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

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## Abstract

**:**

_{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

^{5}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.

_{2}-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-SiO

_{2}-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 SiO

_{2}, the broadband and incident-angle-modulation near-infrared polarizers based on the SnSe-SiO

_{2}-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

_{m}(m = b or c) along b- or c-axis of SnSe is defined as [22]:

_{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.

_{p,m}(m = b or c) along b- or c-axis of SnSe is the reciprocal of α

_{m}[22], i.e.,

_{0}, is empirically calculated by:

_{p}represents the depth that the light penetrates with its intensity ${I}_{{d}_{p}}$, reducing to about 37% of I

_{0}, combining the Equations (2) and (3):

#### 2.2. Net Phase Shift of s Light

_{air-SnSe}is the refractive angle at the interface between air and SnSe, which can be converted from Snell’s law [22]:

_{air}(= 1) is the complex refractive index of air and N

_{c}is the SnSe c-axis’s complex refractive index.

_{2}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-SiO

_{2}-Si multilayers, utilizing the transfer matrix method [37], i.e.,

_{2}-Si multilayers.

#### 2.3. Theoretical CSalculations by the 4 × 4 Matrix Method

**T**can be expressed in matrix form [22,38]:

_{s}and R

_{p}of the SnSe-SiO

_{2}-Si multilayers can be thus calculated with the matrix elements of

**T**[39]:

_{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

## 3. Results and Discussion

#### 3.1. Optical Anisotropy of SnSe

_{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:

_{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]), ReS

_{2}(Δn

_{ReS2}= 0.06 [45]), and α-MoO

_{3}(Δn

_{α-MoO3}= 0.11 [46]).

_{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 10

^{5}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

_{2}-Si multilayer structure (Figure 2a) as a reflective polarizer at incident angle θ. As a lossless dielectric material, SiO

_{2}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.

_{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.

_{2}-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 SiO

_{2}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 SiO

_{2}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]:

_{0}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.

_{2}(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)-SiO

_{2}(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:

_{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). 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:

_{2}-Si multilayer structure, the optimized thickness of SnSe and SiO

_{2}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

_{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 SiO

_{2}’s electric field intensity is dominated in I

_{s}of Figure 4b. However, SiO

_{2}is a lossless dielectric material, as previously mentioned. As a consequence, there is no P

_{abs,s}in SiO

_{2}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 × 10

^{20}W/m

^{3}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.

_{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. According to the analysis above, R

_{s}in Figure 3c is continuously and relatively high through the entire concerned wavelength range.

_{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]

_{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]:

_{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

_{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-SiO

_{2}-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-SiO

_{2}-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.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mueller Matrix Spectroscopic Ellipsometry Analysis

_{q}, η

_{q}, E

_{0,q}, and E

_{g,q}are, respectively, the qth oscillator’s amplitude, damping coefficient, peak transition energy, and bandgap energy. Specifically, the qth Tauc–Lorentz oscillator is composed of the real part ε

_{r,q}and imaginary part ε

_{I,q}in Equation (A2):

_{i,q}[22,43] while the ε

_{r,q}can be converted from ε

_{i,q}by Kramers–Kronig relation [22] of Equation (A4):

## Appendix B. Thickness-Dependent Bandgap of SnSe

_{VBM}remains almost unchanged and the conduction band minimum (CBM) energy E

_{CBM}decline with the increase of the number of layers. Therefore, the bandgap energy Eg decreases with the increase of the number of layers, and the differences of Eg between 5 L SnSe and bulk SnSe is only 0.02 eV. The relation between Eg and number of layers share the same trend with the reported results [58].

**Figure A1.**Band structure of 2D SnSe (

**a**–

**e**) from l layer to 5 layers (1 L–5 L) and (

**f**) bulk SnSe calculated by first-principle, (

**g**) bandgap energy E

_{g}, conduction band minimum (CBM) energy E

_{CBM}, and valence band maximum (VBM) energy E

_{VBM}varied with the number of layers.

