| In 1967, the left-hand material proposed by Veselago, a scientist of Soviet Union. He speculated that if the dielectric permittivity and the magnetic are negative values, the vector of wave vector, electric-field intensity, and magnetic-field intensity will form left-hand rule. The left-hand material have many peculiar properties such as the reversal Doppler shit, the reversal Snell rule, the reversal Cervenkov radiation. Therefore, many scientists have interest in negative refraction research at the end of last century. In 2000, Pendry proposed to build a super lens using the negative material slab which touches off much more interest about negative refraction material. Since then, scientists get many new results both in theory and experiment. In 2001, Smith et al presented experimental verification of a negative index of refraction at microwave frequencies on a structured material which consists of a two-dimensional array of repeated unit cells of copper strips and split ring resonators on interlocking strips of standard circuit board material. Later, Eleftheriades gained negative refraction and observed sub-wave imaging on microwave circuit.The superlens based on visible light negative refraction material can surmount diffraction limit, so it can used to produce super sensitive single-molecular detector. This detector can used to detect trace contamination, the dangerous biological and chemical pharmacy and protein molecules which show early pathological changes in blood. The superlens build by negative refraction material have resolution hundreds of times better than normal lens. The negative refraction material can used to make new optical storage system that the storage capacity is several orders of magnitude higher than DVD.Although the left hand material based on metal structure successfully achieve negative refraction at microwave frequencies, it can't obtain negative refraction at high frequencies especially visible light frequencies. With the study of the photonic crystal's structure become deeply, the scientist find the negative refraction of electromagnetic wave in photonic crystal. The photonic crystal could have small absorption at visible light frequencies, because it can make up with dielectric material. Because of the improvement of modern micro-fabrication technology, the photonic crystal will be a effective way to gain negative refraction at visible light frequencies. In addition, metal-dielectric composite, another material which would be get visible light negative refraction become a latest research focus. This paper will use equal-frequency-surface method to study the negative refraction about 2-D photonic crystal find out the relation between 2-D photonic crystal negative refraction and crystal lattice structure, dielectric style, radius. Also this paper will utilize effective medium theory to calculate the sub-wavelength resolution of metal-dielectric composites. After careful analysis, there are some conclusions as follows:2-D Photonic Crystal1. The crystal lattice structure of photonic crystal is square lattice; the incidence electromagnetic wave is TE mode.(1) The dielectric-air 2-D PC with square lattice has a negative refraction frequency band at the second energy band. When R=0.3a, the normalize frequency range of negative refraction is 0.260-0.309. Keeping the substrate dielectric unchanged, altering the radius of air column, from R = 0.10a to R = 0.50a, the negative refraction frequency bandwidth increase along with the increasing of air column radius. When R = 0.35a, the square lattice 2-D PC has maximum bandwidth of 0.050, and then gradually decreased. When R = 0.10a and R = 0.50a, the square lattice 2-D PC doesn't have negative refraction. That is, when the photonic crystal material in a high proportion of single material, the negative refraction is not easy to achieve. With the increasing radius of the air column, the negative refraction initial normalized frequency of square lattice 2-D PC is also increasing, which means that the larger of the air column radius is, the greater the lattice constant of square lattice 2-D PC need to obtain visible light negative refraction. Further simulation illustrate that negative refraction normalized frequencies range of square lattice 2-D PC is 0.268-0.319 while R = 0.3257a. It shows when the volume fraction of the air column achieves 1/3 the square lattice 2-D PC has the biggest negative refraction bandwidth 0.051.(2) Keeping the air column radius R = 0.3257a unchanged, changing the substrate dielectric permittivity, it show that the larger the substrate dielectric permittivity is the larger, the smaller the initial normalized frequency of square lattice 2-D PC's negative refraction is. In other words, it has much more difficulty to fabricate square lattice 2-D PC when the substrate dielectric permittivity becomes larger because the crystal lattice constant becomes smaller. Compared with the changes of air column radius, the negative refraction frequencies bandwidth is less sensitive to the changes of the substrate dielectric permittivity. The square lattice 2-D PC has the greatest negative refraction frequencies when the substrate dielectric permittivity is 10.(3) For air-dielectric column square lattice 2-D PC, it can not get negative refraction. For dielectric-dielectric column square lattice 2-D PC, it may obtain negative refraction when the difference of the two dielectric materials is big enough.2. The crystal lattice structure of photonic crystal is triangular lattice; the incidence electromagnetic wave is TE mode.