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Multimode Nonlinear Fiber Optics: Massively Parallel Numerical Solver, Tutorial and Outlook

Logan G. Wright, Zachary M. Ziegler, Pavel M. Lushnikov, Zimu Zhu, M. Amin Eftekhar, Demetrios N. Christodoulides, Frank W. Wise

arXiv:1708.05324v2physics.optics

TL;DR

Multimode fibers offer additional spatial degrees of freedom for optical technologies, but their many coupled modes make nonlinear propagation computationally demanding. The paper develops and demonstrates a GPU-accelerated numerical solver for the GMMNLSE, then uses it to examine multimode propagation and discuss broader physical and technological directions.

  • Problem

    Multimode nonlinear propagation involves many coupled spatial, temporal, and spectral degrees of freedom, while existing calculations may require restricting modes, time windows, or propagation lengths because of computational cost.

  • Method

    The paper introduces the GMMNLSE and a massively parallel numerical solver that uses graphical processing units for multimode-fiber propagation modeling.

  • Results

    The solver is demonstrated with multimode-fiber examples, including graded-index and step-index propagation, nonlinear energy transfer, and four-wave-mixing control.

  • Takeaways & Limitations

    Multimode nonlinear fiber optics provides additional spatial and temporal degrees of freedom for telecommunications, imaging, and high-power laser sources, although practical applications still require major development.

Abstract

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Building on the scientific understanding and technological infrastructure of single-mode fibers, multimode fibers are being explored as a means of adding new degrees of freedom to optical technologies such as telecommunications, fiber lasers, imaging, and measurement. Here, starting from a baseline of single-mode nonlinear fiber optics, we introduce the growing topic of multimode nonlinear fiber optics. We demonstrate a new numerical solution method for the system of equations that describes nonlinear multimode propagation, the generalized multimode nonlinear Schrodinger equation. This numerical solver is freely available, and includes a number of multimode fiber analysis tools. It features a significant parallel computing speed-up on modern graphical processing units, translating to orders-of-magnitude speed-up over the split-step Fourier method. We demonstrate its use with several examples in graded- and step-index multimode fibers. Finally, we discuss several key open directions and questions, whose answers could have significant scientific and technological impact.

I. INTRODUCTION

Multimode fibers add spatial degrees of freedom that may address capacity and performance limits in optical technologies, but their coupled dynamics create substantial conceptual, experimental, and computational challenges. The paper introduces the GMMNLSE and a freely available GPU-based solver to support analysis of multimode nonlinear propagation.

  • Motivation: Multimode fibers support many transverse eigenmodes, enabling spatial division multiplexing as a potential response to the capacity limitations of single-mode systems.The modes have different spatial shapes and propagation constants and can interact inside the fiber.
  • Motivation: Multimode architectures spread guided light over larger areas, supporting high-power fiber lasers and offering routes toward new wavelengths and higher peak powers.Fiber sources still lag some solid-state systems in peak power or wavelength coverage.
  • Applications: Multimode waveguides may enable compact endoscopes, beam focusing and scanning, high-bandwidth telecommunications, imaging, and high-power sources.The paper presents these as emerging capabilities whose practical realization still requires development.
  • Challenges: Many coupled modes make multimode physics difficult to understand, while simulations and experiments must resolve coupled spatial, temporal, and spectral degrees of freedom.Numerical calculations are often limited by computational cost, and experiments require multidimensional diagnostics.
  • Paper scope: The paper introduces the GMMNLSE as a model for multimode nonlinear propagation and presents a massively parallel, freely available solver accelerated with GPUs.The solver is demonstrated through concrete multimode-fiber examples and accompanied by analysis tools and discussion of open questions.

III. THE GENERALIZED MULTIMODE NONLINEAR SCHR ¨ODINGER EQUATIONS (GMMNLSE)

The GMMNLSE models multimode pulse propagation as coupled nonlinear Schrödinger-type equations for modal envelopes, incorporating modal dispersion and nonlinear interactions. Its formulation can be interpreted as a 3D nonlinear wave equation, with simplified variants available when selected physical effects are neglected.

