Second harmonic generation in sub-diffractive two-dimensional photonic crystals

Resumen

After it was shown that the diffraction, a fundamental property of waves in homogeneous materials, can be manipulated 'reduced or even eliminated, in spatially modulated materials, it was proved that the nondiffractive propagation regimes (or self-collimation) related to the flat segments from the spatial dispersive curves could also lead to enhancement of the nonlinear parametric processes for very narrow beams (of few wavelengths width). There are two different effects that improve the efficiency of the parametric amplification in these regimes: 1) the beam does not diffractively spread due to self-collimation and 2) the phase matching holds simultaneously for all wave vectors belonging to the flattened segments. The goal of the work presented in this thesis is to find realistic configurations (that are convenient and experimentally available) where the idea of parametric amplification of narrow beams could be successfully implemented. For obtaining self-collimation regimes we used two-dimensional photonic crystals and we considered the degenerated case of the second harmonic generation process. The thesis starts with an overview, in Chapter 1, of the theoretical background of the problems treated along the thesis. It contains a short description of basic phenomena such as dispersion of materials, diffraction, parametric nonlinear processes, a brief characterization of photonic crystals and the main computational techniques used for obtaining the results presented in the thesis. The main body of the thesis, composed of Chapters 2, 3 and 4, describe the original work and it contains three related studies on second harmonic generation in two-dimensional photonic crystals. For starting, in Chapter 2 we considered the case of a bulk two-dimensional photonic crystal made of an ideal dispersionless material and, after a systematic study we found a configuration that simultaneously provides phase matching and nondiffractive propagation regimes for both fundamental and second harmonic waves and a sufficiently large parametric coupling of the two modes. Next, in Chapter 3, we introduce the parameters of a real dispersive material, AlGaAs, and a realistic configuration, the planar waveguide in our calculations. First, we calculate the dispersion surfaces in the case of a bulk two-dimensional photonic crystal made in AlGaAs in order to illustrate the changes introduced by the dispersion of the material. Then, we also introduced the planar configuration, considering both asymmetric and symmetric waveguide situations and using the thickness of the slab as a new parameter for tuning the structure. After a systematic search using different geometries (square and rhombic), different radii for the holes and combining different wave-guiding modes it appears that achieving phase matching and self-collimation for both waves simultaneously under realistic conditions is more difficult than in the ideal case considered previously. However, one configuration appears to be promising where the fundamental and second harmonic waves belong to different guided modes. The dispersion surfaces calculated in this case show that both waves propagate without diffraction at phase matching. Moreover, the position of the corresponding photonic bands with respect to the light cone shows that the out-of-plane losses vanish for both frequencies in this configuration. As described in Chapter 4, an additional factor that could be used to enhance the nonlinear interaction is to extend the phase matching condition for a wider spectral range of components. This requires that the fundamental and second harmonic dispersion curves have similar slopes at the phase matching frequencies and wave vectors. After a systematic search we found a configuration that offers both broad angular phase matching (given by the simultaneous self-collimation regimes of the two waves) and broad spectral range phase matching (given by the similar slopes of the dispersion curves at the phase matching). The nonlinear FDTD (finite-difference time-domain) simulations confirm the nondiffractive propagation for both waves and the increasing of conversion efficiency with respect to that obtained in homogeneous material using plane wave source. The immediate continuation of this work is the fabrication of the structures presented trying to verify the theoretical results experimentally. Besides, we plan to extend the results presented also for three-dimensional structures, like in woodpiles or opals made from, or filled by, quadratic nonlinear materials, and this is one of the main directions for continuing the work presented in this thesis.