1. Introduction

A polarizing beam splitter (PBS) is an essential optical element for separating different polarization beams, and has been widely used in the fields of optical switching networks [1–3], data storage systems [4,5], and imaging systems [6,7]. In practical applications, an ideal PBS is desired to have high extinction ratios, a wide angular bandwidth, a broad wavelength range, and a compact size of effective packaging. Conventional PBSs, such as, Nicol prisms and Wollaston prisms are made of naturally anisotropic materials, which require a large thickness and inevitably lose some energy [8]. A good approach to reducing the energy loss is the use of a multilayer dielectric PBS, which is based on light interference effects. The interference mechanism may provide low energy loss at several discrete wavelengths and specific incident angles [9,10]. Under the demand for a more efficient PBS, researchers have focused more on PBS gratings owing to their great feasibility and efficient performance [11]. However, their inevitable narrow angular bands hinder further applications [12].

With the development of nanotechnology, many artificial metamaterials with smart optical properties have been proposed and provide new ways to manipulate electromagnetic waves [13–16]. Among these applications, PBS has attracted much research attention for its practical necessity. It has been demonstrated that PBS can be realized through different methods, such as, birefringence, diffraction of nano-wire grating, and transformation optics [17–19]. By using the large birefringence of anisotropic metamaterials induced by opposite amphoteric refraction, a slab PBS was theoretically investigated under a harsh condition [20] and a broadband PBS with a high polarization extinction ratio was proposed through a subwavelength grating [19]. Based on the diffraction of the grating, a special diffractive PBS was fabricated for operation in reflection [21]. Zheng et al. further presented a unique 1-D metasurface PBS with two-layer grating for producing strong negative refraction for transverse magnetic (TM) waves and efficient refraction for transverse electric (TE) waves [22]. Besides, self-assembly and nanofabrication techniques have been introduced to improve the performances of the wire-grid PBS [23–25]. Alternatively, the flexible PBS has also been designed via transformation optics [26,27]. However, all of these designed PBSs have disadvantages, such as, relative narrow spectral bandwidth, narrow angular band, and inevitable energy loss because of the impendence mismatch at the optical interfaces. Such deficiencies are not only caused by the imperfection of their microscopic structures and fabrications but also from the intrinsic inability of their theoretical mechanisms to overcome these challenges. Therefore, it is necessary to build a proper theoretical model of a perfect PBS with wide angular and wavelength bandwidths with a relatively low dissipation.

This paper proposes a theoretical approach to realize a perfect polarizing beam splitter (PPBS) with a broad frequency and angular bands between two isotropic media. By using the multi-beam interference and Fresnel formulas, the generalized theoretical conditions of the anisotropic metamaterial film are deduced for an omnidirectional PPBS for white light. The proposed PPBS promises total reflection and transmission to TM and TE waves, respectively, at an arbitrary angle of incidence for the entire visible range. In addition, this PPBS can perform excellent beam splitting irrespective of lying between different media or not. When lying between different media, the PPBS can be regarded as a perfectly impedance-matched layer for TE waves incident from any direction for the total refraction; however, it makes TM waves totally reflected. The required material of the PPBS is both temporally and spatially dispersive, characterized by its particular dispersion surface differentiated from the traditional medium. Furthermore, the numerical simulations show the validity of the theory and reveal the specific features of such new material. It is verified that several hundred nanometers’ thickness already offers great performance of beam splitting for the visible light. This theoretical work offers a guideline for designing an ideal PPBS, which has potential applications in complex optics systems, such as, image processing, optical interconnections, weak light detection, and light energy harvesting.

2. Theoretical design

Let us consider that two dispersionless and lossless isotropic media with permittivities and permeabilities ${\epsilon }_{\text{i}},{\mu }_{\text{i}}$ (i = 1, 2) are mediated by a layer of anisotropic metamaterials characterized by tensorial permittivity and permeability. In the coordinate system collinear with the principal axis of the metamaterial, the permittivity and permeability tensors are diagonalized and have the following forms [28]:

 

Fig. 1 (a) A PPBS with thickness d is placed between medium 1 and medium 2. (b) Illustration of anti-reflection for TE wave in the PPBS.

