Highly multimode solitons in step-index optical fiber

A soliton is a wave packet that maintains its shape owing to the balance of nonlinear and linear dispersive or diffractive effects.¹ Single-mode fibers (SMFs) are an ideal platform for studying solitons governed by the Nonlinear Schrödinger Equation (NLSE).^2,3 The study of solitons in SMFs has, in return, greatly facilitated the understanding of a wide variety of phenomena, including supercontinuum generation,^4,5 mode-locked lasers,^6,7 and optical communication.⁸ In multimode (MM) fibers, pulse propagation involves both spatial and temporal degrees of freedom,⁹ enabling more complex solitons.¹⁰.

Theoretical predictions of solitons in MM fibers date back to the 1980s.^10–15 Grudinin et al. inferred soliton formation from early experiments in MM graded-index (GRIN) fibers. An input pulse underwent fission along with stimulated Raman scattering to yield a Stokes-shifted soliton in the fundamental mode of the fiber.¹⁶ Systematic studies of solitons in MM fibers have only been undertaken since 2013.^17–19 Renninger and Wise reported theoretical and experimental observations of solitons with energies in the lowest few modes of GRIN MM fibers and their Raman shifting.²⁰ Later, Wright et al. and Zitelli et al. provided more detailed studies for the generation of MM solitons.^17,18 Zitelli et al. proposed the walk-off soliton theory and showed that the generated multimode soliton pulse duration and energy only depend on the coupling conditions and linear dispersive properties of the fiber.¹⁷ All the studies mentioned above are based on GRIN MM fibers.

Step-index MM fibers are the most common type of MM fibers, which motivates the investigation of MM solitons in them. However, the large modal dispersion typical of step-index fibers presents a major challenge to the generation of MM solitons. Single-mode solitons, which can be understood with the ordinary nonlinear Schrödinger equation, have been generated in MM step-index fibers.²¹ Zitelli et al. demonstrated the formation of solitons involving a few low-order modes in step-index fibers, and the energy eventually transferred to the fundamental mode.²² It is still an open question whether highly multimode solitons can be generated in step-index fibers, and if so, what conditions lead to their formation. Some properties of step-index fibers should be conducive to the experimental investigation of MM solitons. The transverse modes of step-index fibers overlap more than the modes of GRIN fibers, which will facilitate the observation of speckled spatial profiles that would be the signature of a superposition of higher-order modes. In addition, Raman beam cleanup,²³ which hinders the observation of MM solitons in GRIN fibers, does not occur in step-index fibers. An understanding of MM solitons in step-index fibers can be expected to underpin the understanding of complex nonlinear phenomena, such as supercontinuum generation and spatiotemporal mode-locking in MM fiber lasers.²⁴ 

Here, we report theoretical and experimental studies of highly multimode solitons in step-index fibers. We exploit fission and Raman scattering of high-power pulses to generate MM solitons in step-index fibers with modest modal dispersion. Numerical simulations agree qualitatively with the experimental results and help reveal that the observed solitons are spatio-temporally simple but can be spatio-spectrally complex. Solitons with peak powers as high as 370 kW are generated.

We simulate MM soliton formation and propagation by solving the generalized multimode nonlinear Schrödinger equation²⁵ using a parallel mode-based algorithm.²⁶ Specifically, we model the scalar-wave propagation through a 10 m step index fiber, with a core diameter of 105 µm (numerical aperture NA = 0.1). This fiber supports 105 modes per polarization (210 in total) for wavelengths around 1550 nm. The modal delay with respect to the fundamental mode increases linearly with mode number, with an average difference between adjacently indexed modes of 103 fs/m (see the supplementary material). These parameters are chosen to mirror the fiber used in the experiments described below. A Gaussian pulse at 1550 nm with a full-width at half-maximum (FWHM) duration of 500 fs is launched into the fiber. The pulse energy (200 nJ) is evenly distributed across modes 16 through 25, with a randomly assigned phase for each mode. Under statistically equivalent launching conditions, qualitatively similar results are observed. Under these conditions, the dispersion length is $∼3$ m and the nonlinear length is $∼9$ cm when calculated using the fundamental mode. To keep the computation time reasonable, only the first 30 modes are included in the simulation.

