Open Access
Add to BibSonomy
Abstract
We propose a high fabrication tolerance and broadband silicon polarization beam splitter (PBS) by cascading point-symmetric 3-dB couplers designed using fast quasiadiabatic dynamics (FAQUAD). The FAQUAD strategy only requires access to a single control parameter, and it can shorten the conventional adiabatic designs, maintaining the broadband characteristic and robustness to fabrication errors. The 3-dB FAQUAD coupler for the TM0 mode is made invisible to the TE0 mode due to a large difference in their adiabaticity parameters. Moreover, the system of cascaded point-symmetric devices further enhances the bandwidth and fabrication tolerance of the FAQUAD PBS. The total length of the FAQUAD PBS is 89.4 µm. We find that the FAQUAD PBS exhibits > 10 dB extinction ratio (ER) for the input TM0 mode and the input TE0 mode over a bandwidth of 240 nm and 260 nm, with excess losses below 0.4 dB and 0.021 dB, respectively. The PBS also exhibits excellent fabrication tolerance, showing an ER > 24 dB and an excess loss < 0.19 dB for the TM0 mode, and an ER > 16.8 dB and an excess loss < 0.024 dB for the TE0 mode for waveguide width deviation from −100 nm to 100 nm. For silicon thickness variation from 210 nm to 230 nm, an ER > 15.3 dB and excess loss < 0.2 dB for the TM0 mode, and an ER > 15.9 dB and excess loss < 0.015 dB for the TE0 mode are observed.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nowadays, silicon photonics plays a significant role and draws a great amount of interests from many researchers in the field of photonics integrated circuits (PICs). The platform based on silicon-on-insulator (SOI) is very common today due to its advantages of high index-contrast that gives rise to very strong waveguide modal confinement, enabling high-density integration of optical components; and compatibility with commercial complementary metal-oxide-semiconductor (CMOS) process [1], enabling direct application of the same semiconductor processes to PICs. However, the high index-contrast of SOI also brings some drawbacks, one of which is the strong polarization dependence caused by structural birefringence that induces polarization dependent dispersion and loss, limiting the performance of devices in optical communications [2]. Therefore, efficient polarization diversity components such as the polarization beam splitters (PBSs), which are fundamental polarization-manipulation components used to separate the fundamental TE mode (TE0 mode) and the fundamental TM mode (TM0 mode) to different output ports of devices in PICs, are demanded. In the past few years, many types of PBSs have been proposed, such as designs using multimode interferences (MMIs) [3–5], Mach-Zehnder interferometers (MZIs) [6,7], photonic crystal (PhC) structures [8,9], grating structures [10,11], and symmetric or asymmetric directional couplers (DCs) [12,13]. The PBSs based on MMIs and MZIs have the advantage of simple structure, but MMIs usually struggle with large excess loss, sensitivity to wavelength and long device length. MZIs usually have long device length, and the bandwidth is also limited. Ultra-short PBSs can be realized by PhC structures or grating structures, but precise fabrication of PhCs and gratings is challenging. DCs usually have compact size, and the structure is relatively simple to fabricate. Nevertheless, DCs are sensitive to the wavelength because of the phase mismatch. Recently, a PBS designed by point-symmetric cascaded DCs with asymmetric phase control sections has been proposed [14]. The DC with asymmetric phase control sections has improved wavelength sensitivity [15], leading to broadband response of the PBS. However, this PBS requires precise fabrication to implement the phase control sections.
