2D-Heterostructure Photonic Crystal Formation for On-Chip Polarization Division Multiplexing

: Herein, we offer a numerical study on the devising of a unique 2D-heterostructure photonic crystal (PC) that can split two orthogonally polarized light waves. The analysis is performed via a two-dimensional ﬁnite element method (2D-FEM) by utilizing the COMSOL Multiphysics software. The device consists of two discrete designs of PC formation. The ﬁrst PC formation is optimized so that it permits both TE- and TM-polarization of light to transmit through it. Whereas, the second PC formation possesses a photonic bandgap (PBG) only for TE-polarized light. These two formations are combined at an angle of 45 ◦ , resulting in a reﬂection of self-collimated TE-polarized light at an angle of 90 ◦ owing to the PBG present in the second PC formation. While permitting the self-collimated TM-polarized light wave to travel uninterrupted. The proposed device has a small footprint of ~10.9 µ m 2 offering low transmission loss and high polarization extinction ratio which makes it an ideal candidate to be employed as an on-chip polarization division multiplexing system.


Introduction
In 1987, Yablonovitch [1] and John [2] suggested photonic crystals (PCs), an artificial dielectric structure that uses periodic and random variations in the refractive index to maneuver the light flow. On a microscopic scale, it is possible to customize the propagation of the electromagnetic (EM) field in these constructions. Within the PC, EM waves scatter, and destructive interference occurs at specific wavelengths, forming a photonic bandgap (PBG) such as the energy bandgap of electron waves in a semiconductor [3]. Controlling light propagation may be feasible due to the possibility of generating a PBG. Furthermore, devices with tiny footprints are possible. These remarkable characteristics might have an advantage to the development of compact integrated circuits. PCs are utilized in several notable applications such as polarizers [3][4][5][6], power splitter [7,8], solar cell [9], logic gates [10][11][12], optical diode [13], laser [14], sensor [15], and self-collimation [16,17], etc. For a detailed study on PC applications, we suggest a timely review to the readers [18].
Multiplexing is a promising resolution in response to the enduring demand for increased transmission capacity [19][20][21][22][23][24]. Silicon (Si)-based photonic integrated circuits (PICs) may be utilized in data centers, high-performance computing, and optical communication networks since they are well-suited with CMOS technology, which allows for largescale and cost-effective integration [25,26]. Multi-dimensional multiplexing technologies, such as wavelength division multiplexing (WDM) [27], polarization division multiplexing (PDM) [28], mode division multiplexing (MDM) [29], etc. have gathered a lot of consideration in research and practical applications to expand the interconnection or communication capacity. PDM is one of them, having the potential to double the transmission capacity and the capability to be linked with other multiplexing or advanced modulation format technologies, supporting its widespread use in commercially available optical communication systems. On the Si-based technology, numerous polarization-handling modules or PDM signal-processing devices have been effectively implemented [30][31][32]. The benefit of PDM is that the communication capacity is increased since separate signals can be dispersed over orthogonal states of polarization of the same light [33]. The two polarization channels are isolated at the receiver end and are individually identified. In an ideal world, at each end of the fiber link, the only requirement is to install a transceiver and a related polarization MUX/DEMUX, while the remaining network remains unaffected.
Dielectric waveguides (WGs) are often larger since they involve a large radius of curvature to prevent severe bending loss [34]. Optical WGs established on the PC formation vary from typical WGs in that they provide vigorous optical confinement as well as adaptable light-guiding [35][36][37]. The propagation of an EM wave via a steep bend with minimum bending loss is one of the remarkable uses of PC WGs [36]. This promotes the development and production of PC-based WGs. However, significant bending loss arises due to reflections at the acute bending corner, particularly when the bending angle is bigger than 90 • . Subsequently, a significant study has been performed on the enhancement of bends to reduce reflection loss [38,39].
Usually, PC-WG structures are utilized to create polarization filters which makes them unattractive in the scenarios where polarization splitting is desired. A compact TM-mode pass filter created on a rectangular-air hole 1D-PC WG is proposed which delivers a polarization extinction ratio (PER) of >30 dB [40]. In [41], the PC-WG polarizer is numerically investigated by engineering the defect modes in the PBG structures. However, the device works as a polarization-maintaining device where one polarization can be completely blocked based on the radius of the defect rods. In [6], a T-shaped PBS based on 2D PC is proposed which is composed of three WGs offering a PER of 30 dB for both modes. PBSs based on resonant-coupling mechanism has also been reported in the past which offers good device performance. However, the design configuration is complicated which makes it difficult to implement the device without avoiding fabrication errors [5,42]. Moreover, a TM-polarization pass filter realized on a subwavelength grating WG has been demonstrated with PER of~27 dB over a device length of 9 µm [43].
In this paper, we presented a 2D-PC heterostructure based on a SOI platform for polarization beam splitting application. Two PCs, known as PC-1 and PC-2 formations with different optical properties are joined at 45 • so that light with TE-polarization is reflected at the boundary between PC-1 and PC-2 formation, while self-collimated light having TM-polarization can travel without suffering from diffraction. By employing such a trait, a polarization beam splitter (PBS) is realized which can be employed as an on-chip PDM system. The proposed device is compact with a single interface between the two PC formations. This fact facilitates the fabrication process without significant errors due to its simple lattice structure. Moreover, the optimized PC formation allows the propagation of a collimated TE-and TM-polarized light without forming any WG arrangement. The simulations are conducted via 2D-FEM where the geometry of the PC formation is adjusted to achieve the best performance. The "electromagnetic waves frequency domain (emw)" is employed as the physics interface and the "frequency domain" was included in the study. The sub-domains in the PC formation are split into triangular mesh elements with a "physics-controlled mesh". The meshing is based on the processing power of the system used. We picked a "fine mesh" size that gives realistic solutions established on our system's processing speed. Scattering boundary conditions (SBCs) are assigned to the simulation window's outer borders.

