, which may be caused by the modes that play the role of the background noise.
Fig. 1. Spatial evolution upon the propagation distance z. Isosurfaces represent points at 70$\%$ of the local maximum intensity value, (a) containing the first 10 modes of the fiber in the initial field, (b) containing the first 5 modes of the fiber in the initial field. (c) Modal energy exchange between mode3, LOMs and HOMs. Dotted curves represent the case without HOMs initially.
In the beam self-cleaning process, the output beam profile is influenced by the modal energy exchange. Thus, to understand the formation of the difference between the output beam profiles of two cases above, it is necessary to study the corresponding modal energy exchange. In the next discussion, the propagated modes are divided into three parts: the mode3 (LP$_{11}$ mode), the low-order modes (LOMs, including mode1, mode2, mode4 and mode5), and the high-order modes (HOMs, including the last 5 modes). As shown in Fig. 1(c), the solid curves and dotted curves severally represent the modal energy exchange where the HOM number is 5 and 0 in the initial field. After the significant energy flow back and forth between the mode3, LOMs and HOMs, substantial energy of LOMs is coupled into the mode3 and HOMs. In the case without the HOMs, the energy previously concentrated into mode3 gradually flows to LOMs again, leading to the obvious longitudinal decline of the mode3. However, in the case with the HOMs, the modal energy distribution is almost stable upon the rest of propagation. It is inferred that the HOMs may promote the irreversible energy flow from LOMs to the LP$_{11}$ mode.
The BSC on LP$_{11}$ mode is also sensitively depend on the initial modal energy distribution. According to V. Couderc et al., the initial fraction of power coupled into the LP$_{11}$ mode which is expected to prevail after the nonlinear propagation should be the highest among all modes [22]. Thus, the initial fraction of the LP$_{11}$ mode would impact the modal energy flow and the output beam profile in this process. Figure 2 shows the relationship between the initial and the output fraction of mode3. With the initial fraction rising, the output fraction increases, but seems to saturate when the initial fraction is about 15$\%$. The inset (i) and (ii) respectively represent the output beam profiles corresponding to the initial modal distribution with 5$\%$ and 17.5$\%$ of mode3, suggesting that a suitable initial fraction of LP$_{11}$ mode is the requirement of BSC on LP$_{11}$, but it is not always the highest among all excited modes.
Fig. 2. The output fraction versus the initial fraction of mode3 with initial peak power of 200kW. The fixed initial fraction of HOMs is 5$\%$, the initial fraction of mode3 vary between 5$\%$ and 30$\%$, and the rest of the initial peak power is divided equally between the each modes of LOMs. Inset: (i) and (ii) correspond to the output beam profiles where the initial fraction of mode3 are 5$\%$ and 17.5$\%$ respectively.
2.2 Output modal distribution upon the power
To deeply comprehend the impact of input power on the modal energy flow, we study the correlation between the output modal energy distribution and the initial peak power, with the use of the fiber and initial pulse mentioned in Section 2.1. The first 10 modes of the fiber are considered, with the initial distribution represented in Fig. 3(a). The initial proportion of the mode3 is fixed to 17.5$\%$ which is within the saturation region (see Fig. 2), and the distribution of HOMs is illustrated in inset (i) (remarked as Input 1). Figure 3(b) exhibits the evolution of the output modal energy distribution upon the initial peak power. A few kilowatts of the initial peak power can lead to the energy significantly flowing into the mode3. With the continuously increasing of the power, the output proportion of mode3 rises slow and reaches the highest values of 52$\%$ with the power of 200kW, and then begins decline accompanied by the gradual rise in the output proportion of mode1. The output proportions of mode3 and mode1 are equivalent at the power of about 735kW, after which the mode1 becomes the quantitatively dominant mode. On the whole, the output proportion of LP$_{11}$ mode can stay high enough for a wide extent of the initial peak power, approximately from 10 to 300 kW. Moreover, the output proportions of mode3 and mode1 upon the power exhibit a kind of competitive relation between two modes, with the critical value of initial peak power, P$_{cr}$=735kW. In the nonlinear regime of the power below the P$_{cr}$, the mode3 is dominant quantitatively, while the mode1 occupies the most of energy in the range of the power beyond the P$_{cr}$.
