Broadband and Incident-Angle-Modulation Near-Infrared Polarizers Based on Optically Anisotropic SnSe

Abstract

Optical anisotropy offers an extra degree of freedom to dynamically and reversibly regulate polarizing optical components, such as polarizers, without extra energy consumption and with high modulating efficiency. In this paper, we theoretically and numerically design broadband and incident-angle-modulation near-infrared polarizers, based on the SnSe, whose optical anisotropy is quantitatively evaluated by the complete dielectric tensor, complex refractive index tensor, and derived birefringence (~|Δn|_max = 0.4) and dichroism (~|Δk|_max = 0.4). The bandwidth of a broadband polarizer is 324 nm, from 1262 nm to 1586 nm, with an average extinction ratio above 23 dB. For the incident-angle-modulation near-infrared polarizer, the high incident angles dynamically and reversibly modulate its working wavelength with a maximum extinction ratio of 71 dB. Numerical simulations and theoretical calculations reveal that the considerable absorption for p light and continuously and relatively low absorption of s light lead to the broadband polarizer, while the incident-angle-modulation one mainly arises from the blue shift of corresponding wavelength of p light’s minimum reflectance. The proposed novel design of polarizers based on SnSe are likely to be mass-produced and integrated into an on-chip system, which opens up a new thought to design polarizing optical components by utilizing other low-symmetry materials.

1. Introduction

Owing to ultralow lattice thermal conductivity, tin (II) selenide, i.e., SnSe, crystals possess high thermoelectric figure of merit [1], which surpass many other thermoelectric materials [2,3,4]. Besides, SnSe is also a competitive candidate for solar cells [5,6], due to its suitable bandgap (about 0.86 eV for both theoretical [7,8] and experimental [1] studies), high absorption coefficient (at the level of 10⁵ cm⁻¹ [9]), and relative abundance of composed elements [10]. More attractively, SnSe exhibits giant optical anisotropy, originating from its low-symmetry lattice structure analogous to the puckered structure of black phosphorus (BP) [11], and its symmetry of space group (${\mathrm{D}}_{2h}^{16}$ − Pcnm [12]) is even lower than that of BP (${\mathrm{D}}_{2h}^{18}$ − Cmca [13]). As reported, SnSe displays giant anisotropic absorption measured by angle-resolved polarized optical absorption spectroscopy, and the anisotropic ratio of absorption reaches up to 1.2 [8]. Optical anisotropy not only brings much richer physics such as birefringence [14,15] and dichroism [16,17], but also offers an extra degree of freedom to modulate its optical properties for polarization-related devices. However, most research on SnSe has been focused on the qualitative identification of optically anisotropic phenomena, and there remains a research gap and challenge to accurately acquire the complete dielectric tensor and the complex refractive index tensor to quantitatively evaluate the optical anisotropy of SnSe and then to rationally guide design of polarization-tunable optical components.

As optical components separating the polarization state, polarizers have been widely applied to display and imaging [18,19], optical interconnections and communications [20,21], and optical measurement [22,23]. In general, there are three main methods to produce polarizers. The first is to utilize some absorbing molecules (such as Iodine molecules [24] or liquid crystal molecules [18]), absorbing light along the preferred orientation, to realize the separation of the polarization state; the polarizers made by these absorbing molecules are frequently adopted to the field of display and imaging. However, on account of the instability chemicals, such polarizers may not be capable of working in the long term [25]. In addition, these polarizers are also subject to the problem of non-recyclability and environmental pollution [25]. Considering the demand for high levels of integration in optical interconnections and communications, the polarizers based on optical fibers [26,27] or optical gratings [28,29] are often highly integrated or directly integrated to platform of silicon on insulator (SOI) or lithium niobate on insulator (LNOI). However, such integrated polarizers based on optical fibers or optical gratings need to be elaborately designed, leading to expensive cost of design and fabrication. Although polarizers based on the simple-structure multilayer structure [30,31] is relatively low-cost, these polarizers seem difficult to become broadband, due to the limit of destructive interference. The last kind of polarizers are based on birefringent materials [32,33], with their applications in optical measurement, which separate the polarization state depending on the birefringence of materials or destructive interference in the multilayer structure. These simple-structure and low-cost polarizers generally do not require complex design. However, the polarizers based on birefringent materials are often fabricated with bulk single crystal, and there still remains a difficult and long way to implement miniaturization and integration of these polarizers as a consequence. Furthermore, once all the three kinds of polarizers have been produced, it is scarcely possible to dynamically and reversibly modulate their performance. Although polarizers made by liquid crystal can be tuned by external voltage to dynamically and reversibly regulate the polarization state [34,35], there still remain the tough problems of extra energy consumption and low modulating efficiency. Therefore, it is challenging to design low-cost and broadband polarizers with long-term stability and to efficiently and dynamically modulate their performance without extra energy consumption.

