Negativerefraction Effect for Both TE and TM Polarizations in Twodimensional Annular Photonic Crystals
 Author: Wu Hong, Li Feng
 Publish: Current Optics and Photonics Volume 2, Issue1, p47~52, 25 Feb 2018

ABSTRACT
We systematically investigated the negativerefraction 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 squarelattice airhole photonic crystals, annular photonic crystals have a much wider common frequency band between TE1 and TM2, which is quite beneficial to finding the overlap between the negativerefraction regions belonging to TE1 and TM2 respectively. Further analyses of equifrequency surfaces and the electricfield 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 negativerefraction region between TE1 and TM2, but also revealed some ways to improve the performance of a polarization beam splitter based on the negativerefraction effect in an annular photonic crystal.

KEYWORD
Annular photonic crystals , Negative refraction , Polarization beam splitter

I. INTRODUCTION
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 negativerefraction effect at microwave frequencies in millimeterpatterned 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,
i.e . the photoniccrystal (PC) route, has attracted growing interest. A number of recent theoretical [5, 6] and experimental [7, 8] works have demonstrated that the negativerefraction effect is also possible in PCs. The physical principles that allow negative refraction in PCs arise from the dispersion characteristics of wave propagation in a periodic medium, which can be described well by analyzing the equifrequency surface (EFS) of the band structure. To realize the negativerefraction effect in a PC, the shape of the EFS at the operating frequency must be convex, which ensures that the direction of the group velocity of the refractive wave is on the same side of the normal direction as the incident wave. Using this general criterion, the negativerefraction region can usually be found for a normal twodimensional (2D) PC (airholetype and pillartype PCs) not only in the first band, but also in the second or higher bands.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 airholetype 2D PCs possess good negative behavior for the TE polarization, while a pillartype 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 [1113]. 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
et al . have demonstrated the possibility of polarizationindependent negativerefraction effect in the second band of APC structures [10].Especially, in our previous work [14] the negativerefraction 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 TE1 and TM2. Thus, the common negativerefraction region (CNRR) between TE1 and TM2 bands can be found. Unlike the polarizationindependent effect occurring in the same band for two polarizations, as reported in Refs. [9] and [10], in the CNRR between TE1 and TM2, 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 negativepositive refraction [15, 16]. However, it remains unclear how the filling factor of the APC affects this polarizationbeamsplitting effect. In this paper, we have systematically investigated the formation of CNRR between TE1 and TM2 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 TE1 and TM2 in normal airhole PCs and APCs respectively. Finally, a brief summary will be given in Sec. 5.
II. MODEL AND METHOD
As a model system, we consider a squarelattice APC, as illustrated in Fig. 1. The air rings with inner radius r and outer radius R are arranged in a dielectric material (
ε = 13). The popular free software “MPB”, which treats Maxwell’s wave equation as the Hermitian eigenvalue problem with the planewave expansion (PWE) technique, is utilized to obtain the photonic band structures and EFSs of the PCs, and the finitedifference timedomain (FDTD) technique is adopted to simulate the energy distribution in the PCs and in free space.III. THE FORMATION OF CNRR BETWEEN TE1 AND TM2 IN NORMAL PCs
Since the APCs studied in this paper are directly developed from the normal airhole PCs, for comparison we begin the discussion by first studying the formation of CNRR between TE1 and TM2 in normal squarelattice airhole PCs. By analyzing the EFSs for several frequencies, we have calculated the negativerefraction frequency range at TE1 and TM2 in several PCs with different values of airhole radius. Figures 2(a)~2(c) show the dispersion diagrams when
R is equal to 0.35a , 0.47a and 0.5a , and the calculated results are marked by blue and yellow strips for the TE and TM polarizations, respectively. Those marked regions have lower limits that are obtained by investigating the shape of EFSs, and upper limits that are directly found by computing the upper edges of TE1 and TM2. It is found that when R increases, the negativerefraction regions (NRRs) for both polarizations move to higher frequencies, and the separation of NRRTE and NRRTM is gradually suppressed. In particular, whenR increases to 0.5a , overlap occurs within the frequency range of 0.4160.421 2πc/a , which means that the CNRR between TE1 and TM2 is available in the normal squarelattice PC with the largest filling factor.However, with further investigation of the band structures in Figs. 2(a)~2(c), we can see that as
R increases, overlap of TM2 and TM3 in the ΓK direction always exists. Additionally, whenR increases from 0.35a to 0.47a we can see that the NRRTE can be obviously widened by the enlarged TE1 frequency range, which will be beneficial to obtaining a wide CNRR between TE1 and TM2. However, whenR = 0.5a , the NRRTE is unfortunately cut down by the shape distortion of TE1 in the vicinity of the band edge, so the bandwidth of the CNRR shown in Fig. 2(c) is relative narrow. To check the negativerefraction effects, we chose the frequency 0.416 2πc/a as an example to perform numerical simulations in the airhole PC withR = 0.5a . Figure 3 shows the calculated EFSs and the corresponding electricfield distribution when a slit beam impinges upon the PC slab with angle of incidence θ_{in} = 10°. For TE polarization, the EFS of TE1 is convex around Γ (see Fig. 3(a1)). When the incident angle is 10°, one refracted beam (represented by R_{TE1}) will be excited. However, since the frequency 0.416 2πc/a is very close to the edge of TE1, an obviously low transmission prevents the formation of clearly negative refraction (see Fig. 3(a2)). Hence, to reduce reflection loss, shape distortion of the bands, which will cut down the NRRTE, should be avoided.On the other hand, for TM polarization we find that, owing to the overlap of TM2 and TM3, when the incident angle is 10° one incident wave will correspond to two refractive waves (represented by R_{TM2} and R_{TM3} in Figs. 3(b1) and 3(b2) respectively). This means that the PC cannot support singlebeam negative refraction for TM polarization (see Fig. 3(b2)). Thus, to guarantee singlebeam behavior, the CNRR should be modified by deleting the overlap region, which will be referred to as singlebeam CNRR. This means it will be a tough task to find satisfactory dispersion characteristics for normal airhole PCs, when only the radius of the air holes can be adjusted.
IV. THE FORMATION OF SINGLEBEAM CNRR BETWEEN TE1 AND TM2 IN APCs
Now let us turn our attention to APCs. Apparently, compared to normal airhole 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