Modelling of PC-1 Formation
The first PC formation is denoted as PC-1, which is constituted of a regular assortment of circular air holes etched in the silicon-on-insulator (SOI) platform, as exhibited in Figure 1a. The lattice constant and radius of the air hole is represented as a and R, respec- tively. Where a is the center-to-center gap between two air holes. The PC-1 formation is tailored to produce a homogeneous media where light having TE-and TM-polarization can travel through PC-1 devoid of any PBG. The TE-and TM-polarized plane waves of width 2 µm are connected at the input port (P1) of the PC formation, while the output is gathered at the output port (P2). The subsequent formula can be used to determine the transmission loss (TL): where P out and P in are the output and input power, respectively which is obtained by utilizing the line integration built-in function in the COMSOL Multiphysics software. The TL of TE-and TM-polarized light at the operational wavelength = 1550 nm is plotted for the lattice constant (a) ranging between 400 and 450 nm, as shown in Figure 1b. Five different values of radius (R) of the circular air hole are selected, i.e., R = 0.25a, R = 0.3a, R = 0.35a, R = 0.4a, and R = 0.45a. The selection of the geometric parameters is based on the operating wavelength and the ease of precisely fabricating the PC formations. R should always be smaller than a, otherwise, two air holes can overlap each other, and the device will not work efficiently as desired. The values R = 0.4a and R = 0.45a must be avoided as TE-polarized light suffer heavy TL due to scattering or the presence of PBG. The TL of <0.79 dB is acquired for both the polarization at the optimum value of a = 420 nm and R = 0.35a. The first PC formation is denoted as PC-1, which is constituted of a regular assortment of circular air holes etched in the silicon-on-insulator (SOI) platform, as exhibited in Figure 1a. The lattice constant and radius of the air hole is represented as a and R, respectively. Where a is the center-to-center gap between two air holes. The PC-1 formation is tailored to produce a homogeneous media where light having TE-and TM-polarization can travel through PC-1 devoid of any PBG. The TE-and TM-polarized plane waves of width 2 μm are connected at the input port (P1) of the PC formation, while the output is gathered at the output port (P2). The subsequent formula can be used to determine the transmission loss (TL):  The PC-1 formation is analyzed for the determination of the collimation beam region and PBG region in the wavelength range of 1400 to 1850 nm. The geometric parameters The PC-1 formation is analyzed for the determination of the collimation beam region and PBG region in the wavelength range of 1400 to 1850 nm. The geometric parameters variables such as R is fixed at 0.35a, whereas a is varied between 400 and 450 nm. The transmission spectrum for TE-and TM-polarized light in PC-1 is shown in Figure 1c. There is no TM-PBG for the whole wavelength range, however, at a > 420 nm, the light is not collimated anymore. For TE-polarized light, there exists a strong PBG which depends on the lattice constant (a) of the PC-1. Therefore, the selection of geometric parameters should be wisely adapted for the wavelength range of interest. To visualize the propagation of light in PC-1 formation, the normalized electric field mapping of TE-and TM-polarized light at λ = 1550 nm is represented, as shown in Figure 2a,b, respectively. The structural variables such as a and R of the PC-1 are fixed at 420 nm and 0.35a, respectively. The PC-1 formation is capable of propagating self-collimated TE-and TM-polarized light, which indicates that there is no PBG in the designed PC-1 formation. variables such as R is fixed at 0.35a, whereas a is varied between 400 and 450 nm. The transmission spectrum for TE-and TM-polarized light in PC-1 is shown in Figure 1c.
There is no TM-PBG for the whole wavelength range, however, at a > 420 nm, the light is not collimated anymore. For TE-polarized light, there exists a strong PBG which depends on the lattice constant (a) of the PC-1. Therefore, the selection of geometric parameters should be wisely adapted for the wavelength range of interest. To visualize the propagation of light in PC-1 formation, the normalized electric field mapping of TE-and TM-polarized light at λ = 1550 nm is represented, as shown in Figure 2a,b, respectively. The structural variables such as a and R of the PC-1 are fixed at 420 nm and 0.35a, respectively. The PC-1 formation is capable of propagating self-collimated TE-and TM-polarized light, which indicates that there is no PBG in the designed PC-1 formation.

