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The mode property and light propagation in a tapered silicon-on-insulator (SOI) nanowire with angled sidewalls is analyzed. Mode hybridization is observed and mode conversion between the TM fundamental mode and higher-order TE modes happens when light propagates in a waveguide taper which is used very often in the design of photonic integrated devices. This mode conversion ratio is possible to be very high (even close to 100%) when the taper is long enough to be adiabatic, which might be useful for some applications of multimode photonics. When the mode conversion is undesired to avoid any excess loss as well as crosstalk for photonic integrated circuits, one can depress the mode conversion by compensating the vertical asymmetry in the way of reducing the sidewall angle or introducing an optimal refractive index for the upper-cladding. It is also possible to eliminate the undesired mode conversion almost and improve the desired mode conversion greatly by introducing an abrupt junction connecting two sections with different widths to jump over the mode hybridization region.
© 2015 Optical Society of America
1. Introduction
Silicon photonics has been attracting lots of intention in the recent years and great progresses have been achieved by utilizing the silicon-on-insulator (SOI) platform with very thin top silicon layer [1]. The popular singlemode SOI nanowire is designed usually to have a cross section of ~500 × ~220nm2 and various functionality elements based on SOI nanowires have been realized, including arrayed-waveguide gratings (AWGs) [2–6 ], microring resonators (MRRs) [7,8 ], etc. One might note that most of these silicon photonic integrated devices demonstrated were designed to work for TE polarization only and there are very few silicon nanphotonic integrated devices working for TM polarization. The possible reasons include the following parts. First, it is well known that an SOI nanowire usually has very high birefringence so that silicon nanophotonic integrated circuits are usually strongly polarization-sensitive and it is not easy to achieve the design working for both polarizations simultaneously [9,10 ]. Second, for the ~500nm × ~220nm SOI nanowires used popularly [11], the guided TE-polarization mode has stronger optical confinement than the guided TM-polarization mode, and consequently it allows smaller bending radii, which helps realize ultra-compact devices (e.g., MRRs).
On the other hand, one should realize that TM-polarization mode is also very useful for many applications due to its unique advantages over TE-polarization mode [12]. First, for a ~500nm × ~220nm SOI nanowire, TM-polarization mode is less sensitive to the variation of the waveguide width than TE-polarization mode, which makes the TM-type silicon nanophotonic integrated circuit more tolerant to the dimension deviation due to the fabrication errors. Furthermore, this also helps reduce the scattering loss due to the sidewall roughness and consequently it is possible to make TM-polarization mode have lower loss than TE-polarization mode [13]. Second, TM-polarization mode has much stronger evanescent field than TE-polarization mode, which helps achieve higher sensitivity for the application of optical sensing [14]. Therefore, it is still desired to develop silicon nanophotonic integrated circuits working for TM polarization.
As it well known, an adiabatic taper is an indispensable basic element for designing various photonic integrated circuits [15–22 ]. However, one has to be careful when designing TM-type silicon nanophotonic integrated circuits because there might be some undesired mode conversions between the TM fundamental (TM0) mode and the higher-order modes of TE polarization (e.g., TE1 mode) in an adiabatic taper, as suggested in [23–26 ]. According the results presented in [23,24 ], those mode conversions might happen very efficiently when light propagation in an adiabatic taper based on an SOI nanowire with vertical asymmetry. Such a mode conversion will introduce a significant excess loss as well as some channel crosstalk due to the excited higher-order modes (e.g., in AWG demultiplexers [27]). For example, a submicron SOI rib waveguide, which has been used very popularly for silicon-based integrated optoelectronics [28–33 ], is a typical structure with vertical asymmetry even when having the same material for the upper-cladding and the under-cladding. Consequently, strong mode conversion was observed between the TM0 mode and the higher-order mode of TE polarization in a regular adiabatic taper as well as a bi-level adiabatic taper [23]. Fortunately, it is still possible to remove the undesired mode conversion effect by carefully designing the taper parameters (e.g., the etching depth, the width and length of the taper, etc). Another classical example of an optical waveguide with vertical asymmetry is an SOI nanowire whose upper-cladding is not made of the same material as the insulator layer (SiO2) [24]. Definitely, when the silicon core is rectangular, this mode conversion can be avoided easily by choosing a SiO2 upper-cladding to make the SOI nanowire symmetric vertically if there is not any specific requirement for the upper-cladding [24]. However, the situation becomes quite different for an optical waveguide with a non-rectangular core region, which often happens in practice because the sidewall of the core is usually not vertical perfectly due to the fabrication process [34–36 ]. Figure 1 shows the cross section of a low-loss SOI nanowire fabricated by IMEC foundry as an example [37]. It can be seen that the silicon core has non-vertical sidewalls and the angle is about 8°. People have already noticed that the sidewall angle plays an important role for light propagation in an optical waveguide [34–37 ]. For example, the theoretical simulation has indicated the possibility of significant polarization crosstalk even in planar SiO2 waveguide when the sidewalls are not exactly vertical [34]. The experimental and theoretical results have shown that significant polarization crosstalk occurs in an ultra-sharp SOI-nanowire bend due to the angled sidewalls [36]. In [37], it has been shown that the sidewall angle has some influences on the bending loss. Nevertheless, for SOI nanowires, when the sidewall is not vertical, TE polarization mode can be still supported well in theory and various photonic integrated devices have been demonstrated to work for TE polarization successfully [37]. However, when using SOI nanowires with angled sidewalls to realize photonic integrated devices working for TM polarization, there might be some problems due to the possibility of the mode conversion between the TM0 mode to the TEi mode. In this paper, we give a theoretical analysis on how the angled sidewalls influences the light propagation of TM-polarization modes in an adiabatic taper, which will be helpful for the design of silicon photonic integrated circuits for TM polarization.
