Simulation of a High-Performance Polarization Beam Splitter Assisted by Two-Dimensional Metamaterials

It is challenging to simultaneously consider device dimension, polarization extinction ratio (PER), insertion loss (IL), and operable bandwidth (BW) to design a polarization beam splitter (PBS) that is extensively used in photonic integrated circuits. The function of a PBS is to separate polarizations of light, doubling the transmission bandwidth in optical communication systems. In this work, we report a high-performance PBS comprising two-dimensional subwavelength grating metamaterials (2D SWGMs) between slot waveguides. The 2D SWGMs exhibited biaxial permittivity by tailoring the material anisotropy. The proposed PBS showed PERs of 26.8 and 26.4 dB for TE and TM modes, respectively, and ILs of ~0.25 dB for both modes, with an unprecedented small footprint of 1.35 μm × 2.75 μm working at the wavelength λ = 1550 nm. Moreover, the present structure attained satisfactory PERs of >20 dB and ILs of <0.5 dB within an ultrabroad BW of 200 nm.


Introduction
To accomplish photonic integrated circuits (PICs) [1][2][3][4], a popular material platform is the silicon on insulator (SOI). There are two primary merits of using the SOI platform: matured semiconductor fabrication technology and the high-index contrast of the refractive index, thereby significantly rendering the devices compact. However, the SOI platform's highly birefringent property leads to significant polarization dependence, which is undesirable for an optical-fiber network. To resolve this problem, polarization management components, including polarization rotators [5][6][7][8], polarizers [9][10][11][12][13], and polarization beam splitters (PBSs) [14][15][16][17][18][19][20], were proposed. Among them, PBSs are the most popular for separating two orthogonal polarizations, the transverse-electric (TE) and the transversemagnetic (TM), because of effectively using the two polarizations in transmitting optical signals. The overall performance of a PBS is evaluated by several criteria, such as device footprint, polarization extinction ratio (PER), insertion loss (IL), operable bandwidth (BW), and fabrication tolerances. Most PBSs are based on the directional coupler (DC) type because they can be flexibly designed by adopting diverse structures (e.g., silicon (Si) strip, plasmonic waveguide, multimodal interference effect, and slot waveguides) depending on the selected priorities of the device size, PER, IL, and BW. The authors in [14] proposed a DC-type PBS on a coupler consisting of Si strips. In [15], the authors reported an asymmetric DC-type PBS comprising a Si strip and a hybrid plasmonic waveguide (HPW). They also used an MMI coupler between a Si strip and an HPW to obtain a device footprint of 1.8 × 2.5 µm 2 [16]. Yue et al. [17] designed a PBS consisting of horizontal slot waveguides [21,22] to enhance TM mode coupling compared to that of a PBS consisting of an Si-strip coupler [14], remarkably shrinking the coupling length from a few hundreds to a few tens of micrometers. However, the PERs of the TE and TM modes are >20 dB over a narrow range of about 18 nm BW [17]. In experiments, the PBS [17] exhibited PERs of 16.8 for TE mode and 14.1 dB for TM mode [18]. Kim et al. [19] reported a 7.5 µm long PBS Figure 1a-c show the 3D diagram with the TE (E x ) and TM (E y ) mode profiles in the incident plane, the top view, and the cross-section in the xy plane, respectively, of the proposed PBS. Slot waveguides on a SiO 2 substrate comprised a SiO 2 slot layer sandwiched with high-index Si strips in which the TE channel was connected to a bent waveguide with a radius of curvature (R) at the end in order to effectively decouple the two slot waveguides. In the slot waveguides with width W S , the thicknesses of the slot layers and Si layers were t s and h Si , respectively. The height and width of Si strips (copper red) were 2h Si + t s and W S , respectively. The pitch of SWGMs in the x direction was set to Λ x = W Cl + g with duty cycle ρ x = W Cl /Λ x , where W Cl is the width of the Si strips, and g is the gap between Si strips. Edge-to-edge spacing between slot waveguides is s. Likewise, the pitch of SWGMs (c) schematic of calculating the resultant effective permittivity εemt of 2D SWGMs, which is obtained by sequentially estimating the 1D SWGMs in the z (εp) and x directions (εemt) on the basis of EMT between slot waveguides; (d) cross-section in xy plane of the present PBS.