Modelling of PC-2 Formation
The PC-2 formation is tailored such that there exists a PBG for TE-polarization of light, whereas TM-polarized light let go freely within the formation. It is comprised of a periodic collection of square air holes etched in the SOI platform as exhibited in Figure 3a. The side length of the air hole is represented as w. The structure variables of the PC-2 formation are adjusted to create a TE-PBG but agrees on the self-collimated TM-polarized light to propagate in PC-2 deprived of any significant TL. The TL of TE-and TM-polarized light at the operational wavelength = 1550 nm is plotted for the lattice constant (a) ranging between 400 and 450 nm, as shown in Figure 3b. Four different values of side length (w) of the square air hole are selected, i.e., w = 0.5a, w = 0.6a, w = 0.7a, and w = 0.8a. From the figure, it can be seen that the PC-2 formation permits both polarizations to travel through it for w = 0.5a and w = 0.6a. At w = 0.8a, there exists a weak TE-PBG which offers a TL = ~14-15 dB. However, a strong TE-PBG occurs at w = 0.7a, which is evident in Figure 3b. The TL of ~60 and ~0.1 dB is obtained for TE-and TM-polarized light at a = 420 nm and w = 0.7a.

Modelling of PC-2 Formation
The PC-2 formation is tailored such that there exists a PBG for TE-polarization of light, whereas TM-polarized light let go freely within the formation. It is comprised of a periodic collection of square air holes etched in the SOI platform as exhibited in Figure 3a. The side length of the air hole is represented as w. The structure variables of the PC-2 formation are adjusted to create a TE-PBG but agrees on the self-collimated TM-polarized light to propagate in PC-2 deprived of any significant TL. The TL of TE-and TM-polarized light at the operational wavelength = 1550 nm is plotted for the lattice constant (a) ranging between 400 and 450 nm, as shown in Figure 3b. Four different values of side length (w) of the square air hole are selected, i.e., w = 0.5a, w = 0.6a, w = 0.7a, and w = 0.8a. From the figure, it can be seen that the PC-2 formation permits both polarizations to travel through it for w = 0.5a and w = 0.6a. At w = 0.8a, there exists a weak TE-PBG which offers a TL =~14-15 dB. However, a strong TE-PBG occurs at w = 0.7a, which is evident in Figure 3b. The TL of~60 and 0.1 dB is obtained for TE-and TM-polarized light at a = 420 nm and w = 0.7a. The PC-2 formation is studied to resolve the collimation beam region and PBG region in the wavelength range of 1400 to 1850 nm. The geometric parameters variables such as w are fixed at 0.7a, whereas a is varied between 400 and 450 nm. The transmission spectrum for TE-and TM-polarized light in PC-2 is presented in Figure 3c. There is no TM-PBG for the whole wavelength range, however, at a >420 nm, the light is not collimated anymore. For TE-polarized light, there exists a strong PBG that begins at 1400 nm and extends to 1700 nm depending on the selection of a. We aim to design PC-2 in such a way that there exists a TE-PBG for the S, C, and L telecommunication bands so that the device can help split the polarization in a heterostructure arrangement. The norm. electric field mapping of TE-and TM-polarized light at λ = 1550 nm is presented in Figure 4a,b, respectively. When