2. Structures and analyses
Figure 2 shows the cross section of the SOI nanowire width angled sidewalls considered in this paper. It is assumed that the SOI nanowire has a 220nm-thick top Si layer (i.e., h co = 220nm), which is used very popularly [13,14 ]. The refractive indices of Si, and SiO2 are assumed as n Si = 3.476, and n SiO2 = 1.444, respectively.
The sidewall angle θ could be positive or negative. When θ>0, the top width w co for the core region is less than the bottom width w′co. Otherwise (θ<0), one has w co>w′co. If the sidewall angle θ = 0, the silicon core region becomes rectangular. Here a finite-difference method (FDM) mode-solver (from Fimmwave, Photon Design, UK) is used to calculate the mode field profiles and the effective indices for all eigenmodes. For the simulation of the light propagation in the defined photonic waveguides, FIMMPROP (Photon Design) employing an eigenmode expansion and matching method [38] is then used.
2.1. Mode hybridization and conversion in SOI nanowires with angled sidewalls
Figure 3(a)-3(e) show the calculated effective indices for all the guided-modes supported in the SOI nanowires with different sidewall angle (θ = 0°, 2°, 8°, 15°, and −15°) as the core width w co increases from 0.2μm to 2μm. Here the upper-cladding is chosen as SiO2 to be the same as the under-cladding material. Thus, when θ = 0°, the SOI nanowire becomes a regular strip waveguide, which is symmetrical in the vertical direction.
Fig. 3 Calculated effective indices for the eigen-modes of SOI nanowires with different sidewall angles θ. (a) θ = 0°; (b) θ = 2°; (c) θ = 8°; (d) θ = 15°; (e) θ = −15°. Here the total height of the Si layer is H = 220nm, and the cladding is SiO2.
From Fig. 3(b)-(d), it can be seen that the results for the cases of θ≠0 (e.g., θ = 2°, θ = 8°, 15°) are very different from that for the case of θ = 0°, especially in the regions remarked by the dotted-circles [see Fig. 3(a)]. In these marked regions, the curves for TMi and TEi + k come to be close (in which k is an odd integer, and k>0) while these two close curves keep anti-crossing and there is a small gap between them. This gap will become larger when the sidewall angle θ increases. The sidewall angle θ is also possible to be negative (i.e., θ<0), in which case there is also some vertical asymmetry and thus the small gap due to the mode hybridization also appears as shown in Fig. 3(e). From Fig. 3(d)-(e), one can also see that the negative and positive sidewall-angles have similar influence on the mode hybridization effect. In contrast, when θ = 0° the two curves for TMi and TEi + k just cross each other and there is no any gap between them. The reason for such a significant difference is that the SOI nanowire with angled sidewalls (e.g., θ = 8°) has vertical asymmetry, which introduces the mode hybridization in the remarked regions. Mode hybridization in other SOI nanowires with vertical asymmetry has also been observed theoretically and experimentally in previous literatures [23,24 ]. In the following parts, we focus on the analysis for the case of θ = 8° as an example and the result for the case with a different sidewall angle is similar.
