An inverse taper on silicon is proposed and designed to realize an efficient mode converter available for the connection between multimode silicon nanophotonic integrated circuits and few-mode fibers. The present mode converter has a silicon-on-insulator inverse taper buried in a 3 × 3μm2 SiN strip waveguide to deal with not only for the fundamental mode but also for the higher-order modes. The designed inverse taper enables the conversion between the six modes (i.e., TE11, TE21, TE31, TE41, TM11, TM12) in a 1.4 × 0.22μm2 multimode SOI waveguide and the six modes (like the LP01, LP11a, LP11b modes in a few-mode fiber) in a 3 × 3μm2 SiN strip waveguide. The conversion efficiency for any desired mode is higher than 95.6% while any undesired mode excitation ratio is lower than 0.5%. This is helpful to make multimode silicon nanophotonic integrated circuits (e.g., the on-chip mode (de)multiplexers developed well) available to work together with few-mode fibers in the future.
© 2015 Optical Society of America
In order to improve the link capacity of optical interconnects, currently it is becoming important to develop some (de)multiplexing technology to enhance the capacity for single wavelength carrier. As a promising option, the mode-division-multiplexing (MDM) technology has been attracting lots of attention in recent years . In this case, multiple guided-modes in a multimode bus waveguide/fiber are utilized simultaneously to enable multiple channels carrying different data in parallel while all mode-channels share the same wavelength emitted from the same laser diode, which lowers the cost greatly [2–4 ]. Particularly, for on-chip optical networks, optical signals propagate along planar optical waveguides within the chip and the guided-modes carrying different data could be converted, transferred, as well as (de)multiplexed efficiently by introducing some specific photonic integrated circuits (PICs), which is not difficult with the mature design and fabrication technologies currently. The key device for MDM systems is mode (de)multiplexers for the realization of mode conversion and (de)multiplexing . In the past years, various on-chip mode (de)multiplexers have been developed successfully by using multimode interference (MMI) couplers , adiabatic mode-evolution couplers [7,8 ], asymmetrical Y-junctions [9–11 ], cascaded Mach-Zehnder Interferometers (MZI) , and asymmetrical directional-couplers (ADCs) [13–18 ]. These mode (de)multiplexers were realized by using a very thin silicon-on-insulator (SOI) wafer which is singlemode in the vertical direction. In this case, the multimode bus waveguide is designed to support multiple modes in the lateral direction by making the silicon core region large enough [7–18 ]. For example, a 4-channel mode (de)multiplexer based on cascaded ADCs  was demonstrated to deal with the TM11, TM21, TM31, and TM41 modes. The field profile for the TMm 1 mode has m peaks in the lateral direction and only one peak in the vertical direction, as shown in Fig. 1(a) . These developed mode (de)multiplexers can even work together with polarization-division-multiplexing (PDM) as well as wavelength-division-multiplexing (WDM) devices to enable a kind of multi-dimensional hybrid (de)multiplexing technology [19–23 ]. These recent progresses makes the MDM technology more and more useful and attractive for ultra-high capacity optical networks.
As it well known, MDM has also been proposed to enhance the capacity of optical fiber communications for a long time . However, it is very difficult to handle many higher-order modes in a traditional multimode optical fiber. In recent years, people have developed a modified MDM link with few-mode fibers, in which only a few modes are supported so that the mode-handling becomes simplified. Currently, the few-mode fiber supporting six guided-modes is becoming one of the most popular candidates [24,25 ]. The six guided-modes are the LP01, LP11a, LP11b modes [see Fig. 1(b)] for TM- and TE-polarizations. For the fiber case, it is still a big challenge to realize mode (de)multiplexers. Most mode (de)multiplexers for few-mode fiber systems were developed with some complicated free-space optical setups [26–28 ], e.g., including two beam splitters, two lens and two phase plates . Another possible approach is using a long period fiber grating to convert a LP01 mode to LP11 mode and the converted modes are then (de)multiplexed with free-space couplers or fiber couplers [29–31 ]. Apparently these setups are usually bulky and require high-precision alignments for the elements. Moreover, a large insertion loss is introduced due to the usage of beam splitters for combining light beams, especially when many modes are involved. All-fiber mode multiplexers based on mode-selective couplers were also reported to (de)multiplex the LP modes [32,33 ]. However, the fabrication for all-fiber mode multiplexers is complicated and one has to control the parameters (e.g., the taper diameter, the length of the coupling region, etc.) very carefully. Therefore, it is still desired to develop some compact integrated-type mode (de)multiplexers for few-mode fiber systems. In , a smart silicon PIC including five two-dimensional grating couplers and four thermal-tunable phase-shifters was demonstrated to realize the mode-multiplexing functionalities for all the six mode-channels in a few-mode fiber. The problem is that such a silicon PIC has a complex configuration as well as large excess loss.