TE-polarized light is incident on the PC-2 formation, there exists a TE-PBG which blocks the light at the input. However, self-collimated TM-polarized light propagates diffraction-free from P1 and exits the formation from P2. This signifies the creation of a The PC-2 formation is studied to resolve the collimation beam region and PBG region in the wavelength range of 1400 to 1850 nm. The geometric parameters variables such as w are fixed at 0.7a, whereas a is varied between 400 and 450 nm. The transmission spectrum for TE-and TM-polarized light in PC-2 is presented in Figure 3c. There is no TM-PBG for the whole wavelength range, however, at a >420 nm, the light is not collimated anymore. For TE-polarized light, there exists a strong PBG that begins at 1400 nm and extends to 1700 nm depending on the selection of a. We aim to design PC-2 in such a way that there exists a TE-PBG for the S, C, and L telecommunication bands so that the device can help split the polarization in a heterostructure arrangement. The norm. electric field mapping of TE-and TM-polarized light at λ = 1550 nm is presented in Figure 4a,b, respectively. When TE-polarized light is incident on the PC-2 formation, there exists a TE-PBG which blocks the light at the input. However, self-collimated TM-polarized light propagates diffraction-free from P1 and exits the formation from P2. This signifies the creation of a TM-polarizer formation that can be individually utilized in TM-polarization maintaining applications.

Polarization Division Multiplexing System
For the PDM system, the two optimized PC formations should be combined such that TE and TM polarization of light can split at 90°. The marriage of PC-1 and PC-2 in a distinctive way leads to the formation of PBS, which can be employed as an on-chip PDM system. Both formations are combined at an angle of 45°, as shown in Figure 5. The optimized values for PC-1 and PC-2 are adopted from the previous analysis, where the TL is

Polarization Division Multiplexing System
For the PDM system, the two optimized PC formations should be combined such that TE and TM polarization of light can split at 90 • . The marriage of PC-1 and PC-2 in a distinctive way leads to the formation of PBS, which can be employed as an on-chip PDM system. Both formations are combined at an angle of 45 • , as shown in Figure 5. The optimized values for PC-1 and PC-2 are adopted from the previous analysis, where the TL is low and light travels without diffraction. For that purpose, we have used a = 420 nm, R = 0.35a, and w = 0.7a for PC-1 and PC-2. When the incident light (TE/TM) travelling from P1 passes the PC-1 and strikes the interface between PC-1 and PC-2, the TM-polarized light can continue its path and exit at P2. As both the formations act as a homogeneous medium and permit the self-collimated TM-polarized light to travel diffraction-free. While TE-polarized light reflects from the interface due to the presence of PBG in PC-2 and experiences a 90 • bend. Consequently, a compact on-chip element is constructed which can be utilized in the PDM system to upsurge the number of transmission channels. The footprint of the device studied in this work is~10.9 µm 2 due to the fact that 13 airholes are periodically arranged in the xy-plane having a = 420 nm.

