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Abstract
Properly designed black phosphorus (BP) ribbons exhibit extreme anisotropic properties, which can be used to fabricate a high-efficiency transmitter or reflector depending on the linear polarization of excitation. In this study, we design a highly efficient and broad-angle polarization beam splitter (PBS) based on extremely anisotropic BP ribbons around the mid-infrared frequency region with an ultra-thin structure, and study its performance by using transfer matrix calculation and finite element simulation. In the broad frequency range of 80.4 terahertz - 85.0 terahertz (THz) and an wide angle range of more than 50°, the reflectivity and transmissivity of the designed PBS are both larger than 80% and the polarization extinction ratios are higher than 25.50 dB for s-polarization light and 20.40 dB for p- polarization light, respectively. Furthermore, the effect of incident angle and device parameters on the behavior of the proposed PBS is examined. Finally, we show that the operation frequency of this PBS can be tuned by the electron concentration of BP, which can facilitate some practical applications such as tunable polarization splitters or filters, and mid-infrared sensors.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A polarization beam splitter (PBS) divides a light beam into two beams of orthogonal polarizations in the reflected and transmitted directions. It is a crucial component in many optical systems such as imaging systems [1,2], photopolarimeters [3,4], optical switches [5,6], and communication devices [7,8]. However, conventional optical PBSs are usually too bulky and heavy to be integrated into an optical system because they are based on the natural birefringence of crystals [9]. With the development of nanotechnology, a series of artificial metamaterials and metasurfaces have been proposed to manipulate electromagnetic waves for realizing negative refractive index [10], perfect lenses [11], cloaking [12], and tunable absorption [13] and emission [14], etc. Further, high-performance PBSs have also been realized using materials such as anisotropic metamaterial slab [15,16], epsilon-near-zero metamaterials [17], anisotropic plasmonic nanostructures [18], all-dielectric phase gradient metasurfaces [19], anisotropic matrix metasurfaces [20], and gratings [9]. Most of them work in a narrow bandwidth in a limited range of incident angle, and they are not compact enough for integration.
More recently, layered two-dimensional (2D) materials such as graphene, transition metal dichalcogenides (TMDs), black phosphorus (BP), and hexagonal boron nitride (hBN) have emerged as a new platform to manipulate light at dimensions smaller than the diffraction limit, which helps to integrate more easily into optoelectronic systems. Many metasurfaces consisting of nanoribbons based on 2D materials in previous studies have been reported to achieve high performance absorbers [21–24]. Meanwhile, several PBSs have been developed based on 2D materials. Chen and He [25] proposed a frequency-tunable circular PBS using a graphene-dielectric subwavelength film, which could be tuned by changing the magnitude of the applied magnetic field or the chemical potential in the THz region, and its efficiency was nearly 80%. Shah et al. [26] proposed a PBS with wide incident angle using nanoscale van der Waals heterostructures with a narrow range of frequency. Zhang et al. [27] proposed an ultra-compact beam splitter and filter based on a graphene plasmon waveguide with a narrow incident angle. Although these PBSs based on the 2D materials were compact and they can be tuned dynamically with high polarization extinction ratios (PER) over a wide angular range or a broad bandwidth, it is difficult to achieve broadband and wide angle at the same time.
BP [28–30], a 2D material with a thickness of 0.53 nm, is remarkably different from graphene and TMDs. In particular, its bandgap ranges from 0.3 eV in bulk to 2.0 eV in monolayer BP, and its carrier mobility can reach up to 104 cm2/V·s. Because the phosphorus atoms form a hexagonal lattice with a puckered structure, they exhibit a strong in-plane anisotropic property. This enables novel polarization-dependent absorption of in-plane light, which is potentially useful for achieving high-performance anisotropic absorption [31–34] and polarization control [35–37] in devices such as absorbers and polarizers. Furthermore, the in-plane properties of BP can be tuned both electrically and mechanically [38,39]. In this study, we have proposed a PBS based on anisotropic BP working in a broad spectral range and a wide incident angle. To achieve extreme anisotropic in-plane properties for improving the performance of the proposed PBS, we cut a single layer of BP into ribbons and stacked it up with silicon dioxide. The ribbon array has been modeled as a metasurface using the effective medium approach. Using numerical calculations and simulations, we obtained a high-efficiency PBS with a wide angular range and broad spectral bandwidth on a scale of 165.90 nm which is less than λ/15 (λ is the operation wavelength). By optimizing the parameters, we observed that the reflectivity and transmissivity are both larger than 80% at the designed frequency of 80.4 THz, and the optimized PBS exhibits high PERs of higher than 25.50 dB for s-polarization light and 20.40 dB for p- polarization light in a frequency bandwidth of 4.6 THz and an incident angular range of more than 50°, respectively. In addition, we have investigated the effect of incident angle and structural parameters on the performance of the proposed PBS and have demonstrated that the PBS can be tuned by the electron concentration of BP.
