1. Introduction

Left-handed materials (LHMs) or negative refractive-index materials (NIMs) with negative permittivity and permeability were initially proposed and theoretically analyzed by Veselago in 1960s [1], and recently have attracted considerable interest in realizing anomalous propagation behaviors of light such as negative refraction [2–5] in optical frequencies by means of deliberately designed artificial nanostructures, which can be applied to fabricate the flat superlens [6] and plano-concave lens [7,8]. In contrast to left-handed materials, photonic crystals (PhCs) composed of synthetic periodic dielectric materials can exhibit an extraordinarily high nonlinear dispersion which causes effects such as negative refraction and self-focusing properties that are determined by the characteristics of their photonic band structures and equal frequency contours (EFCs) [9–11]. In the higher frequency bands of PhCs, positive and negative refractions could occur for the same polarization state because photonic bands with different wave vectors might locate in the same frequency region [12–14]. Based on the similar principle, an acoustic negative birefraction phenomenon has been found in sonic crystal [15]. However, to our knowledge, there has been no report about the dual-negative refraction (DNR) effect at the same frequency and the same polarization state in PhCs. The DNR effect means that one incident wave at the frequency in the overlapping band region can be dispersed by the Bragg-scattering of PhC into two negative refracted waves, which might be difficult to realize for standard PhCs with photonic band gaps. The propagation properties of the DNR effect are closely related to the specific PhC structure. Since two-dimensional (2D) holographic structures can be more easily fabricated and have wide potential use, a holographic PhC with hexagonal lattices is used here to obtain a comprehensive understanding about the DNR effect of PhC.

The plane-wave expansion (PWE) method and the finite-difference time-domain (FDTD) method are employed to calculate the band structure of PhC and simulate the propagation of electromagnetic waves in this PhC, respectively. The perfectly matched layer (PML) is introduced as the absorbing boundary condition to absorb outgoing waves from the computation domain.

2. Structures and calculations

The PhC structure to be studied in this work is shown in Fig. 1(a) , it is a triangular PhC composed of hexagonal dielectric rods, whose preparation method has been proposed in our previous work [16]. Left-handed behaviors and all-angle left-handed negative refraction (AALNR) have been found in its inverse structure. Due to few photonic band gaps, the normal PhC structure is neglected in the foregone researches. Here we systematically investigate the characteristics of DNR effect in this normal PhC, where the lattice constant is a, the dielectric constant ε and the filling ratio f are taken to be 11.56 and 82.2%, respectively. In Fig. 1(a), the black arrow denotes the incident beam with an incident angle θ_i, the normal of the input interface is along the ΓM direction. We first made a numerical calculation of the band structure of this PhC. The lowest four bands for TM polarization (with electric field parallel to the dielectric rods) are illustrated in Fig. 1(b), where the frequency is a normalized quantity a/λ, the inset represents the first Brillouin zone with three high-symmetry lattice points, and the symmetry point Г is at the center of the first Brillouin zone. The second and the third bands overlap in the frequency region from 0.22 to 0.34 with different wave vectors, which may lead to two refractions at the same frequency with different phase velocities and group velocities. Photonic band structure can be modulated by adjusting the dielectric constant contrast or the shape and size of PhC unit cells, which are easily realized by holographic lithography (HL) [17].

 

Fig. 1 (a) Schematic of the 2D triangular PhC with hexagonal dielectric rods in air; (b) Lowest four bands of this PhC for TM polarization. The black line marks the normalized frequency a/λ = 0.32.

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The EFC distributions of TM2 and TM3 bands in this HL PhC are shown in Fig. 2(a) and (b) respectively, where the gray hexagon represents the first Brillion zone. The closed curves centered at the point Γ shrink with increasing frequency, which indicates the PhC may be left-handed [18]. The wave vector diagram for TM2 and TM3 bands at the frequency of 0.32 with three exemplificative incident angles of 10°, 20° and 30° is shown in Fig. 2(c), where the red, green and blue closed curves represent the EFCs of the second band, the third band and air respectively; the horizontal line indicates the interface between the PhC and the free space; the vertical line refers to the normal of the interface; the vertical dashed lines are the k-conservation lines of the parallel components for the wave vectors; the black long arrows indicate three incident wave vectors with different incident angles of 10°, 20° and 30°; and the red and green short arrows denote the group velocity vectors of refracted rays in the PhC. According to the intersections of k-conservation lines and the EFCs of TM2 and TM3 bands, we divide the whole case into three geometrical regions: for the incident angle θ_i < 24°, the k-conservation line intersects with the EFCs of TM2 and TM3 bands simultaneously; for 24°<θ_i < 58°, the k-conservation line only intersects with the EFC of TM3 band; for θ_i >58°, the k-conservation line drops into a partial gap and may induce a total internal reflection in this direction region. As shown in Fig. 2(c), when θ_i = 10° the k-conservation line intersects with the EFCs of TM2 and TM3 bands at the points of A and A’ respectively. By the definition of V _gr = ∇_k ω, the group velocity vector is always oriented perpendicular to the EFC in the frequency-increasing direction, the group velocities point inward from the EFCs and are antiparallel to the wave vectors (i.e., K _r ·V _gr< 0), indicating that two negative refracted waves with the different refractive angles and different eigen-modes are excited synchronously by one incident wave in the PhC. In the case of θ_i = 20°, the refractive angles of two negative refracted rays are changed greatly because the wave vectors and group velocity vectors at the points B and B’ are different from those at the points A and A’. With the incident angle varying from 10° to 20°, the refractive angle of the ray with TM2 mode increases from 17° to 55°; on the contrary, the refractive angle of the ray with TM3 mode decreases from 48° to 20°. When θ_i = 30°, the k-conservation line only intersects with the EFC of TM3 band at the point C’ with an almost downward group velocity vector. These analysis results illuminate that the refraction characteristics of the DNR effect in PhC can be modulated by adjusting the incident angle.