As mentioned above, various on-chip mode (de)multiplexers have been developed by using MMI couplers, directional couplers, Y-junctions as demonstrated in [6–18 ]. It will be very helpful if these on-chip mode (de)multiplexers are available for few-mode fiber systems. However, they are designed to deal with the higher-order modes in the lateral direction, none of which is compatible with the LP11a/b higher-order mode [whose field profile has two peaks in the vertical direction, as shown in Fig. 1(a) and 1(b)] in few-mode fibers. This mode incompatibility causes that these on-chip mode (de)multiplexers reported in [6–18 ] cannot be used for few-mode fiber systems directly. In , Hanzawa et al. designed and fabricated a PLC (planar lightwave circuit)-based mode multi/demultiplexer for the LP01, LP11a, and LP11b modes by using an asymmetric mode coupler and an LP11 mode rotator. In the present paper we propose and design an inverse taper to realize efficient mode conversion from six guided-modes of a multimode SOI nanowire waveguide to the six guided-modes which are compatible with the LP01, LP11a, LP11b modes in few-mode fibers. As it well known, an inverse taper has been demonstrated and used for the efficient coupling between an SOI nanowire and a singlemode optical fiber [36–39 ], which works for the fundamental mode only. In contrast, the inverse taper proposed in this paper is to enable the connection between multimode silicon nanophotonic integrated circuits and few-mode fibers. The present mode converter consists of an SOI inverse taper buried in a large SiN strip waveguide. As an example, an inverse taper is designed optimally to enable the conversion between the six modes (i.e., TE11, TE21, TE31, TE41, TM11, TM12) in a 1.4 × 0.22μm2 multimode SOI waveguide and the six modes (like the LP01, LP11a, LP11b modes in a few-mode fiber) in a 3 × 3μm2 SiN strip waveguide. With the inverse taper designed optimally, one has a mode conversion efficiency of >95.6% and a mode extinction ratio of <0.5% for all the six modes (including the fundamental mode as well as the higher-order modes). In this way, multimode silicon nanophotonic integrated circuits  will be available to work together with few-mode fibers in the future.
2. Principle and structure design
Figure 2(a) shows the schematic configuration of the present mode converter based on an inverse taper for multimode silicon photonic integrated circuits. This mode converter consists of an SOI inverse taper buried in a SiN strip which is covered by a SiO2 upper-cladding [see the cross sections at different longitudinal positions shown in Fig. 2(b)-2(d)]. This looks similarly to the inverse tapers reported in [36–39 ] for singlemode SOI nanowires. One should note that only the fundamental mode conversion was considered previously. Here we need realize low-loss mode conversion for not only the fundamental mode but also the higher-order modes, and consequently the situation becomes very complicated. Here this SiN strip waveguide, whose cross section is shown in Fig. 2(b), is designed to support at least six guided-modes (which are compatible with those six modes supported in a few-mode fiber) by choosing the width w SiN and the height h SiN of the SiN core appropriately. The SOI inverse taper is gradually tapered from a silicon nano-tip [w tip, see Fig. 2(c)] to a silicon mulitmode waveguide [w Si, see Fig. 2(d)]. Theoretically speaking, it is ideal to make the width of the nano-tip to be zero so that there is not any facet reflection as well as mode mismatching losses. However, it is impossible almost in reality due to the limitation of the feature seize of the fabrication processes. Nevertheless, the nano-tip should be narrow enough to achieve low-loss connection from the SiN strip waveguide (without any silicon core) to the silicon waveguide.
In this paper, we assume that the refractive indices of Si, SiN and SiO2 are n Si = 3.455, n SiN = 2.0, and n SiO2 = 1.445, respectively, when operating at the wavelength range around λ = 1550nm. For the SiN strip waveguide, we choose w SiN = h SiN = 3 μm so that the six guided-modes (corresponding to those six LP modes supported in a few-mode fiber) are supported, as shown in Fig. 3 . Here one should note that the growth of this 3μm-thick SiN thin film might not be easy due to tensile stress . Fortunately, thick low-loss SiN layers have been grown by plasma-enhanced chemical vapour deposition  and low-pressure chemical vapour deposition , which will be helpful for the fabrication of the present mode converter with a SiN strip waveguide. It is also possible to choose other material with similar refractive index like Ti2O5 (n = 2.35)  when needed.