2. Structure and theoretical model
The anisotropic optical properties of BP can be described using the semi-classical Drude model in the THz range. The surface conductivities σj can be expressed as follows [33]:
The permittivity of phosphorene can be written as [33]
In the proposed design as shown in Fig. 1, a metasurface composed of a periodical array of subwavelength BP ribbons is attached on a layer of SiO2 to form an anisotropic unit, and a large number of units are stacked to constitute the PBS. The metasurface can be optimized to achieve extreme anisotropic properties, which can improve the performance of PBS. Further, the BP ribbons can be modeled as an equivalent layer using the effective medium approach, which predicts the resonant frequency and conductivity tensor of the metasurface with an error less than 1%. When the unit cell period L<< λ, the strip near-field coupling can be considered as an effective capacitance [41],
Fig. 1. (a) Schematic of phosphorene-assisted PBS. (b) A unit cell of the proposed PBS, the fixed unit cell period L is 40 nm, the W is the strip width. (c) An equivalent layer, where the conductivities along the AC and ZZ direction are $|\sigma _{\textrm{AC}}^{\textrm{eff}}|,|\sigma _{\textrm{ZZ}}^{\textrm{eff}}|$, respectively.
Further, the effective conductivity along AC and ZZ direction can be expressed as follows [41]:
As shown in Fig. 2, the electron concentration ns is set to 5×1013 cm−2. To set the central operating frequency of our PBS in the mid-infrared region, W and L are considered as 35 nm and 40 nm, respectively. Based on the effective medium approach, we modeled the ribbons as an equivalent layer, which is shown in Fig. 1(c). The equivalent conductivities along the AC and ZZ direction are $\sigma _{AC}^{eff},\sigma _{ZZ}^{eff}$, and the thickness of the layer is 0.53 nm, which is approximately equal to the thickness of the monolayer BP. Therefore we can use the 4×4 transfer-matrix method (TMM) proposed by Yeh [42] to calculate the reflection and transmission for the light beam with p(s)- polarization, which is incident on the entire anisotropic structure with an incident angle γ.
Fig. 2. Real (solid line) and imaginary (dashed line) parts of conductivity along with the armchair and zigzag directions of phosphorene for (a) monolayer BP and (b) metasurface composed of BP ribbons. (c) The anisotropic ratio of conductivity. (d) The permittivity of the equivalent layer.
The entire structure can be described in terms of the transfer-matrix as follows [42]:
3. Results and discussion
By properly designing the conductivity tensors with the parameters used in Fig. 2, we achieved an extremely anisotropic metasurface. As shown in Fig. 2(c), σAC and σZZ of the monolayer BP are 5.6 + i185.8 µS and 1.2 + i39.8 µS, while the effective conductivity along x and y directions are 6.2 + i0.013 mS and 1.05 + i34.8 µS at 80.4 THz, respectively. Thus, the maximum anisotropic ratio $abs(\sigma _{\textrm{AC}}^{\textrm{eff}})/abs(\sigma _{\textrm{ZZ}}^{\textrm{eff}})$ can reach up to 177. In our optimized design, this anisotropic metasurface is placed on a layer of SiO2 to form a unit, and 30 units stacked in z direction constitute the device, as shown in Figs. 1(a) and 1(b). The refractive index of SiO2 is 1.46, and the thickness is set to 5 nm.
