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Abstract
In this paper, we design and demonstrate polymer-based polarization beam splitters realized with parabolic graded-index (GI) directional couplers at 1550 nm. We design and investigate these beam splitters by using the beam propagation method (BPM) and the RSoft CAD BeamPROP solver is used for model design and characterization. We study the splitting ratio of 50:50 and 100:0 of these beam splitters for both transverse electric (TE) and transverse magnetic (TM) modes of the polarized light. In particular, the 50:50 beam splitters exhibit an output efficiency of 95% due to the absence of polarization. However, in the presence of the polarization effect, the polarization dependency is analyzed and manipulated successfully, and the output efficiency increases by approximately 98% at an incident angle of 31$^\circ $ due to the polarized launch field. Moreover, the 100:0 beam splitters act as the switching devices realized with parabolic GI polymer directional couplers. The insertion losses of the GI beam splitters are 0.162 dB and 0.186 dB, while the excess losses are 0.148 dB and 0.149 dB for TE and TM modes, respectively. Also, the polarization extinction ratio (PER) of the 100:0 beam splitters is >25 dB and >20 dB for TE and TM modes, respectively.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Due to the growing bandwidth requirement for high-performance computing systems and high-speed servers, optical interconnects have intensive advantages over electrical interconnects due to the high-speed transmission signals with less attenuation, low power consumption, low crosstalk, and high-speed data transmission rates [1–3]. For the development of next-generation high-performance computing systems, various configurations of short-reach optical interconnects, particularly waveguides [4], beam splitters [5], couplers [6], modulators [7], etc. are required. Polymer optical waveguides are one of the candidate technologies to develop these optical devices because they have drawn intense attention and have demonstrated high-density wiring at a low cost, higher flexibility, easy integration into printed circuit boards (PCBs), simplicity of the fabrication process, and error tolerance [8–10]. Meanwhile, multimode short-reach optical waveguides have received considerable attention due to their low propagation loss, high bandwidth, and low crosstalk [11]. Additionally, single-mode optical waveguides also exhibit low propagation loss and high-bandwidth distant products for PCBs [12]. Hence, various configurations of single-mode waveguides are required to connect silicon photonics chips with single-mode fibers (SMFs) [13].
Over the past couple of years, researchers demonstrated polymer optical waveguides realized with step-index (SI) profiles [14] by applying conventional photolithography-based fabrication techniques. In the SI core waveguides, the refractive index is uniform throughout the core, and light travels through these waveguides following the principles of total internal reflection. The SI waveguides exhibit high loss due to the surface roughness at the core-cladding boundary, thus exhibiting high crosstalk due to the high-density wiring. In contrast, the index profiles of the GI cores are parabolic, and this profile gradually decreases from the center to the periphery. The light propagates through the GI core waveguides in a sinusoidal way, and the GI core exhibits high-bandwidth data transmission and reduces propagation loss and inter-channel crosstalk [15–17] as well.
Researchers have been trying to develop compatible and compact beam splitters [18,19] realized with single-mode waveguides for the development of photonic integrated circuits (PICs). In particular, a beam splitter is capable of splitting two orthogonal modes and can be demonstrated in several configurations of single-mode waveguides [20]. The polarization dependence is one of the primary issues of the single-mode waveguides, whether these waveguides are coupled with silicon photonics chips and other optical components. For example, in 2006, X. Ao et al. [21] demonstrated the polarization dependence beam splitters using photonic crystal structures. The structural design and fabrication processes of these photonic crystal devices are relatively complex. In 2017, H. Wu et al. [22] proposed polarization-dependence beam splitters realized with asymmetrical directional couplers, and these devices are fabricated with incompatible air cladding to realize vertical symmetry and are also sensitive to fabrication variations. In 2019, X. Xu et al. [23] proposed polymer directional couplers with low excess loss and good coupling ratios. The polarization dependence property of these polymer directional couplers was not investigated. In 2016, J. S. Penades et al. [24] and in 2021, C. Deng et al. [25] demonstrated the polarization beam splitters realized with metamaterials-based single-mode waveguides, and these beam splitters exhibit excellent optical properties, particularly low insertion loss, and good PER. The polarization dependence property is a major concern of the single-mode waveguides fabricated with conventional lithography techniques [26–28] and these waveguides exhibit SI core profiles. The polarization-dependent loss becomes more significant because of stresses induced in the polymer materials during the fabrication processes due to the volume shrinkage after curing, resulting in a mismatch between core and cladding materials [29]. In addition, the GI core polymer waveguides fabricated with the Mosquito method [16] exhibit low polarization dependence loss compared to SI core waveguides. The polarization dependence properties of the GI core polymer directional couplers have never been investigated. Hence, the manipulation of a polarization dependence property is highly appreciated.