Intuitively, we can improve the bandwidth and the fabrication tolerance by applying adiabatic designs to the couplers, making the mode evolving through the device adiabatically [16]. However, the couplers based on adiabatic mode evolution have the shortcoming of very long device lengths, and they usually have the characteristic of polarization independence [17], which is an unfavorable condition for realizing PBSs using the point-symmetric network. Many quantum control theories have been used to implement light manipulation in waveguide structures recently, thanks to the analogies between quantum mechanics and wave optics [18]. The fast quasiadiabatic dynamics (FAQUAD), a member of the family of shortcuts to adiabaticity (STA) protocols, has been proposed to speed up the slow adiabatic evolution in the context of quantum control [19]. FAQUAD has been applied to a number short and robust silicon photonics devices, such as asymmetric Y-junction mode (de)multiplexer [20], polarization splitter-rotator [21], and 3-dB coupler [22]. By this technique, we can effectively shorten the adiabatic couplers used in the point-symmetric network, and the broadband characteristic and robustness to fabrication errors can be well maintained. Moreover, using the large difference in the adiabaticity parameters for the TE0 and the TM0 modes, the coupler can be made invisible to the TE0 mode but functions as a 3-dB coupler for the TM0 mode. In this paper, we first design a 3-dB coupler for the TM0 mode using FAQUAD, then a FAQUAD PBS is designed by cascading point-symmetric 3-dB FAQUAD couplers [14,23]. The total device length is 89.4 µm, and the FAQUAD PBS shows 10 dB extinction ratio (ER) for the input TM0 mode and the input TE0 mode over a bandwidth of 240 nm and 260 nm with excess losses below 0.4 dB and 0.021 dB, respectively. The fabrication tolerance is also investigated, and the FAQUAD PBS exhibits an excellent robustness against fabrication imperfection.
2. 3-dB coupler for the TM0 mode using FAQUAD
Figure 1(a) shows the cross-sectional view of the 3-dB coupler designed for the TM0 mode. An SOI substrate with 220-nm-thick top silicon layer and a 2-µm-thick buried oxide layer for device design is used, and the cladding layer is silica. Figure 1(b) shows the top view of the 3-dB coupler, and the Z axis represents the light propagation direction. The middle section of the 3-dB coupler in Fig. 1(b) is consisted of two tapered waveguides, evenly dividing the input light into the output ports of the 3-dB coupler. The widths of the top waveguide and the bottom waveguide are varied from W1 = 0.4 µm to W0 = 0.5 µm and W2 = 0.6 µm to W0 = 0.5 µm, respectively. L is the 3-dB coupler length, W(z) is the variable taper width varying from 0 µm to 0.1 µm, and the gap G1 is fixed at 0.5 µm. The beginning and end sections of the coupler are S-bend waveguides, where the gap is varied from G0 = 1.5 µm to G1 = 0.5 µm, avoiding the coupling between adjacent waveguides. We choose the length of the S-bend waveguides to be Ls = 8.7 µm to minimize the bending loss.
The operating principle of adiabatic mode evolution devices is to keep a guided mode in the instantaneous eigenmode of the device and to prevent this guided mode from coupling to other eigenmodes [17]. This operating principle leads to the characteristics of insensitivity to wavelength variations and high fabrication tolerance in adiabatic devices. For the 3-dB adiabatic coupler in Fig. 1, we employ the finite difference method (FDM) solver of a commercial software (FIMMWAVE, Photon Design) [24] to analyze the eigenmodes of the 3-dB coupler. Figure 2 shows the relation between the effective indices and the taper function W(z) for the first four eigenmodes of the 3-dB coupler, and the insets in Fig. 2 show the electric fields in the y direction for the 3rd and the 4th eigenmodes at W(z) = 0 µm (input ports) and 0.1 µm (output ports). If the widths of the two tapered waveguides vary slowly enough, the mode evolution would go along one of the eigenmodes without crossing into another eigenmode. For this purpose, the parameters of the adiabatic devices usually change very slowly, resulting in a long device length. For instance, as shown in Fig. 3, when the TM0 mode is input from the bottom (top) port of a 3-dB adiabatic coupler [W(z) is a linear function], the third (fourth) eigenmode will be excited and preserved throughout the coupler [along the yellow (purple) line in Fig. 2] if the coupler length L is long enough, and the input TM0 mode is 50/50 divided ultimately. We simulate the 3-dB adiabatic coupler by employing a full vectorial eigenmode expansion method (EME) [24], and we can see from the simulation that the light power can be split into 50/50 when the coupler length L is 150 µm (insets of Fig. 3).
Fig. 2. The relation between the effective indices and W(z) for the first four eigenmodes of the 3-dB coupler. The electric fields in the y direction for the 3rd and the 4th eigenmodes at W(z) = 0 µm and 0.1 µm are shown in the insets.
Fig. 3. The relation between 3-dB adiabatic coupler length L and transmission of (a) the 3rd eigenmode (b) the 4th eigenmode. The insets show the light distribution in the 3-dB adiabatic coupler with a coupler length L of 150 µm.