The reflectivity and transmissivity of PBS for p- and s- polarized light beam are shown in Figs. 3(a) and 3(b), where the PBS works in an incident angle of 45°. As shown in Fig. 3(a), Rss is nearly zero and Tss is more than 98% in the entire frequency band of 70.0-90.0 THz. Meanwhile, Rpp is more than 60% and Tpp is less than 20% in the same frequency region. Especially, in the region I indicated by blue color in Fig. 3(a) with frequency ranging from 80.4 THz to 85.0 THz, Rpp increases from 80% to 90%, while Tpp is almost negligible (less than 1%). The PER for s-polarized light beam defined as 10 log10(|Tss|/|Tpp|) [43] is high as [21.44 dB∼37.4 dB] and for p-polarized light beam defined as 10log10(|Rpp|/|Rss|) is high as [26.4 dB∼34.96 dB] in the range [80.4 THz, 85.0 THz]. Thus, highly efficient PBS with a wide frequency band can be obtained in the region from 80.4 THz to 85.0 THz. Further, the reflectivity and transmission curves are simulated by using COMSOL as shown in Fig. 3(a), where the results are marked with ‘(S)’. In the simulation, the BP is considered as a conductive medium. It is evident that the simulation results are consistent with the numerical results, which further verifies the validity of our equivalent index model.
Fig. 3. (a) Numerical and simulated results for the dependence of frequency on the reflectivity and transmissivity of the PBS when p-(s-) polarized light beam is incident at γ = 45°. (b) Dependence of incident angle of the p-(s-) polarized light beam on the reflectivity and transmissivity of PBS, where the operation frequency is 80.4 THz. (c) Reflectivity and transmission of PBS as a function of the frequency at γ = 45°, where ribbons of monolayer BP are not constructed. (d) Reflectivity and transmission as a function of the incident angle at the operation frequency of 80.4 THz, where ribbons of monolayer BP are not constructed.
As shown in Fig. 3(b), we plot the effect of the incident angle on performance of the designed PBS at the operation frequency of 80.4 THz. Obviously, Tpp and Rss are zero in the range of incident angle [0°, 80°] in the region II indicated by green color. Meanwhile, Rpp is more than 50% and Tss is more than 98%. It should be noted that in the region I (Fig. 3(b)), the proposed PBS can achieve a much better performance in the range [0°, 50°], where both Rss and Tpp are more than 80%. The PER for s-polarized light beam is high as [34.63 dB∼40.88 dB] and for p-polarized light beam is high as [33.79 dB∼37.91 dB] in the angle range [0°,50°]. Thus, a high-performance PBS with wide angular bandwidth can be realized at 80.4 THz. Therefore, the designed PBS has potential applications in infrared remote sensing (∼3.7 µm) because this frequency band is one of the ideal atmospheric windows for infrared detection. In Table 1, the performances of PBSs based on various layer 2D materials reported in Refs. [25–27] are listed for comparing with this work. These PBSs based on 2D materials have excellent performance in scale and efficiency such as hBN-graphene based PBS, while the PBS based on BP ribbon in this work has advantages in terms of working range and incidence angle.
Table 1. Comparison of the PBS performance based on various 2D materials
To validate the efficiency of the proposed PBS, the transmissivity and reflectivity for p- and s-polarized light beam are calculated as shown in the Figs. 3(c) and 3(d), where the ribbons are replaced by the monolayer BP. Obviously, a PBS cannot be designed in such a case because the transmissions Tpp and Tss don’t show a big difference. However, when extreme anisotropy is introduced by the ribbons instead of the complete monolayer BP, the reflection Rpp of p-polarized light is enhanced and its transmission Tpp is suppressed to approximately zero, as shown in Figs. 3(a) and 3(b). Therefore, the excellent performance of the proposed PBS is a result of the extreme anisotropy, which is induced by the BP ribbons in this frequency band.
