Skip to main content
Engineering LibreTexts

3: Nonlinear Pulse Propagation

  1. Last updated
  2. Save as PDF
  • Page ID
    44645

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\dsum}{\displaystyle\sum\limits} \)

    \( \newcommand{\dint}{\displaystyle\int\limits} \)

    \( \newcommand{\dlim}{\displaystyle\lim\limits} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    There are many nonlinear pulse propagation problems worthwhile of being considered in detail, such as pulse propagation through a two-level medium in the coherent regime, which leads to self-induced transparency and solitons governed by the Sinus-Gordon-Equation. The basic model for the medium is the two-level atom discussed before with infinitely long relaxation times \(T_{1,2}\), i.e. assuming that the pulses are much shorter than the dephasing time in the medium. In such a medium pulses exist, where the first half of the pulse fully inverts the medium and the second half of the pulse extracts the energy from the medium. The integral over the Rabi-frequency as defined in Eq.(2.3.2) is than a mutiple of \(2\pi\). The interested reader is refered to the book of Allen and Eberly [1]. Here, we are interested in the nonlinear dynamics due to the Kerr-effect which is most important for understanding pulse propagation problems in optical communications and short pulse generation.

    • 3.1: The Optical Kerr Effect
      This page discusses the refractive index in isotropic and homogeneous media, highlighting that it can vary quadratically with electric field intensity, resulting in the optical Kerr effect. The refractive index is expressed as \(n \approx n_0(\omega) + n_{2,L}|A|^2\), with \(n_{2,L}\) indicating the coefficient for positive changes. It also includes a table of refractive indices and their intensity-dependent values for different materials.
    • 3.2: Self-Phase Modulation (SPM)
      This page covers intensity-dependent refractive index effects on pulse propagation, particularly self-phase modulation (SPM). It explains how the self-phase shift, related to instantaneous intensity, changes the phase and spectrum of the pulse while maintaining its intensity profile.
    • 3.3: The Nonlinear Schrödinger Equation
      This page covers the Nonlinear Schrödinger Equation (NSE), focusing on the interplay of dispersion and self-phase modulation in wave pulses. It details how positive dispersion causes spreading while negative dispersion enables solitary waves. Key concepts include fundamental solitons, higher-order solitons, and breather solutions, all maintaining their structure during interactions.
    • 3.4: Universality of the NSE
      This page explains the derivation of the Nonlinear Schrödinger Equation (NSE), emphasizing the importance of dispersion and self-phase modulation. It identifies key effects required for the NSE in isotropic and homogeneous media like glass and gases. The page also discusses the relevance of the NSE in various phenomena, including self-focusing, Langmuir waves, and molecular interactions. Notably, self-focusing is highlighted as essential for Kerr-Lens Mode Locking and will be explored further.
    • 3.5: Soliton Perturbation Theory
      This page explores the perturbation effects on solitons in optical fibers as described by the nonlinear Schrödinger equation (NSE), emphasizing how higher-order dispersion and nonlinear effects alter soliton solutions. It introduces a new perturbation theory via the linearized NSE to analyze soliton dynamics, energy, and eigenfunction orthogonality.
    • 3.6: Soliton Instabilities by Periodic Perturbations
      This page examines the significance of periodic perturbations in solitons, particularly in ultrashort pulse lasers and optical communication, emphasizing the importance of a long soliton period to mitigate energy loss from resonant interactions. It also discusses Kelly sidebands, which arise from phasematching in soliton and continuum interactions, particularly in long cavity Ti:sapphire lasers.
    • 3.7: Pulse Compression
      This page covers the dynamics and techniques of pulse compression in optics, emphasizing positive and negative dispersion impacts on self-phase modulation. It reviews various methods such as prism and grating compressors, addressing limitations and losses. The importance of double-chirped mirrors for effective dispersion compensation and high-energy pulse generation is highlighted.
    • 3.8: Appendix- Sech-Algebra
      This page discusses the hyperbolic secant function, \( \text{sech}(x) = \frac{1}{\cosh(x)} \), and its interrelations with other hyperbolic functions through key identities. It presents its derivatives, notably that the first derivative is \( \frac{d}{dx} \text{sech}(x) = -\tanh(x) \text{sech}(x) \), and highlights that the integral of \( \text{sech}(x) \) over its entire domain is equal to \( \pi \).
    • 3.9: Summary
      This page covers the Nonlinear Schrödinger Equation's derivation, significance in optical media, and soliton solutions essential for long-distance communication. It discusses perturbation theory for soliton behavior, pulse compression, and challenges in optical systems.

    This page titled 3: Nonlinear Pulse Propagation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Franz X. Kaertner (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.