Reflectivity Of Semi-Infinite One-Dimensional Photonic Crystals

Photonic crystal is one of artificial crystal materials, dielectric function of which changes periodically. It is grabbing some serious attention as its photonic band gap, and it is considered as the most promising photoelectric material. Photonic crystals can be composed of not only ordinary materials widely exist in nature, but also double-negative materials and single-negative materials.The study of propagation of light in periodic layered media plays an important role in optics, optoelectronics, optical engineering. There are a lot of ways to study periodic layered membrane theory, such as transfer matrix method, iterative method by Fresnel’s formula, Green function and so on. But it will be complex to calculate with those methods as the number of medium layer increasing, the thickness of medium growing, and when the layer number is close to semi-infinite. In this paper, a concise formula for calculating the reflection of the wave incident on a semi-infinite periodic structure is presented explicitly. And it is used to study reflection properties of one-dimensional photonic crystals which are composed of ordinary materials, double-negative materials and single-negative materials.In chapter 2, a formula for calculating the refection and transmission coefficient of wave on a finite periodic structure is presented by using transfer matrix method. Further more, a simple formula for calculating the refection coefficient of wave on a semi-infinite periodic structure is given by using Bloch theorem.In chapter 3, we calculate the reflection of the wave incident on a semi-infinite one-dimensional photonic crystals composed of ordinary materials. We find that the reflection curves of the semi-infinite photonic crystals become smooth compared with the corresponding finite photonic crystals. It is simply the average of the rapidly oscillated reflection of the finite photonic crystals. The reflection of TE wave depends on the difference between A?and B?, while the reflection of TM wave depends on the difference between A?and B?. When permittivities and permeabilities of two layers of semi-infinite photonic crystals are different, but refractive indexes are the same, The reflection curves of TE and TM waves overlap. For certain ? and ?, the parameter region),(BAdd in which the omnidirectional reflection for both TE and TM waves can be achieved is obtained.In chapter 4, the reflection of the wave incident on a semi-infinite one-dimensional photonic crystals composed of metamaterials is calculated. Numerical calculations were carried out for one-dimensional photonic crystals which are composed of ordinary materials, double-negative materials and single-negative materials. It is found that low frequency forbidden band of photonic crystals composed of positive-negative refractive index materials becomes broader compared with photonic crystals composed of ordinary materials. At certain conditions, waves of all frequencies will be reflected, except for some discrete frequency waves. But waves can be propagated by the way of resonant tunneling in finite photonic crystals. If the metamaterials are single negative, that is, only ? or ? is negative, then the refractive index become pure imaginary and the field in the single layer becomes evanescent. The wave can not be propagated when A? and B? have the same sign(positive or negative) meanwhile A? and B? have also the same corresponding sign(negative or positive). However, under the condition that the A? is positive and A? is negative, meanwhile, the B? is negative and B? is positive, the field can be connected in a tunneling way, that is, connected by the tails of the evanescent fields, which leads to the non-decay propagation in the semi-infinite 1D photonic crystals. In particular, when the conditions AB?? ??, AB?? ?? and BAdd ? are satisfied, the transfer matrices are exactly equal to unitary matrix, independence of the frequency and the incident angle. The semi-infinite 1D photonic crystals becomes completely transparent, even without phase delay due to the evanescent fields.