1. Introduction

Metamaterials have been designed and fabricated to perform novel properties, such as negative refractive index [1,2], superlensing [3], perfect absorption [4] and cloaking [5]. The progress of nanotechnology supports the fabrication of subwavelength structures with these novel properties at microwave to ultraviolet wavelengths [6]. The design and fabrication of various metal-dielectric stratified metamaterials has recently attracted much research attention [7]. A multilayer composed of alternatively arranged metal and dielectric films acts as a hyperbolic metamaterial [8] or a metamaterial with a negative real index of refraction [9]. A multilayer with a high fraction of metal layers is not a nonmagnetic material. The equivalent relative permittivity ε and permeability μ can be tailored and measured separately [10]. Instead of refractive index alone, the equivalent optical constants of such a multilayer are admittance ${\text{η}}_{\text{eq}}$ and refractive index ${\text{N}}_{\text{eq}}$. Admittance ${\text{η}}_{\text{eq}}$ and refractive index ${\text{N}}_{\text{eq}}$ are related to both ε and μ by ${\text{N}}_{\text{eq}}=\sqrt{\text{εμ}}$ and ${\text{η}}_{\text{eq}}=\sqrt{\epsilon /\text{μ}}$ [10]. The mutual independence of ${\text{η}}_{\text{eq}}$ and ${\text{N}}_{\text{eq}}$ make the design of optical coatings flexible. A multilayer with admittance matching with air and a refractive index with a large extinction coefficient has been designed and fabricated as an ultra-thin light absorber, which is equivalent to a metamaterial that performs as a low-reflectivity metal [11].

For a metal-dielectric multilayer with a low filling factor of metal, the effective medium approximation is valid and the effective permittivity can be represented by an anisotropic tensor. For an incident p-polarized electromagnetic wave, the associated two principal components of permittivity have opposite signs, corresponding to metal-like and dielectric-like optical characteristics, respectively. Therefore, the isofrequency dispersion becomes a hyperbolic form [8], and the whole layered structure exhibits various extraordinary optical properties, such as negative refraction, and has novel applications in nano-resonators [12] and diffraction-free focusing [13] and imaging [14]. Those applications depend on high transmission of p-polarized light, implying that hyperbolic metamaterial exhibits strong polarization-dependent transmission/reflection. This work develops a compact polarization beam splitter that comprises metal and dielectric, sandwiched between two prisms. Unlike the traditional MacNeille polarizing beam splitter [15] and similar polarization beam splitting cube [16] that comprise all dielectric thin films. The polarization beam splitter that is proposed here requires only three or five layers to achieve high-performance beam splitting. Compared with the MacNeille polarizing beam splitter composed of dielectric thin films, the polarization beam splitter proposed here takes much less number of thin films to achieve polarization beam splitting.

The strategy for achieving polarization beam splitting is to induce p-polarized transmission and increase s-polarized reflection as much as possible. High p-polarized transmission requires admittance matching and low dissipation within the metal film. High s-polarized reflection requires an extremely large or small equivalent admittance relative to the admittance of the cover medium. Both will be designed through a visual method by tracing the variation of equivalent admittance in the normalized admittance diagram.

2. Design Method

The admittance tracing method has been widely used in designing optical thin films [17]. When an electromagnetic wave propagates in a medium, the admittance is defined as the ratio of the magnitude of the magnetic field to that of the electric field. The admittance of a non-magnetic medium can be normalized to the refractive index. At oblique incidence, the admittance is further normalized to ${\text{η}}_{\text{p}}=\text{N}\cos {\text{θ}}_{\text{i}}/\cos \text{θ}$ for p-polarization and ${\text{η}}_{\text{s}}=\text{N}\cos \text{θ}/\cos {\text{θ}}_{\text{i}}$ for s-polarization, where θ and ${\text{θ}}_i$ are the refracted angle in the medium and the angle of incidence in the cover medium with refractive index of ${\text{N}}_{\text{i}}$, respectively. Based on the calculation of characteristic film matrices, any multilayer on a substrate is equivalent to a medium with admittance ${\text{η}}_{\text{eq}}$ adjacent to the cover medium. Therefore, the reflection coefficient can be obtained using the familiar equation $\text{r}=({\text{N}}_{\text{i}}-{\text{N}}_{\text{sub}})/({\text{N}}_{\text{i}}+{\text{N}}_{\text{sub}})$. The constant phase of r and constant reflectance R = |r|^2 form circular loops on the complex plane that correspond to the real and imaginary parts of ${\text{η}}_{\text{eq}}$. The variation of equivalent admittance of films that are grown on a substrate can be simulated by tracing the equivalent admittance on the complex plane. The equivalent admittance of a growing dielectric film follows a circular loop clockwise and the equivalent admittance of a growing metal film with refractive index ${\text{N}}_m=n-ik$ follows one of arc loci toward a terminal point that represents the admittance of a bulk metal, as shown in Fig. 1. All possible loci of a metal film with high ratio k/n are symmetrical to an axis x that males a small angle to the axis $\text{Re}({\text{η}}_{eq})$ [17]. Visually tracing the equivalent admittance in the admittance diagram enables an antireflection coating [18] or high-reflection coating [19] to be designed.