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The dispersion relations of the TM and TE plane waves propagating in the anisotropic metamaterial can be written as [29]

2.1 Conditions of perfect total reflection for TM wave

At the interface of medium 1 and the anisotropic metamaterial, the parallel component of the wave vector is conserved during the refraction process, i.e.$k_{\text{y}}=k_{1\text{y}}$, where $k_{1\text{i}}\left(\text{i=x,y}\right)$ is the corresponding component of the wave vector in medium 1 with a dispersion relation of $k_{1\text{x}}^2+k_{1\text{y}}^2={\epsilon }_1{\mu }_1{k}_0^2$. For the electromagnetic waves propagating in different incident angles, the parallel component of the wave vector $k_{1\text{y}}$ changes in the range of $[0,\sqrt{{\epsilon }_1{\mu }_1{k}_0^2}]$. To achieve perfect total reflection, the propagating waves in medium 1 are totally prohibited in the anisotropic metamaterial and the x component of the wave vector $k_{\text{x}}=\pm \sqrt{{\epsilon }_{\text{y}}\left({\mu }_{\text{z}}{k}_0^2-\frac{k_{\text{y}}^2}{{\epsilon }_{\text{x}}}\right)}$ should be a pure imaginary number at all incident angles. Thus, the optical parameters of the metamaterial should satisfy the relations

In this metamaterial slab, the magnitude of the TM wave attenuates exponentially with distance. After a short thickness, the magnitude of TM is negligible. Especially, hyperbolic metamaterials (HMMs) with${\epsilon }_{\text{y}}<0,{\epsilon }_{\text{x}}>0,{\mu }_{\text{z}}>0$ and${\epsilon }_{\text{x}}{\mu }_{\text{z}}\ge {\epsilon }_1{\mu }_1$ [28] can satisfy the condition of Eq. (3). Such type of HMMs can be realized by a nano-array or metallic layered structures [30,31].

2.2 Conditions of anti-reflection for TE wave

TE waves suffer multiple reflections at interfaces I and II, as shown in Fig. 1(b). For each interface, a general Fresnel formula suitable to metamaterials is presented as

Assuming that the metamaterial is lossless and the component $k_{\text{x}}$ of TE waves is always real for any incident angle, the corresponding amplitude reflection coefficients$r_{1\text{P}},r_{\text{P}2}$must also be real numbers. Therefore, the phase factor in Eq. (6) should satisfy

In terms of Eqs. (2), (4), and (7), a perfect anti-reflection for a TE wave requires

Equation (8a) represents the condition for an omnidirectional PPBS when no half-wave phase shift occurs during reflections. In contrast, Eq. (8b) is valid when the PPBS has an extremal value of the effective wave impendence. There are infinite discrete solutions satisfying Eqs. (7) and (8) for achieving the perfect anti-reflection for TE waves under a certain thickness d.

2.3 Design of omnidirectional polarization beam splitters

To obtain the perfect omnidirectional polarization beam splitting with totally reflected TM waves and perfectly transmitted TE waves, the parameters of the metamaterial should obey Eqs. (3) and (8) simultaneously. Nevertheless, there are still infinite groups of solutions. For an easy fabrication, we assume that the metamaterial is uniaxial and isotropic in the xz-plane, with the relation of ${\epsilon }_{\text{x}}={\epsilon }_{\text{z}}\ne {\epsilon }_{\text{y}}$and${\mu }_{\text{x}}={\mu }_{\text{z}}\ne {\mu }_{\text{y}}$.These requirements put additional constrains on permittivity and permeability. According to Eq. (3), we further choose ${\text{ε}}_{\text{x}}{\mu }_{\text{z}}={\epsilon }_1{\mu }_1$ for simplicity. Now, a group of material parameters that can realize the PPBS for the beams incident from the vacuum is obtained:

The nonlocal nature of the metamaterial implies that it would have a nontrivial dispersion in the wave vector space. Figure 2(a) shows the equal frequency contours (EFCs) of the surrounding medium 1 and 2, as well as the metamaterial of the PPBS. EFCs of medium 1 and 2 are circles whose radii are proportional to their refractive indices, while the EFC of TE waves in PPBS is along a straight line of $k_{\text{x}}=\frac{\pi }{2d}$, which is only determined by the slab width. The direction of the refracted wave in each medium can be easily obtained according to the conservation of the parallel wave vector component$k_{\text{y}}$.