Polarization Division Multiplexing System
For the PDM system, the two optimized PC formations should be combined such that TE and TM polarization of light can split at 90°. The marriage of PC-1 and PC-2 in a distinctive way leads to the formation of PBS, which can be employed as an on-chip PDM system. Both formations are combined at an angle of 45°, as shown in Figure 5. The optimized values for PC-1 and PC-2 are adopted from the previous analysis, where the TL is low and light travels without diffraction. For that purpose, we have used a = 420 nm, R = 0.35a, and w = 0.7a for PC-1 and PC-2. When the incident light (TE/TM) travelling from P1 passes the PC-1 and strikes the interface between PC-1 and PC-2, the TM-polarized light can continue its path and exit at P2. As both the formations act as a homogeneous medium and permit the self-collimated TM-polarized light to travel diffraction-free. While TE-polarized light reflects from the interface due to the presence of PBG in PC-2 and experiences a 90° bend. Consequently, a compact on-chip element is constructed which can be utilized in the PDM system to upsurge the number of transmission channels. The footprint of the device studied in this work is ~10.9 μm 2 due to the fact that 13 airholes are periodically arranged in the xy-plane having a = 420 nm.  The employment of the proposed device is easy where an unpolarized light can be coupled at the input port and the device splits the light into two orthogonal polarized lights. The orthogonal states of polarization (SOP) of the same light can be used for data transmission resulting in an expansion in the transmission capacity. The polarization multiplexing approach may be explained by comparing it to microwave communication systems. When two orthogonally polarized radio frequency signals are sent through wireless connections, the operator bandwidth can be increased. Two antennas with distinct polarization and orientation are utilized at the receiver to distinguish the signals. In optical systems, the two orthogonal SOP are detected at the receiver, letting each of the modulated signals to be acquired separately.
The norm. electric field mapping of TE, TM, and TE + TM-polarized light at 1550 nm is shown in Figure 6. It is evident that when the light having TE-polarization is given at the input (P1), the self-collimated TE-polarized light propagates in PC-1 and reflects at 90 • after colliding with PC-2 and eventually being collected at P3, as shown in Figure 6a. Self-collimated polarized light travels throughout both formations and is collected at P2, as shown in Figure 6b. If TE + TM polarized light is launched from P1, TE-polarized light is bent downwards and collected at P3, while TM-polarized light travels straight and is received at P2, as shown in Figure 6c. This gives rise to the opportunity of utilizing this formation for a compact on-chip PDM scheme.
ization and orientation are utilized at the receiver to distinguish the signals. In optical systems, the two orthogonal SOP are detected at the receiver, letting each of the modulated signals to be acquired separately.
The norm. electric field mapping of TE, TM, and TE + TM-polarized light at 1550 nm is shown in Figure 6. It is evident that when the light having TE-polarization is given at the input (P1), the self-collimated TE-polarized light propagates in PC-1 and reflects at 90° after colliding with PC-2 and eventually being collected at P3, as shown in Figure 6a. Selfcollimated polarized light travels throughout both formations and is collected at P2, as shown in Figure 6b. If TE + TM polarized light is launched from P1, TE-polarized light is bent downwards and collected at P3, while TM-polarized light travels straight and is received at P2, as shown in Figure 6c. This gives rise to the opportunity of utilizing this formation for a compact on-chip PDM scheme.

Results and Discussion
PBS is a more challenging device than a polarizer since it must work equally effectively for both TE-and TM-polarized light. The transmission coefficient for both polarization outputs, which should be as near unity as feasible, and cross talk and back reflection, which should be as low as possible, are the most important factors to evaluate. We used only one interface, as opposed to the PBS formation described in [44], which may minimize back-reflected light and increase transmissivity. PBSs based on PC heterostructures are worth investigating further. The polarization extinction ratio (PER) is an essential figure of merit that ought to be taken care of while devising the PBS formation. The power of a specific polarization in the predicted output port versus the power of the same polarization in the other output port is defined as PER. Since it is a logarithmic ratio, the unit is in decibels which can be established for TE-and TM-polarized light by utilizing the following formula: dB 10 and dB 10 , respectively. Where PTM and PTE are the output power of the light having TM-polarization and the light having TE-polarization, respectively. The TL and PER for the 2D-heterostructure PC are calculated for S, C, and L telecommunication bands, as shown in Figure 7. The TL for the whole wavelength spectrum is in