Herein, we quantitatively evaluate the optical anisotropy of SnSe by the Mueller matrix spectroscopic ellipsometry, and then rationally design the broadband and incident-angle-modulation near-infrared polarizers based on SnSe-SiO₂-Si multilayer structure. Confirmed by the polarization-resolved reflectance spectra, the optical anisotropy of SnSe is then quantitatively characterized by the Mueller matrix spectroscopic ellipsometry to accurately acquire its complete dielectric tensor, complex refractive index tensor and derived birefringence and dichroism. The SnSe-SiO₂-Si multilayer structure is constructed with its blue shift of corresponding wavelength of p light’s minimum reflectance and high reflectance of s light, suggesting significant potential to design high-performance and dynamically and reversibly modulated polarizers. By optimizing the thickness of SnSe and SiO₂, the broadband and incident-angle-modulation near-infrared polarizers based on the SnSe-SiO₂-Si multilayer structure are afterwards proposed by theoretical calculations and numerical simulations. Finally, the contour maps of electric field intensity and absorbed power of p and s light are numerically simulated to explore the mechanism for these broadband and incident-angle-modulation near-infrared polarizers.

2. Materials and Methods

2.1. Absorption Coefficient and Penetration Depth

The absorption coefficient α_m (m = b or c) along b- or c-axis of SnSe is defined as [22]:

$${\alpha }_m=\frac{4\pi k_m}{\lambda }, m=b \mathrm{or} c,$$

(1)

where k_m (m = b or c) is the extinction coefficient along b- or c-axis of SnSe and λ is the wavelength with the unit of cm.

The penetration depth d_p,m (m = b or c) along b- or c-axis of SnSe is the reciprocal of α_m [22], i.e.,

$$d_{\mathrm{p}, m}=\frac{1}{{\alpha }_m}, m=b \mathrm{or} c.$$

(2)

According to Beer’s law [22], the light intensity I after penetrating the sample, compared with the initial incident light intensity I₀, is empirically calculated by:

$$\frac{I}{I_0}=\exp (-\alpha d),$$

(3)

where d is the thickness of the sample. Therefore, the penetration depth d_p represents the depth that the light penetrates with its intensity $I_{d_p}$, reducing to about 37% of I₀, combining the Equations (2) and (3):

$$\frac{I_{d_p}}{I_0}=\exp (-\alpha d_{\mathrm{p}})=1/e\cong 37\%.$$

(4)

2.2. Net Phase Shift of s Light

The net phase shift of s light ${\varphi }_{\mathrm{net}}^{\mathrm{s}}$ involves the propagation phase shift in the SnSe layer and reflective phases of SnSe’s upper and lower interfaces (${\varphi }_{\mathrm{upper}}^{\mathrm{s}}$ and ${\varphi }_{\mathrm{lower}}^{\mathrm{s}}$), calculated by [36]:

$${\varphi }_{\mathrm{net}}^{\mathrm{s}}=2{\varphi }_{\mathrm{prop}}^{\mathrm{s}}+{\varphi }_{\mathrm{lower}}^{\mathrm{s}}-{\varphi }_{\mathrm{upper}}^{\mathrm{s}}=2\times \frac{2\pi d_{\mathrm{SnSe}}{n}_y\cos {\theta }_{\mathrm{air}-\mathrm{SnSe}}}{\lambda }+{\varphi }_{{\mathrm{SnSe}-\mathrm{SiO}}_2-\mathrm{Si}}^{\mathrm{s}}-{\varphi }_{\mathrm{SnSe}-\mathrm{air}}^{\mathrm{s}}$$