The spatially averaged $(x̂,ŷ)$ intensity of the optical field is shown as a function of time delay with respect to the pulse peak at the fundamental mode at 1550 nm (τ) vs propagation (z) in Fig. 1(a) and as a function of wavelength (λ) vs propagation in Fig. 1(b). In the beginning of the fiber (0–1 m), the pulse undergoes compression in time. Simultaneously, the spectral bandwidth rapidly broadens to $∼60$ nm. With further propagation (1–10 m), the pulse undergoes temporal fission, which creates multiple MM solitons. These solitons have 100 fs duration, which corresponds to $∼12$ cm dispersion length. They maintain their temporal shape and duration over more than 70 dispersion lengths in the fiber. Concurrently—in the spectral domain—stimulated Raman scattering creates a constantly shifting peak. The continuously redshifting Raman peak decreases the group velocities of the modes in the solitons, thus introducing a significant time delay to the solitons. The residual light at 1550 nm corresponds to the energy dispersed away and a few single-mode or few-mode solitons. This nonlinear evolution starkly contrasts with the linear pulse evolution, where the wave packet in each mode broadens and temporally “walks off” (see the supplementary material) from other modes owing to modal dispersion. While the observed behavior is reminiscent of soliton self-frequency shifting (SSFS) in SMFs, it is important to keep in mind that this is a multimode process. Specifically, intermodal cross-phase modulation produces frequency shifts that compensate modal dispersion as part of soliton formation. At the end of the fiber, a MM Raman soliton with a duration about 100 fs and peak power as high as 700 kW is generated [Figs. 1(c) and 1(d)]. We will focus on the Raman soliton, because it can be easily isolated for experimental study.

The mode-decomposed properties of the soliton with the largest time delay in Fig. 1(c) are shown in Fig. 2. In the temporal domain [Fig. 2(a)], the mode profiles have the same shape, the same time delay, and the same pulse width (100 fs). These properties are exhibited by the soliton when it forms and continue until the end of the fiber. In the spectral domain [Fig. 2(b)], the mode profiles are centered at different wavelengths, with up to 25 nm of separation. The mode-dependent shifts in the spectrum compensate for the large modal delays to keep the soliton together in time. The time-averaged modal populations [Fig. 2(c)] appear to be a random distribution of the launched modes. In contrast to previously reported MM solitons,^17,18 the low-order modes (modes 1–15) have negligible populations. Raman beam cleaning causes power to transfer to the fundamental mode in GRIN fibers,²³ which explains why lower-order modes eventually dominate MM solitons in GRIN fibers. The Raman solitons consist of fixed groups of modes that are coupled. Energy is exchanged among the modes in the soliton, but this is negligible on the scale of the dispersion and nonlinear lengths (⁠$∼10$ cm). On the other hand, the modal occupations change substantially on the scale of meters, so energy exchange is a feature of these multimode solitons for longer propagation distances (see the supplementary material).

Figure 3 illustrates the spatial properties of the MM soliton presented in Figs. 1 and 2. The primary feature is that the profile is speckled, as expected for a pseudo-random superposition of 8 high-order transverse modes. The time-integrated spatial intensity pattern at the output of the fiber [Fig. 3(a)] is almost the same as the spatial intensity pattern at the peak of the pulse [Fig. 3(b)]. The speckled spatial profile of the soliton does not fluctuate significantly within the temporal envelope of the pulse. This is quantitatively reflected in the intensity contrast, defined as $C=⟨I(x,y)2⟩/⟨I(x,y)⟩2−1$⁠, where I(x, y) is the spatial intensity pattern and ⟨⋯⟩ indicates spatial averaging.²⁷ The intensity pattern at the peak of the pulse [Fig. 3(b)] has higher contrast (C = 0.76) than the time-integrated intensity pattern [Fig. 3(a)] (C = 0.74). Nevertheless, the two values are similar and both are much higher than the contrast of the time-integrated intensity of an equivalent linear propagation (C = 0.39) (see the supplementary material). As such, the intensity contrast is useful for determining if a MM pulse propagating in the supplementary material is a soliton. For context, a uniform intensity pattern has C = 0, while a monochromatic Rayleigh speckle pattern has C = 1. The lack of temporal variation can be seen in Fig. 3(c), which shows the spatiotemporal profile of the pulse. The spectral variation of the pulse can be seen in the spatio-spectral profile of Fig. 3(d).