In order to evaluate how much the mode hybridization is, we define the following mode polarization ratio γx:
where Ex and Ey are x-component and y-component of the electrical field for any eigen mode considered. For a typical TE mode, the x-component Ex is much significant than the y-component Ey and consequently the ratio γx is close to 100%. In contrast, for an typical TM mode, one will have γx≈0. When the mode becomes hybridized, one has 0<γx <100%.Figure 4 shows the calculated the ratios γx of all guided modes for the case of θ = 8° as the waveguide width w co varies. The curves are labeled with numbers, which are consistent with those shown in Fig. 3(a). From Fig. 4, it can be seen that the ratio γx for the mode TEi + k (i≥0 and k>0) have a transition between 100% and 0 in the regions locating at around some specific waveguide widths w co0 = 0.62, 0.8, 1.4, 1.7, and 2.0μm. These transition regions are consistent with those labeled by the circles in Fig. 3. In these transition regions, the hybridized modes have comparable x-component and y-component of the electrical field. As an example, we consider the mode hybridization region around w co0 = 0.62μm and calculate the mode profiles (Ex and Ey) for the two hybridized modes (mode #2 and #4) when choosing w co0 = 0.628μm, as shown in Fig. 5 . In this case one has γx = ~50% (which indicates a very strong mode hybridization). It can be seen that the x-component (Ex) and the y-component (Ey) are comparable. As a result, the mode conversion might happen when light propagates in an adiabatic taper whose end-widths w 1 and w 2 are chosen so that w 1<w co0<w 2, where w co0 is the specific waveguide width for the mode hybridization region (e.g., w co0 = 0.628μm here). In order to show the mode conversion in an adiabatic taper [see Fig. 6(a) ], as an example we simulate the light propagation along a taper whose end-widths are chosen as w 1 = 0.56μm, and w 2 = 0.70μm for the case when the TM0 mode is launched. Figure 6(b) shows the calculated mode conversion ratio for the mode conversion from the TM0 mode to the TE1 mode as the taper length increases. It can be seen that both TM0 and TE1 modes are excited simultaneously when the taper is short not to be adiabatic. On the other hand, when the taper length L tp is long enough to be adiabatic (e.g., L tp>200μm here), the efficiency of the TM0→TE1 mode conversion is as high as 100% almost. This indicates that the launched TM0 mode is converted to the TE1 mode at the output end completely, which can be seen clearly from the simulated light propagation in the designed taper, as shown in Fig. 6(c). One should note that the mode conversion has some dependence on the variation ∆w of the waveguide core width introduced by the fabrication error. For example, the simulation results for the case with ∆w = + 40nm (i.e., w 1 = 0.6μm, and w 2 = 0.74μm) are also shown in Fig. 6(b) and 6(d). It can be seen that the dependence of the TM0→TE1 mode conversion ratio on the taper length has some slight change when the waveguide width has a variation of ∆w = + 40nm. When the taper length is long enough (e.g., here L tp>100μm here), the TM0→TE1 mode conversion ratio becomes insensitive to the width variation ∆w and the mode conversion still happens with high efficiency. This is observed directly from the simulated light propagation in the 200μm-long taper, as shown in Fig. 6(d). This indicates this type of adiabatic taper is robust for the applications of polarization rotation (as demonstrated in [23–26 ]), which might be useful for on-chip polarization-handling systems as well as mode-division-multiplexed optical interconnect systems [39]. For this kind of applications, one can further strengthen this kind of mode hybridization and mode conversion by e.g. making the material of the upper-cladding different from that of the under-cladding to enhance the vertical asymmetry of the optical waveguide if needed. On the other hand, such a mode conversion should be avoided in some cases for a silicon photonic integrated devices designed to work for TM polarization because it introduces a significant excess insertion loss as well as some channel crosstalk.
Fig. 4 Calculated mode polarization ratio γx = E x 2/(E x 2 + E y 2) for SOI nanowires with θ = 8°. Here the total height of the Si core layer is hco = 220nm, and the cladding is SiO2.
Fig. 5 Field profiles (Ex and Ey) for modes #2 and #4 of a SOI nanowire with θ = 8°, w co = 0.628μm, and ncl = 1.444 (SiO2). (a) mode #2; (b) mode #4. The total height of the Si core layer is h co = 220nm .
Fig. 6 (a) Schematic configuration of a taper based on an SOI nanowire with angled sidewalls; (b) Calculated mode conversion ratio when light propagates along a taper whose width varies from w 1 to w 2; (c) Simulated light propagation in the designed taper with w 1 = 0.56μm and w 2 = 0.70μm when the TM0 mode is launched. (d) Simulated light propagation in the designed taper with w 1 = 0.60μm and w 2 = 0.74μm when the TM0 mode is launched. Here the sidewall angle θ = 8°.