  • Model formulation: The GMMNLSE is a system of coupled NLSE-type equations describing the temporal electric-field envelope for each spatial mode.The paper uses a simplified version of the equation to reduce descriptive complexity while retaining a useful multimode model.
  • Model formulation: The modal dispersion terms arise from Taylor-expanding each mode’s propagation constant around a reference frequency and transforming the result into time.Higher-order dispersion can be retained through order N_d.
  • Nonlinear interactions: The nonlinear terms include Raman and Kerr effects through modal coupling coefficients, with scalar modal profiles sufficient under the single-polarization assumption.The formulation also neglects spontaneous processes that could couple into the modeled modes.
  • Model simplifications: Self-steepening, stimulated Raman scattering, and higher-order dispersion may be omitted to obtain simpler equations that still describe many multimode phenomena.These reductions correspond respectively to replacing the self-steepening factor by 1, setting f_R to 0, and retaining only group-velocity dispersion.
  • Alternative perspective: The GMMNLSE can be viewed as a 3D NLSE for an electric-field envelope evolving along z in a transversely inhomogeneous refractive index.This perspective represents diffraction, index inhomogeneity, and a local nonlinear index shift in one equation.

A. Linear propagation in multimode fibers

Linear multimode propagation is governed by modal eigenproperties and dispersion, which determine phase evolution, interference, and temporal pulse behavior. The paper introduces these effects through the first terms of the GMMNLSE and a simplified second-order-dispersion treatment.

  • Linear propagation: The first terms of the GMMNLSE describe the main features of linear propagation in multimode fibers.For simplicity, the paper sets N_d = 2 in the introductory equations.
  • Modal basis: Multimode-fiber eigenmodes are orthonormal electromagnetic field patterns with distinct propagation constants, while quasi-degenerate modes have nearly equal propagation constants.The modes are commonly represented as linearly polarized LP modes with two possible transverse polarizations.
  • Dispersion: Chromatic dispersion is incorporated by Taylor-expanding each eigenmode’s frequency-dependent propagation constant and expressing the result in the time domain.Pulse propagation is examined in a reference frame usually associated with the fundamental mode, after removing its global longitudinal phase.

1) Propagation constant mismatch

Propagation-constant mismatch produces multimode interference and periodic spatial evolution, while the mode-dependent mismatch also determines how pulses separate in time. In parabolic-index fibers, equally spaced mode groups make this evolution especially regular.

  • Propagation-constant mismatch is responsible for multimode interference, or mode beating.
  • In a parabolic-index fiber, equally spaced propagation constants organize modes into groups with mismatch δβ(p) determined by the integer difference between groups.
  • Over short distances, the mismatch term alone determines the modal evolution because the other two terms can be ignored.
  • When modal pulses overlap, the full multimode field undergoes periodic spatial evolution whose beating depends on the mismatch term.
  • The lowest-order linear evolution is more complex in step-index fibers because δβ(p) is not uniform, but it remains multimode interference.

2) Modal dispersion

Modal dispersion gives each mode a different group velocity, causing pulses to separate during propagation, while chromatic dispersion broadens pulses within each mode. Graded-index fibers minimize modal dispersion and can therefore support strong nonlinear interactions with ultrashort pulses.

  • Modal dispersion: Different modal group velocities cause a multimode pulse to break into sub-pulses whose separation from the lowest-order mode grows linearly with fiber length.
  • Chromatic dispersion: Chromatic dispersion broadens short pulses during propagation within each individual mode, as in single-mode fiber.
  • Chromatic dispersion: In many studied situations, the chromatic-dispersion parameter β2 is similar across the relevant modes.
  • Chromatic dispersion: Higher-order multimode waveguide dispersion can be unusually strong, eliminating the single-mode trade-off between dispersion engineering and effective area.
  • Graded-index: Graded-index fibers minimize single-core modal dispersion, with within-group walk-off parameters about 10-100 times smaller than in comparable step-index fibers.
  • Graded-index: This low modal dispersion allows strong nonlinear intermodal interactions with pulses as short as ∼100 fs.

2) Step-index

Step-index fibers have irregular mode structure and strong modal dispersion, which limits interactions between short pulses in different modes. Their mode properties nevertheless support weak disorder sensitivity and opportunities for other nonlinear behaviors.

  • Step-index: Step-index fibers have less regular mode structure and usually much stronger modal dispersion than graded-index fibers.
  • Step-index: Radial symmetry and large propagation-constant differences from nearby modes can make high-order LP0N modes weakly affected by disorder over long distances.
  • Step-index: Strong modal dispersion makes short-pulse interactions between different modes difficult to observe, especially when any mode has anomalous dispersion.
  • Step-index: To date, step-index multimode dynamics have mainly involved broadband intermodal four-wave mixing processes.
  • Step-index: Longer pulses, particularly in the normal-dispersion regime, are expected to support a wider range of processes in step-index fibers.

