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Related Experiment Video

Updated: Jan 19, 2026

Application of the Linear Momentum Equation
01:15

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Linear Schrödinger equation with temporal evolution for front induced transitions.

Mahmoud A Gaafar, Hagen Renner, Alexander Yu Petrov

    Optics Express
    |September 13, 2019
    PubMed
    Summary

    This study introduces a linear Schrödinger equation (LSE) approach for tracking temporal pulse evolution in photonic crystals, enabling analysis near vanishing group velocities. This method accurately describes front-induced transitions (FITs) and pulse reflection dynamics.

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    Area of Science:

    • Nonlinear optics
    • Waveguide theory
    • Photonic crystals

    Background:

    • Nonlinear Schrödinger equation (NLSE) is standard for pulse propagation in nonlinear waveguides.
    • Front-induced transitions (FITs) involve dielectric constant perturbation, allowing for a linear Schrödinger equation (LSE) approach.
    • Standard spatial evolution tracking fails for systems with diverging dispersion near photonic crystal band edges.

    Purpose of the Study:

    • To adapt the LSE for describing pulse propagation in optical systems with vanishing group velocities, such as near photonic crystal band edges.
    • To investigate the temporal evolution of pulse spatial profiles as an alternative to spatial evolution tracking.
    • To analyze signal pulse reflection from static and counter-propagating fronts.

    Main Methods:

    • Application of the LSE with temporal evolution tracking for systems near the band edge.
    • Simulation of intraband indirect photonic transitions.
    • Comparison of numerical results with theoretical predictions from the phase continuity criterion.

    Main Results:

    • The LSE with temporal evolution successfully describes pulse dynamics near vanishing group velocities.
    • The model accurately simulates intraband indirect photonic transitions.
    • Numerical results align with theoretical predictions for complete transitions.

    Conclusions:

    • The LSE with temporal evolution offers a powerful alternative for analyzing pulse propagation in challenging optical systems.
    • This approach is particularly effective for systems with vanishing or near-zero group velocities.
    • The method provides a comprehensive description of FITs and pulse reflection phenomena.