In this paper, we design and demonstrate 3-dimensional (3D) polymer-based directional couplers as an application of the beam splitter using the BPM. The proposed directional couplers are comprised of straight and bending waveguides realized with parabolic GI core profiles. We simulate the GI directional couplers to evaluate the output efficiency and the splitting ratio between the outputs. For the applications of the directional couplers, we demonstrate and characterize the 50:50 and 100:0 polarization beam splitters, taking the polarization dependency into account. Since the efficiency and splitting ratio of the beam splitters are varied due to the polarization effect, we manipulate the polarization dependency for the polarization beam splitters. We also investigate the splitting ratio of the beam splitters for the TE and TM modes realized with the polarization effect.
2. Design and simulation
When two waveguides are placed close together, and the input signal propagates through one waveguide, light can enter into another waveguide, following the evanescent principle, which is known as mode-coupling theory [30]. In this paper, we propose polymer-based polarization beam splitters, which can work following the mode coupled theory. We demonstrate a schematic of the 3D polarization beam splitter shown in Fig. 1. The details of the beam splitters are as follows: two straight input and output waveguides are connected to another two S-bend waveguides, and these two S-bend waveguides are also connected to another two straight waveguides with a separation gap between them at the middle section. The coupling region is formed with two straight waveguides having a separation gap between them at the middle of the beam splitters.
Fig. 1. Schematic design of a symmetric $2 \times 2$ directional coupler: (a) 3D top view, (b) 3D view with the parameters indication, (c) input cross-section, and (d) output cross-section.
We numerically design and simulate the polarization beam splitters by using the BPM. The polarization behavior of these beam splitters is realized by using a 2 × 2 unitary matrix described by $\hat{U}$ using the Eq. (1) [31]:
For the polarization beam splitters, we use the transformer mode, which can transform the polarized launch field by using a Jones Matrix, T [33,34]. We use the linear polarization effect at the input to obtain the TE and TM modes into the thru and cross ports, respectively, as reported in Fig. 1(b), and the linear polarization effect is demonstrated by the following equation:
We designed these polarization beam splitters with the organic-inorganic hybrid resins named Sunconnect, which was provided by Nissan Chemical Corporation Ltd., Japan. The core and cladding materials are NP-005 (n = 1.575) and NP-211 (n = 1.567), respectively at 1550 nm. These materials exhibit low absorption loss because the concentration of carbon-hydrogen (C-H) bonding per unit volume in these materials is comparatively lower than those of other organic polymer materials. This beam splitter can be fabricated by applying the mosquito method, as demonstrated in [35,17]. In the fabrication process, the liquid cladding monomer is coated inside the silicon frame on the glass substrate, and the core monomer is inserted into the cladding monomer with the needle scan by the tabletop robot. Since the core and cladding monomers are in a liquid state, we cure these liquid monomers by using UV exposure followed by post-baking. It is important to say that to form GI parabolic single-mode waveguides, we need to diffuse the core monomer into the cladding monomer, and this phenomenon can be happened during the core monomer dispensing by the needle scan. The UV light is used to cure the monomers immediately after dispensing the core monomer. Otherwise, the dispensed core monomer may diffuse more into the cladding monomer and expand the core diameter, which may not satisfy the single-mode condition. The advantage of the mosquito method is that it can fabricate 3D optical waveguides. Hence, we can fabricate a 3D polymer beam splitter, as illustrated in Fig. 1.
