Introduction

Metamaterials composed of subwavelength microstructures generally can be homogenized to effective media characterized by local effective parameters, i.e., without spatial dispersion ^[1,2,3,4]. However, interestingly, the effective parameters of some metamaterials containing metallic components are found to be nonlocal, i.e., spatially dispersive, even when the microstructures are in deep-subwavelength scale ^{[5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]}. Therefore, traditional effective medium theories (EMTs) predicting the local effective parameters fail to homogenize such metamaterials. As we know, the traditional EMTs usually require relatively uniform fields inside and between the microstructures ^[30,31]. But in metallic structures, surface plasmons may be excited, resulting in drastically varying fields inside and between the microstructures. As a consequence, the nonlocality may appear in the effective parameters, that brings rich and colorful effects and applications beyond the local framework of metamaterials. In particular, the nonlocality becomes even stronger in the zero-index media with the permittivity or/and permeability near zero ^{[20,21,22,23,24,25,26,27,28,32,33,34]}, yielding unique properties and applications like parabolic dispersion ^{[20,21,22,23]}, additional extraordinary waves ^[24,25], all-angle negative refraction and subwavelength imaging ^[23], etc.

On the other hand, polarization beam splitters (PBSs), which separate two orthogonal polarizations of light into different propagation directions, are very important and widely used in polarization manipulation, optical communication, data storage, image processing and display. Traditional PBSs are based on either Brewster effect or natural crystal birefringence, which requires a large thickness to obtain enough walk-off distance between the two polarizations as the birefringence of a natural material is always small ^[35]. With the development of photonic crystals, metamaterials and metasurfaces, novel methods have been proposed ^{[35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55]}. For instance, based on photonic crystals, polarization-dependent dispersions ^{[35,37,38,39,40]} or bandgaps at different wavelength range for two polarizations ^[41,42,43] can be achieved to split transverse electric (TE) and transverse magnetic (TM) polarizations. The strong anisotropy in metamaterials can mitigate the design of PBSs ^[47,48]. Gradient metasurfaces can introduce different additional phases to different polarizations to separate the two polarizations ^[49,50]. In addition, broadband PBSs and polarization rotators can be realized based on multi-wave interference in multilayered meta-grating structures ^[53,54].

In this paper, we propose to use a simple metal-dielectric multilayered structure with nonlocal effects to realize three kinds of PBSs. We find that when average permittivity of the multilayer is near zero, the nonlocal effect become dramatic. This leads to a great difference in dispersions of TE and TM polarizations, enabling the separation of two polarizations.

1 Dispersions and polarization beam splitters

To begin with, we investigate dispersions of the metal-dielectric multilayered structure, as illustrated in Fig. 1(a). Such a multilayer is a periodic stacking of metal and dielectric layers with a lattice constant of a. Based on transfer matrix methods, the dispersions for TE and TM polarizations can be, respectively, expressed as,

图1 (a)第一类偏振分光器的示意图. (b)TE偏振(蓝色曲线)和TM偏振(绿色曲线)下的等频率曲线. [(c)和(e)]TE偏振(左图)和TM偏振(右图)电磁波正入射时的电场振幅分布图, 其中(c)中εm=-2, (e)中εm=-2+0.2i. 多层结构的厚度为w=0.20λ. (d)沿着(c)中虚线上的电场振幅分布图, 其中实线和虚线分别代表TE偏振和TM偏振, 蓝色、红色和黑色曲线分别对应于多层结构厚度为w=0.15λ、w=0.20λ和w=0.25λ的情形. (f)沿着(e)中虚线上的电场振幅分布图, 其中实线和虚线分别代表TE偏振和TM偏振, 蓝色、红色和黑色曲线分别对应于金属相对介电常数为为εm=-2、εm=-2+0.02i和εm=-2+0.2i的情形. 其它相关参数为εd=6, fm=0.75和a=λ/6.