 

Fig. 2 (a) Huygens construction for direction of refraction wave. Circles represent isotropic dispersion relation of medium 1 and medium 2, the red line denotes a special dispersion relation of the PPBS,${\theta }_1$is incident angle,${\theta }_2$is the refraction angle in the PPBS, and${\theta }_3$is the refraction angle in medium 2. (b-d) The theoretically derived parameters of ${\epsilon }_{\text{x}}={\epsilon }_z$ (b), ${\mu }_{\text{x}}={\mu }_{\text{z}}$ (c), and ${\mu }_{\text{y}}$ (d) for obtaining the perfect polarization beam splitter when the incident angle lies in the range $[-90°,90°]$ and the wavelength lies in the visible light spectrum; both medium 1 and 2 are air.

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To exhibit the spatial and temporal dispersion of the PPBS, the values of the metamaterial components ${\epsilon }_{\text{x}}={\epsilon }_{\text{z}}$,${\mu }_{\text{x}}={\mu }_{\text{z}}$and${\mu }_{\text{y}}$varying with the incident angle and with the wavelength in the visible light spectrum from 400 nm to 760 nm are calculated and shown in Figs. 2(b)–2(d), where both medium 1 and 2 are air and the thickness d equals 0.5μm. It can be seen that the optical parameters${\mu }_{\text{y}}$and${\epsilon }_{\text{x}}$change slightly with the wavelength in the visible range. In addition, in the angle range of $[-45°,45°]$, the proposed metamaterial also manifests relative weak spatial dispersion with the wave vector, which is benefit for metamaterial designing and realizing.

3. Simulations

Note that Eq. (9) does not require medium 1 and medium 2 to be the same. Figure 3 shows the simulation results of an air–substrate interface with${\text{ ε}}_1=1,{\mu }_1=1,{\epsilon }_2=4,{\mu }_2=1$, and M = 2. Figure 3(a) shows the transmittance of TE and TM waves with a wavelength of 400 nm with the incident angle in the range $[-85°,85°]$. Figure 3(b) shows the transmittance of TE and TM waves incident at 30° within the visible light spectrum. It is clearly shown that TE waves have 100% transmittance, while TM waves are totally reflected at all tested incident angles and wavelengths. By using the anisotropic metamaterial with the optical parameters${\mu }_{\text{y}}=0.077$,${\mu }_{\text{x}}={\mu }_{\text{z}}=5.8$, ${\epsilon }_{\text{x}}={\epsilon }_{\text{z}}=0.17$,${\epsilon }_{\text{y}}=-2$ satisfied with Eq. (9), the performance of the PPBS are demonstrated as Figs. 3 (c)-3(d) for 400 nm light incident at an angle of 30°. Figure 3(c) shows the distributions of the electric field$E_{\text{z}}$for a plane TE wave with the wavelength of 400 nm, exhibiting that the TE wave passes through medium 2 without any reflection. Figure 3(d) shows the distribution of the magnetic field Hz for a TM incident wave, revealing that the TM wave is totally reflected by the metamaterial slab. Therefore, the validity of the PPBS has been verified when placed between two different media; in this case, the PPBS plays the role of filling the impedance mismatch of the media on the two sides. The results indicate that the designed PPBS is a perfect choice for wide-angle polarization imaging systems.

 

Fig. 3 Optical characteristics of the PPBS in the materials with${\epsilon }_1=1,{\mu }_1=1,{\text{ε}}_2=4,{\mu }_2=1$. (a) Transmittance of the PPBS for TE and TM waves at different incident angles at a wavelength of 400 nm. (b) Transmittance of the PPBS for the visible-range TE and TM waves at an incident angle of 30°. (c) Electrical field distribution when the incident wave is a plane TE wave polarized in the $\text{Z}$ direction at a wavelength of 400 nm and an incident angle of 30°, demonstrating 100% transmittance. (d) Magnetic field distribution when the incident wave is a 400 nm plane TM wave polarized in the xy plane with an incident angle of 30°.

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For comprehensiveness, Fig. 4 also examines the optical transportation as the PPBS is placed in a homogenous medium, where${\epsilon }_1={\epsilon }_2=1,{\mu }_1={\mu }_2=1$, and M = 2 are chosen to simulate a PPBS placed in vacuum. It is also clearly shown that the TE wave has 100% transmittance, while the TM wave is totally reflected for an arbitrary angle of incidence and within the entire range of the visible spectrum. In Figs. 4(c) and 4(d), the optical parameters of the metamaterial are chosen as${\mu }_{\text{y}}=0.12$,${\mu }_{\text{x}}={\mu }_{\text{z}}=8.7$,${\epsilon }_{\text{x}}={\epsilon }_{\text{z}}=0.12$,${\epsilon }_{\text{y}}=-2$to satisfy with Eq. (9) for a perfect PPBS.