## Appendix C. Optimization of the Thickness of SnSe and SiO_{2}

_{2}(250 nm) is selected to preliminarily figure out the variation of reflectance of p and s light and extinction ratio at the incident angle of 75° in Figure A2. As the thickness of SiO

_{2}layer is 250 nm, the alternately bright and shaded stripes emerge in both reflectance of s and p light (Figure A1a,b, respectively) with the increase of SnSe’s thickness, due to the interference effect [59,60] in multilayer structure. The bright stripes becomes weaker and weaker when the SnSe’s thickness get thicker and thicker, which possibly originates from the stronger absorption of the thicker SnSe layer. In addition, by reason of both the extinction coefficient along the b- and c-axis of SnSe decreasing with the increase of near-infrared wavelength in Figure 2e, the absorption instead of interference in thicker SnSe layer is dominant in the shorter wavelength region of Figure A2a,b. Therefore, the stripes become shorter in the thicker SnSe layer and the shorter wavelength region in Figure A2a,b, respectively. More importantly, the optical anisotropy of SnSe brings the delayed reflectance of p or s light, which may lead to the high extinction ratio ER in Figure A2c. We select two representative thicknesses of SnSe layer (d

_{SnSe}= 670 and 280 nm), corresponding to the maximum vaule of extinction ratio ER

_{max}(above 50 dB) of Figure A2c to further illustrate the variation of reflectance of p and s light in Figure A2d. The wavelength corresponding to the s light’s maximum reflectance R

_{s,max}(about 0.75) perfectly matches that of p light’s minimum reflectance R

_{p,min}in both two kinds of thickness of SnSe layer. Therefore, the maximum extinction ratio results from delaying reflectance of p or s light to make their maximum and minimum correspond to the same wavelength. Besides, the p light’s minimum reflectance wavelength increases with the d

_{SnSe}, indicating modulating the d

_{SnSe}is a potential and effective regualtion method.

**Figure A2.**The contour maps of s and p light’s reflectance R

_{s}and R

_{p}and extinction ratio ER varied with the thicknesses of SnSe, selecting the the representative thicknesses of SnSe layer corresponding to the maximum extinction ratio ER

_{max}to show their R

_{s}and R

_{p}, respectively corresponding to the SiO

_{2}’s thickness of (

**a**−

**d**) 250, (

**e**−

**h**) 750, and (

**i**−

**l**) 1250 nm.

_{2}’s thickness of 750 nm in Figure A2e–h is almost the same as that of SiO

_{2}’s thickness of 250 nm, and the major difference is the representative thicknesses of SnSe layer corresponding to the ER

_{max}, other than the increased interference within the same thickness of SnSe layer in Figure A2e,f. Excitingly, the R

_{p,min}wavelength is from about 1400 nm to 1690 nm (marked area) at SnSe’s thickness of 450 nm, offering significant potential to realize the broadband modulation.

_{2}’s thickness is 1250 nm, the interference levels increase with the SiO

_{2}’s thickness, and there are two, even three times, of destructive interference in most the same thickness of SnSe layer in Figure A2i,j. Although the R

_{p,min}wavelength is broadband at SnSe’s thickness of 390 nm in Figure A2l, its bandwidth (50 nm) is much smaller than that of SiO

_{2}and SnSe’s thicknesses of 750 and 450 nm, respectively.

_{2}’s thickness is further shortened to 25 nm, and SiO

_{2}’s thickness is selected as 675, 700, and 725 nm to acquire longer bandwidth of R

_{p,min}wavelength.

_{2}’s thickness is 675 nm, and the two representative thicknesses of SnSe layer (d

_{SnSe}= 400 and 315 nm), corresponding to the maximum alue of extinction ratio ER

_{max}, are selected to illustrate their reflectance R, exticntion ratio ER, and insertion loss IL at these two thicknesses of SnSe layer in Figure A3b,c, respectively. In Figure A3b, the R

_{s}is above 0.6 and the IL is consequently below 2 dB in the whole concerned wavelength region. The p light can be separated over broadband since R

_{p}low than 0.003 within the wavelength region from 1262 to 1586 nm. Therefore, the ER maintains a high level of about 23 dB at such wavelength region with its maximum ER

_{max}over 60 dB. When the thickness of SnSe layer becomes thinner (d

_{SnSe}= 315 nm), the R

_{s}gets much higher, and the IL is lower than 1.5 dB from the wavelength of 1000 nm to that of 1200 nm as a result in Figure A3c. However, the R

_{p}reaches its minimum at the wavelength of about 1100 nm accompanied by the ER

_{max}over 70 dB, which means p light can be separated at a specific wavelength.

_{2}’s thickness of 700 and 725 nm are similar to that of SiO

_{2}’s thickness of 675 nm, and Table A1 and Table A2 summarize their differences.

**Figure A3.**The contour maps of extinction ratio spectrum varied with the thickness of SnSe for the SiO

_{2}’s thickness of (

**a**) 675, (

**d**) 700, and (

**g**) 725 nm, and the two representative thicknesses of SnSe layer (d

_{SnSe}= (

**b**) 400 and (

**c**) 315 nm for SiO

_{2}’s thickness of 675 nm, (

**e**) 430 and (

**f**) 290 nm for that of 700 nm, and (

**h**) 455 and (

**i**) 280 nm for that of 725 nm) corresponding to the maximum value of extinction ratio ER

_{max}.