Fig. 1 (a) Schematic of the first kind of PBS. (b) The EFCs for TE (blue lines) and TM (green lines) polarizations. [(c) and (e)] Amplitude of electric fields for TE (left) and TM (right) polarized waves under normal incidence when the relative permittivity of metal is (c) εm=-2, (e) εm=-2+0.2i. The thickness of the multilayered structure is w=0.20λ. (d) The distribution of amplitude of electric fields along the dashed lines in (c) for TE (solid lines) and TM (dashed lines) polarizations when the thickness is set to be w=0.15λ (blue lines), w=0.20λ (red lines) and w=0.25λ (black lines). (f) The distribution of amplitude of electric fields along the dashed lines in (e) for TE (solid lines) and TM (dashed lines) polarizations when the relative permittivity of metal is εm=-2 (blue lines), εm=-2+0.02i(red lines) and εm=-2+0.2i(black lines). The relevant parameters are εd=6, fm=0.75 and a=λ/6.

and coskya=cosfdpyacoshfmqya-12εmpyεdqy-εdqyεmpysinfdpyasinhfmqya， (2)

where py=εdk02-kx2, qy=kx2-εmk02. k0 (=2π/λ0) and λ0 are the wave number and wavelength in free space, respectively. εd (fd) and εm (fm) are the relative permittivity (filling ratio) of the dielectric and metal layers, respectively. Here we have the relation fd+fm=1.

Assuming that the lattice constant is much smaller than the working wavelength, thus, the trigonometric functions in Eqs. (1-2) can be simplified by using the relation sinx≈x and cosx≈1-12!x2. Then, Eqs. (1-2) are reduced to,

and kx2ε⊥+ky2ε∥≈k02 ， (4)

where ε∥ (=εdfd+εmfm) and ε⊥ (=εdεm/εdfm+εmfr) are the average permittivities in the xz plane and y direction, respectively. Actually, they correspond to the local effective parameters predicted by traditional EMTs ^[30,31].

In particular, if ε∥=0, then Eqs. (1-2) indicate that the equal frequency contours (EFCs) of TE and TM polarizations are a point and a line, respectively. Evidently, they are not physical. In fact, in the case of ε∥=0, one higher order term of the trigonometric functions should be considered, i.e. sinx≈x-13!x3 and cosx≈1-12!x2+14!x4. Thus, Eqs. (1-2) are rewritten as,

and ky≈±afdεd23k02-kx2ε⊥ . (6)

Equations (5-6) indicate that the EFCs of TE and TM polarizations, respective, are a small circle and two branches of parabolas (see Fig. 1(b)), which are quite different from those obtained from Eqs. (3-4). This reveals that the traditional EMTs fail to describe the multilayered structure with ε∥=0, and the effective parameters are not local any more. Especially for the TM polarization, the dispersion is drastically different from the local one, which actually is caused by the strong nonlocality induced by surface plasmons at metal/dielectric surfaces ^{[20,21,22,23,24,25,26,27,28]}.

In the numerical calculations, the relevant parameters are εd=6, εm=-2, fm=0.75 and a=λ/6. From the EFCs in Fig. 1(b), we can see that TE polarized waves under normal incidence onto the yz plane of the multilayered structure can be transmitted, while TM polarized waves would be reflected due to the bandgap, as illustrated in Fig. 1(a). Bases on such a difference, TE and TM polarizations can be separated. Thus, one kind of PBSs can be realized.

To verify such PBSs, numerical simulations are performed by using finite-element software COMSOL Multiphysics. In the left and right figures of Fig. 1(c), TE and TM polarized waves are normally incident from air onto the multilayered structure, whose thickness is w=0.20λ in the x direction. The amplitude of the incident electric field is 1V/m. The simulation results show that most TM polarized waves can propagate through the multilayered structure, while TE polarized waves are mostly reflected. Furthermore, in Fig. 1(d), we plot the amplitude of electric fields for TE (solid lines) and TM (dashed lines) polarizations along the dashed lines in Fig. 1(c). It is seen that the transmission decreases as the increase of the thickness of the multilayered structure. And the transmission contrast of the two polarizations can be enlarged by the increasing the thickness to block the TM polarization. For example, when the thickness is increased to w=0.25λ, the transmission coefficient for the TM polarization is around 0.1 (see Fig. 1(d)). On the other hand, the loss of transmission for the TE polarization is mainly due to the impedance mismatch between air and the multilayered structure, which can be relieved by using appropriate antireflection coatings ^[56,57,58].