(5)

In Equation (5), the first term ${\varphi }_{\mathrm{prop}}^{\mathrm{s}}$ is the propagation phase shift of s light in the SnSe layer, and the θ_air-SnSe is the refractive angle at the interface between air and SnSe, which can be converted from Snell’s law [22]:

$$N_{\mathrm{air}}\sin \theta =N_c\sin {\theta }_{\mathrm{air}-\mathrm{SnSe}},$$

(6)

where N_air (= 1) is the complex refractive index of air and N_c is the SnSe c-axis’s complex refractive index.

The second term ${\varphi }_{\mathrm{lower}}^{\mathrm{s}}$ is the reflective phase for s light from the SnSe layer to the SiO₂ layer, and then to Si substrate, which can be acquired by calculating the argument of amplitude reflection coefficient of s light ${\varphi }_{{\mathrm{SnSe}-\mathrm{SiO}}_2-\mathrm{Si}}^{\mathrm{s}}$ for the corresponding SnSe-SiO₂-Si multilayers, utilizing the transfer matrix method [37], i.e.,

$$r_{{\mathrm{SnSe}-\mathrm{SiO}}_2-\mathrm{Si}}^{\mathrm{s}}=\left|r_{{\mathrm{SnSe}-\mathrm{SiO}}_2-\mathrm{Si}}^{\mathrm{s}}\right|{\varphi }_{{\mathrm{SnSe}-\mathrm{SiO}}_2-\mathrm{Si}}^{\mathrm{s}}.$$

(7)

In Equation (7), $r_{{\mathrm{SnSe}-\mathrm{SiO}}_2-\mathrm{Si}}^{\mathrm{s}}$ is the amplitude reflection coefficient of reflected s light for SnSe-SiO₂-Si multilayers.

The third term ${\varphi }_{\mathrm{upper}}^{\mathrm{s}}$ is the reflective phase for s light from SnSe layer to air and equals to the argument of the reflected s light’s amplitude reflection coefficient from SnSe layer to air ${\varphi }_{\mathrm{SnSe}-\mathrm{air}}^{\mathrm{s}}$, i.e.,

$$r_{\mathrm{SnSe}-\mathrm{air}}^{\mathrm{s}}=\left|r_{\mathrm{SnSe}-\mathrm{air}}^{\mathrm{s}}\right|{\varphi }_{\mathrm{SnSe}-\mathrm{air}}^{\mathrm{s}}.$$

(8)

According to Fresnel equations [22], the reflected s light’s amplitude reflection coefficient from SnSe layer to air $r_{\mathrm{SnSe}-\mathrm{air}}^{\mathrm{s}}$ can be expressed as:

$$r_{\mathrm{SnSe}-\mathrm{air}}^{\mathrm{s}}=\frac{N_y\cos {\theta }_{\mathrm{air}-\mathrm{SnSe}}-N_{\mathrm{air}}\cos \theta }{N_y\cos {\theta }_{\mathrm{air}-\mathrm{SnSe}}+N_{\mathrm{air}}\cos \theta }.$$

(9)

2.3. Theoretical CSalculations by the 4 × 4 Matrix Method

The 4 × 4 transfer matrix T can be expressed in matrix form [22,38]:

$$T=\left[\begin{array}{cccc}{T}_{11}& T_{12}& T_{13}& T_{14}\\ T_{21}& T_{22}& T_{23}& T_{24}\\ T_{31}& T_{32}& T_{33}& T_{34}\\ T_{41}& T_{42}& T_{43}& T_{44}\end{array}\right].$$

(10)

The reflectance of s and p light R_s and R_p of the SnSe-SiO₂-Si multilayers can be thus calculated with the matrix elements of T [39]:

$$\begin{array}{l}{R}_{\mathrm{s}}=r_{\mathrm{ss}}^2+r_{\mathrm{sp}}^2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={(\frac{T_{21}{T}_{33}-T_{23}{T}_{31}}{T_{11}{T}_{33}-T_{13}{T}_{31}})}^2+{(\frac{T_{11}{T}_{23}-T_{13}{T}_{21}}{T_{11}{T}_{33}-T_{13}{T}_{31}})}^2\end{array}$$