To quantify device adiabaticity, we can define the adiabaticity parameter C(z) by [25,26]:
where $|{m,n} \rangle$ are the eigenmodes of an optical waveguide, βm and βn are the propagation constants of the mth and the nth eigenmode, and the dot denotes derivative with respect to z. Based on the adiabatic theorem, C(z) must be small enough to satisfy the adiabatic condition, that is, $C(z) \ll 1$ [25,26]. First, we substitute the 3rd and the 4th eigenmodes of optical waveguide $|{3,4} \rangle$ (TM supermodes) and propagation constants β3 and β4 into Eq. (1), and the blue-dashed line in Fig. 4(a) shows the C(z) of the 3-dB adiabatic coupler. We can see that C(z) is not homogeneously distributed and increases rapidly when W(z) is larger than 0.06 µm, meaning that a 3-dB adiabatic coupler needs to be long enough to compensate for this large C(z) in order to satisfy the adiabatic condition. That is why adiabatic devices are usually very long. Therefore, if we can reduce the value of C(z) when W(z) is above 0.06 µm, the device length would not need to be so long to compensate for the large C(z). In other words, we can shorten the 3-dB adiabatic coupler.Fig. 4. (a) Device adiabaticity parameter C(z) for the TM supermodes of the 3-dB adiabatic coupler and the 3-dB FAQUAD coupler. (b) Variation of W with z for the 3-dB FAQUAD coupler. (c) Transmission of the 4th eigenmode as a function of device length L for the 3-dB adiabatic coupler and the 3-dB FAQUAD coupler. The inset shows the light distribution in the L = 36 µm 3-dB FAQUAD coupler.
The FAQUAD approach is to engineer a single control parameter, here we use the waveguide width W, to make the mode evolution as quick as possible while maintaining the adiabaticity [19]. By imposing a small constant ε and using the chain rule, Eq. (1) can be expressed as:
Next, we analyze the relation between transmission of the 2nd eigenmode and device lengths L for the same 3-dB FAQUAD coupler by launching the TE0 mode into the top input. Figure 5(a) shows the adiabaticity parameter C(z) for the TE supermodes of the 3-dB FAQUAD coupler, and we can find that the C(z) for the TE supermodes is not redistributed homogeneously. Instead, C(z) becomes extremely large when W(z) is above 0.099 µm, that is, the adiabatic condition is violated. In other words, the device does not function as a mode evolution device for the TE mode, but works as a mode coupling device instead. Since the mode coupling length for the TE mode is much longer than the FAQUAD coupler length, the TE0 mode goes through directly as shown in Fig. 5(b), making the 3-dB coupler invisible to the TE0 input. From inset of Fig. 4(c) and Fig. 5(b), we can see that when the TM0 mode is launched into the coupler, the light is split 50/50; and when the TE0 mode is launched into the coupler, the light goes through the coupler directly. Based on the results we mention above, we can design a PBS which splits the TE0 and TM0 modes into different output ports by cascading point-symmetric 3-dB FAQUAD couplers, which we discuss in the next section.
Fig. 5. (a) Device adiabaticity parameter C(z) for the TE supermodes of the 3-dB FAQUAD coupler. (b) Light distributions in the 3-dB FAQUAD coupler for the TM0 mode input and (c) the TE0 mode input.
3. PBS by cascaded point-symmetric 3-dB FAQUAD couplers
Figure 6 shows the schematic of cascaded point-symmetric devices, where the left device is the original device and the right device is the point-symmetry device. O1 and O2 are the output light power of the through port and the cross port of the original device, respectively. O3 and O4 are the output light power of the through port and the cross port of the system of cascaded point-symmetric devices, respectively. And we assume that there is no loss in the system. Applying the transfer matrix, the original device can be expressed as [14,23]
The 3-dB FAQUAD coupler we designed previously can be used as the original device in Fig. 6. O1 and O2 are equal or close to 0.5 when we launch the TM0 mode into the 3-dB FAQUAD coupler. Figure 7(a) shows the relation between transmission and wavelength for O1 ∼ O4 of the 3-dB FAQUAD coupler using the TM0 mode input, and we can see that for a wavelength range from 1.5 µm to 1.6 µm, the transmissions of O1 and O2 are within 0.37 ∼ 0.54 and 0.43 ∼ 0.56, respectively. If we cascade two point-symmetric 3-dB FAQUAD couplers, we can obtain the relation between transmission and wavelength for O3 and O4 from Eq. (9), as shown in Fig. 8(b), and the transmissions of O3 and O4 are within 0 ∼ 0.04 and 1 ∼ 0.96 with enhanced robustness.