Moreover, we consider material loss effects on the performance of the PBS. In Fig. 1(e), we re-simulate the distributions of electric-field amplitudes under both TE (left inset) and TM (right inset) polarizations when the relative permittivity of the metal component is set to be εm=-2+0.2i. Compared with Fig. 1(c), we can see that although the transmission is decreased a bit due to material losses, the transmission contrast of the two polarizations is still quite large. In Fig. 1(f), we plot the electric-field amplitudes along the dashed lines in Fig. 1(e) for TE (solid lines) and TM (dashed lines) polarizations when the relative permittivity of metal component is εm=-2 (blue lines), εm=-2+0.02i (red lines) and εm=-2+0.2i (black lines). These results reveal that the PBS can still work well in the existence of material losses.

It is worth noting that the thickness of such a PBS is much smaller than the working wavelength, as we have shown above. Such an ultrathin PBS may be more useful in compact optical devices compared with the previously proposed PBSs ^[47].

Now, we rotate the multilayered structure, so that the waves are incident onto the xz plane. Then, we can get different refractive behaviors of the two polarizations, as illustrated in Fig. 2(a). Such a difference in the refractive behaviors enables another kind of PBSs. Therefore, we can obtain different kinds of PBSs by using the same multilayered structure when waves are incident onto the different surfaces.

图2 (a)第二类偏振分光器的示意图. (b)TE偏振(蓝色曲线)和TM偏振(绿色曲线) 下的等频率曲线. (c)TE偏振(左图)和TM偏振(右图)电磁波以10°入射角照射时的平均能流分布图, 其中颜色和箭头分别代表平均能流的大小和方向. 该多层结构由5个结构单元和额外的一层金属层组成. (d)沿着(c)中虚线上的平均能流大小分布图, 其中蓝色和绿色曲线分别代表TE偏振和TM偏振, 实线和虚线对应于无吸收情形εm=-2和有吸收情形εm=-2+0.02i.其它相关参数与图1中相同.

Fig. 2 (a) Schematic of the second kind of PBS. (b) The EFCs for TE (blue lines) and TM (green lines) polarizations. (c) Snapshots of the magnitude (color) and direction (arrows) of the time-averaged power flow for TE (left) and TM (right) polarized waves under an incident angle of θ=10 deg. The multilayered structure is composed of 5 unit cells and an additional metal layer. (d) The distribution of time-averaged power flow along the dashed lines in (c) for TE (blue lines) and TM (green lines) polarizations. The solid and dashed lines in (d) denote the lossless case with εm=-2 and the lossy case with εm=-2+0.02i, respectively. The relevant parameters are the same as those in Fig. 1.

Specifically, positive and negative refractions occur for the TE and TM polarizations, respectively, as seen from the EFCs in Fig. 2(b). The arrows denote the directions of group velocities of incident and refracted beams. Simulation results under an incident angle of θ=10 deg are presented in Fig. 2(c), showing positive refraction for the TE polarization (left) and negative refraction for the TM polarization (right). In Fig. 2(c), the color (arrows) denotes the magnitude (direction) of the time-averaged power flow. We notice that almost all the incident waves are transmitted through the multilayered structure irrespective of the polarizations. Moreover, in Fig. 2(d), we plot the time-averaged power flow along the dashed lines in Fig. 2(c). The solid and dashed lines in Fig. 2(d) denote the lossless case with εm=-2 and the lossy case with εm=-2+0.02i, respectively. We can clearly see a large walk-off distance in the x direction for the two polarizations, thus making it possible to separate the two polarizations. We note that such a walk-off distance can be further enlarged by increasing the thickness of the multilayered structure, or increasing the incident angle.