(11)

$$\begin{array}{l}{R}_{\mathrm{p}}=r_{\mathrm{ps}}^2+r_{\mathrm{pp}}^2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={(\frac{T_{33}{T}_{41}-T_{31}{T}_{43}}{T_{11}{T}_{33}-T_{13}{T}_{31}})}^2+{(\frac{T_{11}{T}_{43}-T_{13}{T}_{41}}{T_{11}{T}_{33}-T_{13}{T}_{31}})}^2\end{array}$$

(12)

In Equations (11) and (12), r_ss represents the amplitude reflection coefficient of reflected s light, while r_sp is that of reflected p light induced by incident s light. Similarly, r_ps and r_pp respectively show the amplitude reflection coefficient of reflected s and p light with the incidence of p light.

2.4. Numerical Simulations Using the Finite-Difference Time-Domain (FDTD) Method

The FDTD is a power way to simulate the reflectance and distribution of the electric field and absorbed power of a multilayer structure [40,41]. A very high mesh accuracy of 8 has been adopted to well match the theoretical calculation results. The incident plane wave is along the negative direction of the z-axis that is parallel to the vibration direction of the p light’s vertical component. Owing to the oblique incidence, the Bloch boundary conditions have been employed along x and y directions, which are respectively parallel to the vibration direction of the p light’s horizontal component and s light. The perfectly matched layer (PML) boundary condition has been utilized along the z direction.

3. Results and Discussion

3.1. Optical Anisotropy of SnSe

Figure 1a illustrates the low-symmetry orthorhombic lattice structure of layered SnSe. Analogous to BP’s structure, the van der Waals (vdW) force connects each layer along the a-axis of SnSe [7], and the Sn and Se atoms alternately arrange and covalently constitute the zigzag and armchair structure along SnSe’s b- and c-axis, respectively [7,10]. This low-symmetry structure leads to the optical anisotropy of SnSe, identified by polarization-resolved reflectance spectra in Figure 1b. With a step of 30°, the reflectance spectra of SnSe vary distinctly with the polarization angle, suggesting the giant optical anisotropy. To clarify the relation between the reflectance and the polarization angle, the polar coordinate curve at peak wavelength of 520 nm is demonstrated in the insert of Figure 1b. The ratio of the maximum to the minimum is 1.26, indicating the SnSe’s giant optical anisotropy once again.

To quantitatively evaluate the optical anisotropy of SnSe, the Mueller matrix spectroscopic ellipsometry has been adopted to accurately acquire the complete dielectric tensor and the complex refractive index tensor of SnSe, demonstrated by Figure 1c,d, respectively. Details about ellipsometric analysis can be found in Appendix A. Since SnSe’s low-symmetry structure belongs to the orthorhombic crystal system, its dielectric tensor and complex refractive index tensor are diagonal and can be mutually converted by Equation (13) [38]:

$$\left[\begin{array}{ccc}{\epsilon }_a& 0& 0\\ 0& {\epsilon }_b& 0\\ 0& 0& {\epsilon }_c\end{array}\right]=\left[\begin{array}{ccc}{N}_a^2& 0& 0\\ 0& N_b^2& 0\\ 0& 0& N_c^2\end{array}\right]$$

(13)

where the subscript a, b, and c denote the dielectric functions ε and the complex refractive indices N along the a-, b-, and c-axis of SnSe, respectively. Furthermore, the dielectric function ε is composed of real part ε_r and imaginary part ε_i, i.e., ε = ε_r − iε_i, while the refractive index n and the extinction coefficient k constitute the complex refractive index N (= n − ik). Therefore, the Equation (13) can be further expressed as:

$$\left[\begin{array}{ccc}{\epsilon }_{\mathrm{r}, a}-\mathrm{i}{\epsilon }_{\mathrm{i}, a}& 0& 0\\ 0& {\epsilon }_{\mathrm{r}, b}-\mathrm{i}{\epsilon }_{\mathrm{i}, b}& 0\\ 0& 0& {\epsilon }_{\mathrm{r}, c}-\mathrm{i}{\epsilon }_{\mathrm{i}, c}\end{array}\right]=\left[\begin{array}{ccc}{(n_a-\mathrm{i}{k}_a)}^2& 0& 0\\ 0& {(n_b-\mathrm{i}{k}_b)}^2& 0\\ 0& 0& {(n_c-\mathrm{i}{k}_c)}^2\end{array}\right]$$