To better understand the physical mechanism, we can consider the entire device as an anisotropic slab and the approximate permittivity can be calculated using the effective medium equation discussed in [44]. The basic mechanism of the proposed PBS can be analyzed based on the Fresnel formula. The general Fresnel formula of the amplitude reflection for the front interface is ${r_{}} = ({Z_{\textrm{2eff}}} - {Z_{\textrm{1eff}}})/({Z_{\textrm{2eff}}} + {Z_{\textrm{1eff}}})$[45], where Z1eff and Z2eff are the effective wave impedance of the vacuum and the device, respectively. Zmeff is defined as Zmeff=Em/Hm, m = 1,2 for vacuum and the anisotropic slab. Em and Hm are the parallel components to the incident interface of the electric and magnetic fields. For s-polarized light, the real part of the permittivity of the anisotropic slab is around 1.0 and its imaginary part is less than 0.06, which can be ignored in the entire frequency region. Thus, as shown in Fig. 4(a), the impedance for vacuum and the device is very close and the reflection coefficient for the front interface is less than 0.04. In this case, the impedance of the designed system is properly matched with that of vacuum, so the reflectivity of the anisotropic slab is very low in a wide range of incident angles. In other words, the transmissivity of PBS for s-polarized light is very high. When p-polarized light is incident, the anisotropic slab has a high refractive index in the region from 70.0 THz to 80.4 THz, which leads to a high impedance difference between the slab and vacuum. From 80.4 THz to 90.0 THz, the anisotropic slab behaves like metal because the real and imaginary parts of the permittivity are negative and positive, respectively. As shown in Fig. 4(b), the difference of the impedance for vacuum and the anisotropic slab is very huge and the reflection coefficient for the front interface is more than 0.566. In contrast to the case of s-polarized light, the impedance of the designed system is not well-matched with that of vacuum for p-polarized light. In this region, it acts like a metal reflector. Therefore, a high reflectivity for p-polarized light is observed.
Fig. 4. (a)/(b) the effective wave impedance of the vacuum and the anisotropic slab when the incident angle of the s- (p-) polarized light beam is 45°.
To verify the numerical results as well as the above analysis for the physical mechanism of the PBS, we derived the diagrams for electrical/magnetic field distribution under different incident angles and frequencies of the incident light beam using COMSOL. In our simulation, the parameters used are based on the effective relative permittivity equation in [44]. As shown in Figs. 5(a) and 5(b), the s-polarized light exhibits nearly 100% transmittance, while the p-polarized light exhibits almost 100% reflectance at 80.4 THz for the incident angles of 20°, 40°, and 60°. It can be seen that E1≈E2 (E1 is the electric field in vacuum and the E2 is the electric field in the slab) when s-polarized light is incident, so Z2≈Z1. However, when p-polarized light is incident, the magnetic field in the slab is almost zero, therefore it in vacuum is much larger than that in the slab, which leads Z2>>Z1. Thus, the field distributions prove that the proposed PBS is based on variable impedance at the interface with respect to polarization, which is attributed to extreme anisotropy. Figure 5(c) clearly indicates that the s-polarized light beam can almost pass unobstructed through our designed device at 75.0 THz, 83.0 THz, and 85.0 THz. Figure 5(d) demonstrates that, at 75.0 THz, only a weak part of the p-polarized light can cross the device and most of it is reflected. At 83.0 THz, the p-polarized light does not propagate to the right side of the device and total reflection occurs, and at 85.0 THz, the p-polarized light does not exhibit total reflection. These results indicate that the field distribution is in excellent agreement with the transmittance and reflectance curves obtained through numerical calculations, as shown in Fig. 3(a). These results also provide a useful platform for developing wide-angle and wide-frequency-band polarization-based imaging systems.
Fig. 5. (a) Normalized electrical field distribution of PBS (b) Normalized magnetic field distribution of PBS, in the (a) and (b), the incident angles are 20°,40°, and 60° when the frequency of the p-polarized light beam is 80.4 THz. (c) Normalized electrical field distribution of PBS (d) Normalized magnetic field distribution of PBS, in the (c) and (d), the operation frequencies are 75.0 THz, 83.0 THz, and 85.0 THz when the incident angle of the p-polarized light beam is 45°.