Fig. 3. (a)The initial modal distribution of the propagating modes. Inset: (i) and (ii) correspond to the two different initial distribution of HOMs remarked as Inputs 1 and 2 severally. (b) and (c) The output modal distribution upon the power, corresponding to the initial distribution with Inputs 1 and 2 respectively.
Since HOMs may influence the modal energy flow (see section 2.1), the initial distribution of HOMs is likely to impact the output modal distribution. Figure 3(c) shows the output modal energy distribution upon the initial peak power, corresponding to the initial energy distribution illustrated in Fig. 3(a) but with distribution of HOMs represented in the inset (ii) (remarked as Input 2 which is slightly different from the Input 1). It is perceived that the output modal distributions of two cases gradually become different from each other as the power rises. These change of the similarity can be observed directly via the output beam profiles of the total field, as exhibited in the Fig. 4. The panels (a) and (b) represent the output beam profiles in the cases of Inputs 1 and 2 severally. With the increase in power, the discrepancy between the output beam profiles of two cases becomes more and more perceptible. Additionally, in the situation with the power higher than the $P_{cr}$, though the fundamental mode possesses the highest proportion of total energy, the output beam profile still exhibits the double-peak intensity spots. The low-intensity band between two peak-intensity spots fades away, leading to a indistinct pattern which exhibits the double-peak intensity spots with the background of relatively high intensity.
Fig. 4. The output beam profiles of the total field with different initial peak power (a) in the case of Input 1 (b) in the case of Input 2.
To figure out the evolution of the discrepancy between two cases, we carry a further investigation on the evolution of the HOMs. In Fig. 5, panels (a) and (b) demonstrate the output modal energy distributions and the output beam profiles of HOMs corresponding to the Input 1 and Input 2 respectively. When the initial peak power is 200kW, two different initial energy distributions of HOMs evolve into the same distribution after propagating in the 12m-long fiber, and their output beam profiles are the same. This evolution of HOMs exhibits the behavior of the Attractor, which has been reported by Hong et al. [29]. However, when the power gradually approaches the P$_{cr}$, the output energy distributions of Inputs 1 and 2 are not exactly alike any longer. In the range of the power above the P$_{cr}$, the discrepancy of them becomes obviously much wide, although the initial difference between two Inputs is tiny. Such evolution, in which the initial conditions with tiny difference will lead to the large discrepancy between the corresponding results, exhibits the behavior of the chaos [30]. It is an interesting discovery that the behavior of the evolution of HOMs changes from Attractor to chaos upon the input power.
Fig. 5. The modal energy distributions and beam profiles at the output,in the case of Input 1 (a) and Input 2 (b).
3. Transition of the behavior of HOMs upon the power
To deeply understand the transition of the evolution of HOMs upon the power, we would like to consider HOMs as a system including variables $E_{H1}$, $E_{H2}$, $E_{H3}$, $E_{H4}$ and $E_{H5}$. These variables correspond to the proportions of each mode of HOMs respectively. The space, constructed by the proportions of modal energy of HOMs, is the five-dimensional phase space $\mathcal {H}$ for the system. In $\mathcal {H}$, each point stands for the each state of the system, corresponding to each distribution of HOMs. With the given initial distribution, as the HOMs evolve upon the propagation distance $L$, the system also evolves from the corresponding given initial state $S_{z=0}$ into the state $S_{z=L}$, along the exclusive orbit in $\mathcal {H}$.
Based on the above constructed phase space of HOMs, $S_{z=0}^{(1)}$ and $S_{z=0}^{(2)}$, representing for Inputs 1 and 2 respectively, are considered to be two neighboring initial states in $\mathcal {H}$. The distance of two corresponding orbits, $d$, is measured by the 2-norm, $d=\Vert S_{z}^{(1)}-S_{z}^{(2)}\Vert _2$ [29]. The evolution of $d$ upon the propagation distance z with different initial peak power is illustrated in Fig. 6(a). In the region of power below $P_{cr}$, $d$ decreases to the value under $d_0$ (the initial distance of two orbits) soon along the propagation, or gradually declines and becomes stable after the fluctuation in the range below $d_0$. On the contrary, when the power is higher than $P_{cr}$, $d$ increases rapidly to the value above $d_0$, even exhibits the strong fluctuation, indicating that the adjacent orbits are separate from each other and unpredictable.