The cross-sectional size of the GI directional coupler is 8 ${\times} $ 8 µm2. Since the core profile of this polarization beam splitter is GI parabolic, we incorporate the parabolic index profile in the simulation solver realized with the power-law approximation [36] using the following equation:
here, the parameters a and g are the radius and exponential coefficient, respectively. The coefficient, g defines the shape of the index distribution. If the value of g is large or infinite, the index distribution approaches SI-type. However, when g is 2, the refractive index is completely parabolic. We demonstrate the 3D polymer-based polarization beam splitter realized with a parabolic GI profile because the GI core exhibits a relaxed core size compared to the SI counterpart. Since the core size of this beam splitter is single-mode, the core size should follow the single-mode conditions. It is evident that in [17], the core size of the single-mode GI waveguide is less than 14 µm at 1550 nm, and hence the GI waveguide has a relaxed core size of approximately 7 µm compared to the SI waveguide. We demonstrate the index profiles of the GI and SI cores, as in Fig. 2, in which the index profile of the SI core (red color) is uniformly distributed throughout the core, as shown in Fig. 2(a). On the other hand, Fig. 2(b) illustrates that the GI core intensity gradually decreases from center to periphery, and hence the red color is gradually shifted to other colors before touching the core-cladding boundary. The index profile of this GI beam splitter is shown in Fig. 2(c), where the highest peak intensity is at the center of the core and the intensity gradually decreases from the core center to the periphery and exhibits the parabolic GI profiles.Fig. 2. Index profiles of the polymer directional couplers: (a) SI index profile and (b) GI index profile, and (c) parabolic profiles of the GI beam splitter.
The main parameters for the optimization of the 3D polarization beam splitters are as follows: S-bend waveguides, the coupling length, and the separation gap between the two straight waveguides in the middle section, as presented in Fig. 1. First, we optimize the S-bend waveguides, which are directly connected to the coupling region from both ends of the polarization beam splitter, as shown in Fig. 1. We study the S-bend waveguides as a function of the bending radii, and the output intensity is presented in Fig. 3. Since the bending radius and bending length are very closely related to each other, the bending radius varies automatically as the bending length changes. Thus, according to Fig. 3, the highest output intensity is obtained at the bending radius of 92,000 µm when the bending length is1685 µm, which is not included in Fig. 3.
Next, in order to optimize the separation gap and coupling length of the GI beam splitters, we conduct the simulation varying the coupling length and the separation gap simultaneously, and the simulation results are presented in Fig. 4(a) and Fig. 4(b). Figure 4(a) shows the light intensity optimization for the thru port, while the cross-port optimization is presented in Fig. 4(b). The color bar on the right side of Figs. 4(a) and 4(b) shows the intensity distribution throughout the thru and cross port. It is evident that when the light intensity in the thru port is maximum, the cross port exhibits the minimum one, except for the rectangular area indicated by black dotted lines. The dotted line-based rectangular region suggests that both ports receive very nearly equal amounts of light with a uniform splitting ratio. We further optimize the coupling length and the separation gap, which are placed in the black dotted area, as shown in Figs. 4(a) and 4(b), and we present the simulation results in Figs. 4(c) and 4(d), respectively. From Fig. 4(c), we can see that the optimal coupling occurs at the length of 3100 µm for the two symmetric straight waveguides at the coupling region, while the suitable separation gap between the two waveguides for the 50:50 beam splitter is 29.35 µm, as shown in Fig. 4(d), and the beam splitting ratio is 47:48.
Fig. 4. Optimization for the coupling length and separation gap simultaneously: (a) thru port and (b) cross port, (c) coupling length optimization, and (d) separation gap optimization.
After performing a couple of preliminary investigations, we find a solution for the optimized 3D symmetric GI beam splitters. The optimized parameters for the GI beam-splitters having the splitting ratios of 50:50 and 100:0 are demonstrated in Table 1, where all the parameters of both splitters are the same except for the separation gap. The 50:50 beam splitters can act as a 100:0 beam splitter by changing the separation gap only, and the separation gap for 100:0 beam splitters is slightly smaller than that of the 50:50 splitters.
Table 1. The parameters for polymer-based GI polarization beam splitters
3. Results and discussion
3.1 50:50 polarization beam splitters
3.1.1 Performance analysis
We examine the polarization beam splitter, which can act as a 50:50 beam splitter, and the simulation results suggest that the light intensity is uniformly distributed into the two outputs, as shown in Fig. 5. We insert the light into a waveguide, and after the coupling region, the TE mode remains the same waveguide and propagates into the thru (output 1) port, while the TM mode coupled into another waveguide and goes out from the cross (output 2) port. Hence, the beam splitter shows that the two outputs exhibit the same splitting ratio, as in Fig. 5(a) when the separation gap is 29.35 µm. We investigate that the field intensity propagates into the thru and cross ports is 0.4783 a.u. and 0.488 a.u., which are very close to the 50:50 splitting ratio for both TE and TM modes, respectively. The 3D peak values of the 50:50 splitters are shown in Fig. 5(b), which indicates that the two outputs have almost the same peak value. We also introduce the energy density patterns for the 50:50 beam splitters in Fig. 5(c), which exhibit how the light (red color) is distributed from the center to the periphery of the core, and this typical property demonstrates the GI core. We also examine how the output intensity propagates through the GI beam splitters, as shown in Fig. 5(d). The simulation results confirm that the 50:50 beam splitters exhibit the same splitting ratio (50:50) between the two outputs, and the output efficiency for the 50:50 beam splitters is 95%.