To further study the characteristics of the designed PBS, we calculated the reflectivity and transmissivity for PBS in the entire frequency band for the incident angle in the range of [−90°, 90°]. Figures 6(a) and 6(b) show the reflectivity and transmissivity when the incident light is s-polarized. When the s-polarized light beam is incident, reflectivity is less than 1% in the region between the dashed lines shown in Fig. 6(a), and the transmissivity is more than 90% in the area surrounded by the dash-dotted lines and more than 90% in the area surrounded by the dashed lines, as shown in Fig. 6(b). Meanwhile, the PERs in this region are in the range of [25.48 dB∼37.36 dB]. It is evident that the transmissivity of s-polarized light beam is high in the entire frequency region, and there is almost no reflection when the incident angle is between −65° and 65°. This is because the impedance on both sides of the incident interface is well-matched. Further, although the imaginary part of the refractive index is not zero (less than 0.04), it causes a definite absorption, but it has a negligible impact on transmittance. Correspondingly, Figs. 6(c) and 6(d) show the reflectivity and transmissivity when the incident light is p-polarized. When the p-polarized light beam is incident, the response of the proposed device is obviously the opposite. Here, the reflectivity is more than 80% in the region surrounded by the dashed lines and more than 90% in the area surrounded by the dash-dotted lines, as shown in Fig. 6(c), while the transmissivity in the region surrounded by dashed lines is less than 1%, as shown in the Fig. 6(d). As discussed above, this is because the difference of the impedance between the vacuum and the slab is very big. For incident angle between −50° and 50°, reflectivity (more than 80%) is acceptable, and the transmissivity is almost negligible in the frequency region [80.4 THz, 85 THz]. Meanwhile, the PERs in this region are up to [20.44 dB∼39.94 dB]. Comparing Figs. 6(a)–6(b) and Figs. 6(c)–6(d), we can infer that the p-polarized light beam cannot penetrate through the device, while almost the entire s-polarized light beam propagates through the device in the region I, as shown in the Figs. 6(c)–6(d). Thus, a broadband and high-PER PBS with wide range of incident angle can be realized.
Fig. 6. (a)/(b) Reflectivity and transmissivity when the incident light is s-polarized. (c)/(d) Reflectivity and transmissivity when the incident light is p-polarized.
We now investigate the effect of the width of BP ribbon on the performance of PBS. All the parameters of the polarizer except the width of the BP ribbon are considered to be the same as that in Fig. 3(a). Here, we set W as 34, 35, and 36 nm. As shown in Figs. 7(a) and 7(c), the variation of W has a negligible impact on the reflectivity and transmissivity when s-polarized light beam is incident at any frequency and any incident angle. This is because the variation of W (from 34 nm to 36 nm) has a little impact on the conductivity along the ZZ direction, as indicated in Eq. (5). Thus in this case, impedance matching conditions for the interface can still be satisfied for s-polarization. As shown in Fig. 7(b), the reflectivity and transmissivity spectra exhibit a redshift, which is due to the redshift of conductivity along the ZZ direction when the p-polarized light beam is incident at 45°. This is because when the width of BP ribbon is varied, the effective capacitance also varies, as indicated in Eq. (4). Further, it strongly affects the conductivity along the AC direction of the metasurface, as indicated in Eq. (6). It should be noted that the width of bandgap for Tpp is almost unchanged. In the frequency region of the bandgap, the value of Rpp is negligibly changed with the increase in W. As shown in the Fig. 7(d), at the operation frequency of 80.4 THz, the difference between Rpp and Tpp increases as width increases in the incident angle range of [−50°, 50°]. This is because at this operating frequency, as the width of BP ribbon increases, the spectral lines are red-shifted, causing the metallic characteristics of the structure to be substantially powerful, which can lead to a strong reflection effect. Therefore, by changing the surface structure parameters of BP ribbon, it is possible to obtain a broadband and efficient PBS in different operation frequency bands.
Fig. 7. (a)/(b) Reflectivity and transmissivity of PBS with the width of the BP ribbon set as 34 nm, 35 nm, and 36 nm for s-(p-) polarization, where the incident angle is 45°. (c)/(d) Reflectivity and transmissivity of PBS with the width of the BP ribbon set as 34 nm, 35 nm, and 36 nm for s-(p-) polarization, when the operation wavelength is 80.4 THz.