(1) Different from square lattice, the triangular lattice 2-D PC has two negative refraction bands while R=0.3a, located in the second energy band and eighth energy band, respectively. Their normalized negative refraction frequencies bands are 0.3271-0.3585 and 0.6054-0.6247. Keeping the substrate dielectric unchanged, altering the radius of air column, from R = 0.10a to R = 0.50a, the negative refraction initial normalized frequency increase along with the increasing of air column radius. When R = 0.45a, the triangular lattice 2-D PC has maximum bandwidth of the first negative refraction frequencies band. With the same radius of air column, the negative refraction initial normalized frequency of triangular lattice 2-D PC located in the second energy is bigger than that of square lattice. In other words, it is easier to gain visible light negative refraction used triangular lattice compared with square lattice. Triangular lattice 2-D PC also has greater negative refraction frequencies bandwidth than square lattice at the second energy band. In the second negative refraction frequencies band triangular lattice 2-D PC has much bigger initial normalized frequency than the first negative refraction frequencies band.(2) Keeping the air column radius R = 0.45a unchanged, changing the substrate dielectric permittivity, it show that, with the substrate dielectric permittivity increasing, the first negative refraction frequencies band of triangular lattice 2-D PC increases until the substrate dielectric permittivity achieves 10, and then gradually decreased. The initial normalized frequency of first negative refraction frequencies band becomes smaller along with substrate dielectric permittivity increasing.(3) For air-dielectric column triangular lattice 2-D PC, it can easily get negative refraction. Its normalized negative refraction frequency band is 0.2893-0.3275 while R = 0.45a. For dielectric-dielectric column triangular lattice 2-D PC, it may obtain negative refraction when the difference of the two dielectric materials is big enough. Same to the dielectric- air column triangular lattice 2-D PC, there are also second negative refraction frequencies band about air-dielectric column triangular lattice and dielectric-dielectric column triangular lattice 2-D PC.3. The crystal lattice structure of photonic crystal is honeycomb lattice; the incidence electromagnetic wave is TE mode.(1) Keeping the substrate dielectric unchanged, altering the radius of air column, the negative refraction frequency bandwidth increase slowly along with the increasing of air column radius. When R = 0.35a(the volume fraction of air column is 0.3), the square lattice 2-D PC has maximum bandwidth, and then gradually decreased. Compared with triangular lattice 2-D PC, the initial normalized negative refraction frequency of honeycomb lattice 2-D PC is significantly smaller. Even though its maximum initial normalized frequency is smaller than the initial normalized frequency of triangular lattice 2-D PC. Its greatest bandwidth of first negative refraction band is narrower than triangular lattice 2-D PC.(2) Keeping the air column radius R = 0.30a unchanged, changing the substrate dielectric permittivity, it show that the initial normalized frequency of first negative refraction frequencies band becomes smaller along with substrate dielectric permittivity increasing. The honeycomb lattice 2-D PC has the greatest negative refraction frequencies when the substrate dielectric permittivity is 4.(3) The honeycomb lattice 2-D PC has a second negative refraction frequencies band while the radius of air column is from 0.10a to 0.30a. It has a third negative refraction frequencies band when the radius of air column is from 0.10a to 0.20a.Metal-dielectric composites1. Under the conditions of different shapes Ag nanoparticles, Ag-SiO2 granular composites can achieve negative dielectric constant. When Lz=1/3(in other words Ag particles are spheres), Ag-SiO2 granular composites the widest absorption band. When Ag particles are ellipsoid, the absorption band become narrower, at the same time the absorption loss is low. When Lz=1/3, the superlens based on Ag-SiO2 granular composites have the resolution ofλ/16. When Lz=0.2 and Lz=0.8, both the resolution of superlens based on Ag-SiO2 granular composites areλ/39. When the thickness of composites is twice of the thickness of host medium, the superlens based on Ag-SiO2 granular composites has the best resolution. Along with the thickness of Ag-SiO2 granular metal-dielectric composites and host medium increase, the resolution becomes lower. When the thickness of Ag-SiO2 granular metal-dielectric composites achieves 40 nm, the lens based on composites can not have sub-wavelength resolution.2. The resolution of superlens based on Ag-SiO2 columnar metal-dielectric composites isλ/ 48. It is better than the resolution of Ag-SiO2 granular metal-dielectric composites slab lens. Its working wavelength located on infrared frequency rather than visible light frequency.In this paper many new conclusions are obtained. These conclusions can used to select the parameters to fabricate the 2-D photonic crystal which can achieve negative refraction in visible light frequency range such as crystal lattice constant, crystal lattice type, dielectric material type and so on. These conclusions will provide theoretical guidance to fabricate negative refraction photonic crystal. The conclusions about metal-dielectric composites may used to manufacture slab superlens based on metal-dielectric composites. These conclusions can also used to confirm the slab lens'thickness, working wavelength, resolution and so on.
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