Fig. 6. (a) The distance $d$ of the two orbits upon the propagation with different initial peak power. (b) The relationship between the largest Lyapunov exponent and the initial peak power.
Lyapunov exponent [30,31] is the useful parameter to demonstrate the behavior of the system, which reads as follows:
The beam self-cleaning on LP$_{11}$ mode may be regarded as the self-organized phenomenon. It can be observed when the power and the initial fraction of LP$_{11}$ mode reach the suitable values. As a case of beam self-cleaning, it originates from a universal unstable Attractor, and once the critical state of the Attractor (the double-peak intensity patter) is reached, the given initial field will self-organize into a steady state [1,14]. In the situation with the initial peak power under the $P_{cr}$, the HOMs who play the role of the background noise can be considered a perturbation of the whole initial field. Accordingly, when the whole initial field self-organizes into the steady state, the HOMs with different initial distribution will evolve into the same steady state. In other words, there is the existence of Attractor in the phase space $\mathcal {H}$ in which the two adjacent orbits will converge together. However, once the power reaches above $P_{cr}$, the Attractor becomes the strange Attractor, and the field will develop into the chaotic state which is sensitively dependent on the initial conditions. Figure 7 shows the output spectrum in the situation with initial peak power of 800kW. Due to the conjunction of strong stimulated Raman scattering (SRS) and inter-modal four-wave mixing (IFWM), the spectrum broadens obviously. On the one hand, in a parabolic fiber, SRS lead to energy flow into the fundamental mode which has the largest mode overlap factor [14,32], resulting in a blur of the output beam profile like the LP$_{11}$ mode (see Fig. 4). On the other hand, because of the effect of the strong SRS and IFWM, the energy concentrated into HOMs is high enough and more than that in the case with relatively low initial peak power. As a result, the HOMs, which originally perform a role of background noise, create the perceptible disturbance to the evolution of the whole field. By exploring for the transition of the Attractor, we can comprehend more deeply about the characteristic of beam self-cleaning on LP$_{11}$: the critical power, $P_{cr}$ is not only the turning point of the quantitatively dominant mode converting from LP$_{11}$ mode to LP$_{01}$ mode, but also the transition point of the Attractor.
Fig. 7. The output spectrum corresponding to the initial peak power of 800kW. SRS leads to the red shifting in spectrum. A series of output beam profiles with different frequency detuning shows that different wavelengths are carried by different modes, and prove the emergency of IFWM.
4. Influence of the fiber numerical aperture
Furthermore, to verify whether the BSC on LP$_{11}$ is an universal phenomenon in MMF, we study the modal energy variation versus the NA. A 90 fs Gaussian pulse, with initial peak power of 200kW, center wavelength of 1064nm and containing the first 10 modes, was launched into the 12m-long parabolic GRIN-MMF. The NA varies from 0.2 to 0.28. To avoid the coincidence, two random initial energy distributions are considered, as illustrated in Fig. 8(a) and (d), and remarked as Inputs 3 and 4 severally. The correlation between the modal energy variation and NA is demonstrated in Fig. 8(b) and (e). $\Delta E$ is the energy difference between the output and input energies. The result shows that in the fiber with 0.2NA and 0.28NA, the energy tends to flow into the mode3 (LP$_{11}$ mode) no matter what the initial modal energy distribution is, suggesting that it is possible to achieve the beam self-cleaning on the given transverse modes of the fiber by tailoring the suitable parameter of the fiber.
Fig. 8. The fraction of the energy variation versus NA. (a) and (d) exhibit two random initial modal energy distributions remarked as Inputs 3 and Input 4 respectively. (b) and (e) show the fraction of the energy variation after the propagation versus NA, corresponding to the initial distribution of Inputs 3 and 4, severally. (c) and (f) demonstrate the output energy distribution versus NA, corresponding to the initial distribution of Inputs 3 and 4, respectively.
Besides, the pulses with different initial distribution propagating in the same fiber may also lead to two completely different output beam profiles. The output modal energy distributions of the Inputs 3 and 4 are shown in Fig. 8(c) and (f), respectively. Attention should be pay to the situation of fiber with 0.24NA. For the Input3, mode3 occupies the highest proportion of the total energy among all modes at the output. As for the Input4, the energy of mode3 flows to other modes, and the mode1 is dominant at the output, leading to the bell-shape beam profile.
