Research On The Complex Photonic Band Structure Of Photonic Crystal

In the past two decades, "photonic crystals" have attracted more and more people's attention. A photonic crystal, proposed simultaneously by E. Yablonovitch and S. John, is an artificial "band gap" material in which the dielectric constant is arranged periodically. Analogous to the semiconductors considered in solid state theory, photonic crystals are called semiconductors of photons. A photonic crystal is characteristic of "photonic band gap", in which the propagation of electromagnetic wave is highly prohibited. The defects modes can be introduced into the photonic band gap by fabricating carefully designed defects in perfect photonic crystal, and these defects modes are of more plentiful applications, such as photonic crystal fiber, micro resonant cavity etc.. Additionally, a photonic crystal can act as a conductor of photons. Some abnormal dispersive phenomena have been found in certain frequency range in some photonic crystals of special structures, such as super prism effect and negative index phenomena etc. In a word, photonic crystals give us a possibility to successfully control photons, and will pave the road for optical integration.There are two main ways to describe the property of a photonic crystal, the photonic band structure and the complex photonic band structure. Analogous to the band structure of electrons in traditional semiconductor, the photonic band structure gives the dispersive relation of propagating modes in a photonic crystal. From the band structure, we can see the frequency scope of photonic band gaps, but we know nothing about the evanescent modes in the band gap, such as the decay constant of an evanescent mode. The traditional plane wave expansion method (PWM) is the most classical way to calculate the band structure of a photonic crystal. Because PWM is based on the infinite and periodic assumptions and it gives nothing about the evanescent modes, it can not be used to describe the light behavior in semi-infinite and finite systems of photonic crystals. The complex photonic band structure can be regarded as an extension of traditional photonic band structure, which gives the decay constants of evanescent modes in the photonic band gaps in addition to the dispersive relation of the propagating modes. Because the eigen modes contain both the propagating modes and the evanescent modes, the complex photonic band structure and its eigen modes can be applied to semi-infinite and finite systems of photonic crystals.The aim of my dissertation is to give a method to calculate the complex photonic band structure of a photonic crystal and discuss its applications to semi-infinite and finite systems of photonic crystals systematically. The main content of the dissertation is given below:1. A brief introduction about the research progress of photonic crystal is given in chapter 1.In this chapter, we give a brief introduction about photonic crystals, such as concept, fabrication methods and applications, research methods, etc.2. A detailed description of two methods used in my dissertation, plane wave expansion method (PWM or PWEM) and plane-wave-based transfer matrix method (TMM), is given in chapter 2.In the calculation of complex photonic band structure, PWM is used to verify the dispersive relation of propagating modes and the plane-wave-based TMM is used to verify the decay constants of evanescent modes in photonic band gap. In chapter 5, the plane-wave-based TMM is used to verify the validity of our TMM for finite system of photonic crystals. We programmed for these two methods in Matlab language, and checked the programs with the data from references.3. A plane-wave-based approach for complex photonic band structure is given in chapter 3.Start from Maxwell's equations, we deduced the eigen equations of TE mode and TM mode for complex photonic band structure of two-dimensional (2D) photonic crystals. We have discussed the properties of eigen equations, and found that the eigen equations have some other virtues in addition to the calculation of complex photonic crystals, such as its application to photonic crystals composed of dispersive material, its convenience to get the equi-frequency curve (surface for three-dimensional (3D) photonic crystals) for a fixed frequency value. We have calculated the complex photonic band structures of 2D photonic crystals (including square lattice and triangular lattice), and applied coordinates transformation method to overcome the difficulty encountered in the calculation of complex photonic band structure alongΓ- M direction. The calculated results have been verified with PWM and plane-wave-based TMM. For 2D photonic crystal of square lattice, the equi-frequency curves have been calculated, and the convergence property of present approach has been discussed. We have also deduced the eigen equation for complex photonic band structure of 3D photonic crystals. The complex photonic band structure of a 3D photonic crystal of simple cubic lattice (dielectric spheres in air) has been calculated and verified with PWM and plane-wave-based TMM.4. We have applied the eigen modes for complex photonic band structure to semi-infinite systems of photonic crystals in chapter 4.Based on the group velocity of propagating mode, the refraction phenomena at the interface of photonic crystal have been discussed, and the results have been verified with Notomi's method and finite difference time domain (FDTD) simulation. We have found that it is very convenient to analyze the abnormal dispersive phenomena with group velocity of propagating mode. Compared with Notomi's method and FDTD simulations, present method is convenient to get generalized conclusion.We have described the boundary conditions at the interface of a photonic crystal with the eigen modes from eigen equation for complex photonic band structure, and applied the boundary conditions to get the spectra for electromagnetic wave propagating through the interface of semi-infinite photonic crystal. Two different systems are considered, one is composed of homogeneous media and semi-infinite photonic crystal, and the other is a heterostructure composed of two different semi-infinite photonic crystals. The calculated results are verified with the data from references. For the system composed of homogeneous media and semi-infinite photonic crystal, a "single mode approximation model" is given to get the reflection coefficients and the phase shift of reflected wave relative to the incident wave.5. We have applied the eigen modes for complex photonic band structure to finite systems of photonic crystals in chapter 5.Based on the boundary conditions described with eigen modes, a TMM for single slab of photonic crystal is proposed. We have discussed the spectra for electromagnetic wave propagating through a single slab of photonic crystal in three different structures. The first considered structure is composed of a single slab of photonic crystal sandwiched by homogeneous media, the second one is composed of a single slab sandwiched by homogeneous media and a different semi-infinite photonic crystal, and the third one is composed of a single slab sandwiched by two different semi-infinite photonic crystals. The calculated results for the first structure are verified with plane-wave-based TMM.Based on the transfer matrix for single slab of photonic crystal, an extended transfer matrix method (ETMM) for multiple slabs of photonic crystals is given. The calculated spectra for multiple slabs of photonic crystals are verified with plane-wave-based TMM.A quantum well structure of photonic crystals is a special double-heterostructure of photonic crystals, in which we have discussed the resonant modes. For the quantum well structure with finite barrier thickness, ETMM for multiple slabs of photonic crystals can be used to calculate the resonant modes. For the quantum well structure with infinite barrier thickness, we have to resort to other method. Here, based on the standing wave property of resonant modes, the eigen equation for resonant modes is deduced. Using the eigen equation, we have discussed the resonant modes in some different quantum well structures, and the results are verified by ETMM for multiple slabs of photonic crystals. For symmetric quantum well structure, we give a method to distinguish the parity of the resonant modes. For quantum well structure with homogeneous media well slab, a simple method is given to calculate the resonant modes, which is based on the phase shift of reflected wave relative to incident wave, and this method is very convenient to calculate resonant modes in a quantum well structure with barriers of different lattice constants.In brief, we have proposed a plane-wave-based approach to calculate the complex photonic band structure of photonic crystals, and applied it to semi-infinite and finite systems of photonic crystals. From the discussion, we can see that although the eigen equations are dependent on the infinite and periodic assumptions, the complex photonic band structure and its eigen modes contain enough information to describe the light behavior in semi-infinite and finite systems of photonic crystals.