**Table A1.**Performances of separating p light over broadband with thicker SnSe’s thickness (d

_{SnSe}= 400, 430 and 455 nm) respectively at the SiO

_{2}’s thickness of 675, 700, and 725 nm.

Thickness of SiO_{2} [nm] | Thickness of SnSe [nm] | Wavelength Region ^{1} [nm] | Bandwidth [nm] | Maximum Extinction Ratio ER_{max} [dB] | Average Insertion Loss IL_{ave} [dB] |
---|---|---|---|---|---|

675 | 400 | 1262−1586 | 324 | 62 | 1.7 |

700 | 430 | 1325−1642 | 317 | 52 | 1.7 |

725 | 455 | 1383−1688 | 305 | 66 | 1.8 |

^{1}Corresponding to wavelength region of R

_{p}< 0.003.

**Table A2.**Performances of separating p light at specific wavelength with thinner SnSe’s thickness (d

_{SnSe}= 315, 290 and 280 nm) respectively at the SiO

_{2}’s thickness of 675, 700, and 725 nm.

Thickness of SiO_{2} [nm] | Thickness of SnSe [nm] | Wavelength ^{1} [nm] | Maximum Extinction Ratio ER_{max} [dB] | Insertion Loss IL [dB] |
---|---|---|---|---|

675 | 315 | 1100 | 71 | 1.1 |

700 | 290 | 1043 | 68 | 1.2 |

725 | 280 | 1027 | 53 | 1.1 |

^{1}Corresponding to the wavelength of ER

_{max}.

_{ave}and the IL at the wavelength of ER

_{max}are lower than 2 dB, suggesting the high s light’s reflectance R

_{s}(above 0.63). Although the ER

_{max}for the SiO

_{2}’s thickness of 675 nm is slightly lower than that for the SiO

_{2}’s thickness of 725 nm, its bandwidth of R

_{p}< 0.003 is longest among three kinds of thickness of SiO

_{2}in Table A1. Additionally, its ER

_{max}in Table A2 is also the highest. Hence, compared with the other two kinds of thicknesses of SiO

_{2}(d

_{SiO2}= 700 and 725 nm), the SnSe-SiO

_{2}-Si multilayer structure with the SiO

_{2}’s thickness of 675 nm displays relatively highest performances.

_{max}within it at the thickness of SiO

_{2}d

_{SiO2}, from 725 to 500 nm, with the interval of 25 nm in Figure A4 to find out the optimal d

_{SiO2}. With the reduction of SiO

_{2}’s thickness, the bandwidth at first increases to 325 nm at the SiO

_{2}’s thickness of 675 nm, and then gradually decreases until the SiO

_{2}’s thickness of 500 nm. Therefore, the bandwidth corresponding to the SiO

_{2}’s thickness of 675 nm is the longest, whose ER

_{max}is 62 dB, and is the second highest among these thicknesses of SiO

_{2}. As shown in Table A1, the insertion losses of three kinds of SiO

_{2}’s thickness (d

_{SiO2}= 675, 700, and 725 nm) are relatively small and almost the same; hence, the insertion losses may not significantly influence the performances of separating p light over broadband and are not taken into consideration for the thinner thickness of SiO

_{2}.

**Figure A4.**The bandwidth and the maximum extinction ratio ER

_{max}within the corresponding wavelength region at the thickness of SiO

_{2}d

_{SiO2}from 725 to 500 nm with the interval of 25 nm.

_{max}shares the same trend as the bandwidth in Figure A5, and the ER

_{max}at the SiO

_{2}’s thickness of 675 nm is the highest as a result.

**Figure A5.**The maximum extinction ratio ER

_{max}at the thickness of SiO

_{2}d

_{SiO2}from 725 to 500 nm with the interval of 25 nm for separating p light at specific wavelength.

_{2}’s thickness is selected as 675 nm, and the corresponding SnSe’s thickness is 400 and 315 nm for separating p light over broadband and at specific wavelength respectively.

## Appendix D. Distribution of Horizontal Component of p Light’s Electric Field Intensity Ix

**Figure A6.**The contour maps of horizontal component of p light’s electric field intensity I

_{x}in SnSe (d

_{SnSe}= 315 nm)-SiO

_{2}(d

_{SiO2}= 675 nm)-Si multilayer structure at the incident angle of (

**a**) 70°, (

**b**) 75°, and (

**c**) 80° at the corresponding ER

_{max}wavelength for incident-angle-modulation near-infrared polarizer.