 

Fig. 1 Admittance loci of (a) a dielectric film and (b) a metal film with = fractive index $\text{N}=n-ik$ in normalized admittance diagram with (c) constant reflectance circles (pink) and constant phase circles (blue).

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At oblique incidence, the admittances of the thin film and the substrate depend on the cover medium and vary strongly with the angle of incidence. Figure 2 plots the ${\text{η}}_{\text{s}}$ and ${\text{η}}_{\text{p}}$ of an SiO₂ film with refractive index N = 1.46 as a function of angle of incidence ${\text{θ}}_{\text{i}}$ when the cover medium is a glass with refractive index ${\text{N}}_{\text{i}}$ = 1.520. The admittance shown in Fig. 2 is dependent on the angle of incidence and refractive index of each medium. The refractive indexes of both BK7 prism and SiO2 are almost wavelength independent and the curves in Fig. 2 were plotted with refractive indexes at a wavelength of 550 nm. Since SiO₂ is a material with a lower refractive index than that of the cover medium, ${\text{η}}_s$ decreases as ${\text{θ}}_{\text{i}}$ increases. However, ${\text{η}}_{\text{p}}$ increases toward infinity as ${\text{θ}}_{\text{i}}$ increases. At the Brewster angle ${\text{θ}}_{\text{B}}$ = 43.8°, ${\text{η}}_{\text{p}}$ is 1.520, yielding a reflectance of zero. At the critical angle ${\text{θ}}_{\text{c}}={\sin }^{-1}(\text{N}/{\text{N}}_{\text{i}})$, ${\text{η}}_s$ and ${\text{η}}_{\text{p}}$ are zero and infinite, respectively. In this work, the polarization beam splitter is designed near the critical angle ${\text{θ}}_{\text{c}}$ to have the SiO₂ film with a large value of ${\text{η}}_{\text{p}}$ and a small real value of ${\text{η}}_s$. Although both values result in high reflection, a three-layered arrangement DMD = SiO₂-Ag-SiO₂ sandwiched between two glass prisms with ${\text{N}}_{\text{i}}={\text{N}}_{\text{sub}}=1.520$ will have p-polarized equivalent admittance matching to ${\text{N}}_{\text{i}}$, yielding low reflectivity if the thickness of each film is properly chosen. However, the s-polarized equivalent admittance is still too small to cause high reflectivity.

 

Fig. 2 P(S)-polarized admittance as a function of angle of incidence for a Glass- SiO₂-Glass structure.

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Figure 3 shows the admittance diagram of a Prism-${\text{SiO}}_2(150\text{nm})$-Ag (11 nm)- ${\text{SiO}}_2(150\text{nm})$-Prism system. In the simulation, the refractive index of each film is derived from measurements made on the deposited 100nm-thick ${\text{SiO}}_2$ film and 100nm-thick $\text{Ag}$ film. The ${\text{SiO}}_2$ films and Ag film were deposited by electron beam evaporation and sputtering evaporation, respectively. The refractive indices of both films were measured using a commercial ellipsometer (J. A. Woollam M2000).

 

Fig. 3 P(S)-polarized admittance diagram of Prism-SiO₂ (150 nm)-Ag (11 nm)-SiO₂ (150 nm)-Prism system at ${\theta }_i$ = 74° for wavelengths of 500 nm, 650 nm, and 800 nm.