Fig. 7. The relation between transmission and wavelength of (a) O1 and O2, (b) O3 and O4 of the 3-dB FAQUAD coupler using the TM0 mode input.
It is worth mentioning that the 3-dB FAQUAD coupler itself has high fabrication tolerance. Figure 8(a) shows the relation between transmission and Δw for O1 and O2 of the 3-dB FAQUAD coupler using the TM0 input, and we can see that for a Δw range from −100 nm to 100 nm, the transmissions of O1 and O2 are both within 0.46 ∼ 0.5. If we cascade two point-symmetric 3-dB FAQUAD couplers, we can obtain the relation between transmission and Δw for O3 and O4 shown in Fig. 7(b), and the transmissions of O3 and O4 are nearly 0 and 1, respectively. From Figs. 7(b) and 8(b), we clearly see that the bandwidth and fabrication tolerance are enhanced by using the system of cascaded point-symmetric devices.
Fig. 8. The relation between transmission and Δw of (a) O1 and O2, (b) O3 and O4 of the 3-dB FAQUAD coupler using the TM0 mode input.
In addition, we can find from Eq. (9) that if O1 and O2 are equal or close to 1 and 0, respectively (when the TE0 mode is launched into the 3-dB FAQUAD coupler), O3 is equal or close to 1 and O4 is equal to or close to 0; that is, the TE0 mode will output from the through port without coupling to the cross port. Hence, we can design a broadband and high fabrication tolerance PBS by cascading point-symmetric 3-dB FAQUAD couplers.
Figure 9(a) shows the schematic of FAQUAD PBS, and the parameters of the FAQUAD PBS are summarized as follows: W0 = 0.5 µm, W1 = 0.4 µm, W2 = 0.6 µm, G0 = 1.5 µm, G1 = 0.5 µm, L = 36 µm, Ls = 8.7 µm, and the total length of the FAQUAD PBS is 89.4 µm. The FAQUAD PBS is simulated at a wavelength of 1.55 µm, and Figs. 9(b) and 9(c) show the simulated light distribution in the FAQUAD PBS. We can see that when the TM0 mode is launched into the FAQUAD PBS, it outputs from the cross port; and when the TE0 mode is launched into the FAQUAD PBS, it goes through directly without coupling and exits from the through port. Therefore, the TE0 and TM0 modes of the input light are successfully split.
Fig. 9. (a) Schematic of FAQUAD PBS. The light distribution in the FAQUAD PBS using cascaded point-symmetric 3-dB FAQUAD couplers with (b) the TM0 mode input and (c) the TE0 mode input. The total length of the device is 89.4 µm.
4. Performance analysis
The bandwidth of the FAQUAD PBS is simulated. The ER for the TE0 mode and the TM0 mode are expressed as:
Figure 10 shows the wavelength dependence of the transmission of the PBSs using the TM0 and TE0 inputs. We can see that when the TM0 mode is launched into the FAQUAD PBS, the ERTM0 is above 10 dB and the excess loss is below 0.4 dB from 1.46 µm to 1.7 µm, and when the TE0 mode is launched into the FAQUAD PBS, the ERTE0 is above 10 dB and the excess loss is below 0.021 dB from 1.43 µm to 1.69 µm. The FAQUAD PBS can provide >10 dB ER for TM0 mode and TE0 mode over a bandwidth of 240 nm and 260 nm, respectively.
Fig. 10. Transmission of the FAQUAD PBS as a function of wavelength for (a) the TM0 mode input and (b) the TE0 mode input.
Next, the fabrication tolerance of the FAQUAD PBS is simulated. We investigate the fabrication tolerance of the FAQUAD PBS by adding a Δw to the width of FAQUAD PBS, and the wavelength is fixed at 1.55 µm. Figure 11 shows the relation between transmission and Δw, and we can see that the PBS exhibits an excellent performance. The ERTM0 is above 24 dB and the excess loss is below 0.19 dB; and the ERTE0 is above 16.8 dB and the excess loss is below 0.024 dB for Δw from −100 nm to 100 nm. As far as we know, the observed fabrication tolerance of the FAQUAD PBS is much better than the other PBS designs (most of the PBS designs have an extinction ratio of around 12 dB for Δw from −50nm to 50nm [3–14,16,28–31]). The results in Figs. 10 and Fig. 11 indicate that the FAQUAD PBS has large bandwidth and has excellent robustness to fabrication error.