Fig. 5. (a) Simulation result of the optical field distribution of the GI profile, (b) the 3D intensity peak of the 50:50 beam splitter, (c) Energy-density profiles of the 50:50 beam splitters, and (d) light propagation through the two outputs.
3.1.2 Polarization Effect
We analyze the polarization effect by using the transformer mode of a polarizer via Jones Matrix [35], which can modify the polarization state of the lunch field. This analysis reveals linear polarization when we insert the light into an input of one waveguide with a fully vectorial approach based on the linear polarization effect. This investigation indicates that the TE mode propagates along the thru port, while the TM mode goes out from the cross port, as shown in Fig. 6(a). The output intensity due to the linear polarization effect is shown in Fig. 6(b), in which the red color curves (solid lines) indicate the polarization effect, whereas the polarization effect is absent by the black dotted lines. Figure 6(b) explains that the input power of the beam splitters is very close to 1.0 a.u., while the polarization effect is absent. However, due to the insertion of the polarization effect, the input power reduces to 0.6349 a.u. Hence, we need to use an alternative technique to obtain a high coupling efficiency at the input light using polarization manipulation, which will be discussed in the next section.
Polarization manipulation demonstrates the solutions to improve the coupling efficiency of the input light for the polarization beam splitters [37]. The linear polarized light is inserted into input by varying a couple of incident angles realized with a Jones Matrix, this study improves the coupling efficiency shown in Fig. 7. Hence, we insert the fully polarized light into a waveguide at the different incident angles, and we observe that at the incident angle of 0 degrees, the input intensity is 0.6 a.u. and the output efficiency is minimum of about 56%, while at 31 degrees, the input intensity and the output efficiency are maximum in values, which are nearly equal to the input intensity and the output efficiency obtained by the unpolarized light. At this 31-degree incident angle, the coupling loss is minimum, and the output efficiency improves to 98% for the GI polarization beam splitters. Thus, polarization manipulation ensures the minimum coupling loss and maximum output efficiency for the symmetric polarization beam splitters, as shown in Fig. 7.
3.2 100:0 polarization beam splitters
We demonstrate the 100:0 polarization beam splitters, which act as the switching devices realized with TM or TE modes. In particular, we convert the 50:50 beam splitter into a 100:0 beam splitter by modifying the separation gap only, and the rest of the parameters remain the same, as shown in Table 1. The separation gaps for the 100:0 beam splitters are 27.6 µm and 26.22 µm for TE and TM modes, respectively, which are determined similarly as for the 50:50 beam splitters, as shown in Fig. 4. When the light is inserted into a single input, the output intensity propagates either in the TM or the TE mode due to the mode coupling property. Figures 8(a) and 8(b) show the field intensity distribution along the cross and thru ports, respectively, and this phenomenon is known as a function of a switching device. Figures 8(c) and 8(d) introduce the 3D intensity peak along the propagation direction for both TM and TE modes, respectively. Additionally, Fig. 8(e) exhibits that the input light propagates through the TM mode (cross port), while the light goes through the TE mode is blocked, and the TE-polarized light goes out from the thru port, while the TM mode is blocked as shown in Fig. 8(f). Since the propagation length is the same for both TE and TM modes, we can change the phase shift of the input signal by varying the separation gap for the 100:0 polarization beam splitters at 1550 nm, and the polarization beam splitters act as the switching devices.
Fig. 8. Simulation results of the field distribution for the 100:0 beam splitters: (a) TM and (b) TE modes, 3D peak intensity distribution: (c) TM and (d) TE modes, and field-intensity distribution through the 100:0 beam splitter: (e) TM and (f) TE modes.
