Reflection Of 1D Semi-infinite Photonic Crystals Containing Single Negative Materials

Metamaterials in which only one of the two parametersεandμis negative,which have imaginary refraction index and special electromagnetic properties, are called single negative materials. When the metamaterials are introduced into photonic crystals, new types of photonic band gaps appear. Since the properties of such photonic band gaps are different from those of the Bragg gap which leads to potential applications, photonic crystals containing metameterials have become a hot issue in present research. In this thesis, by means of numerical stimulations and theoretical analysis, on the basis of reflection of semi-infinite structure and band theory, we discuss the characteristic of the band gap in photonic crystal containing single-negative materials, properties that light propagate through the structure, characteristic of defect modes in the structure. The major contents and most important results are given as follows.In chapter 2, by solving Maxwell equation, we get the transfer matrix which is suitable for photonic crystals containing positive refraction indices, negative refraction indexes, and single-negative materials for monochromatic plane wave. Also work out the formula which can calculate the reflection of the limited photonic crystal. On the basis of it, using the Bloch theory, we can get the relationship between the electromagnetic wave function of the incident plane, and then get the reflection formula of the semi-infinite photonic crystal.In chapter 3, using the reflection formula of semi-infinite photonic crystal which is got in chapter 2, we can calculate reflection of semi-infinite photonic crystals stacked by two kinds of single-negative materials. The reflection curves of the semi-infinite photonic crystals become smooth compared with the corresponding finite multi-layered structures. It is the result of the average of the rapidly oscillated reflection in pass band of the finite structure and we can estimate the reflection in pass band in this way.Photonic crystal constituted by single negative materials with constant value, choosing the certain area of the ratio between the thickness of two materials, the allowed energy band exits in the area of low frequency. The reason is that evanescent filed can be combined in this region. The condition that effective phase matched and wave impedance is satisfied when changes dielectric constant of one until it matches with that of the other. All the modes can pass through the structure.Photonic crystal constituted by single negative dispersive materials, changing the incident angle, studying on the reflection of the semi-infinite structure we can get a zero effective phase gap, an angular gap, especially the omnidirectional Bragg gap in high frequency. Zero effective phase gap is insensitive to the variance of the angle and polarization. Angular gap exits when the incident angle is not equal to zero. The bigger the angle is, the wider the angular gap is. The omnidirectional Bragg gap can be got by properly choosing the parameters.Studying on the reflection of the semi-infinite photonic crystals, the width of the zero effective phase gap is changing with the ratio of two constitutes while not changing with the scale of two constitutes. Single negative materials with transmission line model, using the band gap theory, the frequency corresponding to the edge of the gap can be calculated and they are not changing with the variance of the thickness. Angular gap is insensitive to the ratio of the two constitutes.For the omnidirectional Bragg gap, the area of the ratio of two constitutes to form this omnidirectional Bragg gap is the same in different polarization. The electromagnetic field according to the band edge of the zero effective phase gap means it originates from the interaction of the evanescent waves. The electromagnetic field is localized in the interface of the single negative materials. And the changing tendency of electric field and magnetic field in the structure is the same. The electromagnetic field according to the band edge of Bragg gap means it originates from the interaction of the propagating wave and forms standing wave in the structure. The electromagnetic field localized inside high(low) reflective index material. So the Bragg gap depends greatly on the scaling. Comparing with the zero effective gap, the changing tendency of electric field and magnetic field in the structure is opposite. In the different polarization the intension of electromagnetic field according to the band edge of the angular gap changes greatly.Studying on the transmission of phase when light propagates in the photonic crystals containing single-negative dispersive materials. We find that direction of light propagation does matter with the average reflection, if the average reflection is zero, the phase is zero. If the average reflection is positive, light propagates in z direction, or propagates in the opposite direction.In chapter 4, we discuss the multi-channel filter in the fractal structure containing single-negative materials. Using the transfer matrix method, we can calculate the transmission spectrum in the fractal structure. Comparing with transmission spectrum in the 1^st and the 2^nd fractal structure, the defect mode is similar because the structure is similar. We mainly consider the 2^nd fractal structure. The magnetic fields of two defect modes are calculated by numerical method. The multi-channels can be independently modulated by changing the thickness of the defect. If the defect modes are in the zero effective gap, the independently modulation of the omindirectional multi-channel filter in the photonic crystal containing single-negative materials can be got.