We systematically investigated the negative-refraction effect for both TE and TM polarizations in annular photonic crystals. Since two polarization waves are excited in different bands, they result in different refractive angles, and so polarization beam splitters can be made of annular photonic crystals. It was found that, in comparison to normal square-lattice air-hole photonic crystals, annular photonic crystals have a much wider common frequency band between TE-1 and TM-2, which is quite beneficial to finding the overlap between the negative-refraction regions belonging to TE-1 and TM-2 respectively. Further analyses of equifrequency surfaces and the electric-field distribution of annular photonic crystals with different parameters have not only demonstrated how the filling factor of annular cells affects the formation of the common negative-refraction region between TE-1 and TM-2, but also revealed some ways to improve the performance of a polarization beam splitter based on the negative-refraction effect in an annular photonic crystal.
Since the pioneering works of Veselago [1] and Pendry [2] devoted to the imaging properties of a slab with simultaneous negative permittivity and permeability, there have been many efforts in the study of the negative-refraction effect at microwave frequencies in millimeter-patterned metallic materials [3, 4]. However, when infrared or visible light is considered, the limitations of the damping constants of metals must be faced. To overcome this difficulty at optical frequencies, the purely dielectric route,
It is well known that an electromagnetic wave can be decomposed into transverse magnetic (TM) polarization and transverse electric (TE) polarization for a 2D PC structure. However, investigations have shown that air-hole-type 2D PCs possess good negative behavior for the TE polarization, while a pillar-type 2D PC favors the TM polarization. Alternatively, many recent works have demonstrated that good negative behavior for both TE and TM polarizations can simultaneously be realized in annular PC (APC) structures. [9, 10] These novel systems have an unusual composition of annular dielectric rods in air or annular air voids in a dielectric background; this can be regarded as a combination of the two normal PC types [11-13]. Usually, researchers always focus their attention on the optical characteristics in the same band for the two polarizations. For instance, Zhang has indicated that absolute negative refraction can be realized in the first band of 2D composed PCs for both polarizations [9]. Jiang
Especially, in our previous work [14] the negative-refraction effect in APCs has been applied to design a novel kind of polarization beam splitter, when TE and TM polarizations were excited in different bands. Owing to the depressed band of the TM polarization induced by the inclusion of dielectric rods within the APC, a structure with some special filling factors will have a wide common frequency band between TE-1 and TM-2. Thus, the common negative-refraction region (CNRR) between TE-1 and TM-2 bands can be found. Unlike the polarization-independent effect occurring in the same band for two polarizations, as reported in Refs. [9] and [10], in the CNRR between TE-1 and TM-2, both polarization beams undergo negative refraction, but the corresponding refractive angles are different. As a result, the TE and TM polarization waves can be separated efficiently. Through an appropriate set of design parameters, the proposed polarization beam splitter can work within a wider normalized frequency range than a splitter based on negative-positive refraction [15, 16]. However, it remains unclear how the filling factor of the APC affects this polarization-beam-splitting effect. In this paper, we have systematically investigated the formation of CNRR between TE-1 and TM-2 in APCs, arranged as follows: In Sec. 2 we will present the main model and methods used in this study. For comparison, in Secs. 3 and 4 we will discuss the formation of CNRR between TE-1 and TM-2 in normal air-hole PCs and APCs respectively. Finally, a brief summary will be given in Sec. 5.
As a model system, we consider a square-lattice APC, as illustrated in Fig. 1. The air rings with inner radius r and outer radius R are arranged in a dielectric material (
III. THE FORMATION OF CNRR BETWEEN TE-1 AND TM-2 IN NORMAL PCs
Since the APCs studied in this paper are directly developed from the normal air-hole PCs, for comparison we begin the discussion by first studying the formation of CNRR between TE-1 and TM-2 in normal square-lattice air-hole PCs. By analyzing the EFSs for several frequencies, we have calculated the negative-refraction frequency range at TE-1 and TM-2 in several PCs with different values of air-hole radius. Figures 2(a)~2(c) show the dispersion diagrams when
However, with further investigation of the band structures in Figs. 2(a)~2(c), we can see that as
On the other hand, for TM polarization we find that, owing to the overlap of TM-2 and TM-3, when the incident angle is 10° one incident wave will correspond to two refractive waves (represented by RTM-2 and RTM-3 in Figs. 3(b1) and 3(b2) respectively). This means that the PC cannot support single-beam negative refraction for TM polarization (see Fig. 3(b2)). Thus, to guarantee single-beam behavior, the CNRR should be modified by deleting the overlap region, which will be referred to as single-beam CNRR. This means it will be a tough task to find satisfactory dispersion characteristics for normal air-hole PCs, when only the radius of the air holes can be adjusted.
IV. THE FORMATION OF SINGLE-BEAM CNRR BETWEEN TE-1 AND TM-2 IN APCs
Now let us turn our attention to APCs. Apparently, compared to normal air-hole PCs, the extra dielectric cylinders centered in the air holes give us more freedom to adjust the band structures of APCs. To find possible structures of APCs with satisfactory dispersion relationships, it is necessary to investigate different values of the radii
For example, Fig. 4 presents the band structures of an APC (
To have an overall observation of the dispersion characteristics of such APC systems, the single-beam NRR-TE and single-beam NRR-TM regions for different values of
Hence, compared to normal air-hole PCs, such an APC shows more satisfactory dispersion features, which are quite beneficial to the formation of single-beam CNRR. Obviously, such particular dispersion of APCs is attributed to the phase contribution from the extra dielectric rods. When the radius of the dielectric rods is carefully chosen, the APCs can offer complete separation of TM-2 and TM-3 bands and bring substantial overlap between the single-beam NRR-TE and single-beam NRR-TM, which cannot be obtained in normal air-hole PCs.
To find other satisfactory structures of APCs similar to that in Fig. 4, we have also made a survey of the bandwidth of single-beam CNRR for APCs with different values of radii
On the other hand, when
To check the negative-refraction effects, we chose the frequency 0.264 2
In summary, we have systematically demonstrated the formation of single-beam CNRR between TE-1 and TM-2 in all-dielectric annular photonic crystals with a square lattice. The key physical mechanism of this special effect lies in the fact that band structures of APCs for TM polarization can be pulled down to a lower frequency region, compared to normal air-hole PCs, and APCs with some special filling factors can offer a complete separation of TM-2 and TM-3 bands, which cannot be attained in normal air-hole PCs. These band properties bring a substantial overlap between the single-beam NRR-TE and NRR-TM. Analysis shows that the single-beam CNRR appears when