3.2.1 Loss analysis
For the TE and TM mode splitting ratio, the insertion loss (IL) is another critical characteristic of the polarization beam splitters. Since these polarization beam splitters are comprised of straight and bending waveguides, the insertion losses are the combination of coupling loss, bending loss, and propagation loss. We calculate the insertion losses for both TE and TM modes by using the equations [38] as follows:
Both the TE and TM modes exhibit different insertion losses for various wavelengths. We investigate the insertion losses in a wavelength range of 1400-1650 nm along the propagation direction, and the output losses are presented in Fig. 9(a). The loss curves exhibit that the losses of the GI splitters are 0.162 dB and 0.186 dB for TE and TM modes, respectively.
Fig. 9. (a) Insertion losses with the wavelength variation, and (b) excess losses according to the wavelength variation.
The excess loss (EL) for this directional coupler is measured by the equation [39] in dB units as follows:
where, $\sum {P_i}$ and $\sum {P_0}$ are the sum of input and output power, respectively. The excess losses of the polarization beam splitters for both TE and TM modes are reported in Fig. 9(b), which shows the variation of excess loss in a wavelength range of 1330-1630 nm. For our optimized parameters, the dominant wavelength is 1550 nm at which the excess losses are 0.148 dB and 0.149 dB for TE and TM modes, respectively.3.2.2 Polarization extinction ratio
The PER is another important parameter in which light is confined in a principal polarization mode. In particular, the PER is the ratio of the power in the principal polarization mode (TE/TM) to the power in the orthogonal polarization mode (TE/TM) after propagation through a beam splitter. We calculate the PER of the polarization beam splitter for both TE and TM modes by the following equations [40]:
andSince the beam splitter is extremely sensitive to the separation gap between the symmetric straight waveguides, we simulate these devices by varying the separation gap and calculating the PER at 1550 nm for both the thru and cross ports. The recorded data is plotted in Fig. 10, which reveals that the PER is greater than 25 dB and 20 dB for both TE (corresponds to the separation gap of 27.6 µm) and TM (corresponds to the separation gap of 26.22 µm) modes, respectively.
4. Conclusion
In this paper, we have successfully designed and demonstrated the GI core-based polarization beam splitters realized with organic-inorganic hybrid polymer materials. The polarization beam splitters exhibit the splitting ratio of 50:50 and 100:0 for both TE and TM modes. For the 50:50 beam splitter, the output efficiency is 95%, when the polarization is absent. However, due to the insertion of the polarization effect, the output efficiency reduces to 56%. Alternatively, polarization manipulation improves the output efficiency of the polarized beam splitters, and the efficiency approaches 98% when we set the incident angle of the launch field to 31$^\circ $. In addition, for the 100:0 beam splitting ratio, the polarization beam splitter act as a switching device. The insertion losses of the 100:0 beam splitters are 0.162 dB and 0.186 dB for both TE and TM modes, respectively, and the excess losses are 0.148 dB and 0.149 dB for TE and TM modes, respectively. Also, the PER of the 100:0 beam splitters is $> 25\; $dB and $> 20\; $dB for both TE and TM modes, respectively. Therefore, these numerical investigations would be helpful to develop parabolic GI profile-based polarization beam splitters for the next-generation optical computing system.
Acknowledgments
We would like to acknowledge the support of Khulna University Research and Innovation Center for pursuing this research.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the corresponding author upon reasonable request.
References
1. H. Cho, P. Kapur, and K. C. Saraswat, “Power comparison between high-speed electrical and optical interconnect for interchip communication,” J. Lightwave Technol. 22(9), 2021–2033 (2004). [CrossRef]
2. D. A. B. Miller, “Rationale and Challenges for Optical Interconnects to Electronic Chips,” Proc. IEEE 88(6), 728–749 (2000). [CrossRef]
3. C. Xie, “Datacenter optical interconnects: Requirements and challenges,” in Proc. IEEE Opt. Interconnects Conf., pp. 37–38, (2017).
4. S. Yakabe, H. Matsui, Y. Kobayashi, Y. Saito, K. Manabe, and T. Ishigure, “Multi-Channel Single-Mode Polymer Waveguide Fabricated Using the Mosquito Method,” J. Lightwave Technol. 39(2), 547–556 (2021). [CrossRef]
5. M. K. Alam, N. Afsary, M. O. F. Rasel, and T. Ishigure, “Multimode Interference based Y-Branch Polymer Optical Waveguide Splitter: Design and Investigation,” in Proceedings of International Conference on Electronics, Communications and Information Technology, ICECIT 2021, 2021.
