Research On Band Gap And Imaging Characteristics Of Two-dimensional Photonic Quasicrystals

Photonic quasicrystals (PQCs), also known as quasi-periodic photonic crystals(QPCs), are a special class of photonic crystals (PCs). These have properties thatinclude long-range orientational order, structural self-similarity, and rotationalsymmetry, but no translational symmetry. With regard to their photonic band gaps(PBG), focusing, and imaging, PQCs have many advantages over traditionalperiodic photonic crystals (PPCs). There have been a number of studies on thetheoretical, experimental, and applied research on PPCs in the past26years, butrarely on PQCs. For this dissertation, theoretical research on PBG, refraction, andimaging for2D PQCs has been conducted to provide guidance in experimental andapplied research on the relevant physical characteristics. The main research contentsof this dissertation are as follows.Using the plane wave expansion (PWE) method, a study was conducted on thePBGs of different unit cells in the basic structural unit of a decagonal PQC withdifferent scatterer radii and different substrate relative permittivity values, and twoconstruction cases (viz., air cylinders arranged in a dielectric medium, array ofdielectric cylinders in air). Assuming Bragg-like scattering effect in the2D PQC asthe elastic collision in physics, a qualitative relationship of PBG impact factors wasestablished showing a dependence on the scatterer radius. Based on theelectromagnetic theory, this relationship has been transformed into a qualitativerelationship for the PBG impact factors in terms of the relative permittivity of thesubstrate. The different PBGs of the decagonal PQC and2D triangular-lattice PCand2D square-lattice PC with given substrate relative permittivity and differentscatterer radii have been analyzed. A similar analysis was performed with fixedscatterer radius and different substrate relative permittivity as well as for the twoconstruction cases. For comparison, the analysis was extended to the different PBGsof the germanium-based2D PCs and the silicon-based2D PCs. The conclusionsobtained were that the2D PQC generates a PBG and complete band gap more easilythan the2D PC, and a germanium-based2D PC is more likely to generate a bandgap than a silicon-based2D PC.Based on the decagonal PQC, a2D PQC wedge-shaped prism model has beendesigned. A problem arose requiring solution in that the exit position for themaximum-power outgoing light beam is offset from the intersection of the incidentbeam and the prism slope. Moreover, the real refraction angle of the outgoing beamcan not be detected accurately using previous detection equipment with a singlesemicircular track. Our solution uses a double-straight-line-track detection method that can be used to detect the refractive characteristics of the2D PQC wedge-shapedprism. According to this method, the refraction angle, the effective refractive index(ERI), and the offset at the exit position can be calculated and analyzed using thefinite-difference time-domain (FDTD) method for a given wavelength with twopolarization modes, at different incident positions in the prism, and with differentincident beam widths. A conclusion drawn is that the refraction characteristic of the2D PQC wedge-shaped prism is irregular. The fundamental reason, which is pointedout, is the non-uniformity of the arrangement of the scatterers in the prism whichinduce irregularities in the Bragg-like scattering effect.On the basis of the decagonal PQC, a2D PQC flat lens model has beendesigned. The relationship between the band structure and the focusing wavelengthhas been developed. The focusing and imaging characteristics of thedielectric-cylinder-type2D PQC flat lens have been analyzed using the FDTDmethod. This included the dependence of image power and quality and therelationship between object distance and image distance, on parameters such aswavelength, object distance, scatterer radius, substrate refractive index, thickness,and width of lens. The relationship between object distance and image distance andthe wavelength dependence of the focusing and imaging characteristics for the2DPQC flat lens have been obtained. A spectroscopic double-focusing effect and a ringphoton localization phenomenon have been found in the2D PQC flat lens for agiven point light source with TM polarization mode. A critical-state width analysismodel of the2D PQC flat lens performing unstable focusing and a focus model ofthe2D PQC flat lens performing stable focusing have been proposed. Different lensthicknesses have different critical-state width ranges for unstable focusing and widthlimits for stable focusing. An approximate relationship between the minimum lenssize and wavelength of the point light source is proposed. The fundamental reasonand the physical mechanism underlying focusing and imaging of the2D PQC flatlens for a point light source have been pointed out. The2D PQC flat lens with threescatterers has been designed and the relationship between its focusing and imagingcharacteristics and scatterer radius has been analyzed.