R andr .For example, Fig. 4 presents the band structures of an APC (
R = 0.47a ) when different values ofr are considered; the calculated singlebeam NRRs (obtained by removing the overlapping regions in NRRs) are also marked in the figures. It is found that asr increases, the singlebeam NRRTE is gradually pulled down to lower frequencies, and the bandwidth is gradually reduced by the suppressed band structures. Different from the monotonic behavior of the singlebeam NRRTE, the singlebeam NRRTM experiences more complex properties, which should be investigated in detail as follows: Whenr increases from 0.1a to 0.2a , since the TM2 band falls to a greater degree than the TM3 band, the separation of these two bands is gradually enhanced. Particularly, whenr = 0.2a the singlebeam NRRTM is available within the frequency range of 0.2770.304 2πc/a ,. Asr varies from 0.2a to 0.26a , despite that both the TM2 and TM3 bands have slight drops, the large radius of air holesR guarantees the absolute separation, and in turn the bandwidth of singlebeam NRRTM remains at the maximum value whenr is smaller than 0.26a . On the other hand, whenr increases from 0.26a to 0.35a , the gradually falling TM3 band causes increasing overlap of the TM2 and TM3 bands, so that the singlebeam NRRTM will be reduced, even disappearing whenr = 0.35a .To have an overall observation of the dispersion characteristics of such APC systems, the singlebeam NRRTE and singlebeam NRRTM regions for different values of
r are summarized in the NRR maps shown in Fig. 5. The corresponding bandwidths of the CNRR occurring in Fig. 5 have also been scanned. Since the singlebeam NRRTE always covers the entire area of singlebeam NRRTM, the behavior of the singlebeam NRRTM is believed to play a critical role in the formation of CNRR. It is clearly seen that whenr is chosen within the range 0.13a 0.34a , the APC systems can support the singlebeam NRRTM, and then the singlebeam CNRR is available in this situation. In particular, without considering the slight fluctuation on the top of the scanned curve in Fig. 5, the optimum values ofr for broad singlebeam CNRR will be in the range 0.2a 0.26a , and the corresponding bandwidths are all over 0.026 2πc/a .Hence, compared to normal airhole PCs, such an APC shows more satisfactory dispersion features, which are quite beneficial to the formation of singlebeam 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 TM2 and TM3 bands and bring substantial overlap between the singlebeam NRRTE and singlebeam NRRTM, which cannot be obtained in normal airhole PCs.
To find other satisfactory structures of APCs similar to that in Fig. 4, we have also made a survey of the bandwidth of singlebeam CNRR for APCs with different values of radii
R andr , and Fig. 6(a) shows the corresponding results for several values ofR asr changes from 0 to 0.36a . In general, for a particular type of squarelattice 2D PC, the dispersion features of the PC depend largely on the filling factor. A relatively large filling factor in airhole PCs can push band structures to higher frequencies, while the opposite is true for the case of dielectricrod PCs. Since the APCs are a combination of these types, which means annular air holes highly favor TE modes while dielectric rods favor TM modes, the singlebeam CNRR induced by TE1 and TM2 may prefers APCs with relative large radii of both air holes and dielectric rods. As expected, the singlebeam CNRR appears whenR increases from 0.44a andr is no less than 0.1a . Moreover, asR increases the range ofr supporting singlebeam CNRR also increases.On the other hand, when
R increases from 0.44a to 0.49a , the scanned curves gradually change from a single peak to double peaks. For cases whereR is less than 0.47a , the curves each have a single peak, and the corresponding peak values increase with the increment ofR . However, a slight trough appears on the top of the curve for R = 0.47a , and the trough deepens asR continues to grow. Finally, the trough turns into a gap forR = 0.49a . To indicate the formation of a trough, Fig. 6(b) shows the NRR map forR = 0.48a , and the band structures of an APC withR = 0.48a andr = 0.23a (the corresponding parameters at the bottom of trough) have also been presented in Fig. 6(c). The blue and yellow areas correspond to the singlebeam NRR for TE and TM polarizations, respectively. In particular, between the blue and yellow areas there is a partially overlapping region, which is marked by the green area. Comparing Fig. 6(b) to Fig. 5, the singlebeam NRRTM area is almost in the same place, while the singlebeam NRRTE area moves to a higher position whenR increases from 0.47a to 0.48a . Additionally, comparing Fig. 6(c) to Fig. 4(b), since the overlapping area between TE1 and TE2 becomes larger owing to the increase inR , the singlebeam NRRTE area becomes narrower. As a result, the leaflike NRRTM area cannot be fully covered by the NRRTE area, and a green overlapping area with two peaks appears.To check the negativerefraction effects, we chose the frequency 0.264 2
πc/a as an example to perform numerical simulations of an APC withR = 0.47a andr = 0.25a , which are the optimal values ofR andr given by Fig. 6(a). Figure 7 shows the calculated EFSs and the corresponding electricfield distributions when a slit beam impinges upon the APC slab with angle of incidence θ_{in} = 20°. It can be clearly seen that the energy flux of the refractive wave follows the negativerefraction law for both polarization waves. In particular, when the incident angle is 20° the refractive angle is calculated to be 5° for TE and 45° for TM, which are in good agreement with estimation from the EFCs in Fig. 7(a). Similarly, in our previous work [14], the negativerefraction effects in an APC withR = 0.47a andr = 0.188a have been studied and verified to separate the polarization beams efficiently. Without providing an exhaustive parameter search, the bandwidth of CNRR in Ref. [14] is about 0.022 2πc/a , which is obviously smaller than the optimal bandwidths in this paper. Hence, we believe that other APC structures with optimal parameters provided in this paper might be used to design polarization beam splitters of wider working bandwidth.V. SUMMARY
In summary, we have systematically demonstrated the formation of singlebeam CNRR between TE1 and TM2 in alldielectric 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 airhole PCs, and APCs with some special filling factors can offer a complete separation of TM2 and TM3 bands, which cannot be attained in normal airhole PCs. These band properties bring a substantial overlap between the singlebeam NRRTE and NRRTM. Analysis shows that the singlebeam CNRR appears when
R andr are no less than 0.44a and 0.1a respectively. Moreover, through an appropriate set of design parameters, the optimal bandwidth of singlebeam CNRR can be achieved as over 0.026 2πc/a , which is obviously larger than the bandwidth of CNRR used in Ref. [14]. Further numerical results revealed that APC structures with the optimal parameters provided in this paper might be used to design an efficient polarization beam splitter based on the negativerefraction effect, and the wider bandwidth of singlebeam CNRR might provide a betterperforming polarization beam splitter than that in Ref. [14].