In the experimental setup, an erbium-doped fiber amplifier (Calmar model Cazadero) generates pulses with a temporal duration of $∼500$ fs FWHM, a spectral bandwidth of 8.3 nm FWHM, and a maximum pulse energy of 1 µJ (Fig. 4). The pulses are coupled into a 10 m step-index MM fiber (Thorlabs FG105LVA) using a lens with focal length of 25 mm. Using a variable attenuator and a 3D stage to control the spot’s location on the fiber input face, we can generate a diverse range of input conditions. The fiber is loosely coiled, with a diameter of $∼35$ cm. The beam radius at the focus is 15 µm. Since the fundamental mode radius is 40 µm and the beam is launched with a deviation from the fiber center, high-order modes (HOMs) are excited. We observe that different excitation conditions always produce superpositions of HOMs. This is likely due to linear mode coupling caused by the clamps that hold the fiber in place and the coiling of the fiber. The output facet of the fiber is directly imaged using an InGaAs camera, via two lenses in a configuration with 11× magnification. Using a flip-mirror, we measure the spatially integrated temporal profile using an autocorrelator based on 2-photon photocurrent in a Si detector. The wavelength response of the detector covers all the wavelengths generated in the experiment. The beam is also coupled into a MM fiber and sent to an optical spectrum analyzer with a wavelength range of 1–2.5 µm to record the spatially integrated power spectrum.

Temporal and spectral characteristics of the output pulses for different input pulse energies are shown in Fig. 4. At the lowest energy (16 nJ), the pulse propagation is essentially linear, as indicated by the lack of spectral broadening. As the input pulse energy increases to 160 nJ, the spectrum broadens symmetrically and the autocorrelation narrows. At 160 nJ, the spectrum becomes asymmetric owing to Raman scattering and the pulse autocorrelation narrows further. Given the emergence of Raman effects in the spectrum, soliton fission has likely occurred based on the numerical simulations. As the input pulse energy increases further, the pulse duration remains around 100 fs. For selected experiments, the autocorrelation was measured out to a delay of 20 ps, but a second peak was not observed (see the supplementary material). The dispersive waves are immersed in the background because they have low peak powers and the speckle patterns observed several picoseconds apart are not correlated in space. As is evident in Fig. 4(b), the first Raman peak in the spectrum continuously shifts to longer wavelengths and a second Raman peak forms at an input energy of 520 nJ. We find that the output spatial beam pattern [for example, shown in the inset of Fig. 4(b)] and spectrum are very sensitive to the excitation conditions. Spatial intensity patterns observed for different input pulse energies are shown in the supplementary material.

MM solitons can be isolated by filtering out the $∼1550$ nm spectral peak with a long-pass filter with a cutoff at 1600 nm. A representative example is shown in Fig. 5, where the input pulse energy is 320 nJ and the Raman peak is centered at 1650 nm. Before filtering [Fig. 5(b)], the output intensity has low speckle contrast (C = 0.17) due to the temporal integration over multiple incoherent waves by the camera. After filtering away the dispersive waves [Fig. 5(c)], a marked increase in contrast is observed (C = 0.42), which is consistent with the numerical results for a soliton. The pulse after filtering has 44 nJ energy, a pulse width of 120 fs, and no discernible secondary temporal structures [Fig. 5(d)] (see the supplementary material for a measurement with a larger temporal range). The peak power of the MM soliton is $∼370$ kW. We choose this example because the presence of the second Raman peak at higher input pulse energies makes it difficult to ascertain the peak power of each soliton.