Mode Characteristic and Coupling Effect with Anisotropic SWGMs
Before analyzing the proposed design, the fabrication processes are schematically illustrated in Figure 2. (1) A negative photoresist (PR) film (purple) was deposited to pattern the lower Si strips with a hard mask defining the patterns of the proposed structure on a SiO2 substrate (blue); then PR exposure and development were conducted. (2) The proposed structure was formed by etching SiO2 and lifting off the PR film. (3) A Si film was deposited by chemical vapor deposition, and the Si layer was planarized by chemical mechanical polishing (CMP). (4) Depositing a positive PR (green) to pattern the SWGMs with another mask was followed by PR exposure and development, SiO2 slot layer deposition using thermal oxidation, and a CMP was conducted. (5) Similar to Step 4 but with a different mask. (6) After SiO2 reactive ion etching, the PR film was lifted off. (7) The upper Si strips and SWGMs were deposited, and a CMP was carried out. (8) A positive PR was deposited with the same mask as that in Step 1, followed by PR exposure and development. (9) After Si etching, the positive PR film was lifted off to reach the desired (c) schematic of calculating the resultant effective permittivity ε emt of 2D SWGMs, which is obtained by sequentially estimating the 1D SWGMs in the z (ε p ) and x directions (ε emt ) on the basis of EMT between slot waveguides; (d) cross-section in xy plane of the present PBS.
Before analyzing the proposed design, the fabrication processes are schematically illustrated in Figure 2. (1) A negative photoresist (PR) film (purple) was deposited to pattern the lower Si strips with a hard mask defining the patterns of the proposed structure on a SiO 2 substrate (blue); then PR exposure and development were conducted. (2) The proposed structure was formed by etching SiO 2 and lifting off the PR film. (3) A Si film was deposited by chemical vapor deposition, and the Si layer was planarized by chemical mechanical polishing (CMP). (4) Depositing a positive PR (green) to pattern the SWGMs with another mask was followed by PR exposure and development, SiO 2 slot layer deposition using thermal oxidation, and a CMP was conducted. (5) Similar to Step 4 but with a different mask. (6) After SiO 2 reactive ion etching, the PR film was lifted off. (7) The upper Si strips and SWGMs were deposited, and a CMP was carried out. (8) A positive PR was deposited with the same mask as that in Step 1, followed by PR exposure and development. (9) After Si etching, the positive PR film was lifted off to reach the  The working principle of the present PBS is that we propagated TE mode with major electric field Ex along the bar, while TM mode was coupled with a major electric field Ey under the phase matching condition (PMC) into the cross, as shown in Figure 1a. Using the EMT [26,27], the proposed 2D SWGMs displayed an equivalent material anisotropy εemt by considering the permittivity of Si, εSi and εp as follows (see Figure 1c): where εxx, εyy, and εzz denote permittivity in the x, y, and z directions, respectively. In this work, εp = diag [εpx, εpy, εpz] denotes the equivalent anisotropic permittivity of the SWGMs consisting of SiO2 strips and air gaps along the z direction, as shown below: where εSiO2 and εair denote the permittivity of SiO2 and air, respectively. Differing from the previous reports [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] using one-dimensional SWGMs, εemt in Equation (1) showed biaxial anisotropy, thus more flexibly tuning the optical characteristics. To design a PBS here, the mode characteristics of the TE and TM modes had to be obtained in advance. After that, the coupling length of a coupled waveguide for a specific mode measuring the distance required to completely transfer power from one waveguide to another can be computed by Lc,i = λ/{2(ni,even − ni,odd)} [48], where i denotes TE or TM. ni,even and ni,odd denote the effective indices of the even and odd modes, respectively. First, we analyzed the mode characteristics of the present design. The relative permittivity of Si and SiO2 was εSi = 12.110 and εSiO2 = 2.085 [49], respectively, at λ = 1550 nm. The working principle of the present PBS is that we propagated TE mode with major electric field E x along the bar, while TM mode was coupled with a major electric field E y under the phase matching condition (PMC) into the cross, as shown in Figure 1a. Using the EMT [26,27], the proposed 2D SWGMs displayed an equivalent material anisotropy ε emt by considering the permittivity of Si, ε Si and ε p as follows (see Figure 1c): where ε xx , ε yy , and ε zz denote permittivity in the x, y, and z directions, respectively. In this work, ε p = diag [ε px , ε py , ε pz ] denotes the equivalent anisotropic permittivity of the SWGMs consisting of SiO 2 strips and air gaps along the z direction, as shown below: where ε SiO 2 and ε air denote the permittivity of SiO 2 and air, respectively. Differing from the previous reports [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] using one-dimensional SWGMs, ε emt in Equation (1) showed biaxial anisotropy, thus more flexibly tuning the optical characteristics. To design a PBS here, the mode characteristics of the TE and TM modes had to be obtained in advance.
