This thesis deals with theoretical investigations of a newly proposed grating structure, referred to as hybrid grating (HG) as well as vertical cavity lasers based on the grating reflectors. The HG consists of a near-subwavelength grating layer and an unpatterned high-refractive-index cap layer. Though both sides of the grating layer are not surrounded by low refractive-index materials as in high-index-contrast gratings (HCGs), the HG can provide a near-unity reflectivity over a broader wavelength range than HCGs, or work as a resonator with a quality (Q) factor as high as 109. The physics behind these reflector and resonator properties are studied thoroughly. A HG structure comprising a III-V cap layer with a gain material and a Si grating layer enables the realization of a compact vertical cavity laser integrated on Si platform, which has a superior thermal property and fabrication feasibility than the HCG-based ones. Furthermore, the concept of cavity dispersion in vertical cavities is introduced and its importance in the modal properties is numerically investigated. The dispersion curvature of a cavity mode is interpreted as the effective photon mass of the cavity mode. In a vertical cavity based on a HCG or HG reflector, this effective photon mass can be engineered by changing the grating parameters, which is not the case in a vertical cavity based on distributed Bragg reflectors (DBRs). This engineering capability enables us to form various photonic heterostructures in lateral directions, which is analogous to electronic quantum wells in conduction or valencebands. Several interesting configurations of heterostructures have been investigated and their potential in fundamental physics study and applications are discussed. For numerical and theoretical studies, a three-dimensional (3D) optical simulator has been implemented, based on the Fourier modal method (FMM). A method to simplify 3D simulations to lower dimensional simulations is suggested, which enables us to perform fast simulations before doing a thorough 3D simulation. Moreover, three different techniques for determining the resonance frequency and Q-factor ofa cavity mode are compared. Based on that, the quasi-normal mode approach with real frequency has been chosen due to its numerical efficiency. In this comparison, the associated computational uncertainty for the resonance frequency and Q-factor is investigated, which shows that the uncertainty in the Q-factor can be several orders of magnitude larger than the uncertainty in the resonance frequency. Next, the HG is shown to possess a near-unity reflectivity in a broad wavelength range, which can be broader than the HCG, since the cap layer introduces more guided mode resonances (GMRs) in the reflectivity spectrum. The fabrication tolerance of the HG is investigated numerically, which shows that the broadband near-unity reflectivity characteristic is prone to common fabrication errors. An experimental demonstration of the HG reflector confirms its broadband reflection characteristics. Furthermore, the physics study of HG as high Q-factor resonator illustrates that the resonance mechanism is similar to the resonances appearing in HCG resonators, and it is quite different from the conventional GMR filters. The effect of fabrication errors and finite size of the structure is investigated to understand the feasibility of fabricating the proposed resonator. Finally, the significance of the cavity dispersion in vertical cavity structure is illustrated. An analytic expression is derived for the dispersion, which shows that the cavity dispersion has contributions from both top and bottom mirrors through their reflectivity phase response as well as the nominal cavity through its thickness. For conventional DBRs, the mirror contribution in dispersion curvature is always positive and negligible, compared to the nominal cavity contribution. However, the HCG or HG contributions can be a specific positive or negative value in different transverse directions, significantly modifying the entire dispersion curvature. The influences of the photon effective mass on the mode confinement, mode spacing and transverse modes are investigated. Particularly, it is shown that the anisotropic dispersion curvature in in-plane heterostructure is responsible for the phenomenonof mode grouping, which is also confirmed by experimental results. Furthermore, in Si-integrated photonics, a laser source that can output light into a Si waveguide is essential, and it is shown that in HGG-based vertical cavity laser the light can be coupled to an in-plane output waveguide. The design rules for achieving a high out-coupling efficiency into the in-plane waveguide are discussed and the in-plane out-coupling efficiency as high as 68% is achieved in design. Based on this platform, a system of two laterally coupled cavities is proposed and investigated, which exhibits the breaking of parity-time (PT) symmetry in vertical cavity structures. Compared to other types of platform for studying this phenomenon such as ring/disk resonators and photonic crystal cavities, the HCG/HG-based vertical cavities appear to be more feasible for realizing an electrically pumped device, which may pave the way for finding device applications for PT-symmetry breaking phenomenon.
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