(14)

Since the vdW force between the layers is obviously different from the covalent interactions in the layers, a Cauchy model has been adopted to describe the dielectric function along the a-axis of SnSe [42], instead of the Tauc–Lorentz oscillators representing those along the b- and c-axis of SnSe [43] (See Appendix A for more details). Therefore, the dielectric function along the a-axis ε_a and the complex refractive index N_a show total differences from those along the b- and c-axis of SnSe in Figure 1c,d, respectively. Besides, the different arrangement of Sn and Se atoms along the b- and c-axis of SnSe possibly makes the corresponding dielectric functions and complex refractive indices discrepant in the peak positions and intensities, although they share almost the same peak shape. In brief, both dielectric functions and complex refractive indices show many differences along the direction of SnSe’s crystal axes, quantitatively demonstrating the giant optical anisotropy of SnSe. Shown in Figure 1e, we also calculate the birefringence Δn (= n_b − n_c) and the dichroism Δk (= k_b − k_c) to further quantitatively evaluate the optical anisotropy of SnSe. Both the birefringence Δn and the dichroism Δk display giant absolute value with their maximum about 0.4; the maximum absolute value of birefringence outperforms many other low-symmetry materials, such as BP (Δn_BP = 0.15 [44]), ReS₂ (Δn_ReS2 = 0.06 [45]), and α-MoO₃ (Δn_α-MoO3 = 0.11 [46]).

Moreover, the extinction coefficient k along the b- and c-axis of SnSe, i.e., k_b and k_c, descend gradually to zero above the wavelength of 1200 nm (marked area in Figure 1d), and this feature of weak absorption can be also observed from the absorption coefficient α and the penetration depth d_p along the b- and c-axis of SnSe in Figure 1f. Both α_b and α_c keep a high level of 10⁵ cm^±1 in visible light region, analogous to the reported result [9], while they drop sharply to zero in the near-infrared region. The penetration depth d_p,b and d_p,c are also calculated to highlight the weak absorption feature, since they rapidly ascent in the near-infrared region. Besides, both the absorption coefficient α and the penetration depth d_p along the b- and c-axis of SnSe remain anisotropic, due to the anisotropic extinction coefficient k along the b- and c-axis of SnSe.

3.2. Design of Broadband and Incident-Angle-Modulation Near-Infrared Polarizers

The weak absorption feature of materials, in most cases, gives rise to the destructive interference in their layer structure [47,48]. Hence, the polarization state can be separated by delaying the destructive interference of p or s light, utilizing optically anisotropic materials, such as BP [30,31]. Inspired by this idea, we constructed the SnSe-SiO₂-Si multilayer structure (Figure 2a) as a reflective polarizer at incident angle θ. As a lossless dielectric material, SiO₂ adds an extra degree of freedom to modulate the performance of the polarizer by altering its thickness (d_SiO2), other than that of SnSe (d_SnSe). Additionally, the silicon (Si) substrate is selected due to its compatibility with integrated optoelectronics and on-chip system [28]. For convenience, the vibrational direction of s light and the horizontal component of p light keep parallel to the c- and b-axis of SnSe, respectively, all the time. The complex refractive index tensor adopted in the next theoretical calculations and numerical simulations is obtained from the SnSe single crystal sample, since we believe that several-hundred-nanometer-thickness SnSe’s properties may be more likely to that of bulk SnSe, which is proved by the thickness-dependent bandgap in Figure A1 of Appendix B.

As shown in Figure 2b, the reflectance of s light retains a high level (almost above 0.6) in the whole concerned wavelength range at high incident angle of 70°, 75°, and 80°. For p light, however, the reflectance is always below about 0.1, indicating that p light is probably able to be separated. More excitingly, the corresponding wavelength of minimum reflectance for p light is blue shift with increase of the incident angle, which offers significant potential to realize dynamical and reversible modulation.