As an example, here we consider an SOI wafer with a 220nm-thick top silicon layer as it is very popular for silicon nanophotonic integrated circuits . In order to determine how small the tip width should be, first we calculate all the six mode coupling between the SiN strip waveguide [see Fig. 2(b)] and the SiN waveguide with a silicon nano-tip [see Fig. 2(c)] for all the six modes. Figure 4 shows the calculated coupling ratio from the lunched i-th mode in the SiN strip to the i-th mode in the SiN waveguide with a silicon nano-tip (i = 1, …, 6). From this figure, it can be seen that the dependences on the tip width are different for different modes. For example, in order to achieve a coupling ratio of >99%, modes #1, #2, and #5 can tolerate a tip width as large as 0.11μm. However, for modes #4, and #6, the tip width should be as small as 0.045μm, which is pretty difficult for the fabrication. According to the feature seize of the fabrication processes [36–39 ], here we choose the tip width w tip as 60nm. In this case, all the six mode coupling ratios between the SiN strip waveguide [see Fig. 2(b)] and the silicon nano-tip waveguide [see Fig. 2(c)] are higher than 98.4%, which indicates that the mode coupling loss is negligible.
Then it is desired to efficiently convert the guided-modes of the silicon nano-tip waveguide to the guided-modes of the silicon mulitmode waveguide through an optimally designed inverse taper which varies the silicon core width from w tip to w Si. In order to design the inverse taper appropriately, first we give a detailed analysis for the guided-modes supported in the inverse taper as the silicon core width w Si varies. A finite-difference method (FDM) mode-solver with the grid size of ∆x = 20nm and ∆y = 20nm is used for the calculation of the mode field profiles as well as the effective indices for all guided-modes. Figure 5(a) shows the calculated effective index n eff for the six lowest-order modes when the core width varies from 0.05μm to 1.6μm. Figure 5(b) and the inset in it show the enlarged view so that those guided-modes whose effective indices are close can be distinguished.
From Fig. 5(a)-5(b), it can be seen that there are some special regions locating around w Si = w Si0 (e.g., w Si0 = 0.572μm, 0.73μm, …) remarked by the dotted-circles. In these marked regions, the two curves for the i-th mode and the j-th mode come to be close while these two close curves keep anti-crossing and there is a small gap between them. These regions with gaps indicate that there is mode hybridization happening and the electrical-field components Ex and Ey are comparable for the modes (see the insets). The mode hybridization is due to the vertical asymmetry of the cross section of the optical waveguide, which is similar to those reported previously [45,46 ]. It is well known that mode conversion will happen when light propagates in an adiabatic taper whose end-widths w 1 and w 2 are chosen to be w 1<w Si0<w 2, where w Si0 is the specific waveguide width for the mode hybridization region. For example, for the region around w Si0 = 0.73μm, when choosing w 1 = 0.6μm and w 2 = 0.8μm, the mode conversion from the TM11 mode to the TE21 mode happens when light propagates along this taper if the taper is long enough to be adiabatic. Similar phenomena for the mode conversion have also been observed theoretically and experimentally in other SOI nanowires previously [45,46 ]. Figure 6(a) shows the field profiles for all the guided-modes (mode #1~6) when w Si = 60nm. It can be seen that these six modes are similar to those six modes supported in the SiN waveguide [shown in Fig. 3(a)-3(f)]. As indicated by Fig. 5(a), these six modes will then be converted to the corresponding guided-modes [i.e., the TE11, TE21, TE31, TM11, TE41, and TM21 modes shown in Fig. 6(b)] in the multimode silicon waveguide, when the silicon width w Si of inverse taper varies from 60nm to ~1.6μm slowly. On the other hand, a compact inverse taper is desired to save physical space. Therefore, it is very important to determine how slow the taper variation could be. Definitely a non-linear inverse taper is preferred to be compact. Since there are six modes involved in the present case, it can be imaged that the current situation is more much complicated than the conventional inverse taper working for the fundamental mode only.