With the combination of the results and analysis mentioned above, one would conclude that the BSC on LP$_{11}$ mode is not universal as the BSC on LP$_{01}$ mode. The BSC should be observed only if the relative conditions are satisfied simultaneously. It is a process in which the output beam profile is determined by the modal energy exchange that is influenced by the combined action of different conditions. Firstly, the nonlinear coupling coefficients play the primary role in the modal energy flow. In the fibers with different parameters (including core diameter and NA), they are not exactly alike. Thus, for the same pulse propagation in the fibers with different parameters, the modal energy exchange may be not alike, which probably leads to the different output beam profiles. The fundamental mode has the largest mode overlap factor among all modes [14,32]. Therefore, the BSC on LP$_{01}$ mode is more universal than the BSC on LP$_{11}$ mode or other modes. Secondly, the impact of the initial modal distribution may be understood with the thermalization of modal distribution [18,19]. The thermalization process can be accelerated by the small random mode coupling effects [18]. By comparing the cases of Input3 and Input4 in fiber with 0.24NA, their different initial distributions lead to different mode coupling and the different output distribution. The distribution of Input4 is beneficial to the thermalization process, causing that most of energy flows into the fundamental mode and the bell-shaped beam profile is obtained after the propagation.
Since our simulation illustrates that the BSC on LP$_{11}$ mode is not a common phenomenon, it is necessary to discuss a possible experimental setting for its observation. A femtosecond fiber laser works as the pump, delivering 90fs pulses at 1064nm. The laser beam goes through a half-wave plate and a polarization beam splitter which are used to adjust the peak power of the input beam. A focusing lens, which is employed to injected the laser beam into the fiber, is mounted on the kinematic support offering control over the angular movement of the incident beam. The parabolic GRIN MMF with 0.2NA and core diameter of 50 $\mu$m is loosely coiled on the table forming rings of about 15 centimeter diameter, and placed on the 3-axis precision translation stages. The CCD cameras for output beam profiles monitoring and the optical spectrum analyzer is needed. According to the previous research [21–24] and our simulation results, the initial fraction of power coupled into the LP$_{11}$ mode should be not excessively low, which is achieved by carefully adjusting the input tilt angle of the beam. With properly tuning the translation stages and the kinematics mount, the incident external angle is fixed to be close to 2.5$^{\circ }$. Then, the input power is varied form linear regime to nonlinear regime so that the output beam profile evolution upon the power can be observed. In addition, with the proper fixed input power, by gradually and carefully varying the input tilt angle of the input beam, the output beam profile evolution upon the varying modal distribution may be observed possibly.
5. Conclusion
In summary, we systematically study the femtosecond-pulsed beam self-cleaning on LP$_{11}$ mode by focusing on the modal energy flow with the effects of different factors such as initial proportion, initial peak power, distribution of HOMs and the numerical aperture of the fiber. With the suitable input power, as the mode anticipated prevailing after the propagation in fiber, although the LP$_{11}$ mode is not included in the excited modes with the highest initial power proportion, it can be in the dominant position among all modes at the output. The output fraction of LP$_{11}$ mode can preserve high enough for a wide range of initial peak power, approximately from 10 to 300kW. With the increase of the initial peak power, at the output, the quantitatively dominant mode converts from LP$_{11}$ mode into LP$_{01}$ mode, and the corresponding turning point is the $P_{cr}$ of 735kW. It is found that the evolution of HOMs shows the behavior of Attractor and chaos in different ranges of initial peak power, and the $P_{cr}$ is also the turning point of the transition from Attractor to chaos. This is to comprehend the beam self-cleaning on LP$_{11}$ mode from the perspective of nonlinear dynamics, with the help of phase space and Lyapunov exponent. It may provide experience of research on the theory of spatial beam self-cleaning. Further more, our simulation results show the beam self-cleaning on LP$_{11}$ mode with kilowatts peak power, which may contribute to the application such as spatial division multiplexing, nonlinear medical microscopy and micro-machining.
Funding
Natural Science Foundation of Guangdong Province (2022A1515010817); National Natural Science Foundation of China (11874019, 92050101).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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