Figure 2c illustrates the contour map of reflectance of p light varied with the incident angle to explain the blue shift phenomenon. Under a low incident angle (below about 20°), the horizontal component of p light, parallel to the b-axis of SnSe, plays a dominant role. Due to the fact that high refractive index of materials make their reflectance independent of the incident angle [49,50], the reflectance of p light remains unchanged within a low incident angle, due to the high refractive index along SnSe’s b-axis n_b (≈3) in the near-infrared region. With the increase of higher incident angle, blue shift of corresponding wavelength of p light’s minimum reflectance comes up as a possible result of the refractive index along the a-axis n_a, taking a more and more critical part in affecting the p light’s reflectance.

We also calculated the contour map of reflectance of s light, with the incident angle from 0° to 80° in Figure 2d, marking out the net phase shift of 2π. The net phase shift of 2π is the condition of destructive interference for the multilayer structure [36], and the corresponding wavelength of net phase shift of 2π perfectly matches with corresponding wavelength of minimum reflectance wavelength in Figure 2d. Therefore, for s light, destructive interference takes place in this SnSe-SiO₂-Si multilayer structure. The high reflectance of s light in Figure 2b is probably a result of constructive interference. When the incident angle is below 60°, the s light’s reflectance is independent of incident angle, the same as p light’s circumstance under low incident angle, owing to the high refractive index along the c-axis [49,50]. However, reflectance of s light increases as the incident angle is beyond 60°. Figure 2e,f, respectively, demonstrate the contour maps of s light’s electric field intensity I_s and absorbed power P_abs,s, varied with the depth and the incident angle at the representative constructive interference wavelength (1164 nm) to interpret the increase of s light’s reflectance. In Figure 2e, the electric field intensity I_s of SiO₂ and SnSe maintain a consistently high level at low incident angle, while they decrease with the high incident angle. As to the absorbed power P_abs,s, the absorption takes place almost totally in the SnSe layer, since SiO₂ is a lossless dielectric material. Furthermore, the absorbed power P_abs,s in the SnSe layer shares the same trend as the electric field intensity I_s, since P_abs,s is proportional to I_s in Equation (15) [51,52]:

$$P_{\mathrm{abs},\mathrm{s}}=\frac{1}{2}\omega {\epsilon }_0{I}_{\mathrm{s}}{k}_c$$

(15)

where ω is the angular frequency, expressed as ω = 2πc/λ (c: light speed in vacuum), and ε₀ is the dielectric constant in vacuum. k_c is the extinction coefficient along SnSe’s c-axis at the wavelength of 1164 nm, which can be regarded as a constant. Therefore, the absorption of the SnSe layer reduces with the increase of high incident angle, which leads to the high reflectance of s light.

By optimizing the thickness of SnSe and SiO₂ (details in Appendix C), the broadband and incident-angle-modulation near-infrared polarizers based on optically anisotropic SnSe are acquired in Figure 3. For the broadband near-infrared polarizer based on SnSe (d_SnSe = 400 nm)-SiO₂ (d_SiO2 = 675 nm)-Si multilayer structure, its reflectance of s light is above 0.6 at the incident angle of 75° within the whole concerned wavelength region in Figure 3a, which corresponds to the insertion loss IL below 2.1 dB. Herein, the IL of s light is defined as:

$$IL=10{\log }_{10}\frac{1}{R_{\mathrm{s}}}$$

(16)

More importantly, the p light’s reflectance R_p approaches zero in the almost concerned wavelength region in Figure 3a, with bandwidth of 324 nm for R_p < 0.003 (also shown in

Table 1. Comparation of performances of near-infrared polarizers.