When trying to have an optimal design for the inverse taper, we notice that there are some mode-hybridization regions, which should be considered carefully for the taper design . In order to locate these mode hybridization regions, we calculate a hybridization ratio γ defined byFigure 7 shows the calculated ratios γx of all the six guided-modes as the waveguide width w Si varies. From Fig. 7, it can be seen very clearly that the curves for the ratios γx of some guided-modes have a transition between 100% and 0 in the regions around w co = 0.572, 0.73, 0.83, 0.99, and 1.29μm, where the mode hybridization regions locate. In these regions, the hybridization modes have comparable x-component and y-component of the electrical field. According to our previous paper for the design of adiabatic tapers [45,46 ], the taper should varies very slowly around the mode hybridization region. When light propagates along such an adiabatic taper, the
As shown in Fig. 5(a) and Fig. 7, the mode evolution is very complicated when the waveguide width varies from 50nm to 1.6μm. In order to simplify the taper design and realize a compact inverse taper, we divide the whole taper into a series of cascaded linearly tapered segments whose widths w Si are defined by wi 1<w Si< wi 2, where wi 1, and wi 2 are the minimal value and the maximal value for the width of the i-th segment. In this way, it becomes easy to design these segments optimally one by one. For the i-th segment with given widths (wi 1 and wi 2), we calculate the mode excitation ratio ηpq for the q-th guided-modes at the output end of the i-th segment when the p-th guided-mode is launched at the input end of the i-th segment. Here the mode excitation ratio ηpq is defined as the power percentage of the q-th guided-mode at the output end when the p-th guided mode is launched at the input end of the i-th segment. From the dependence of the mode excitation ratio ηpq on the length Li of the i-th segment, one can determine the minimal length Li , p _0 for guaranteeing a conversion efficiency of >99% when the p-th guided-mode is launched. Definitely the designed taper for the i-th segment is required to work for all the six guided-modes (p = 1, 2, ..., 6) launched at the input end. Therefore, finally the length Li should be chosen as the maximum of all the lengths Li , p _0 (p = 1, 2, ..., 6), i.e., Li = max(Li , p _0).
For the present case, the inverse taper is divided into 15 segments whose widths (wi 1, wi 2) are chosen according to the locations of the mode-hybridization regions, and a commercial software (FIMMPROP, Photon Design, UK) employing an eigenmode expansion and matching method  is used to simulate the light propagation in the present structure. Here we monitor the mode excitation ratio ηjq, which gives the power percentage of the q-th guided-mode at the output end when the j-th guided mode is launched at the input end of the i-th segment. Figure 8(a)-8(o) show the calculation mode excitation ratio ηjq when the j-th guided-mode is launched at the input end of the i-th segment, i = 1, ..., 15, respectively. In order to make the figure readable, here we only show the result when it is launched with the j-th guided-mode which needs the longest taper length to guarantee a mode excitation ratio of ηjq>99% for the desired mode. The results shown in Figs. 8(a)-8(o) give the calculated mode excitation ratio ηjq (q = 1, 2, ..., 6) for the cases of (wi 1, wi 2, j) = (0.06μm, 0.1μm, 3), (0.1μm, 0.2μm, 3), (0.2μm, 0.3μm, 3), (0.3μm, 0.4μm, 4), (0.4μm, 0.568μm, 6), (0.578μm, 0.67μm, 6), (0.67μm, 0.78μm, 6), (0.78μm, 0.81μm, 4), (0.81μm, 0.84μm, 4), (0.84μm, 0.94μm, 4), (0.94μm, 1.08μm, 5), (1.08μm, 1.24μm, 4), (1.24μm, 1.27μm, 6), (1.27μm, 1.32μm, 6), and (1.32μm, 1.4μm, 6), respectively.
From Fig. 8(a)-8(o), one can easy determine the length of the i-th segment as Li = 270, 180, 126, 10, 170, 20, 60, 85, 350, 250, 375, 10, 80, 300, 15μm for i = 1, ..., 15, respectively, to guarantee a mode excitation ratio of ηjq>99% for the desired guided-mode. Meanwhile, the mode excitation ratios for the undesired modes are ηjq<1%. The parameters are summarized as shown in Table 1 . By combining all of these designed segments, one has an nonlinearly tapered structure as shown in Fig. 9 . Fimmprop tool is used to simulate the light propagation in the whole structure when different modes are launched at the input end and evaluate the mode excitation ratio ηjq in the designed nonlinear inverse taper. The Fimmprop simulation shows the present inverse taper cannot enable mode excitation ratios ηjq for all the desired modes to be sufficiently high as required (e.g., 99%) and there are some notable undesired mode excitation. The reason is that there are some accumulated mode conversions when all the segments are cascaded.