 

Fig. 4 Optical characteristics of the perfect polarization beam splitter in air with ${\text{ε}}_1=1,{\mu }_1=1,{\text{ε}}_2=1,{\mu }_2=1$. (a) Transmittance of 400 nm TE and TM waves at an incident angle range of [-85°, 85°]. (b) Transmittance of TE and TM waves within the visible light spectrum at an incident angle of$30°$. (c) Magnitude distribution of $E_{\text{z}}$ when a plane TE wave with a wavelength of 400 nm is incident on the PPBS at an angle of 30°. (d) Magnitude distribution of $H_{\text{z}}$ when a 400 nm plane TM wave is incident on the PPBS at an angle of 30°.

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Additionally, the transportation behavior of the Gaussian beams with a wavelength of 400 nm through the designed slab at an incident angle of 30° is carefully shown in Fig. 5. When the Gaussian beam is incident on the PPBS from vacuum, the TE-polarized component perfectly transmits through the PPBS without any reflection, as shown by the E_z field and the energy flow in Figs. 5(a) and 5(c). However, the TM-polarized component of the Gaussian beam suffers total reflection on the PPBS surface according to Fig. 5(b), and Fig. 5(d) shows that no energy penetrates medium 2. The results indicate that the PPBS can work for not only plane waves but also Gaussian beams.

 

Fig. 5 Polarization splitting of Gaussian beams induced by a PPBS placed in vacuum. The distribution of (a) electrical field component$E_{\text{z}}$and (c) the x component of energy flow for a TE-polarized Gaussian beam incident on the PPBS. The distribution of (b) magnetic field component$H_{\text{z}}$and (d) the x component of energy flow for a TM-polarized Gaussian beam incident on the PPBS. For both TE and TM beams, the waist is 0.2$\text{ μm}$, the wavelength is 400 nm, and the angle of incidence is$30°$. (see Visualization 1)

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The thickness of the metamaterial PPBS determines whether the TM waves can penetrate medium 2, and influences the effective packaging. Placing the PPBS in air and choosing$M=2$, we can obtain the reflectivity of TM wave varying with the thickness of PPBS and the incident angle, as shown in Fig. 6(a). We find that nearly perfect total reflection is achieved for TM waves within a wide-angle range when d is larger than 200 nm. Figure 6(b) demonstrates the reflectance of TM wave changes with d when a 400 nm electromagnetic wave is incident on the PPBS at an angle of${30}^{°}$. The reflectance first increases quickly with d, and then, approaches nearly 100% when d is larger than 200 nm. The result shows that a perfect reflection performance can be achieved as d is comparable to the wavelength. Figure 6(c) shows the energy attenuation of the TM wave inside a 200-nm-thick PPBS, revealing that the energy density drops to zero quickly after entering the PPBS. To quantify the efficiency of polarization splitting, we introduce the transmission polarization ratio as

 

Fig. 6 (a) Reflectance of TM waves varying with PPBS thicknesses d and incident angle θ. (b) Reflectance of TM wave changing with the thickness d for a 400 nm plane wave incident on the PPBS at$30°.$(c) Energy attenuation of TM wave inside a 200-nm-thick PPBS. (d) Polarization ratio varying with the incident angle for a 400 nm circularly polarized beam passing through the PPBS (M = 2).

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4. Conclusions and discussions

In summary, a perfect polarizing beam splitter is proposed based on the multi-beam interference approach, which renders TM and TE waves totally reflected and transmitted, respectively, for omnidirectional incidence. The required material parameters of the PPBS are shown to be hyperbolic and spatially dispersive. The nontrivial spatial dispersion of the PPBS is reflected in its EFC, which is along the straight line of a constant$k_{\text{x}}$. Numerical simulations further verify the efficiency of such PPBS in a large range of wavelengths and incident angles. In addition, the designed PPBS cannot only work in a homogenous background medium but can also be used as a functional coating structure on the interface between two different materials, revealing wide application prospects. In fact, the idea of omnidirectional and wideband transparent has recently been turned into reality through artificial structures possessing a straight-line-shaped EFC [33]. Thus, we believe that our model of PPBS is not merely of theoretical interests and practical structures for this model can be realized based on the existent technology.