After that, the coupling length of a coupled waveguide for a specific mode measuring the distance required to completely transfer power from one waveguide to another can be computed by L c,i = λ/{2(n i,even − n i,odd )} [48], where i denotes TE or TM. n i,even and n i,odd denote the effective indices of the even and odd modes, respectively. First, we analyzed the mode characteristics of the present design. The relative permittivity of Si and SiO 2 was ε Si = 12.110 and ε SiO 2 = 2.085 [49], respectively, at λ = 1550 nm. The chosen parameters were W Si = 400 nm, W Cl = 75 nm, g = 50 nm (i.e., ρ x = 0.6), h Si = 150 nm, L a = 100 nm, L SiO2 = 150 nm (i.e., ρ z = 0.6), s = 550 nm, and four Si strips between slot waveguides. First, we obtained the anisotropic ε p = diag [1.285, 1.285, 1.206] according to Equations (5) and (6). Next, the resultant biaxially anisotropic ε emt = diag [1.851, 2.815, 2.805] could be obtained according to Equations (2)-(4). With the obtained ε emt in the region between slot waveguides and using COMSOL Multiphysics 6.0 (COMSOL Inc., Burlington,VT, USA), the coupling length of TM mode L c,TM and the coupling-length ratio of TE and TM modes L c,TE /L c,TM versus t s for the present PBS and the conventional SWs [17,18] are shown in Figure 3. The larger value of L c,TE /L c,TM implies that more TE power is retained in the bar while TM power is completely transferred to the cross. L c,TE was far larger than L c,TM due to better TE mode confinement; thus, a PBS's length is determined by a shorter L c,TM . For conventional SWs [17], L c,TM (dull blue dotted line) varied from 82.42 to 32.19 µm for t s for 10 and 60 nm, respectively, and L c,TE /L c,TM (copper red dotted line) was only 9.16 at t s = 60 nm. In contrast, the proposed structure not only shrunk L c,TM (dull blue solid line) to 10.69 and 3.15 µm for t s = 10 and 60 nm, respectively, but also improved L c,TE /L c,TM (copper red solid line) to 891 at t s = 60 nm. Results demonstrate that the device length and PER of a PBS based on a two-SW can be significantly reduced and improved, respectively, by locating the proposed 2D SWGMs between the two SWs.  (4). With the obtained εemt in the region between slot waveguides and using COMSOL Multiphysics 6.0 (COMSOL Inc., Burlington,VT, USA), the coupling length of TM mode Lc,TM and the coupling-length ratio of TE and TM modes Lc,TE/Lc,TM versus ts for the present PBS and the conventional SWs [17,18] are shown in Figure 3.  Further 3D calculations of mode and propagation characteristics are outlined in the next section. For observing the mode coupling, the even TE and TM field contours of the present structure are exhibited in Figure 4a,b, respectively, and those of the SWs are exhibited in Figure 4c  Further 3D calculations of mode and propagation characteristics are outlined in the next section. For observing the mode coupling, the even TE and TM field contours of the present structure are exhibited in Figure 4a,b, respectively, and those of the SWs are exhibited in Figure 4c,d, respectively.  Further 3D calculations of mode and propagation characteristics are outlined in the next section. For observing the mode coupling, the even TE and TM field contours of the present structure are exhibited in Figure 4a,b, respectively, and those of the SWs are exhibited in Figure 4c,d, respectively. By adding 2D SWGs in the middle region of conventional SWs, the decaying rates of evanescent wave of the TE and TM modes are considerably suppressed (i.e., larger Lc,TE/Lc,TM) and enhanced (i.e., shorter Lc,TM), respectively. In addition, the relative field amplitudes of the TE and TM modes along the line denoted in the inset of Figure 4e are shown in Figure 4e,f, respectively. The above results can be explained by the decay constant of the TE mode kTE that can be adjusted by the εzz/εxx ratio according to the dispersion relation [39], and the εzz > εxx condition is always fulfilled. Therefore, 2D SWGMs significantly reduce crosstalk compared to an isotropic cladding with εzz/εxx = 1. In contrast, the decay constant of the TM mode, kTM, depends on the εzz/εyy < 1 ratio, leading to weaker mode confinement.

