The Study Of Cylindrical Nonlinear Optics

Cylindrical nonlinear optics, which describes cylindrical electromagnetic waves’propagation in media with different types of nonlinearity and inhomo-geneity, is a fundamental branch of nonlinear optics and becomes a burgeoning research area recently. For the features of propagation and polarization, there are many differences between cylindrical electromagnetic waves and traditional plane electromagnetic waves. For example, electromagnetic modes in a cylin-drical cavity depends on the polarization direction of the electromagnetic field; The propagation features of cylindrical electromagnetic waves in a nonlinear medium also depends on the polarization direction of the electromagnetic field; There is a new type of electromagnetic modes:amplitude modes, in which the amplitudes, rather than the frequencies, of the electromagnetic waves are lim-ited by the boundary condition. Due to the novel characteristics of cylindrical electromagnetic wave, the researches of cylindrical electromagnetic waves in inhomogeneous and nonlinear media are of great interest and meaningful for theory and application, and the studies of cylindrical nonlinear optics can al-so deepen the understanding of traditional nonlinear optical effects with plane waves, even provide simple methods of dealing traditional nonlinear optical ef-fects.In this thesis, we study the features of cylindrical electromagnetic waves’ propagation in nonlinear and inhomogeneous media, including the following several aspects:1. We study the method of solving cylindrical electromagnetic waves prop-agation in nonlinear and inhomogeneous media, and obtain a series of exact so-lutions which describe cylindrical electromagnetic fields in media with different types of nonlinearity and inhomogeneity.2. We develop a set of effective mathematical methods to analyze the exact solution, and found that most of the nonlinear optical effects, including second-harmonic generation, sum-and difference-frequency generation, op-tical Kerr effect, and linear electrooptic effect, come out quite naturally from the exact solution, and the analytical expressions for these effects can be ob-tained. We also deduce coupled-wave equations which describe the interaction between cylindrical electromagnetic waves and nonlinear inhomogeneous me-dia. Using the coupled-wave equations, we analyze some nonlinear effects, and compare the results obtained from the two different methods, the method of us-ing exact solutions and the method of using coupled-wave equations, and find that descriptions of nonlinear effects by the coupled-wave equations are in good agreement with the exact solutions. We also show that the traditional slowly-varying-envelope approximation, which is widely used in nonlinear optics of plane waves, is inapplicable in cylindrical nonlinear optics.3. We theoretically analyze nonlinear optics in a cylindrical ideal cavi-ty. Nonlinear optical processes under boundary conditions (such as cavity and waveguide) is a fundamental problem in optics, and the traditional view of such processes is that the electromagnetic modes with frequencies of second-and higher-order harmonics are not allowed by the boundary conditions, regardless of the function of polarizability P depends on E. However, both the numer-ical and analytical methods of using the exact solution suggest that there are second-and higher-order harmonics generation in such a cylindrical cavity, and it can be easily verified that the exact solution satisfies the boundary conditions. These electromagnetic oscillations obey a new type of electromagnetic modes: amplitude modes, in which the amplitudes, rather than the frequencies, of the electromagnetic waves are limited by the boundary condition. We furthermore show that the photon state density of the system is needed to be modified when taking account of the amplitude modes.4. Starting from the Maxwell equations, the propagation of cylindrical electromagnetic waves with z polarization and (?) polarization in media are dis-cussed. We find that the equations of z-polarized cylindrical electromagnetic wave are very different from (?)-polarized, which may lead to a series of new properties. For electromagnetic modes of a cylindrical cavity, eigenfrequencies are related to the polarization; For cylindrical electromagnetic waves in nonlin-ear media, nonlinear effects are also related to the polarization.In a word, this thesis is the summary of preliminary exploration of cylin-drical nonlinear optics. These findings may shed light on solving nonlinear partial differential equation and understanding the propagation features of cylindrical electromagnetic wave.