2) Cross phase modulation

Cross-phase modulation is a pure phase effect that can broaden spectra asymmetrically for pulses with different speeds, while four-wave mixing is the nonlinear class that can transfer energy. Phase matching provides multiple control variables in highly multimode fibers.

  • Cross phase modulation: Cross-phase modulation has the form i|An|2Ap and can produce asymmetric spectral broadening when pulses travel at different speeds.
  • Cross phase modulation: Self-phase and cross-phase modulation are pure phase modulations and cannot exchange energy between modes.
  • Four wave mixing: Four-wave mixing is defined here as nonlinear coupling terms that can transfer energy between modes or waves.
  • Four wave mixing: Significant four-wave-mixing energy exchange requires conservation of energy and momentum, with momentum conservation usually expressed as phase matching.
  • Four wave mixing: In highly multimode fibers, adjusting participating spatial modes, frequencies, or modal dispersion provides substantial control over generated modes, frequencies, and bandwidth.
  • Self-steepening: Self-steepening may cause intermode energy transfer, but it has not produced major multimode-propagation features to date.
  • Raman scattering: Raman scattering can transfer energy between modes and is modeled in the GMMNLSE with a phenomenological response function hR.

D. Parallel algorithm for solving the GMMNLSE

The MPA reformulates GMMNLSE propagation so the computationally expensive nonlinear integrand can be evaluated in parallel, while retaining iterative accuracy control. It provides substantial speedups over split-step calculations, especially with GPU acceleration, although performance depends on mode count and implementation parameters.

  • The conventional split-step method scales as O(P^4) with P modes because each nonlinear step requires three nested sums over all modes.This scaling makes split-step GMMNLSE calculations increasingly impractical beyond roughly 10–30 modes.
  • The released package includes MPA and split-step GMMNLSE implementations, GPU functionality, and multimode-fiber analysis tools.The code was checked against analytic expressions, prior group codes, and 3D NLSE predictions.
  • The MPA divides a large step L into M small steps Δz, determined by nonlinear and intermode beat-length scales, to parallelize integrand evaluation.The step sizes satisfy L = MΔz, with L ≪ zNL and Δz ≪ zIM.
  • The MPA iteratively recomputes the nonlinear phase and accepts the large-step result after convergence, with relative error scaling approximately as (L/zNL)^(n+1).In typical simulations, convergence requires at most 2–3 iterations even when L is approximately 1/10 zNL.
  • For a 10-mode graded-index simulation, GPU use produced a significant speedup, while MPA added nearly another order-of-magnitude improvement over GPU-accelerated split-step computation.The optimal M depends on zNL/Δz and on the overhead of parallelization.

E. Examples calculated with the GMMNLSE

The examples are designed both to demonstrate the numerical tools and to use the modular GMMNLSE framework for examining complex multimode pulse propagation. They also emphasize the model’s limits.

  • The examples demonstrate mode calculation, dispersion and coupling-tensor calculation, and representative multimode pulse-propagation studies.
  • The modular GMMNLSE is used to isolate and understand different physical processes in complex nonlinear pulse propagation.
  • The examples explicitly emphasize the limits of the GMMNLSE model.

1) Example 1: linear propagation in a multimode fiber

The multimode examples examine nonlinear pulse evolution and soliton formation in graded-index fibers, including how cross-phase modulation and initial pulse duration affect the outcome. Short pulses form multimode solitons, whereas longer pulses can break up before a clear soliton emerges.

  • Multimode soliton formation: The multimode-solition figure tracks three LP0N modes whose combined pulse evolves over short distances because their modal propagation constants differ.
  • Multimode soliton formation: Cross-phase modulation can mutually trap pulses across modes by shifting their spectra and group velocities toward a common propagation velocity.
  • Multimode soliton formation: A 50-fs, 6-nJ pulse distributed across eight modes forms a multimode soliton and undergoes a soliton self-frequency shift after 15 m.
  • Multimode soliton formation: Kerr self-phase and cross-phase modulation alone are sufficient for multimode soliton formation under the same initial condition.
  • Multimode soliton formation: With all terms included, a 1-ps initial pulse breaks into multiple pulses and shows no clear multimode soliton at 15 m, although a subset begins forming one near 3 ps.

