This work concerns analytical and numerical investigations of cavity enhanced x2 frequency conversion processes, specifically second-harmonic generation (SHG). We focus on how the transverse degrees of freedom affect the dynamics, where the interaction between nonlinearity and diffraction gives rise to spatially modulated structures, patterns. The two main parts of the thesis are the classical model and the quantum mechanical model, the latter being an extension of the former by including the inherent quantum fluctuations of light. From a theoretical point of view the classical dynamics are investigated with an experimental implementation in mind. Thus, we study the internally pumped optical parametric oscillator (IPOPO) as an experimentally more realistic model than the usual SHG model. In the IPOPO a competing process to SHG is taken into account, where the generated second harmonic drives a nondegenerate parametric oscillation. We find that this model may completely stabilize the instabilities normally expected in SHG, but it may also give rise to entirely new phenomena, such as oscillating cavity solitons, intensity spirals and self-pulsing solutions. Especially the self-pulsing is important in the singly resonant cavity setup, where the first experimental observation of the fast oscillating self-pulsing solutions is shown. The IPOPO model confirms very well the oscillation frequencies as well as the regions of stability observed in the experiment. The quantum mechanical investigations concern two different setups. Using a quantum mechanical model of SHG we investigate the effect of the quantum fluctuations on pattern formation in the system. Strong spatial correlations are observed between symmetrical points in the far-field, including cross-correlations between the fundamental and second-harmonic field, and the distinct peaks at the critical wave numbers reveal a quantum image. A microscopical model is suggested as a guide to understanding the processes involved in producing a classical pattern. Finally, the quantum nature of the correlations leads to spatial multimode nonclassical light, which is revealed by twin-beam correlations between symmetrical points in the far-field, and the correlations are shown to survive above the pattern formation threshold. The presence of quantum noise is also investigated in a model consisting of two SHG waveguides in a cavity. Due to interaction of evanescent waves the waveguides are spatially coupled, and this gives rise to strong violations of the standard quantum limit in the twin-beam correlations. Particularly strong quantum correlations are observed when several instabilities are competing, and complete suppression of the noise is found near bistable transition points.
Optical cavities; Quadratic optical nonlinearities; Quantum imaging and spatial behaviour of nonclassical light; Squeezing and sub-Poissonian statistics; Nonlinear optics; Quantum optics; Pattern formation