Polarizers	Extinction Ratio ER [dB]	Insertion Loss IL [dB]	Working Wavelength (Range) [nm]	References
Graphene fiber	~19 ± 2.5	~5	1530−1630 (100)	[26]
Graphene microfiber	31	-	1550	[53]
Subwavelength grating on SOI platform	>35	<0.6	1260−1390 (130) & 1520−1600 (80)	[28]
Subwavelength gratings on LNOI platform	>30	<3.1	1550	[29]
Hybrid plasmonic grating on LNOI platform	20	<2.3	1470−1700 (230)	[54]
Long-period grating on LNOI waveguide	20	<2	1430−1700 (270)	[55]
Broadband polarizer based on optically anisotropic SnSe	>23 (Maximum: 62)	<2.1 (Average: 1.7)	1262−1586 (324)	This work
Polarizer dynamically and reversibly modulated by high incident angle	21 & 71 & 35	1.7 & 1.1 & 1.1	1184 & 1100 & 955	This work

). Therefore, the corresponding extinction ratio ER is over 23 dB from the wavelength of 1262 nm to 1586 nm with the maximum extinction ratio ER_max of 62 dB in Figure 3b and Table 1. In this paper, the mentioned ER can be defined as:

$$ER=10{\log }_{10}\frac{R_{\mathrm{s}}}{R_{\mathrm{p}}}$$

(17)

Compared to other near-infrared polarizers in Table 1, the proposed broadband polarizer possesses competitive ER and IL, as well as the longest bandwidth. Moreover, the numerical simulations on the reflectance of p and s perfectly match the theoretical results calculated by the 4 × 4 matrix method [22,38], which confirms the validity and reliability of this broadband near-infrared polarizer.

With regard to the incident-angle-modulation near-infrared polarizer based on SnSe-SiO₂-Si multilayer structure, the optimized thickness of SnSe and SiO₂ are, respectively, 315 nm and the same as that of the broadband near-infrared polarizer (d_SiO2 = 675 nm). It can be found in Figure 3c that the s light’s reflectance R_s keeps at a high level within all incident angles. For p light, the corresponding wavelengths of p light’s minimum reflectance are blue shift with the increase of the incident angle, which means that the incident angle is capable to dynamically and reversibly tune the working wavelength. The numerical simulations confirm this incident-angle-modulation polarizer as well in Figure 3c. In Figure 3d and Table 1, the corresponding wavelengths of p light’s minimum reflectance are, respectively, 1184, 1100, and 955 nm at the incident angle of 70°, 75°, and 80°, corresponding to the ER_max of 21, 71, and 35 dB and the IL of 1.7, 1.1, and 1.1 dB. Such performances of incident-angle-modulation polarizer are superior to most of the high-performance near-infrared polarizers in Table 1.

3.3. Mechanism for Broadband and Incident-Angle-Modulation Near-Infrared Polarizers

As illustrated in Figure 4, we have numerically simulated the contour maps of electric field intensity and absorbed power of p and s light (I_p and I_s; P_abs,p and P_abs,s) to explore the mechanism for the broadband near-infrared polarizer at first. The electric field intensity in the SnSe layer dominates the I_p in Figure 4a, while SiO₂’s electric field intensity is dominated in I_s of Figure 4b. However, SiO₂ is a lossless dielectric material, as previously mentioned. As a consequence, there is no P_abs,s in SiO₂ layer of Figure 4d. Furthermore, the I_p in the SnSe layer maintains a relatively high level within the whole concerned wavelength region in Figure 4a. Eventually, the P_abs,p in the SnSe layer nearly keeps a high level of 1 × 10²⁰ W/m³ through the entire concerned region in Figure 4c, contributing to the R_p approaching zero over broadband in Figure 3a. For s light, however, SnSe’s I_s in Figure 4b is nearly zero, and thus the P_abs,s of SnSe layer is approximately one order of magnitude lower than P_abs,p in the SnSe layer within the whole wavelength range. Consequentially, the s light’s reflectance R_s stays at a high level for the entire concerned range in Figure 3a, generating Figure 3b’s high-performance broadband near-infrared polarizer with comparatively low IL and high ER.

To investigate the mechanism for incident-angle-modulation near-infrared polarizer, the contour maps of electric field intensity and absorbed power of s and p light (I_s and I_p; P_abs,s and P_abs,p) with high incident angle from 70° to 80° have been numerically simulated in Figure 5. Despite the fact that I_s in longer wavelength is much higher than that in shorter wavelength in Figure 5a,e,i, P_abs,s displays nearly homogeneous distribution through the entire wavelength range. According to Equation (15), the k_c in Figure 1d shows a declining trend during the near-infrared region, which leads to homogeneously distributed P_abs,s in almost concerned wavelength region. In addition, the average SnSe’s P_abs,s at specific incident angle (70°/75°/80°) in Figure 5b,f,j is about 2−4 times smaller than corresponding P_abs,p of Figure 5d,h,l in

Table 2. The average SnSe’s I_p and P_abs,p, with their ratio to I_s and P_abs,s, respectively, at high incident angle and ER_max wavelength.