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**Figure 1.**The optical anisotropy of SnSe. (

**a**) The low-symmetry lattice structure of SnSe. (

**b**) The polarization-resolved reflectance spectra with insert of the corresponding polar coordinate curve at peak wavelength of 520 nm. (

**c**) The dielectric tensor, (

**d**) the complex refractive index tensor, and (

**e**) the derived birefringence Δn and dichroism Δk of b-c plane of SnSe. (

**f**) The absorption coefficient α and the penetration depth d

_{p}along the b- and c-axis, respectively, of SnSe.

**Figure 2.**(

**a**) SnSe-SiO

_{2}-Si multilayer structure with the incident angle θ. (

**b**) The reflectance of s and p light at high incident angle of 70°, 75°, and 80°, where the thickness of SiO

_{2}(d

_{SiO2}) and the thickness of SnSe (d

_{SnSe}) are, respectively, 200 and 300 nm. The contour maps of reflectance of (

**c**) p and (

**d**) s light varied with the incident angle. The contour maps of s light’s (

**e**) electric field intensity I

_{s}and (

**f**) absorbed power P

_{abs,s}varied with the depth and the incident angle, selecting the maximum reflectance wavelength (1164 nm) at normal incidence.

**Figure 3.**The broadband near-infrared polarizer based on SnSe-SiO

_{2}-Si multilayer structure with SnSe’s thickness of 400 nm and SiO

_{2}’s thickness of 675 nm: (

**a**) the theoretical and numerical reflectance of p and s light at the incident angle of 75° and (

**b**) the corresponding theoretical extinction ratio ER and insertion loss IL. The incident-angle-modulation near-infrared polarizer based on SnSe-SiO

_{2}-Si multilayer structure with SnSe’s thickness of 315 nm and the same SiO

_{2}’s thickness (d

_{SiO2}= 675 nm): (

**c**) the theoretical and numerical reflectance of p and s light at the incident angle of 70°, 75°, and 80° and (

**d**) the corresponding theoretical extinction ratio ER and insertion loss IL at the same incident angle.

**Figure 4.**The mechanism for the near-infrared broadband polarizer based on SnSe (d

_{SnSe}= 400 nm)-SiO

_{2}(d

_{SiO2}= 675 nm)-Si multilayer structure. The contour maps of the electric field intensity of (

**a**) p and (

**b**) s light (I

_{p}and I

_{s}), and the absorbed power of (

**c**) p and (

**d**) s light (P

_{abs,p}and P

_{abs,s}) at the incident angle of 75°.

**Figure 5.**The mechanism for incident-angle-modulation near-infrared polarizer based on SnSe (d

_{SnSe}= 315 nm)-SiO

_{2}(d

_{SiO2}= 675 nm)-Si multilayer structure. The contour maps of electric field intensity of (

**a**,

**e**,

**i**) s and (

**c**,

**g**,

**k**) p light (I

_{s}and I

_{p}) and absorbed power of (

**b**,

**f**,

**j**) s and (

**d**,

**h**,

**l**) p light (P

_{abs,s}and P

_{abs,p}) at the incident angle of 70°/75°/80°, respectively, and I

_{p}and P

_{abs,p}is at the corresponding ER

_{max}wavelength.

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 |

**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 [°] | ER_{max} Wavelength [nm] | I_{p} [a. u.] | I_{p}/I_{s} | P_{abs,p}/10^{20} [W/m^{3}] | P_{abs,p}/P_{abs,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 |

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**MDPI and ACS Style**

Guo, Z.; Gu, H.; Yu, Y.; Wei, Z.; Liu, S. Broadband and Incident-Angle-Modulation Near-Infrared Polarizers Based on Optically Anisotropic SnSe. *Nanomaterials* **2023**, *13*, 134.
https://doi.org/10.3390/nano13010134

**AMA Style**

Guo Z, Gu H, Yu Y, Wei Z, Liu S. Broadband and Incident-Angle-Modulation Near-Infrared Polarizers Based on Optically Anisotropic SnSe. *Nanomaterials*. 2023; 13(1):134.
https://doi.org/10.3390/nano13010134

**Chicago/Turabian Style**

Guo, Zhengfeng, Honggang Gu, Yali Yu, Zhongming Wei, and Shiyuan Liu. 2023. "Broadband and Incident-Angle-Modulation Near-Infrared Polarizers Based on Optically Anisotropic SnSe" *Nanomaterials* 13, no. 1: 134.
https://doi.org/10.3390/nano13010134