Performance Dependences on Duty Cycle, Wavelength, and Fabrication Tolerance
Device performance dependences on geometry parameters, wavelength response, and fabrication tolerance proceeded by executing 3D simulations. To assess the transmission performance of a PBS, we analyze the PERs and ILs of both modes, which are formulated in Equations (7) and (8) where  By adding 2D SWGs in the middle region of conventional SWs, the decaying rates of evanescent wave of the TE and TM modes are considerably suppressed (i.e., larger L c,TE /L c,TM ) and enhanced (i.e., shorter L c,TM ), respectively. In addition, the relative field amplitudes of the TE and TM modes along the line denoted in the inset of Figure 4e are shown in Figure 4e,f, respectively. The above results can be explained by the decay constant of the TE mode k TE that can be adjusted by the ε zz/ ε xx ratio according to the dispersion relation [39], and the ε zz > ε xx condition is always fulfilled. Therefore, 2D SWGMs significantly reduce crosstalk compared to an isotropic cladding with ε zz/ ε xx = 1. In contrast, the decay constant of the TM mode, k TM , depends on the ε zz/ ε yy < 1 ratio, leading to weaker mode confinement.

Performance Dependences on Duty Cycle, Wavelength, and Fabrication Tolerance
Device performance dependences on geometry parameters, wavelength response, and fabrication tolerance proceeded by executing 3D simulations. To assess the transmission performance of a PBS, we analyze the PERs and ILs of both modes, which are formulated in Equations (7) and (8) where P in denotes the input power, P TE(TM),bar(cro) denotes the TE (TM) mode power at the bar (cross), and P TE(TM),cro(bar) is the TE (TM) mode power at the cross (bar). By adopting R = 3 µm and the same parameters displayed in Figure 4, the y component of the magnetic field (H y ) and total power (|P|) evolutions of TE mode are shown in Figure 5a,c, respectively, and the y component of the electric field (E y ) and total power (|P|) evolutions of TM mode are shown in Figure 5b,d, respectively. In Section 2, the calculated Lc,TM at ts = 60 nm was 3.15 μm on the basis of equivalent permittivity εemt. The practical Lc,TM to obtain optimal performance is device length LD = 2.75 μm, which was shorter than the calculated Lc,TM in Section 2 because the coupling continued a short distance from the entrance of the bent waveguide and then gradually decreased. At device length LD = 2.75 μm, the obtained results were PERTE = 26.81 dB, PERTM = 26.48 dB, ILTE = 0.16 dB, and ILTM = 0.19 dB. By contrast, conventional SWs [17] with Lc,TM = 34.6 μm achieved PERTE = 25.5 dB, PERTM = 14.8 dB, ILTE = 0.07 dB, and ILTM = 0.06 dB. Results show that the proposed PBS not only considerably reduced the proposed PBS's length by about 12 times when compared to the conventional SWs [17], but also achieved superior PERs, particularly for PERTM. Although the ILs of the current design were higher than those of the SWs, values of <0.2 dB were still acceptable. Considering the effect of duty cycles, we show the PERs and ILs versus ρx in Figure 6a. The optimal PERs and ILs of both modes appeared in the interval of ρx = 0.4 to 0.6. Once the condition of ρx > 0.6 had been reached, PERs and ILs dramatically degraded because a larger ρx decreases (increases) the value of εzz/εxx (εzz/εyy) leading to smaller (larger) kTE (kTM), simultaneously increasing the crosstalk of both modes. The higher PERs are also reflected in lower ILs. Although optimal PERs and ILs were at ρx = 0.4 or 0.5 here, the device length was LD = 6.50 or 4.15 μm, respectively. Considering compactness, we chose ρx = 0.6 to investigate the subsequent analyses, and PERs and ILs versus ρz are exhibited in Figure 6b. In Section 2, the calculated L c,TM at t s = 60 nm was 3.15 µm on the basis of equivalent permittivity ε emt . The practical L c,TM to obtain optimal performance is device length L D = 2.75 µm, which was shorter than the calculated L c,TM in Section 2 because the coupling continued a short distance from the entrance of the bent waveguide and then gradually decreased. At device length L D = 2.75 µm, the obtained results were PER TE = 26.81 dB, PER TM = 26.48 dB, IL TE = 0.16 dB, and IL TM = 0.19 dB. By contrast, conventional SWs [17] with L c,TM = 34.6 µm achieved PER TE = 25.5 dB, PER TM = 14.8 dB, IL TE = 0.07 dB, and IL TM = 0.06 