Incident Angle [°]	ERmax Wavelength [nm]	Ip [a. u.]	Ip/Is	Pabs,p/1020 [W/m3]	Pabs,p/Pabs,s
70	1184	0.45	15	2.29	2.75
75	1100	0.32	32	2.57	4.02
80	955	0.18	36	2.83	4.22

. According to the analysis above, R_s in Figure 3c is continuously and relatively high through the entire concerned wavelength range.

As regards p light, the average I_p in the SnSe layer decreases with increase of high incident angle from 70° to 80° at the ER_max wavelength in Figure 5c,g,k and Table 2. However, there seems to be no consistent trend in P_abs,p, and P_abs,p sustains fairly high value at the ER_max wavelength throughout all the high incident angle in Figure 5d,h,l and Table 2. Herein, the I_p is constituted by quadratic sum of horizontal and vertical electric field component E_x and E_z, i.e., [56]

$$I_{\mathrm{p}}={\left|E_x\right|}^2+{\left|E_z\right|}^2.$$

(18)

Since the extinction coefficient along the a-axis (parallel to vibration direction of E_z) k_a is zero in Figure 1d, no absorption takes place along the vibration direction of E_z accordingly. Thus, the P_abs,p can be represented as [51,52]:

$$P_{\mathrm{abs},\mathrm{p}}=\frac{1}{2}\omega {\epsilon }_0{I}_x{k}_b.$$

(19)

In Equation (19), I_x is the horizontal component of p light’s electric field intensity with the relation of I_x = |E_x|². In Figure A6 of Appendix D, SnSe’s I_x at ER_max wavelength also declines with the increase of high incident angle from 70° to 80°. However, the blue shift of ER_max wavelength makes the corresponding k_b increase with the high incident angle since k_b shares the same declining trend as k_c in the near-infrared region of Figure 1d. Consequently, P_abs,p in the SnSe layer holds a considerable value throughout all high incident angle in Figure 5d,h,l and Table 2, producing the lowest R_p and corresponding ER_max.

4. Conclusions

In summary, we have theoretically and numerically designed broadband and incident-angle-modulation near-infrared polarizers based on optically anisotropic SnSe. The giant optical anisotropy of SnSe is quantitatively revealed by the complete dielectric tensor, complex refractive index tensor, and derived birefringence (~|Δn|_max = 0.4) and dichroism (~|Δk|_max = 0.4), determined by the Mueller matrix spectroscopic ellipsometry. We find the weak absorption feature of SnSe in near-infrared region and the blue shift of the corresponding wavelengths of p light’s minimum reflectance in SnSe-SiO₂-Si multilayer structure. Moreover, s light’s high reflectance originates from the constructive interference and reduction of absorption at high incident angle. All these features make the SnSe based multilayer structure become broadband, and incident-angle-modulation near-infrared polarizers with a bandwidth of 324 nm from the wavelength of 1262 nm to 1586 nm, and an average extinction ratio above 23 dB and dynamically and reversibly modulating by high incident angle with a maximum extinction ratio of 71 dB, respectively. The broadband polarizer originates from the results that p light’s absorption is one order of magnitude higher than that of s light throughout the entire concerned wavelength range. The numerical simulations reveal that the continuously and relatively low absorption of s light and the considerable absorption for p light at the high incident angle regulated absorption wavelength lead to high-performance and incident-angle-modulation polarizer in the near-infrared region. The SnSe-SiO₂-Si multilayer structure may be easily fabricated by vacuum evaporation or magnetron sputtering, compared with manufacture process of the optical fibers or optical gratings structures. Such easily fabricated broadband and incident-angle-modulation near-infrared polarizers based on multilayer structure are likely to be mass-produced and can be used in integrated optics in practice. Our research opens up new thought to design other polarizing optical components by utilizing other low-symmetry materials.