3) Example 3: Generation of 1300-nm pulse through self-phase modulation

Self-phase modulation was investigated as a route to generate energetic 1300-nm pulses in GRIN and step-index multimode fibers. The simulations found substantial mode-dependent energy transfer but produced energetic filtered pulses in both fibers.

  • 1300-nm excitation is attractive for biomedical nonlinear microscopy because tissue attenuation is minimized near this wavelength.The study considers nonlinear conversion from common fiber-laser wavelengths toward 1300 nm.
  • Self-phase modulation can create spectral sidelobes that yield nearly transform-limited pulses after bandpass filtering.The filtered sidelobes are targeted because their phase inflections support short pulses with high pulse energy.
  • Strong nonlinear energy transfer was relatively stable within the GRIN fundamental mode but transferred substantial energy into LP02 in the step-index fiber.The step-index fiber was designed with a similar fundamental-mode area; self-focusing explains the different transfer patterns.
  • ∼MW pulses at 1300-nm were obtained from 600-nJ, 200 fs pulses after 3.6 cm in GRIN fiber and 2.8 cm in step-index fiber.The figure reports propagation from the fundamental mode and filtering at 1300 nm.
  • Disordered coupling can phase-match modes with similar propagation constants, while shorter correlation lengths qualitatively change pulse-propagation physics.The disorder spectrum is concentrated near zero longitudinal momentum, and its effects depend on comparison with modal-dispersion and nonlinear length scales.
  • Disordered modal coupling produces diffusive pulse broadening as energy executes a random walk between modes.This differs from the linear-with-length broadening associated with modal dispersion without such coupling.

2) Saturating gain

The paper frames multimode gain as a future direction spanning fiber lasers, amplifiers, and telecommunications. Gain can be incorporated into the multimode propagation model, but practical development remains dependent on unresolved disorder and coupling issues.

  • Multimode gain must account for competition among modes that overlap the same inverted gain medium.At lowest order, gain is added as a mode-dependent term, while saturation requires a power-dependent gain model.
  • A first approximation models ultrashort-pulse gain saturation through the time-integrated power and a Taylor expansion.The full gain-saturation treatment is deferred to future work.
  • Strongly coupled disorder may benefit telecommunications, but experimental demonstrations verifying its nonlinear impact remain severely lacking.Progress requires integrating theoretical and experimental developments with practical and economic considerations.
  • Multimode propagation may support high-power and spatially or spatiotemporally engineered laser output.Recent work is cited as demonstrating this potential in multimode fiber amplifiers and lasers.
  • Parametric gain in multimode transmission amplifiers could expand both spatial and spectral channel counts.Each spectral channel would add N total channels across the spatial degrees of freedom.

3) Wave turbulence and other fundamental nonlinear dynamics

Multimode nonlinear fibers offer a broad setting for fundamental dynamics and signal processing, while extending the platform to resonators, bulk waveguides, and infrared sources. Realizing these directions requires more capable excitation and mode-resolved measurement techniques.

  • Wave turbulence: Multimode pulse propagation combines dispersion and nonlinearity with vortices, disorder, and dissipation, motivating studies of optical wave turbulence.The paper identifies multimode fibers as a setting where these ingredients interact in multiple dimensional arrangements.
  • Signal processing: Intermodal interactions provide degrees of freedom for inline signal processing and routing in spatial-division-multiplexed systems.Such functions may become important because spatial-division multiplexing requires more signal processing than single-mode transmission.
  • Other platforms: Multimode nonlinear dynamics may extend beyond fibers to microresonators, bulk waveguides, and large stiff rod fibers.These platforms trade fiber flexibility or heat dissipation for stability or larger mode areas.
  • Infrared sources: GRIN multimode chalcogenide fibers may support higher-power infrared supercontinuum sources with more convenient pump wavelengths.Earlier few-mode analyses and recent GRIN-fiber developments motivate this direction.
  • Experimental tools: More sophisticated excitation and spatial, temporal, and spectral measurements are needed to study multimode phenomena and develop practical instruments.Spatial-light modulators and accessible mode-resolved or spatiotemporal measurement tools are highlighted as enabling technologies.
  • Conclusion: The GMMNLSE organizes multimode propagation around dispersion, nonlinear modal interactions, and future extensions for disorder and gain.The conclusion presents these features as the basis for applications in telecommunications, imaging, and high-power sources.
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