dB. Results show that the proposed PBS not only considerably reduced the proposed PBS's length by about 12 times when compared to the conventional SWs [17], but also achieved superior PERs, particularly for PER TM . Although the ILs of the current design were higher than those of the SWs, values of <0.2 dB were still acceptable. Considering the effect of duty cycles, we show the PERs and ILs versus ρ x in Figure 6a. The optimal PERs and ILs of both modes appeared in the interval of ρ x = 0.4 to 0.6. Once the condition of ρ x > 0.6 had been reached, PERs and ILs dramatically degraded because a larger ρ x decreases (increases) the value of ε zz/ ε xx (ε zz/ ε yy ) leading to smaller (larger) k TE (k TM ), simultaneously increasing the crosstalk of both modes. The higher PERs are also reflected in lower ILs. Although optimal PERs and ILs were at ρ x = 0.4 or 0.5 here, the The ILs of the TE and TM modes slightly increased, ranging from ρz = 0.2 (LD = 2.9 μm) to 0.8 (LD = 2.55 μm), and PERTE and PERTM slightly varied as ρz varies. Differing from the major effect of ρx, ρz plays a finetuning role on device performance and length. Regarding to the number of Si strips (N), the PERs and ILs versus N (at the condition of ρx =0.6 and ρz = 0.6) are shown in Figure 7. The proposed PBS achieved optimal performance when N = 4 was chosen. Further increasing N resulted in moderately varying performance. The performance of TM mode dramatically worsened as N decreased. This result can be attributed to t Λx being increasingly away from the condition of Λx << λ, causing the increase in scattering loss that resulted from the grating structure [21]. To assess the BW of the present PBS, the PERs and ILs as a function of wavelength are shown in Figure 8a,b, respectively. PERTM (ILTM) significantly decreased (increased) as the wavelength moved away from the target wavelength of λ = 1550 nm because the short Lc,TM was more sensitive than the extremely long Lc,TE to deviation from the PMC. By contrast, the PERTE and ILTE of the shorter (longer) than λ = 1550 nm wavelengths were higher (lower) and lower (higher), respectively. This can be explained by the guided modes with shorter (longer) wavelengths leading to shorter (longer) evanescent wave tails, thus reducing (increasing) crosstalk between waveguides. The working BW of both modes with PERs > 20 dB and ILs < 0.5 dB ranged from λ = 1440 to 1650 nm (>200 nm). The ILs of the TE and TM modes slightly increased, ranging from ρ z = 0.2 (L D = 2.9 µm) to 0.8 (L D = 2.55 µm), and PER TE and PER TM slightly varied as ρ z varies. Differing from the major effect of ρ x , ρ z plays a finetuning role on device performance and length. Regarding to the number of Si strips (N), the PERs and ILs versus N (at the condition of ρ x =0.6 and ρ z = 0.6) are shown in Figure 7. The proposed PBS achieved optimal performance when N = 4 was chosen. Further increasing N resulted in moderately varying performance. The performance of TM mode dramatically worsened as N decreased. This result can be attributed to t Λ x being increasingly away from the condition of Λ x << λ, causing the increase in scattering loss that resulted from the grating structure [21]. The ILs of the TE and TM modes slightly increased, ranging from ρz = 0.2 (LD = 2.9 μm) to 0.8 (LD = 2.55 μm), and PERTE and PERTM slightly varied as ρz varies. Differing from the major effect of ρx, ρz plays a finetuning role on device performance and length. Regarding to the number of Si strips (N), the PERs and ILs versus N (at the condition of ρx =0.6 and ρz = 0.6) are shown in Figure 7. The proposed PBS achieved optimal performance when N = 4 was chosen. Further increasing N resulted in moderately varying performance. The performance of TM mode dramatically worsened as N decreased. This result can be attributed to t Λx being increasingly away from the condition of Λx << λ, causing the increase in scattering loss that resulted from the grating structure [21]. To assess the BW of the present PBS, the PERs and ILs as a function of wavelength are shown in Figure 8a,b, respectively. PERTM (ILTM) significantly decreased (increased) as the wavelength moved away from the target wavelength of λ = 1550 nm because the short Lc,TM was more sensitive than the extremely long Lc,TE to deviation from the PMC. By contrast, the PERTE and ILTE of the shorter (longer) than λ = 1550 nm wavelengths were higher (lower) and lower (higher), respectively. This can be explained by the guided modes with shorter (longer) wavelengths leading to shorter (longer) evanescent wave tails, thus reducing (increasing) crosstalk between waveguides. The working BW of both modes with PERs > 20 dB and ILs < 0.5 dB ranged from λ = 1440 to 1650 nm (>200 nm). To assess the BW of the present PBS, the PERs and ILs as a function of wavelength are shown in Figure 8a,b, respectively. PER TM (IL TM ) significantly decreased (increased) as the wavelength moved away from the target wavelength of λ = 1550 nm because the short L c,TM was more sensitive than the extremely long L c,TE to deviation from the PMC. By contrast, the PER TE and IL TE of the shorter (longer) than λ = 1550 nm wavelengths were higher (lower) and lower (higher), respectively. This can be explained by the guided modes Nanomaterials 2022, 12, 1852 9 of 13 with shorter (longer) wavelengths leading to shorter (longer) evanescent wave tails, thus reducing (increasing) crosstalk between waveguides. The working BW of both modes with PERs > 20 dB and ILs < 0.5 dB ranged from λ = 1440 to 1650 nm (>200 nm). To show our design's superiority, we compared its overall performance with that of SWGM-based PBSs, as shown in Table 1. The footprint of the proposed structure was the smallest PBS compared with the reported PBSs [19,20,[42][43][44][45][46], rendering it more beneficial in constructing a highly dense photonic component. In addition to analyzing performance dependence on duty cycles and number of Si strips, as shown in Figures 6 and 7, respectively, we investigated severe geometries ts and WCl to assess fabrication tolerance. Performance versus variations in Si strip width ΔWCl and slot thickness Δts is shown in Figure 9a,b, respectively. PERTM significantly depended on the two fabrication errors (ΔWCl and Δts) because of the extremely short Lc,TM = 2.75 μm of the TM mode. As shown in Figure 9a, PERTM decreased to ~17 dB (from ~26 dB), and ILTM increased to ~0.4 dB (from ~0.2 dB) while ΔWCl > 5 nm or Δts > 10 nm. Within the variation range of ΔWCl < 3 nm or Δts < 5 nm, PERTM maintained values of >20 dB and ILTM < 0.25 dB. In contrast, PERTE and ILTE showed slight variations in ΔWCl and Δts due to the exceptionally long Lc,TE = 891 μm of TE mode. To show our design's superiority, we compared its overall performance with that of SWGM-based PBSs, as shown in Table 1. The footprint of the proposed structure was the smallest PBS compared with the reported PBSs [19,20,[42][43][44][45][46], rendering it more beneficial in constructing a highly dense photonic component. In addition to analyzing performance dependence on duty cycles and number of Si strips, as shown in Figures 6 and 7, respectively, we investigated severe geometries t s and W Cl to assess fabrication tolerance. Performance versus variations in Si strip width ∆W Cl and slot thickness ∆t s is shown in Figure 9a,b, respectively. PER TM significantly depended on the two fabrication errors (∆W Cl and ∆t s ) because of the extremely short L c,TM = 2.75 µm of the TM mode. As shown in Figure 9a, PER TM decreased to~17 dB (from~26 dB), and IL TM increased to~0.4 dB (from~0.2 dB) while ∆W Cl > 5 nm or ∆t s > 10 nm. Within the variation range of ∆W Cl < 3 nm or ∆t s < 5 nm, PER TM maintained values of >20 dB and IL TM < 0.25 dB. In contrast, PER TE and IL TE showed slight variations in ∆W Cl and ∆t s due to the exceptionally long L c,TE = 891 µm of TE mode. Nanomaterials 2022, 12, x FOR PEER REVIEW 10 of 13 (a) (b) Another essential fabrication error, angled sidewall, frequently occurs in etching processes that typically do not have a perfect 90° sidewall. In the present structure, the crucial parts are SWGMs due to their larger aspect ratio compared to that of the slot waveguides. Therefore, we now discuss the sidewall effect of the SWGMs on device performance. The width difference between the bottom and top of the SWGMs is Wsw, as shown in Figure  10a, and the PERs and ILs of both modes are shown in Figure 10b. PERTE (PERTM) and ILTE (ILTM) showed slight (significant) dependences on Wsw due to the extremely weak (strong) coupling strength induced by the 2D SWGMs. PERTM decreased to about 14.2 (9.8) dB, and ILTM increased to about 0.62 (1.25) dB, while Wsw increased to 5 (10) nm.

