Regimes and Transitions of the Nonlinear Temporal Talbot Effect: Underlying Mechanism for A-Type Breathers, Soliton Crystals, and Soliton Gas
Marina Zajnulina
TL;DR
The paper tackles the challenge of unifying linear temporal Talbot self-imaging with nonlinear Kerr dynamics in a phase-modulated CW frequency comb by deriving a dispersion relation for discrete frequency-comb spatial modes that includes GVD, SPM, XPM, and FWM. Using SRBA, it shows that the temporal Talbot effect seeds higher-order spatial-mode interference which, under nonlinear evolution, leads to four input-power regimes: linear Talbot, A-type breathers, soliton crystals, and soliton gas, with regime transitions controlled by $P_0$ and the phase-modulator depth $m$. The work distinguishes A-type breathers (regular FWM at lower powers) from soliton crystals (cascaded FWM at higher powers) and demonstrates that soliton encoding can occur on individual frequency-comb lines, while higher-order interference governs the overall recurrence and spatio-temporal structure. This provides a concise, theory-driven account of nonlinear temporal Talbot dynamics with potential applications in frequency-comb engineering, high-resolution spectroscopy, and frequency-m multiplexed optical computing, and suggests experimental validation and extensions to more complete dispersive models as future work.
Abstract
A frequency comb generated from a phase-modulated continuous-wave laser is simultaneously subject to the temporal Talbot effect and modulational instability (MI) when propagating through a piece of optical fiber. The temporal Talbot effect refers to the dispersion-driven self-imaging of optical pulses and is, per se, linear in optical field amplitudes. MI is a nonlinear effect. Despite growing interest and a variety of possible applications, a concise theory of the nonlinear temporal Talbot effect that incorporates nonlinear effects is not yet available; the self-imaging of optical patterns under the influence of nonlinearity remains largely unexplored. Here, I derive a dispersion relation for frequency-comb spatial modes. It integrates the contributions of the linear temporal Talbot effect, self-phase modulation, and MI-driven cross-phase modulation and four-wave mixing (FWM) between frequency-comb lines, paving the way to the development of a concise theory of the nonlinear temporal Talbot effect. With the help of Soliton Radiation Beat Analysis, a technique for retrieving the soliton content of optical fields in fibers, I demonstrate that the temporal Talbot effect seeds the spatio-temporal distribution of the field by driving higher-order spatial-mode interference. Nonlinear effects modify this interference, linking Talbot effect to Fermi-Pasta-Ulam-Tsingou recurrence of optical pulses and producing input-power-dependent regime transitions between the linear Talbot effect, A-type breathers, soliton crystals, and soliton gas. I show that A-type breathers arise from regular FWM at lower input powers, whereas soliton crystals emerge from cascaded FWM at higher powers. This study advances the fields of Nonlinear Optics and Wave Theory.