2.2. Elimination of the mode conversion in SOI nanowires with angled sidewalls
According to the analyses given in Fig. 3(a)-(c), it can be seen that mode hybridization happens in the region around some special widths (e.g., w co = 0.62μm) and there is a gap between the two curves in the mode hybridization region when the SOI nanowire is asymmetrical in the vertical direction (e.g., the sidewall angle θ≠0). For the mode hybridization region around w co = 0.62μm, the mode conversion between the TM0 mode and the TE1 mode happens with a mode conversion ratio even close to 100% if the taper length is long enough to be adiabatic [see Fig. 6(b)]. In order to avoid this kind of mode conversion in a taper, a straightforward way is to compensate the vertical asymmetry of the SOI nanowires. When the vertical asymmetry of the SOI nanowire becomes less (e.g., with smaller sidewall angles), one has a smaller gap between the two curves in the mode hybridization region [see Fig. 3(a)-(e)], and thus longer taper is required to be adiabatic according to the coupled mode theory [40]. This indicates that the mode conversion will become less for a taper with a fixed length (e.g., L tp = 40μm).
In order to make the SOI nanowire have less vertical asymmetry, a direct way is to reduce the sidewall angle, which can be realized by optimizing the etching process carefully [37]. Here we simulate the light propagation in a 40μm-long taper with the end-widths w 1 = 0.5μm and w 2 = 1.0μm when the sidewall angle θ varies from −15° to 15° as an example. Figure 7(a) shows the calculated mode conversion ratios when the TM0 mode is launched at the input end. It can be seen that the undesired mode conversion from the TM0 mode to the TE1 mode becomes less when the sidewall angle is smaller as predicted. The sidewall angle should be |θ|<2° to ensure the desired conversion ratio from the TM0 mode to the TM0 mode to be high enough (>98.5%) for the case of L tp = 40μm. As example, Fig. 7(b) shows the simulated light propagation in the designed taper with L tp = 40μm and θ = 2° when the TM0 mode is launched. In this case, the TM0→TM0 conversion ratio is as high as 98.6% and no notable multimode interference is observed in Fig. 7(b), which indicates that the TM0 mode is the dominant mode at the output end as predicted.
Fig. 7 (a) Calculated mode conversion ratio when light propagates along a 40μm-long taper as the sidewall angle varies from −15° to 15°; (b) Simulated light propagation along a 40μm-long taper when θ = 2°. The width varies from w 1 = 0.5μm to w 2 = 1.0μm,and the TM0 mode is launched.
According to the effective index method [41], an SOI nanowire with a positive sidewall (θ>0) can be equivalent as a slab waveguide whose refractive index decreases vertically from the bottom to the top. Therefore, another possible way to compensate the vertical asymmetry of an SOI nanowire with positive angled sidewalls is introducing an upper-cladding whose refractive index (n cl) is optimally higher than that of the insulator layer (n bf). In this way, the vertical asymmetry of the SOI nanowire with positive angled sidewalls will be reduced when the upper-cladding’s refractive index increases from 1.444 (which is the same that of the insulator layer). On the other hand, when the refractive index n cl is very high, the SOI nanowire will become asymmetrical vertically again and mode hybridization happens. Therefore, there is an optimal refractive index for the upper-cladding to compensate the vertical asymmetry of an SOI nanowire with angled sidewalls and thus avoid the mode conversion due to the mode hybridization. Figure 8(a) shows the calculated mode conversion ratios when the TM0 mode is launched and propagates along a taper with different refractive indices for the upper-cladding. Here θ = 8°, w 1 = 0.5μm, w 2 = 1.0μm, and L tp = 40μm. From Fig. 8(a), it can be seen that the optimal refractive index n cl is around n cl = 1.62 to ensure low conversion ratio from the TM0 mode to the TE1 mode. This optimal refractive index is achievable by using the SiON material whose refractive index can be tuned in a large range by changing the recipe for the thin-film deposition process. Figure 8(b) shows the simulated light propagation in the designed taper with L tp = 40μm and n cl = 1.62 when the TM0 mode is launched. In this case, the TM0→TM0 conversion ratio is as high as 99.8% and there is no multimode interference almost, which indicates that the TM0 mode is the dominant mode at the output end as predicted. On the other hand, from Fig. 8(a), one also sees that it is possible to control the mode hybridization and mode conversion if the refractive index can be tuned with a large range.
Fig. 8 (a) Calculated mode conversion ratio when light propagates along a 40μm-long taper as the refractive index ncl of the upper-cladding varies from 1.444 to 1.75; (b) Simulated light propagation along a 40μm-long taper when n cl = 1.62. Here the TM0 mode is launched, and θ = 8°, w 1 = 0.5μm, w 2 = 1.0μm.