 

Fig. 2 EFC plots of (a) TM2 and (b) TM3 bands in the PhC. (c) Wave vector diagram for TM2 and TM3 bands at the frequency of 0.32 with different incident angles of 10°, 20° and 30°.

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Based on the AALNR effect, the inverse PhC slab can be applied to realize the perfect imaging as a flat superlens [16]. What kind of images can be obtained by the DNR effect in this normal PhC? In order to demonstrate the process of focusing imaging, one point source at the frequency of 0.32 is placed in front of the PhC slab and the ray tracing of the point source at this frequency is illustrated in Fig. 3 , where the red lines denote the refracted ray traces with the TM2 mode, the black lines denote the refracted ray traces with the TM3 mode, and the orange rounds and ellipses give the possible regions of focusing imaging in the outgoing space. Despite the different eigen-modes, the outgoing rays originating from the same source are coherent with the same frequency and polarization state, which result in the optical interferences at their foci. Obviously, there are two focusing images with different image distances locating at the symmetry axis in Fig. 3, where the first one is the focus of the rays with small negative refractive angles and the second elongated one is the focus of the rays with large negative refractive angles. The phenomenon of double focusing imaging based on the DNR effect is closely relative to the EFC distribution of the PhC. As shown in Fig. 2(c), the EFCs at the frequency of 0.32 are not circular, which lead to the optical anisotropies. Therefore, the focusing images are elongated to the diamondlike patterns, and the second focusing image with a long image distance is elongated more obviously.

 

Fig. 3 Ray tracing of the point source at the frequency of 0.32 placed in front of the PhC slab.

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3. Simulations and results

In order to verify the analysis results above, we assume the PhC slab has 11 layers of rods in the ΓM direction and a continuous Gaussian plane wave with the frequency of 0.32 is incident onto the input interface at the different incident angles θ_i of 10°, 20° and 30°. Figure 4 gives the instantaneous distributions of the electric field. As shown in Fig. 4(a), when θ_i = 10° the negative refracted waves in this PhC slab locate at the same side of the incident wave with two separated parallel outgoing waves in the output surface, indicating that both of them originate from the same wave. When θ_i = 20°, the DNR effect becomes more obvious with two more widely separated outgoing rays as shown in Fig. 4(b). The outgoing ray with the small refractive angle has a higher intensity quality than the other. From Fig. 2(c) we can infer that the modal density of TM3 is greater than that of TM2, which leads to an intense photonic degeneracy at the TM3 band. For the third case of θ_i = 30°, only one outgoing ray is emitted from the output surface with an almost vertical refracted wave penetrating this PhC slab in Fig. 4(c). Moreover, the FDTD simulations prove that the incident waves with θ_i >58° are totally reflected back to the incident space, like the behavior in total internal reflection.

 

Fig. 4 The FDTD simulations of electric field distributions with different incident angles of (a) θ_i = 10°; (b) θ_i = 20°; (c) θ_i = 30° at the frequency of 0.32.

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Figure 5 gives the electric field distributions of the focusing imaging by utilizing the PhC slabs with the DNR effect. When a continuous point source at the frequency of 0.32 is placed in front of the PhC slab with 8 layers of rods, a novel double focusing imaging phenomenon is obtained in Fig. 5(a), where the first focusing image is an approximate point image near the output surface and the second one is an elongated focusing image with a longer image distance. Since the waves with large incident angles are reflected back to the input space or diffused in the outgoing space, only the waves with limited incident angles can penetrate the PhC slab and focus in the outgoing space. Hence, the intensity qualities of two focusing images are not very high for the partial energy loss. The symmetrical distortions and variations of intensity distribution in the output space arise from the optical interferences of the coherent outgoing rays. At the frequency of 0.28, the EFC of TM2 band in the PhC extends to a concave hexagon and is of comparable size with the corresponding EFC in air. The AALNR condition is approximately satisfied in this case, however the EFC of TM2 band is not circular. Hence only one elongated focusing image with a high brightness is obtained in Fig. 5(b). Besides adjusting the frequency, the double focusing imaging can also be manipulated by changing the shape and size of the PhC unit cell. With a proper modulation, as shown in Fig. 5(c), two double elongated focusing images are obtained at the frequency of 0.35 by utilizing the PhC slab with f = 59.1%.

 

Fig. 5 Electric field patterns of double focusing imaging for the PhC slab with (a) a/λ = 0.32, f = 82.2%, (b) a/λ = 0.28, f = 82.2%, (c) a/λ = 0.35, f = 59.1%.

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4. Conclusions

In conclusion, the dual-negative refraction of PhC based on the overlap of different bands is proposed in this paper for the first time. Based on this effect, a novel double focusing imaging phenomenon with two elongated line images has been achieved by utilizing the PhC slab. These effects can be predicted by analyzing the band structure and the EFCs of the PhC. The negative refraction behaviors and the double focusing imaging have been simulated by FDTD method, which agree well with the theoretical analyses. Since the PhC structure can be engineered easily by HL and have more design flexibility, it is convenient to manipulate the DNR effect and the double focusing imaging by adjusting the incident angle, the frequency of incident wave or the filling ratio of PhC. The DNR effect can be applied to realize the division of wave-front and the optical interferences in optical holography. Because the line focus is the key for irradiation uniformity in photovoltaic solar concentrator to achieve more efficient solar energy conversion, the DNR effect of PhC might bring great potential applications in highly efficient concentrator solar cells with the low-costing mass materials. This DNR effect of PhC can also be extended to other periodic systems. We believe this DNR effect might be important and useful to the designs of new type PhC devices.