Fig. 11. Transmission of the FAQUAD PBS as a function of Δw for (a) the TM0 mode input and (b) the TE0 mode input at 1.55 µm.
We also investigate the fabrication tolerance of FAQUAD the PBS at 1.7 µm and 1.45 µm input wavelength, near the edges of the operating band, as shown in Figs. 12 and 13. We can see from Fig. 12 that for a Δw range from −100 nm to 100 nm, The ERTM0 is above 8.3 dB and the excess loss is below 0.58 dB; and the ERTE0 is above 4.6 dB and the excess loss is below 0.12 dB at 1.7 µm. Furthermore, we can also see from Fig. 13 that for a Δw range from −100 nm to 100 nm, The ERTM0 is above 5.4 dB and the excess loss is below 0.061 dB; and the ERTE0 is above 11.1 dB and the excess loss is below 0.014 dB at 1.45 µm.
Fig. 12. Transmission of the FAQUAD PBS as a function of Δw for (a) the TM0 mode input and (b) the TE0 mode input at 1.7 µm.
Fig. 13. Transmission of the FAQUAD PBS as a function of Δw for (a) the TM0 mode input and (b) the TE0 mode input at 1.45 µm.
Furthermore, we add Δw = 10 nm and −10 nm to the FAQUAD PBS and investigate the device bandwidth, as shown in Figs. 14 and 15. From Fig. 14, we can see that the ERTM0 is above 10 dB and the excess loss is below 0.37 dB from 1.46 µm to 1.69 µm for the TM0 mode input, and the ERTE0 is above 10 dB and the excess loss is below 0.017 dB from 1.4 µm to 1.69 µm for the TE0 mode input. The FAQUAD PBS can provide >10 dB ER for the TM0 mode and TE0 mode over a bandwidth of 230 nm and 290 nm, respectively. From Fig. 15, we can see that the ERTM0 is above 10 dB and the excess loss is below 0.39 dB from 1.46 µm to 1.68 µm for the TM0 mode input, and the ERTE0 is above 10 dB and the excess loss is below 0.02 dB from 1.43 µm to 1.68 µm for the TE0 mode input. The FAQUAD PBS can provide >10 dB ER for the TM0 mode and TE0 mode over a bandwidth of 220 nm and 250 nm, respectively. Therefore, the FAQUAD PBS still maintains its broadband characteristic broadband even if it has a fabrication deviation Δw of ± 10 nm.
Fig. 14. Transmission of the FAQUAD PBS with a Δw = 10 nm width deviation as a function of wavelength for (a) the TM0 mode input and (b) the TE0 mode input.
Fig. 15. Transmission of the FAQUAD PBS with a Δw = −10 nm as a function of wavelength for (a) the TM0 mode input and (b) the TE0 mode input.
To further investigate the fabrication tolerance, we simulate the transmission of the FAQUAD PBS with different silicon thickness, and the wavelength is fixed at 1.55 µm. Figures 16(a) and 16(b) show the transmission of the FAQUAD PBS with different silicon thickness, and we can see that for a silicon thickness range from 210 nm to 230 nm, the ERTM0 is above 15.3 dB and the excess loss is below 0.2 dB; and the ERTE0 is above 15.9 dB and the excess loss is below 0.015 dB.
Fig. 16. Transmission of the FAQUAD PBS with different silicon thickness for (a) the TM0 mode input and (b) the TE0 mode input.
