3.9: Summary

We found, that the lowest order reversible linear effect, GVD, together with the lowest order reversible nonlinear effect in a homogeneous and isotropic medium, SPM, leads to the Nonlinear Schrödinger Equation for the envelope of the wave. This equation describes a Hamiltonian system. The equation is integrable, i.e., it does possess an infinite number of conserved quantities. The equation has soliton solutions, which show complicated but persistent oscillatory behavior. Especially, the fundamental soliton, a sech-shaped pulse, shows no dispersion which makes them ideal for long distance optical communication. Due to the universality of the NSE, this dynamics is also extremely important for modelocked lasers once the pulses become so short that the spectra experience the dispersion and the peak powers are high enough that nonlinear effects become important. In general, this is the case for subpicosecond pulses. Further, we found a perturbation theory, which enables us to decompose a solution of the NSE close to a fundamental soliton as a fundamental soliton and continuum radiation. We showed that periodic perturbations of the soliton may lead to side-band generation, if the nonlinear phase shift of the soliton within a period of the perturbation becomes comparable to \(\pi/4\). Soliton perturbation theory will also give the frame work for studying noise in mode-locked lasers later.

A medium with positive dispersion and self-phase modulation with the same sign can be used for pulse compression. The major problem in pulse compression is to find a compressor that can that exactly inverts the group delay caused by spectral broadening. Depending on bandwith this can be achieved by grating, prism, chirped mirrors, puls shapers, AOPDFs or a combination thereof.

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