Finally, we examine the dynamic tunability of the proposed PBS with varying electron concentration when the incident angle is set as 45°. This result is shown in Fig. 8, which is similar to that obtained by varying the width of BP ribbon. The reflectivity and transmissivity for s-polarization are almost unchanged, as shown in Figs. 8(a) and 8(c), which may be attributed to the fact that the interface impendence is hardly affected with varying electron concentration when s-polarized light beam is incident. When p-polarized light beam is incident, the reflectivity and transmissivity spectra exhibit a blueshift when the electron concentration is increased from 4×1013 cm−2 to 6×1013 cm−2. The initial position of the bandgap of Tpp moves from 71.0 THz to 87.0 THz. Further, the corresponding reflectance also shifts, and simultaneously, the state of reflectance is basically unchanged. It is well known that the electron concentration can be tuned by chemical doping and by varying the Fermi level, temperature, and strain [38,39,46]. Therefore, the operation frequency of the PBS can be dynamically tuned. As shown in Fig. 8(d), the designed PBS is operated at 80.4 THz. When the electron concentration is increased, the reflection decreases, while the transmission increases in the region [0°, 50°]. Because the parameters of PBS are optimal at ns = 5×1013 cm−2, the perfect impedance matching is broken by the variation of electron concentration. Therefore, the transmissivity is no longer less than 1% as shown in the curves of Fig. 8(d) that are marked by ‘*’ and ‘Δ’. It should be noted that under optimal parameters, the effect of decreasing electron concentration on the transmittance is weaker than that of increasing electron concentration, while the effect of decreasing electron concentration on the reflectivity is greater than that of increasing electron concentration. The difference between reflectivity and transmissivity is always greater than 65%. Therefore, in the case of a fixed incident angle, a wider operating frequency can be obtained by adjusting the electron concentration, and the effect of electron concentration on the behavior of PBS at different incident angles facilitates the design of an efficient PBS.
Fig. 8. (a)/(b) Reflectivity and transmissivity of PBS with the electron concentration ns set as 4×1013 cm−2, 5×1013 cm−2, and 6×1013 cm−2 for s-(p-) polarization when the incident angle is 45°. (c)/(d) Reflectivity and transmissivity of PBS with the electron concentration ns set as 4×1013 cm−2, 5×1013 cm−2, and 6×1013 cm−2 for s-(p-) polarization when the operation wavelength is 80.4 THz.
As is well known, black phosphorus is unstable in the air. An additional thin SiO2 layer could be deposited on top layer to protect the BP from oxidation, and the numerical results show that it does not obviously affect the performance of the designed BPS if the thickness is in order of 5-30 nm.
4. Conclusion
We proposed a highly efficient PBS based on extremely anisotropic BP ribbons around mid-infrared frequency with an ultra-thin structure. Using numerical calculation and simulations, we obtained the reflectivity and transmissivity of the PBS, and the results show that the splitting of the polarized light beam is due to the extreme anisotropy of the metasurface formed by BP ribbon. Besides, we explained the physical mechanism of the BP-based PBS by analyzing the field distributions for s-(p-) polarized light beam. It was observed that for the p-polarized light beam, there is a huge difference between the impedance of the vacuum and the device, but for the s-polarized light beam, the interface impedance is well-matched. Therefore, the polarization sensitivity of impedance based on extreme anisotropy is the key to realize PBS. Furthermore, we investigated the reflectivity and transmissivity of PBS from 70.0 THz to 90.0 THz when s-(p-) polarized light beam was incident at an angle in the range of [−90°,90°]. Under optimized parameters, the operation frequency band could reach 4.6 THz, the incident angle could reach 50°, and the power efficiency always remained above 80%. At the same time, the polarization extinction ratios can higher than 25.50 dB for s-polarization light and 20.40 dB for p-polarization light, respectively. Finally, we examined the effect of the width of the BP ribbon on the performance of PBS as well as the effect of electron concentration on its dynamic tunability. Overall, our study is potentially useful for the development of tunable, broad-frequency band, wide-angle, and highly efficient PBS based on BP, which has wide applications in filters, remote detectors, and imaging in the mid-infrared region.
Funding
National Natural Science Foundation of China (11904169, 61675095); Natural Science Foundation of Jiangsu Province (BK20190383); Graduate Research and Innovation Projects of Jiangsu Province (KYLX15_0317).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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