5. Conclusion
In summary, a broadband and fabrication tolerant FAQUAD PBS by cascaded point-symmetric 3-dB couplers has been proposed, successfully splitting the input TM0 mode and the input TE0 mode to the cross port and through port, respectively. The FAQUAD protocol can shorten the conventional adiabatic device and maintain the characteristic of broadband and robustness to fabrication error by controlling only a single system variable. By engineering the system adiabaticity, the TE0 is made invisible to the 3-dB FAQUAD coupler. The structure of cascaded point-symmetric devices can further enhance the bandwidth and fabrication tolerance of the device. The total length of the FAQUAD PBS is 89.4 µm. The bandwidth of the FAQUAD PBS has been investigated, and the FAQUAD PBS can provide >10 dB ER for TM0 mode and TE0 mode over a bandwidth of 240 nm and 260 nm, respectively. We have also investigated the fabrication tolerance of the FAQUAD PBS by adding errors to the waveguide width, and the PBS exhibits an excellent performance with an ERTM0 above 24 dB and an excess loss below 0.19 dB, and an ERTE0 above 16.8 dB and an excess loss below 0.024 dB for a Δw range from −100 nm to 100 nm. Device bandwidths when adding fabrication deviations of Δw = 10 nm and −10 nm to the FAQUAD PBS have also been investigated, and the performance of the FAQUAD PBS still maintains the broadband characteristic. For silicon thickness variations from 210 nm to 230 nm, the ERTM0 is above 15.3 dB and the excess loss is below 0.2 dB; and the ERTE0 is above 15.9 dB and the excess loss is below 0.015 dB.
Funding
Ministry of Science and Technology, Taiwan (105-2221-E-006-151-MY3, 108-2221-E-006-204-MY3, 108-2218-E-992-302).
References
1. T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, “Microphotonics devices based on silicon microfabrication technology,” IEEE J. Sel. Top. Quantum Electron. 11(1), 232–240 (2005). [CrossRef]
2. L. Pavesi and D. J. Lockwood, Silicon Photonics (Springer, 2004).
3. B. M. A. Rahman, N. Somasiri, C. Themistos, and K. T. V. Grattan, “Design of optical polarization splitters in a single-section deeply etched MMI waveguide,” Appl. Phys. B: Lasers Opt. 73(5−6), 613–618 (2001). [CrossRef]
4. Y. Xu, J. Xiao, and X. Sun, “Compact polarization beam splitter for silicon-based slot waveguides using an asymmetrical multimode waveguide,” J. Lightwave Technol. 32(24), 4884–4890 (2014). [CrossRef]
5. M. Yin, W. Yang, Y. Li, X. Wang, and H. Li, “CMOS-compatible and fabrication-tolerant MMI-based polarization beam splitter,” Opt. Commun. 335, 48–52 (2015). [CrossRef]
6. D. Dai, Z. Wang, J. Peters, and J. Bowers, “Compact polarization beam splitter using an asymmetrical mach-zehnder interferometer based on silicon-on-insulator waveguides,” IEEE Photonics Technol. Lett. 24(8), 673–675 (2012). [CrossRef]
7. D. Dai, Z. Wang, and J. Bowers, “Considerations for the design of asymmetrical mach–zehnder interferometers used as polarization beam splitters on a submicrometer silicon-on-insulator platform,” J. Lightwave Technol. 29(12), 1808–1817 (2011). [CrossRef]
8. Y. Shi, D. Dai, and S. He, “Proposal for an ultra-compact PBS based on a photonic crystal-assisted multimode interference coupler,” IEEE Photonics Technol. Lett. 19(11), 825–827 (2007). [CrossRef]
9. P. Rani, Y. Kalra, and R. K. Sinha, “Complete photonic bandgap-based polarization splitter on silicon-on-insulator platform,” J. Nanophotonics 10(2), 026023 (2016). [CrossRef]
10. Y. Tang, D. Dai, and S. He, “Proposal for a grating waveguide serving as both a polarization splitter and an efficient coupler for silicon-on-insulator nanophotonic circuits,” IEEE Photonics Technol. Lett. 21(4), 242–244 (2009). [CrossRef]