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The p-polarized admittance loci at a wavelength of 650nm and an incident angle of 74° exhibit admittance matching. At ${\theta }_i$ = 74°, the large value of ${\text{η}}_{\text{p}}$ = 115.75 causes the locus of the bottom SiO₂ film to follow a nearly upright vertical locus from (1.514, 0) to (1.52, 0.83). The middle metal film follows a curve from (1.52, 0.83) to (1.525, −0.85). Since the terminal of the metal film is almost right under the initial point (1.512, 0), the locus of the upper SiO₂ film brings the equivalent admittance back to the initial point. Figures 3(a)-3(c) plot the admittance loci at wavelengths of 500 nm, 650 nm, and 800 nm. The closed loop expands or shrinks as the wavelength decreases or increases, respectively, keeping the terminal around the initial point. Figure 3(e) plots the s-polarized admittance loci. At ${\theta }_i$ = 74°, the small value of ${\text{η}}_{\text{s}}$ = 0.016 causes the locus of the bottom SiO₂ film end at point A. The locus of the middle metal film follows an arc, stopping at point B. The terminal of the upper SiO₂ film at the point (0.07, −1.87) close to the imaginary axis, results in high reflection for the s-polarized state. The admittance loci at two other wavelengths exhibit the high reflection property.

Figure 4 plots transmittance and reflection as functions wavelength from 450 nm to 850 nm at ${\theta }_i$ = 74 for both polarization states. The p-polarized transmittance increases from $T_p$ = 90.2% at wavelength λ = 450 nm to 96.5% at λ = 525 nm. The transmittance remains T = $\left(97\pm 5\right)\text{\%}$ from λ = 525 nm to λ = 850 nm. The s-polarized reflectance also remains high, ${\text{R}}_s=\left(92\pm 2\right)\text{\%}$ from λ = 450 nm to λ = 850 nm. For our design, the admittance locus of the ultra-thin silver film is located in the region $\text{Re}\left({\text{η}}_{\text{eq}}\right)>1.5$, reducing the dissipation of light in the silver film. The s-polarized transmittance ${\text{T}}_s$ remains above 5% throughout this range of wavelengths.

 

Fig. 4 Simulated transmittance and reflectance spectra of Prism-SiO₂ (150nm)- Ag (11nm)-SiO₂ (150nm)-Prism system.

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To reduce $T_s$ to increase the ratio $T_p$/$T_s$, a five layered structure is introduced as a Prism-D₁-M₂-D₃-M₄-D₅-Prism = Prism-SiO₂ (110 nm)-Ag (11 nm)-SiO₂ (260 nm)-Ag (11 nm)-SiO₂ (110 nm)-Prism system. Figure 5(b) shows the admittance diagrams at angle of incidence ${\theta }_i$ = 74° and a wavelength of λ = 650 nm. The p-polarized loci of D₃-M₄-D₅ and the loci of D₁-M₂-D₃ have similar shapes to that of the aforementioned three-layered structure. The condition of admittance matching holds in the case with five layers. Figures 5(a) and 5(c) show the admittance diagrams at wavelengths λ = 500 nm and λ = 800 nm. However, the s-polarized loci of D₁-M₂-D₃-M₄ form two dips in admittance diagram, so their ends are closer to the imaginary axis than in the case with three layers. Therefore, the s-polarized reflection is increased owing to the one additional dip in admittance diagram. Figure 6 plots transmittance and reflection as functions wavelength from 450 nm to 850 nm at ${\theta }_i$ = 74° for both polarization states. The high p-polarized transmittance remains within $\left(91\pm 3\right)\text{\%}$ from λ = 450nm to λ = 850 nm. The s-polarized reflectance remains high ${\text{R}}_s=\left(96.5\pm 1.5\right)\text{\%}$ from λ = 450 nm to λ = 850 nm. The s-polarized transmittance ${\text{T}}_s$ is suppressed to under 0.14% in this range of wavelengths.

 

Fig. 5 P(S)-polarized admittance diagram of Prism-SiO₂ (110 nm)-Ag (11 nm)-SiO₂ (260 nm)-Ag (11 nm)-SiO₂ (110 nm)-Prism system at ${\theta }_i$ = 74° for wavelengths of 500 nm, 650 nm, and 800 nm.