6. R. Zhang, S. Sun, F. Zhang, K. Chen, L. Liu, and N. Zhu, “Four-mode parallel silicon multimode waveguide crossing scheme based on the asymmetric directional couplers,” Opt. Express 30(13), 22442–22451 (2022). [CrossRef]
7. I. Taghavi, R. Dehghannasiri, T. Fan, A. Tofini, H. Moradinejad, A. Efterkhar, S. Shekhar, L. Chrostowski, N. Jaeger, and A. Adibi, “Enhanced poling and infiltration for highly efficient electro-optic polymer-based Mach-Zehnder modulators,” Opt. Express 30(15), 27841–27857 (2022). [CrossRef]
8. M. Hikita, R. Yoshimura, M. Usui, S. Tomaru, and S. Imamura, “Polymeric optical waveguides for optical interconnections,” Thin Solid Films 331(1-2), 303–308 (1998). [CrossRef]
9. H. Ma, A. K.-Y. Jen, and L. R. Dalton, “Polymer-based optical waveguides: Materials, process, and devices,” Adv. Mater. 14(19), 1339–1365 (2002). [CrossRef]
10. H. Nawata, J. Oshima, and T. Kashino, “Organic-inorganic hybrid material SUNCONNECT for photonic integrated circuit,” Proc. SPIE 10538, 14 (2019). [CrossRef]
11. F. Shi, N. Bamiedakis, P. P. Vasil’ev, R. V. Penty, I. H. White, and D. Chu, “Flexible Multimode Polymer Waveguide Arrays for Versatile High-Speed Short-Reach Communication Links,” J. Lightwave Technol. 36(13), 2685–2693 (2018). [CrossRef]
12. H. Zuo, S. Yu, T. Gu, and J. Hu, “Low loss, flexible single-mode polymer photonics,” Opt. Express 27(8), 11152–11159 (2019). [CrossRef]
13. R. Dangel, Antonio La Porta, Folkert Horst, et al., “Polymer Waveguides Enabling Scalable Low-Loss Adiabatic Optical Coupling for Silicon Photonics,” IEEE J. Select. Topics Quantum Electron. 24(4), 1–11 (2018). [CrossRef]
14. Y. Maeda, H. Takase, K. Tamaki, F. Huang, and Y. Eriyama, “Polysiloxane based optical waveguide equipped with fiber refractive index guides by using a direct photolithography technique,” in Proc. IEEE Conf. Opt. Fiber Commun., 2005, Paper OME33.
15. Y. Koike, Y. Takezawa, and Y. Ohtsuka, “New interfacial-gel copolymerization technique for steric GRIN polymer optical waveguides and lens arrays,” Appl. Opt. 27(3), 486–491 (1988). [CrossRef]
16. R. Kinoshita, D. Suganuma, and T. Ishigure, “Accurate interchannel pitch control in graded-index circular-core polymer parallel optical waveguide using the Mosquito method,” Opt. Express 22(7), 8426–8437 (2014). [CrossRef]
17. O. F. Rasel and T. Ishigure, “Circular core single-mode 3-dimensional crossover polymer waveguides fabricated with the Mosquito method,” Opt. Express 27(22), 32465–32479 (2019). [CrossRef]
18. H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S. Itabashi, “Ultrasmall polarization splitter based on silicon wire waveguides,” Opt. Express 14(25), 12401–12408 (2006). [CrossRef]
19. X. Huang, Y. Liu, Z. Li, Z. Fan, and W. Han, “High-performance and compact integrated photonics platform based on silicon rich nitride–lithium niobate on insulator,” APL Photonics 6(11), 116102 (2021). [CrossRef]
20. H. Qin, X. Shi, and H. Ou, “Exceptional points at bound states in the continuum in photonic integrated circuits,” Nanophotonics 11(21), 4909–4917 (2022). [CrossRef]
21. X. Ao, L. Liu, W. Lech, and S. He, “Polarization beam splitter based on a two-dimensional photonic crystal of pillar type,” Appl. Phys. Lett. 89(17), 171115 (2006). [CrossRef]
22. H. Wu, Y. Tan, and D. Dai, “Ultra-broadband high-performance polarizing beam splitter on silicon,” Opt. Express 25(6), 6069–6075 (2017). [CrossRef]
23. X. Xu, L. Ma, and Z. He, “3D polymer directional coupler for on-board optical interconnects at 1550 nm,” Opt. Express 26(13), 16344–16351 (2018). [CrossRef]