Regarding that it might be inconvenient to change the upper-cladding material for achieving the optimal refractive index or modify the etching recipe for achieving SOI nanowires with vertical sidewalls, it is desired to develop some simple approach. Note that the desired TM0→TM0 mode conversion ratio can be pretty high when the taper length is very short (see Fig. 6). This provides a potential design to avoid the mode conversion due to the mode hybridization by using an ultra-short taper. In [42], the design with abrupt junctions was proposed to avoid the mode conversion in a taper based on SOI nanowires with an air-cladding. This method can also be used for the present case and we consider an abrupt waveguide junction which connects two sections with different core widths (w 1, w 2). Assuming that w 1 = 0.5μm (which is usually chosen for a singlemode SOI nanowire), light propagation in this junction is simulated with the TM0 mode launched at the input end. Figure 9(a) shows the calculated mode conversion ratios as the width w 2 varies. It can be seen that the TM0→TM0 mode conversion ratio is not sensitive to the width w 2 when w 2>0.70μm and there is maximum as high as 98.2% for the TM0→TM0 mode conversion ratio when choosing w 1 = 0.5μm, and w 2 = 0.78μm. Meanwhile, the TM0→TE1 mode conversion ratio becomes very weak (~1.0%). It can be seen that the TM0 mode is the dominant mode excited in the wide section. This can be verified from the simulated light propagation in the designed junction with w 2 = 0.78μm when the TM0 mode is launched from the section with w 1 = 0.5μm, as shown in Fig. 9(b). From this figure, it can be seen that there is no notable multimode interference in the section with w 2 = 0.78μm. Therefore, this design with an abrupt junction provides a good option to jump over the mode hybridization region and thus avoid the undesired mode conversion very well. For those regions without mode hybridization, one can use the regular adiabatic taper, which works very well. In this way, a low-loss taper for TM polarization can be realized for the design of silicon photonic integrated circuits.
Fig. 9 (a) Calculated mode conversion ratio at a junction as the width w 2 varies; (b) Simulated light propagation in the designed junction with w 2 = 0.78μm. Here w 1 = 0.5μm, θ = 8°, and n cl = 1.444.
3. Conclusions
In this paper, we have given a detailed analysis for the mode hybridization and mode conversion when light propagates along a taper based on SOI nanowires with angled sidewalls. For an SOI nanowires with angled sidewalls, mode conversion between the TM0 mode and higher-order TE modes might happens due to the mode hybridization in the regions around some specific waveguide widths. This mode conversion efficiency is possible to be very high (even close to 100%) when the taper is long enough to be adiabatic, which might be useful for some applications of multimode photonics [39, 43 ]. For example, the TM0 → TE1 mode conversion can be utilized to realize polarization splitter-rotators [23–26 ]. Such a kind of mode conversion also possibly enables the data-exchange between these two mode channels for a mode-divison-multiplex optical interconnect link [39]. On the other hand, the mode conversion will become harmful due to the possible excess loss as well as crosstalk. In order to eliminate the undesired mode conversion, intrinsically speaking, one should compensate the vertical asymmetry. One straightforward solution is optimizing the recipe of the etching process to reduce the sidewall angle, which definitely plays a key role for these mode hybridization and conversion discussed in this paper. The simulation result show that the sidewall angle should be less than 2° to guarantee a low mode conversion from the TM0 mode to the higher-order modes of TE polarization in a 40μm-long waveguide taper. We have also proposed another possible solution by introducing an optimal refractive index for the upper-cladding, so that the vertical asymmetry of the SOI nanowire can be less greatly. Another possible solution is introducing an abrupt junction connecting two sections with different widths (w 1, w 2) to jump over the mode hybridization region around w 0 (w 1< w 0<w 2). In this way, the conversion ratio to the desired mode is up to 98.2% while the conversion ratio to the undesired mode is lower than 1%, which provides a good approach to realize a low-loss taper and silicon photonic integrated circuits for TM polarization. The present numerical analysis can be extended to SOI ridge waveguides, which have been used very widely. For SOI ridge waveguides, the vertical asymmetry is further enhanced when there is positive sidewall angle (i.e., θ>0) and thus strong mode hybridization might be introduced. This should be realized and considered carefully in the design of silicon photonic integrated circuits.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (NSFC) (61422510, 91233208, 6141101056); The Doctoral Fund of Ministry of Education of China (No. 20120101110094); The Fundamental Research Funds for the Central Universities.
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