Results and Discussion
PBS is a more challenging device than a polarizer since it must work equally effectively for both TE-and TM-polarized light. The transmission coefficient for both polarization outputs, which should be as near unity as feasible, and cross talk and back reflection, which should be as low as possible, are the most important factors to evaluate. We used only one interface, as opposed to the PBS formation described in [44], which may minimize back-reflected light and increase transmissivity. PBSs based on PC heterostructures are worth investigating further. The polarization extinction ratio (PER) is an essential figure of merit that ought to be taken care of while devising the PBS formation. The power of a specific polarization in the predicted output port versus the power of the same polarization in the other output port is defined as PER. Since it is a logarithmic ratio, the unit is in decibels which can be established for TE-and TM-polarized light by utilizing the following formula: PER P2 (dB) = 10log 10 P TM P TE and PER P3 (dB) = 10log 10 P TE P TM , respectively. Where PTM and PTE are the output power of the light having TM-polarization and the light having TE-polarization, respectively.
The TL and PER for the 2D-heterostructure PC are calculated for S, C, and L telecommunication bands, as shown in Figure 7. The TL for the whole wavelength spectrum is in the range of 1.57-3.5 dB and 0.25-2 dB for TE-and TM-polarized light, respectively. This includes the bending loss suffered by TE-polarized light. The PER calculated at P2 is significantly higher than the one obtained at P3. This is due to the fact that TE-polarized light is entirely reflected from the interface which gives a pure TM-polarized light at P2. The major portion of the TM-polarized light travels to P2, while some part of it reflects at the interface and enters the P3 which reduces the PER at P3. The maximum PER for 1550 nm obtained at P2 and P3 is~40.5 and~6.5 dB, respectively. the range of 1.57-3.5 dB and 0.25-2 dB for TE-and TM-polarized light, respectively. This includes the bending loss suffered by TE-polarized light. The PER calculated at P2 is significantly higher than the one obtained at P3. This is due to the fact that TE-polarized light is entirely reflected from the interface which gives a pure TM-polarized light at P2. The major portion of the TM-polarized light travels to P2, while some part of it reflects at the interface and enters the P3 which reduces the PER at P3. The maximum PER for 1550 nm obtained at P2 and P3 is ~40.5 and ~6.5 dB, respectively.

Conclusions
In conclusion, a compact polarization beam splitter (PBS) device is numerically investigated by utilizing 2D-heterostructure photonic crystals (PCs). The PC formations are designed in a unique way in which PC-1 allows the transfer of self-collimated TE-and TM-polarization of light. PC-2 is constructed such that there exists a photonic band bap (PBG) for TE-polarized light. However, TM-polarized light can travel devoid of any interference. These two distinctive PC formations are combined at an angle of 45°, which reflects the self-collimated TE-polarized light at a 90° sharp bend when it interferes with the PC-2, while allowing the self-collimated TM-polarized light to pass both formations as it does not see any PBG on the way. In this way, both the polarizations are separated at 90° in a compact area. The proposed device offers a small footprint of ~10.9 μm 2 , low transmission loss of 1.57-3.5 dB, and 0.25-2 dB for TE-and TM-polarized light (for S, C, and L band), high polarization extinction ratio of ~40.5 and ~6.5 dB for TM-and TE-polarized light, which makes it likely to be employed in the on-chip polarization division multiplexing systems for doubling the transmission capacity.

Conclusions
In conclusion, a compact polarization beam splitter (PBS) device is numerically investigated by utilizing 2D-heterostructure photonic crystals (PCs). The PC formations are designed in a unique way in which PC-1 allows the transfer of self-collimated TE-and TMpolarization of light. PC-2 is constructed such that there exists a photonic band bap (PBG) for TE-polarized light. However, TM-polarized light can travel devoid of any interference. These two distinctive PC formations are combined at an angle of 45 • , which reflects the self-collimated TE-polarized light at a 90 • sharp bend when it interferes with the PC-2, while allowing the self-collimated TM-polarized light to pass both formations as it does not see any PBG on the way. In this way, both the polarizations are separated at 90 • in a compact area. The proposed device offers a small footprint of~10.9 µm 2 , low transmission loss of 1.57-3.5 dB, and 0.25-2 dB for TE-and TM-polarized light (for S, C, and L band), high polarization extinction ratio of~40.5 and~6.5 dB for TM-and TE-polarized light, which makes it likely to be employed in the on-chip polarization division multiplexing systems for doubling the transmission capacity.