The above two kinds of PBSs are based on the separated parabolic dispersion of the TM polarization. Interestingly, the two branches of parabolas can be tuned to be crossed. In Fig. 3, we show that based on the crossed parabolic dispersion, a new kind of PBSs can be achieved. Figure 3(a) illustrates the designed PBS, showing that the normally incident TE polarized waves will propagate through the multilayered structure without refraction. On the other hand, the TM polarized waves split into two beams having symmetric span angles around surface normal. Therefore, the propagation directions of the two polarizations are separated in the simple multilayered structure. The mechanism of the beam splitting for the TM polarization relies on the crossed parabolas (Fig. 3(b)) as the result of surface plasmon-induced nonlocal effect. This means that this kind of PBS is unique and not realizable in metamaterials described by local parameters.

图3 (a)第三类偏振分光器的示意图. (b)TE偏振(蓝色曲线)和TM偏振(绿色曲线) 下的等频率曲线. (c)TE偏振(左图)和TM偏振(右图)电磁波正入射时的平均能流分布图, 其中颜色和箭头分别代表平均能流的大小和方向. 该多层结构的厚度为w=1.6λ. (d)沿着(c)中虚线上的平均能流大小分布图, 其中蓝色和绿色曲线分别代表TE偏振和TM偏振, 实线和虚线对应于无吸收情形εm=-4和有吸收情形εm=-4+0.04i. 其它相关参数为εd=1, fm=0.2和a=λ/8.

Fig. 3 (a) Schematic of the third kind of PBS. (b) The EFCs for TE (blue lines) and TM (green lines) polarizations. (c) Snapshots of the magnitude (color) and direction (arrows) of the time-averaged power flow for TE (left) and TM (right) polarized waves under normal incidence. The thickness of the multilayered structure is w=1.6λ. (d) The distribution of time-averaged power flow along the dashed lines in (c) for TE (blue lines) and TM (green lines) polarizations. The solid and dashed lines in (d) denote the lossless case with εm=-4 and the lossy case with εm=-4+0.04i, respectively. The relevant parameters are εd=1, fm=0.2 and a=λ/8.

Simulation results are presented in Fig. 3(c), in which the left and right figures correspond to the TE and TM polarizations, respectively. The color (arrows) denotes the magnitude (direction) of the time-averaged power flow. The relevant parameters are εd=1, εm=-4, fm=0.2 and a=λ/8. It is clearly seen that the TE polarized waves can propagate through the multilayered structure (w=1.6λ) without splitting, while the TM polarized wave are symmetrically split into two beams. As a result, there is a wall-off distance in the y direction for the transmitted TE and TM polarized beams, as demonstrated by the distributions of the time-averaged power flow in Fig. 3(d), in which the solid and dashed lines denote the lossless case with εm=-4 and the lossy case with εm=-4+0.04i, respectively. Here, the transmission of this kind of PBSs is not quite high due to the impedance mismatch and material losses, which actually can be relieved by using antireflection layers ^[56,57,58] and optical gain media ^[59,60].

Finally, we note that all the proposed PBSs in the above can be realized in practice. For instance, in 2012, Subramania et al. fabricated a Ag-TiO₂ multilayered structure with a near-zero average permittivity in the optical regime ^[61]. After this work, similar multilayered structures have been fabricated by exploiting Ag and SiN layers ^[62], Au and SiO2 layers ^[63], Ag and SiO2 layers ^[64]. These experimental investigations manifest that practical implementation of the proposed PBSs is realizable.

2 Conclusions

In summary, we have proposed three kinds of PBSs by using metal-dielectric multilayered structures based on different dispersions of TE and TM polarizations. The surface plasmon-induced nonlocality results in parabolic dispersion for the TM polarization when the average permittivity is near zero. Interestingly, the designed PBSs can be ultrathin, which may have important applications in compact optical devices.

References