Conclusions
An ultracompact and broadband PBS comprising slot waveguides assisted by in between 2D SWGMs was proposed to increase the integration density and transmission bandwidth of photonic devices. The proposed 2D SWGMs served as both a barrier to suppress TE mode coupling and bridging to improve TM mode coupling by carefully engineering the duty cycles of the SWGMs. The numerical results of the proposed PBS showed PERs of 26  Another essential fabrication error, angled sidewall, frequently occurs in etching processes that typically do not have a perfect 90 • sidewall. In the present structure, the crucial parts are SWGMs due to their larger aspect ratio compared to that of the slot waveguides. Therefore, we now discuss the sidewall effect of the SWGMs on device performance. The width difference between the bottom and top of the SWGMs is W sw , as shown in Figure 10a, and the PERs and ILs of both modes are shown in Figure 10b. PER TE (PER TM ) and IL TE (IL TM ) showed slight (significant) dependences on W sw due to the extremely weak (strong) coupling strength induced by the 2D SWGMs. PER TM decreased to about 14.2 (9.8) dB, and IL TM increased to about 0.62 (1.25) dB, while W sw increased to 5 (10) nm. Another essential fabrication error, angled sidewall, frequently occurs in etching processes that typically do not have a perfect 90° sidewall. In the present structure, the crucial parts are SWGMs due to their larger aspect ratio compared to that of the slot waveguides. Therefore, we now discuss the sidewall effect of the SWGMs on device performance. The width difference between the bottom and top of the SWGMs is Wsw, as shown in Figure  10a, and the PERs and ILs of both modes are shown in Figure 10b. PERTE (PERTM) and ILTE (ILTM) showed slight (significant) dependences on Wsw due to the extremely weak (strong) coupling strength induced by the 2D SWGMs. PERTM decreased to about 14.2 (9.8) dB, and ILTM increased to about 0.62 (1.25) dB, while Wsw increased to 5 (10) nm.

Conclusions
An ultracompact and broadband PBS comprising slot waveguides assisted by in between 2D SWGMs was proposed to increase the integration density and transmission bandwidth of photonic devices. The proposed 2D SWGMs served as both a barrier to suppress TE mode coupling and bridging to improve TM mode coupling by carefully engineering the duty cycles of the SWGMs. The numerical results of the proposed PBS showed PERs of 26.8 and 26.4 dB for TE and TM modes, respectively, and ILs of ~0.25 dB for both modes, with a compact PBS of 1.35 μm × 2.75 μm. Moreover, the present PBS achieved

Conclusions
An ultracompact and broadband PBS comprising slot waveguides assisted by in between 2D SWGMs was proposed to increase the integration density and transmission bandwidth of photonic devices. The proposed 2D SWGMs served as both a barrier to suppress TE mode coupling and bridging to improve TM mode coupling by carefully engineering the duty cycles of the SWGMs. The numerical results of the proposed PBS showed PERs of 26.8 and 26.4 dB for TE and TM modes, respectively, and ILs of~0.25 dB for both modes, with a compact PBS of 1.35 µm × 2.75 µm. Moreover, the present PBS achieved satisfactory performance while operating in a BW of 200 nm. Fabrication tolerance analyses showed that the PER TM maintained superior values of >20 dB and IL TM < 0.25 dB within the variation range of clad width ∆W Cl < 3 nm or slot thickness ∆t s < 5 nm.