Contributions to the Study of Nonlinear Optics
Doctoral thesis, 1994
Nonlinear optics, the study of how high intensity light propagates through and interacts with matter, is a subject which is rich in scientific problems and offers many technologically important applications. In this thesis, a number of physical and mathematical problems concerning nonlinear Kerr media are addressed. The papers of the thesis can be divided into four categories.
(i) Nonlinear pulse propagation in optical fibers. Soliton pulses can be established in optical fibers but require high peak intensities for short pulse durations. The possibility to obtain low intensity solitons at the zero dispersion wavelength has been examined.
A phenomenon which tends to deteriorate the obtainable compression factor in a fiber grating compressor is optical wave breaking. Analytical estimates of the wave breaking distance for different pulse shapes have been obtained. Approximately wave-breaking free pulses which can be used to enhance the compression factor have also been found.
(ii) Phase-shift-keying coherent optical transmission systems. An approximate analytical expression has been derived for the intensity modulation index, accounting for both group-velocity dispersion and higher order dispersion.
(iii) Optical beam propagation in a Kerr nonlinear medium. Beam propagation has traditionally been analyzed by means of the aberrationless paraxial-ray approximation method. This method sometimes gives poor results and it is demonstrated that a direct variational approach based on trial functions and a subsequent Ritz optimization procedure is a more reliable method for obtaining approximate solutions.
(iv) Analysis. The accuracy of a direct variational approach is known to depend crucially on the choice of the trial functions. It is demonstrated that the ability to preserve invariants can be used as an a priori qualitative assessment of the accuracy.
The nonlinear Schrödinger equation can, according to the inverse scattering transform method, be solved by essentially linear methods. However, the discrete eigenvalues of the Zakharov-Shabat scattering problem, at the heart of the method, can only be found in closed form for a very restricted set of initial conditions. The possibility of using direct variational methods to obtain good approximations to the discrete eigenvalues has been examined.