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Fig. 6 Simulated transmittance and reflectance spectra of Prism-SiO₂ (110 nm)-Ag (11 nm)-SiO₂ (260 nm)-Ag (11 nm)-SiO₂ (110 nm)-Prism system.

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3. Fabrication and Measurement

Three-layered and five-layered structures with the above designs were fabricated in physical vapor deposition systems. The ultra-thin Ag films were deposited in a sputtering system and the SiO₂ films were deposited in an electron-beam evaporation system. The thicknesses of SiO₂ films were controlled by quartz-crystal monitor. The thicknesses of silver films were controlled by controlling the deposition time with respect to the deposition rate. The thickness of each film was measured from cross-sectional SEM images, which are shown in the insets in Figs. 7 and 8. The deposited films were attached to BK7 prisms using optical index matching glue to form a three-layered system of Prism-SiO₂ (150.7 nm)-Ag (12.6 nm)-SiO₂ (145 nm)-Prism and a five-layered system of Prism-SiO₂ (106 nm)-Ag (12.7 nm)-SiO₂ (265 nm)-Ag (10.7 nm)-SiO₂ (119 nm)-Prism. The optical gel (Code 081160, Cargille Laboratories Inc.) was adopted to attach two prisms.

 

Fig. 7 Measured (dotted line) and simulated (solid lines) spectra of Prism-SiO₂-Ag-SiO₂-Prism system vs. wavelength for p-polarized and s-polarized states. Insets: cross-sectional image of layered structure.

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Fig. 8 Measured (dotted line) and simulated (solid lines) spectra of Prism-SiO₂-Ag-SiO₂-Ag-SiO₂-Prism system vs. wavelength for p-polarized and s-polarized states. Insets: cross-sectional image of layered structure.

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The p(s)-polarized reflectance and transmittance spectra were measured at an angle of incidence of 74^∘, as shown in Fig. 7. The reflection at the surface of prism was normalized by the angular spectra of reflectance and transmittance that were measured for a bare prism and two glued prisms, respectively. In the measurement, two linear polarizers (Newport 5524, Irvine, California, USA) and an achromatic waveplate (ThorLabs AQWP05M-600, Newton, New Jersey, USA) were arranged in front of the monochromator (JobinYvon iHR320, Kyoto, Japan) with halogen lamp source (Newport 66882, Irvine, California, USA) to generate p-polarized light and s-polarized light with wavelengths from 450 nm to 850 nm. The transmitted and reflected light intensities were collected by photomultiplier (Hamamatsu R636-10, Hamamatsu, Shizuoka, Japan). The theoretical reflectance and transmittance spectra were calculated by the measured thicknesses and compared with experimental spectra. The discrepancies between the calculated spectra and the measured spectra come from the ultra-thin silver thin films. The refractive index of an ultra-thin silver thin film deposited on a coated surface would be different from that deposited on a smooth glass substrate.

4. Discussion

Although p-polarized admittance matching is achieved in the five-layered structure at the designated wavelength, the compensation effect at wavelength of 500 nm still functions to keep the terminal of admittance around its initial point. The deviation of admittance at a wavelength of 800 nm in the aforementioned case arises from the shortening of the loci of the dielectric layers. Since the possible loci in Fig. 1(b) are symmetrical about the x axis that makes a small angle with respect to the axis Re(η), the shortening of locus D₅ prevents the locus of the metal film M₄ from being nearly symmetrical about that axis Re(η); instead, it stops right under the initial point. A similar situation exists for D₃ and M₂. Therefore, increasing the thicknesses of the dielectric layers can increase the bandwidth of beam splitting. Here a modified five-layered structure is proposed here as Prism-D₁-M₂-D₃-M₄-D₅-Prism = Prism-SiO₂ (150 nm)-Ag (11 nm)-SiO₂ (300 nm)-Ag (11 nm)-SiO₂ (150 nm)-Prism. Figure 9 shows the admittance diagrams at angle of incidence ${\theta }_i$ = 74° for wavelength of 500 nm, 650 nm and 800 nm. The p-polarized admittance matching is achieved well and the s-polarized admittance terminals are close to the imaginary axis to achieve high reflection for the three wavelengths. Figure 10 shows the transmittance and reflection spectra at ${\theta }_i$ = 74° for both polarization states. The p-polarized transmittance is increased from ${\text{T}}_p$ = 86% at λ = 450 nm to 92% at λ = 500 nm. The transmittance remains T = $\left(93.5\pm 1.5\right)\text{\%}$ from λ = 500 nm to λ = 850 nm. The s-polarized reflectance also remains ${\text{R}}_s$>96.5% from λ = 450 nm to λ = 850 nm. The s-polarized transmittance ${\text{T}}_s$ is suppressed to below 0.1% over these wavelengths.