24. J. S. Penades, A. Ortega-Moñux, M. Nedeljkovic, J. G. Wangüemert-Pérez, R. Halir, A. Z. Khokhar, C. Alonso-Ramos, Z. Qu, I. Molina-Fernández, P. Cheben, and G. Z. Mashanovich, “Suspended silicon mid-infrared waveguide devices with subwavelength grating metamaterial cladding,” Opt. Express 24(20), 22908 (2016). [CrossRef]
25. C. Deng, M. Lu, Y. Sun, L. Huang, D. Wang, G. Hu, R. Zhang, B. Yun, and Y. Cui, “Broadband and compact polarization beam splitter in LNOI hetero-anisotropic metamaterials,” Opt. Express 29(8), 11627 (2021). [CrossRef]
26. X. Shi, J. Zhang, W. Fan, Y. Lu, N. Peng, K. Rottwitt, and H. Ou, “Compact low-birefringence polarization beam splitter using vertical-dual-slot waveguides in silicon carbide integrated platforms,” Photonics Res. 10(1), A8 (2022). [CrossRef]
27. Y. H. Min, M. H. Lee, and J. Y. Do, “Polarization dependent loss in polymeric rib channel waveguide,” IEEE Photonics Technol. Lett. 12(11), 483–1485 (2000). [CrossRef]
28. S. Y. Cheng, K. S. Chiang, and H. P. Chan, “Polarization dependence in polymer waveguide directional couplers,” IEEE Photon. Technol. Lett. 17(7), 1465–1467 (2005). [CrossRef]
29. M. F. Hossain, H. P. Chan, M. A. Uddin, and R. K. Y. Li, “Efficient design of polarization independent polymer optical waveguide devices,” inProc. 14th OptoElectron. Commun. Conf., paper WA5, (2009). [CrossRef]
30. M. Weiss and R. Srivastava, “Spectral characteristics of asymmetric directional couplers in graded index channel waveguides analyzed by coupled-mode and normal-mode techniques,” Appl. Opt. 34(6), 1029–1040 (1995). [CrossRef]
31. G. Corrielli, S. Atzeni, S. Piacentini, I. Pitsios, A. Crespi, and R. Osellame, “Symmetric polarization-insensitive directional couplers fabricated by femtosecond laser writing,” Opt. Express 26(12), 15101–15109 (2018). [CrossRef]
32. C. Lee, “Design of polarization-independent coarse wavelength splitters based on ridge-waveguide directional couplers,” Int. J. Opt. 2011, 1–9 (2011). [CrossRef]
33. O. Schwelb, “Evolution of the polarization in codirectional and contradirectional optical couplers,” J. Opt. Soc. Am. 72(9), 1152–1158 (1982). [CrossRef]
34. R. Clark Jones, “A New Calculus for the Treatment of Optical Systems: Description and Discussion of the Calculus,” J. Opt. Soc. Am. 31(7), 488–493 (1941). [CrossRef]
35. K. Soma and T. Ishigure, “Fabrication of a graded-index circular-core polymer parallel optical waveguide using a microdispenser for a high-density optical printed circuit board,” IEEE J. Select. Topics Quantum Electron. 19(2), 3600310 (2013). [CrossRef]
36. R. A. Sammut and A. K. Ghatak, “Perturbation theory of optical fibres with power-law core profile,” Opt. Quantum Electron. 10(6), 475–482 (1978). [CrossRef]
37. W. Bogaerts, D. Taillaert, P. Dumon, D. Thourhout, R. Baets, and E. Pluk, “A polarization-diversity wavelength duplexer circuit in silicon-on-insulator photonic wires,” Opt. Express 15(4), 1567–1578 (2007). [CrossRef]
38. H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S. Itabashi, “Silicon photonic circuit with polarization diversity,” Opt. Express 16(7), 4872–4880 (2008). [CrossRef]
39. Y. Shao, X. Han, Z. Lu, Z. Wu, and J. Teng, “Optimal design of 850 nm 2×2 multimode interference polymer waveguide coupler by Imprint technique,” Photon. Sensors 6(3), 234–242 (2016). [CrossRef]
40. J. Zhang, X. Shi, Z. Zhang, K. Guo, and J. Yang, “Ultra-compact, efficient and high-polarization-extinction-ratio polarization beam splitters based on photonic anisotropic metamaterials,” Opt. Express 30(1), 538–549 (2022). [CrossRef]