[FIG. 1.] Schematic diagram of the squarelattice APC.

[FIG. 2.] The band structures of squarelattice airhole PCs with different parameters. The blue and yellow strips correspond to the negativerefraction regions for TE and TM polarizations respectively.

[FIG. 3.] The EFSs and electricfield distribution when a slit beam impinges upon the PC slab with angle of incidence θin = 10° for (a1, a2) TE and (b1, b2) TM. The radius of the air holes is 0.5a. RTE1, RTM2, and R TM3 represent the negativerefracted beams inside the PC slab respectively.

[FIG. 4.] The band structures of squarelattice APCs with different values of r when R = 0.47a. The blue and yellow strips correspond to the singlebeam negativerefraction regions for TE and TM polarizations respectively.

[FIG. 5.] The singlebeam NRRTE and singlebeam NRRTM areas in APC systems with different values of r when R = 0.47a.

[FIG. 6.] (a) The bandwidth of singlebeam CNRR for an APC, for several values of R as r changes from 0 to 0.36a. (b) The singlebeam NRRTE and singlebeam NRRTM areas in APC systems with different values of r when R = 0.48a. (c) The band structure of an APC with R = 0.48a, r = 0.23a.

[FIG. 7.] (a) The EFSs and electricfield distribution when a slit beam impinges upon the APC slab with angle of incidence θin = 20° for (b1) TE and (b2) TM. The parameters of the APC are R = 0.47a and r = 0.25a. RTE and RTM represent the negativerefracted beams inside the PC slab respectively.