Experimental emulator of pulse dynamics in fractional nonlinear Schrödinger equation

TL;DR

This work experimentally demonstrates a nonlinear Lévy waveguide that emulates pulse dynamics governed by the generalized FNLSE using a mode-locked fiber laser and a pulse shaper to implement fractional dispersion with Lévy index $\alpha$. It shows two regimes: intra-cavity, where stable fractional solitons with heavy tails arise from FGVD-nonlinearity interplay, and extra-cavity, where spectral valleys with multiple lobes are engineered via segmented fractional phases and explained by a three-force model. The valleys support high-dimensional data encoding, demonstrated with five valley modes and data transmitted over ~$100$ km of fiber, highlighting potential for advanced nonlinear spectral shaping and real-time photonic processing. The work provides a practical framework (force model and phase design) to explore spectral-temporal dynamics in fractional nonlinear systems and to extend fractional-derivative concepts to optical encoding and processing applications.

Abstract

We present a nonlinear optical platform to emulate a nonlinear \textit{Lévy waveguide} that supports the pulse propagation governed by a generalized fractional nonlinear Schrödinger equation (FNLSE). Our approach distinguishes between intra-cavity and extra-cavity regimes, exploring the interplay between the effective fractional group-velocity dispersion (FGVD) and Kerr nonlinearity. In the intra-cavity configuration, we observe stable \textit{fractional solitons} enabled by an engineered combination of the fractional and regular dispersions in the fiber cavity. The soliton pulses exhibit their specific characteristics, \textit{viz.}, "heavy tails" and a "spectral valley" in the temporal and frequency domain, respectively, highlighting the effective nonlocality introduced by FGVD. Further investigation in the extra-cavity regime reveals the generation of spectral valleys with multiple lobes, offering potential applications to the design of high-dimensional data encoding. To elucidate the spectral valleys arising from the interplay of FGVD and nonlinearity, we have developed an innovative "force" model supported by comprehensive numerical analysis. These findings open new avenues for experimental studies of spectral-temporal dynamics in fractional nonlinear systems.

Figures (5)

• Figure 1: Simulations for the pulse dynamics in the framework of FNLSE, and its implementation in optics. (a) The pulse-intensity evolution in five FDLs, where the initial pulse is set to be an Lévy distribution, $A\cdot \mathcal{L}_{1.2,1}(t/T_{0})/|\mathcal{L}_{1.2,1}(t/T_{0})|$, see Eq. (\ref{['E-Levy']}). Here, $T_{0}$ is $60.1$ fs, $A=18.76$, and $\alpha =1.2$. The inserted fragmented white line shows the half-pulse width, which is $85$ fs in the output. (b) The pulse amplitude for the produced fractional soliton with $\alpha =1.2$ and the conventional soliton (the dashes red profile) are displayed for comparison. (c) The scheme for the implementation of FNLSE in optics: the extra-cavity setup ($R=0$), and the intra-cavity one ($R>0$). (d) The experimental setup of the intra-cavity is designed to emulate FNLSE. Here the labels are defined as follows: pump-1(2): the pump laser operating at $980$ nm; EDFA-1(2): erbium-doped fiber amplifiers; PC: the polarization controller; SLM: the spatial light modulator; ISO: the fiber isolator; SA: the saturable absorber; SMF: the single-mode fiber.
• Figure 2: The spectral-temporal analysis of the soliton pulses produced by the mode-locked fiber laser. (a) The stability analysis produced by varying LI ($\alpha$) and dispersion length ($L_{\mathrm{dis}}$). Data points indicate the boundary between stable and unstable soliton states; the dashed line represents the fractional-dispersion length (FDL) for $T_{0}=250$ fs. Insets show the measured and simulated (left and right sides, respectively) spectra for specific parameter settings: [$\alpha$, $L_{\mathrm{dis}}$] = [$0,0$], [$1,10$ m], and [$0.05,142$ m], respectively. (b) The comparison of the pulse amplitude produced by the experiment and simulations. The experimental profiles were reconstructed by the FROG system.
• Figure 3: Spectral responses to varying the EDFA's pump power in the extra-cavity regime. (a1)-(a4): Recorded spectra produced by the flat [$\alpha =0,L_{\mathrm{dis}}=0$], second-order [$\alpha =2,L_{\mathrm{dis}}=1.7L_{\alpha ,\mathrm{FDL}}$], fractional-order [$\alpha =1,L_{\mathrm{dis}}=2.42L_{\alpha ,\mathrm{FDL}}$] and [$\alpha =0.2,L_{\mathrm{dis}}=0.88L_{\alpha ,\mathrm{FDL}}$] GVD, respectively. The bottom panels show the temporal intensity reconstructed by the FROG system for the pump power in 240 mW. (b1)-(b4): Showcasing the spectral valleys with multiple lobes, under the action of the segmented fractional phase, as indicated at the top of each panel and defined in Eqs. \ref{['E7']}- \ref{['E10']}, respectively. (c1)-(c4): The corresponding spectra, as obtained in the simulations.
• Figure 4: The application of the spectral valleys to high-dimensional data encoding. (a): The pulse transmission regime for the data encoding from binary to quinary by using five spectral-valley modes. It includes a 'soliton' pulse from a mode-locked fiber laser, the 'write' section operated by a pulse shaper, a 'nonlinear' SPM section, a 'linear' fiber link of the length $\simeq 100$ km, and a 'read' section including a photodetector and oscilloscope. (b): Stretched temporal profiles recorded by the oscilloscope for modes of $\mathrm{\{1\}}$, $\mathrm{\{2\}}$, and $\mathrm{\{3\}}$, respectively. (c): The recorded temporal profiles of {25412, 33142, 35243, 4152, 14313} strings, as realized with $25$ holograms and produced by the oscilloscope for one round trip.
• Figure 5: Simulations and experiments for pulse's spectra under the action of FGVD and nonlinearity. (a) and (d): The normalized temporal intensity and gradient ($-\partial I/\partial t$). (b) and (e): The corresponding temporal phase and gradient ($-\partial \phi /\partial t$). (c) The summary effective force for $B=\pi /2$. (f) The calculated spectral intensity for $C_{0}$ varying between $-3$ to $+3$. (g)-(h): The calculated three forces for $C_{0}=+\pi /4$ and $C_{0}=-\pi /4$, respectively. (i) The measured spectral intensity for $C_{0}$ varying from $-1.8$ to $1.8$.