3: Nonlinear Pulse Propagation

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Page ID: 44645

Franz X. Kaertner
Massachusetts Institute of Technology via MIT OpenCourseWare

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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
3.2: Self-Phase Modulation (SPM)
In a purely one dimensional propagation problem, the intensity dependent refractive index imposes an additional self-phase shift on the pulse envelope during propagation, which is proportional to the instantaneous intensity of the pulse.
3.3: The Nonlinear Schrödinger Equation
If both dispersion and self-phase modulation effect act simultaneously on the pulse, the field envelope obeys the equation the Nonlinear Schrödinger Equation.
3.4: Universality of the NSE
3.5: Soliton Perturbation Theory
The investigation of solitons under perturbations is as old as the solitons itself. Many authors treat the perturbing effects in the scattering domain. Only recently, a perturbation theory on the basis of the linearized NSE has been developed, which is much more illustrative then a formulation in the scattering amplitudes. This was first used by Haus and rigorously formulated by Kaup. In this section, we will present this approach as far as it is indispensable for the following.
3.6: Soliton Instabilities by Periodic Perturbations
Periodic perturbations of solitons are important for understanding ultrashort pulse lasers as well as long distance optical communication systems. Along a long distance transmission system, the pulses have to be periodically amplified. In a mode-locked laser system, most often the nonlinearity, dispersion and gain occur in a lumped manner. The solitons propagating in these systems are only average solitons, which propagate through discrete components in a periodic fashion, as we will see later.
3.7: Pulse Compression
3.8: Appendix- Sech-Algebra
3.9: Summary