We approximately estimate the modal contents of the observed solitons by combining information from spatial intensity correlations with the spectro-temporal properties of the pulses. As indicated above, numerical simulations show that solitons tend to form in a group of modes with adjacent indices, as one might expect intuitively considering the need to overcome the temporal separation of modes in linear propagation. The time–bandwidth product (TBP) can be used as a measure of the range of occupied modes because a wider group of modes will require greater spectral shifts between the modes to compensate for the increase in modal delay. With an approximately constant pulse duration in each mode, this will yield a larger TBP. The soliton of Fig. 5 is a representative example. The time–bandwidth product for the filtered pulse is 0.39. (For reference, the TBP of a transform-limited sech-shaped pulse is 0.315.) In simulations, a 6-mode soliton has a TBP of 0.36 and a 10-mode soliton has a TBP of 0.46 (see the supplementary material). The experimental value (0.39) is between the theoretical values for 6- and 10-mode solitons. To determine the mode indices of the band, we use the spatial intensity correlation function. In step-index fibers, nondegenerate modes have different spatial frequencies: the higher the mode index, the higher the spatial frequency. As a consequence, the length scale of the spatial correlation function of the intensity pattern will be related to the mode content; a narrower spatial correlation peak corresponds to higher-order mode content (see the supplementary material). The half-width at half-maximum (HWHM) of the spatial correlation function of the measured MM soliton in Fig. 5(c) is around 6.2 µm. We compare this to the HWHM of simulated patterns (see the supplementary material for details) with 6 modes and 10 modes in Fig. 5(e). Both approach the experimental value when the index of the highest mode is in the range 22–25. Thus, we estimate that the MM soliton in the experiment contains 6–10 modes, with the highest order mode around 22–25. This estimate is consistent with the results of numerical simulations in Fig. 2. It is worth pointing out, however, that we do not know the exact initial condition in the experiment. Therefore, our estimates are intended to give approximate values for the modal content of the soliton. Direct measurements of the modal contents of the solitons will be a valuable future direction for this work. Simulations with an accurate representation of the actual coupling condition would be valuable, too.

Numerical simulations predict the main experimental features, including the speckled spatial profile, the spectra, and the soliton duration. However, there are some discrepancies between simulated and experimental results. The energy fraction in the Raman solitons is about 14% in the experiments, which is significantly less than in the simulations (⁠$∼50$%). Another observation is that the speckle contrast of MM solitons in the experiments is lower than in simulations. Bending-induced linear mode-coupling may underlie these discrepancies.²⁸ Even the loose coiling employed in the experiments may introduce mode-coupling that is neglected in the simulations. A preliminary experiment was performed with a stronger mode-coupling, by coiling the fiber on a smaller (15 cm diameter) spool. Although similar spectra are observed, the intensity profiles after the long-pass filter have lower contrast than those obtained with loose coiling (see the supplementary material). Linear mode-coupling would be expected to eventually transfer energy to modes that are not part of the soliton. Due to the delay between modes, the camera would record an incoherent sum of the mode intensities, with reduced contrast. More systematic studies will be needed to understand the effects of mode-coupling on MM solitons in step-index fibers.

The MM solitons observed in step-index fibers have 3× higher peak power than solitons observed to date in GRIN fibers.¹⁸ There may be applications of the high-power, femtosecond pulses with speckled intensity profiles that are produced by MM soliton formation. MM solitons in step-index fibers can reach high powers, thanks to the large effective mode areas along with the high power required for cross-phase modulation to compensate for relatively large modal dispersion. It will be interesting to study the limits of high-power and highly multimode solitons in the future.

To summarize, highly multimode solitons are observed in step-index fibers. Numerical simulations and experimental results are consistent with the generation of solitons that consist of superpositions of 6 to 10 high-order transverse modes. The resulting pulses have modal occupancies that are temporally aligned, but yet can be spatio-spectrally complex. Their duration is around 100 fs, and the peak power can exceed 300 kW. The knowledge of solitons in step-index fibers will help provide a framework for a variety of multimode nonlinear phenomena and may be relevant to space-division multiplexing communication systems and applications in imaging.