 

Fig. 9 P(S)-polarized admittance diagram of Prism-SiO₂ (150 nm)-Ag (11 nm)-SiO₂ (300 nm)-Ag (11 nm)-SiO₂ (150 nm)-Prism system at ${\theta }_i$ = 74° for wavelengths of 500 nm, 650 nm, and 800 nm.

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Fig. 10 Simulated transmittance and reflectance spectra of Prism-SiO₂ (150 nm)-Ag (11 nm)-SiO₂ (300 nm)-Ag (11 nm)-SiO₂ (150 nm)-Prism system.

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Since this polarization beam splitter can function only at angles of incidence that are close to the critical angle, which is determined by the dielectric film and the cover medium, a modified design is required to eliminate this limitation. Here, a modified five-layered structure, Prism-D₁-D₂-M₃-D₄-D₅-Prism = Prism-Ta₂O₅ (60 nm)-SiO₂ (150 nm)-Ag (11 nm)-SiO₂ (150 nm)-Ta₂O₅ (60 nm)-Prism, is introduced. Although the refractive index of Ta₂O₅ exceeds that of SiO₂. At the angle of incidence ${\theta }_i$ = 63.5°, the normalized p-polarized admittance of Ta₂O₅ is lower than that of SiO₂. Figure 11(b) plots the admittance loci at a wavelength of 550 nm. Although ${\theta }_i$ is less than 74° in the previous case, the end of loci D₄-D₅ remains right above the initial point (1.52, 0). Therefore, the locus of the middle metal film is still on the right side of $\text{Re}(\text{η})$ = 1.52, ensuring low light dissipation. The loci of the upper two layers D₁-D₂ bring the terminal of all loci back to the point (1.36, −0.05), which is close to the initial point. Figure 12 shows the p(s)-polarized transmittance and reflectance spectra. The p-polarized transmittance is raised from ${\text{T}}_p=$68% at λ = 400nm to 92.5% at λ = 450nm. The transmittance remains T = $\left(93.5\pm 1.5\right)\text{\%}$ from λ = 450nm to λ = 700nm. The s-polarized reflectance remains high, ${\text{R}}_s=\left(95\pm 2\right)\text{\%}$ from λ = 400nm to λ = 700nm. The s-polarized transmittance ${\text{T}}_s$ is less than 2% over wavelengths from 400nm to 700nm.

 

Fig. 11 P(S)-polarized admittance diagram of Prism-Ta₂O₅ (60 nm)-SiO₂ (150 nm)-Ag (11 nm)-SiO₂ (150 nm)-Ta₂O₅ (60 nm)-Prism system at ${\theta }_i$ = 63.5° for wavelengths of 500 nm, 650 nm, and 800 nm.

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Fig. 12 Simulated transmittance and reflectance spectra of Prism-Ta₂O₅ (60 nm)-SiO₂ (150 nm)-Ag (11 nm)-SiO₂ (150 nm)-Ta₂O₅ (60 nm)-Prism system.

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5. Conclusions

This work developed a compact metal-dielectric multilayer as a compact polarization beam splitter. The design method involves tracing the equivalent admittance in a normalized admittance diagram for both p-polarization and s-polarization states. The high p-polarized transmission and high s-polarized reflection over a wide wavelength range are achieved by arranging metal and dielectric thin films between two prisms. A five-layered Ag-SiO₂ multilayer with the proposed design exhibits high p-polarized transmittance of over 93.5% and low s-polarized transmittance of less than 0.1%, yielding an extinction ratio of over 935 at wavelengths from 450 nm to 850 nm. This work can be extended to develop stratified metamaterials with desired polarization-dependent optical properties.