As it well known, one can improve the mode excitation ratio for the desired mode when the taper is long enough. Therefore, we lengthen all the segments for the inverse taper as Li' = ρLi, where ρ is the scale factor (ρ≥1), L i is the original length given in Table 1 for the i-th segment. Figure 10(a)-10(e) respectively show the calculated mode excitation ratios of all the guided-mode when the j-th guided-mode (j = 1, 2, 3, 4, 5, and 6) is launched at the input end of the inverse taper as the scale factor ρ increases from 1.0 to 5. From this figure, it can be seen that the inverse taper with ρ = 1works very well when the first, second, 5-th, and 6-th mode is launched even. The desired mode excitation ratio is higher than 94% while the mode excitation ratio for any other undesired mode is lower than 1.4%. In contrast, when the 3rd mode or the 4-th mode is launched [see Fig. 10(c)-10(d)], the mode excitation ratio for the desired mode is about 89% while the undesired mode excitation ratio P 35 (or P 43) is as large as 9.5% if choosing ρ = 1. Fortunately, when the inverse taper is lengthened by choosing ρ = 1.56, all the six modes work very well. For any launched mode (j = 1, 2, ..., 6), the desired mode excitation ratio is higher than 95.6% while any undesired mode excitation ratio is lower than 0.5%.
Figure 11(a)-11(f) show the simulated light propagation in the designed inverse taper with ρ = 1.56 when the j-th guided-mode is launched, j = 1, 2, ..., 6, respectively. The operation wavelength is 1550nm. The insets in each figure show the launched field at the input end as well as the output field at the output end of the inverse taper. It can be clearly seen that the six modes (TM11, TE11, TE12, TM21, TM12, and TE21) are converted successfully to six guided-modes (TE21, TE11, TE31, TM21, TE41, and TM11) in the SOI 1.4 × 0.22μm2 nanowire waveguide. The output fields shown in the insets also show that the mode conversion is efficient without any significant crosstalk from the undesired modes, as predicted by the results given in Fig. 10(a)-10(f). Such an inverse taper should be useful for the connection between multimode silicon nanophtotonic integrated circuits and few-mode fibers in the future.
We also check the bandwidth for the designed mode converter by simulating the propagation for light with different wavelengths (λ = 1540, 1550, 1560nm). As the mode conversion for mode #1, 2, 5, and 6 is wavelength-insensitive, which allows a broadband operation, we focus on the analysis for modes #3 and #4. Figure 12 shows the calculated mode excitation ratios η 33 when the 3-rd guided-mode is launched at the input end of the inverse taper (the result for the case with the 4-th mode is similar). It can be seen that the performance degrades when the wavelength deviates from the central wavelength λc = 1550nm, which limits the bandwidth of the designed taper. On the other hand, the performance becomes less wavelength sensitivity when choosing longer taper (i.e., larger scale factor ρ). For example, when choosing ρ = 3.4, the desired mode excitation ratio η 33 is higher than 96% when the wavelength ranges from 1540nm to 1560nm, which indicates that a relatively large bandwidth is achievable.
In this paper, we have proposed and designed an inverse taper on silicon to realize an efficient mode converter to enable the connection between multimode silicon nanophotonic integrated circuits and few-mode fibers. The present mode converter consists of an SOI inverse taper buried in a large SiN strip waveguide. This mode converter based on an inverse taper is designed not only for the fundamental mode but also for the higher-order modes. The inverse taper is designed by dividing it into many segments so that each segment can be designed optimally individually. As an example, an inverse taper is designed optimally to enable the conversion between the TE11, TE21, TE31, TE41, TM11, and TM12 modes in a 1.4 × 0.22μm2 multimode SOI waveguide and the TM11, TE11, TE12, TM21, TM12, and TE21 modes (which are like the LP01, LP11a, LP11b modes in a few-mode fiber) in a 3 × 3μm2 SiN strip waveguide. The numerical simulation has shown that the desired mode excitation ratio for all these modes are higher than 95.6% while any undesired mode excitation ratio is lower than 0.5%. One can have a larger bandwidth by choosing a longer taper. With this mode converter, those on-chip mode (de)multiplexers developed well become available for few-mode fiber systems and multimode silicon nanophotonic integrated circuits can work together with few-mode fibers. On the other hand, since the dimension of the SiN strip waveguide is still smaller than a normal few-mode fiber, further mode conversion is needed. It is possible to expand the mode size of the SiN strip waveguide by using an bi-level taper, which was used previously for silicon photonics . Another possible way is to use lensed or tapered few-mode fiber to match the mode size. There has been some work reported for analyzing the mode evolution in tapered few-mode fibers  and hopefully lensed or tapered few-mode fiber will be available in the near future.
This project was partially supported by the Nature Science Foundation of China (No. 61422510, 11374263, and 61431166001), the Doctoral Fund of Ministry of Education of China (No. 20120101110094), and the Fundamental Research Funds for the Central Universities.
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