11. H. Qiu, Y. Su, P. Yu, T. Hu, J. Yang, and X. Jiang, “Compact polarization splitter based on silicon grating-assisted couplers,” Opt. Lett. 40(9), 1885–1887 (2015). [CrossRef]
12. D. Dai, “Silicon polarization beam splitter based on an asymmetrical evanescent coupling system with three optical waveguides,” J. Lightwave Technol. 30(20), 3281–3287 (2012). [CrossRef]
13. D. W. Kim, M. H. Lee, Y. Kim, and K. H. Kim, “Planar-type polarization beam splitter based on a bridged silicon waveguide coupler,” Opt. Express 23(2), 998–1004 (2015). [CrossRef]
14. Z. Lu, Y. Wang, F. Zhang, N. A. F. Jaeger, and L. chrostowski, “Wideband silicon photonics polarization beamsplitter based on point-symmetric cascaded broadband couplers,” Opt. Express 23(23), 29413–29422 (2015). [CrossRef]
15. Z. Lu, H. Yun, Y. Wang, Z. Chen, F. Zhang, N. A. F. Jaeger, and L. Chrostowski, “Broadband silicon photonic directional coupler using asymmetric-waveguide based phase control,” Opt. Express 23(3), 3795–3808 (2015). [CrossRef]
16. Z. Su, E. Timurdogan, E. S. Hosseini, J. Sun, G. Leake, D. D. Coolbaugh, and M. R. Watts, “Four-port integrated polarizing beam splitter,” Opt. Lett. 39(4), 965–968 (2014). [CrossRef]
17. Y. Wang, L. Xu, H. Yun, M. Ma, A. Kumar, E. El-Fiky, and R. Li, “Polarization-independent mode-evolution-based coupler for the silicon-on-insulator platform,” IEEE Photonics J. 10(3), 4900410 (2018). [CrossRef]
18. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3(3), 243–261 (2009). [CrossRef]
19. S. Martínez-Garaot, A. Ruschhaupt, J. Gillet, Th. Busch, and J. G. Muga, “Fast quasiadiabatic dynamics,” Phys. Rev. A 92(4), 043406 (2015). [CrossRef]
20. H.-C. Chung, K. S. Lee, and S.-Y. Tseng, “Short and broadband silicon asymmetric Y-junction two-mode (de)multiplexer using fast quasiadiabatic dynamics,” Opt. Express 25(12), 13626–13634 (2017). [CrossRef]
21. H.-C. Chung and S.-Y. Tseng, “ultrashort and broadband silicon polarization splitter-rotator using fast quasiadiabatic dynamics,” Opt. Express 26(8), 9655–9665 (2018). [CrossRef]
22. Y.-J. Hung, Z.-Y. Li, H.-C. Chung, F.-C. Liang, M.-Y. Jung, T.-H. Yen, and S.-Y. Tseng, “Mode evolution based silicon-on-insulator 3-dB coupler using fast quasiadiabatic dynamics,” Opt. Lett. 44(4), 815–818 (2019). [CrossRef]
23. K. Jinguji, N. Takato, Y. Hida, T. Kitoh, and M. Kawachi, “Two-port optical wavelength circuits composed of cascaded Mach-Zehnder interferometers with point-symmetrical configurations,” J. Lightwave Technol. 14(10), 2301–2310 (1996). [CrossRef]
24. FIMMWAVE / FIMMPROP, Photon Design Ltd, http://www.photond.com.
25. M. Born and V. Fock, “Beweis des adiabatensatzes,” Z. Physik 51(3−4), 165–180 (1928). [CrossRef]
26. L. I. Schiff, Quantum Mechanics (Mcgraw-Hill, 1981).
27. S. Martinez-Garaot, J. G. Muga, and S.-Y. Tseng, “Shortcuts to adiabaticity in optical waveguides using fast quasiadiabatic dynamics,” Opt. Express 25(1), 159–167 (2017). [CrossRef]
28. D. Dai and J. E. Bowers, “Novel ultra-short and ultra-broadband polarization beam splitter based on a bent directional coupler,” Opt. Express 19(19), 18614–18620 (2011). [CrossRef]
29. Y. Ding, H. Ou, and C. Peucheret, “Wideband polarization splitter and rotator with large fabrication tolerance and simple fabrication process,” Opt. Lett. 38(8), 1227–1229 (2013). [CrossRef]
30. B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 µm2 footprint,” Nat. Photonics 9(6), 378–382 (2015). [CrossRef]
31. Y. Zhang, Y. He, J. Wu, X. Jiang, R. Liu, C. Qiu, X. Jiang, J. Yang, C. Tremblay, and Y. Su, “High-extinction-ratio silicon polarization beam splitter with tolerance to waveguide width and coupling length variations,” Opt. Express 24(6), 6586–6593 (2016). [CrossRef]
