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Abstract
The multimode power splitter is a fundamental component in mode-division multiplexed systems. In this paper, we design and characterize a broadband compact dual-mode multimode interferometer (MMI) optical power splitter based on subwavelength grating (SWG) structures. The optimized dual-mode MMI is three times more compact than its conventional mode insensitive MMIs and shows low loss and low crosstalk flat response over 100 nm bandwidth. Characterizations of the fabricated dual-mode splitter show that the total excess loss in the experiment is less than 0.1 dB and 0.65 dB for TE0 and TE1, respectively, and the modal crosstalk is less than -17 dB for both input modes.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The ever-increasing demand for bandwidth density and internet protocol (IP) traffic causes a bottleneck in current communication systems between and within datacenters. This imposes a challenge in bandwidth density, communication speed, and energy consumption as the intra-data center IP traffic rapidly increases up to the range of zettabyte [1]. Complementary metal-oxide-semiconductor (CMOS) compatible silicon on insulator (SOI) technology offers low absorption within a large wavelength range, high index contrast between core and surrounding oxide, and high bandwidth data transmission improving the energy per bit performance of data center interconnects [2]. The bandwidth density can be further increased by employing multiplexing techniques such as polarization division multiplexing (PDM), wavelength division multiplexing (WDM), mode division multiplexing (MDM), or advanced modulation formats such as pulse amplitude modulation (PAM) and parallel signal modulation (PSM). While parallel signal modulation (PSM) has been recently demonstrated [3], the distance between waveguides must be sufficient to reduce the crosstalk between channels. This will lead to an increase in the footprint when scaling. PAM-4 modulation is another modulation format to increase throughput but requires a digital to analog converter adding to the complexity and power consumption of the system. WDM techniques enable the simultaneous transmission of multiple parallel data streams but need multiple individual optical sources for each wavelength or a reliable laser comb adding to the overall system complexity. To further increase the power efficiency of silicon photonic integrated circuits, WDM should be combined with MDM techniques in which each WDM laser source is used to modulate several optical mode channels [4]. All modes share the same waveguide in an MDM-based switch for a more practical footprint. As demonstrated in [5], all modes can share the same phase shifter leading to better power consumption in an MDM-based switch system. Reducing the size and power consumption of next-generation integrated electronic-optical switch interconnects is a serious requirement [6]. In SOI, the effective mode manipulation is strong due to the large refractive index difference between Si core and surrounding oxide. Therefore, the realization of low-loss and low crosstalk mode-selective manipulation in a multimode SOI nanowire is less challenging compared to MDM in optical fibers [7]. In recent years, people have developed various multimode silicon photonics structures including on-chip mode (de)multiplexers [8–11], multimode waveguide bends [12–14], multimode waveguide crossings [15], multimode switches [16–18], and multimode power splitters [19–23]. Among all, the multimode power splitter is the fundamental photonic building block to simultaneously split all the input modes with 3 dB splitting ratio which is essential for multimode switching. Such power splitting is experimentally demonstrated by asymmetric directional couplers in [19,20]; however, such structures are wavelength sensitive due to the limited bandwidth of asymmetric directional coupler (ADC) structures. The adiabatic coupler (AC) can convert the input modes into the corresponding super-modes and realize the multimode power splitting with enhanced bandwidth [21]. However, the adiabatic mode conversion process requires a long coupling length (>800 µm). A waveguide integrated SWG transflector simultaneously reflects and transmits light in [22]. However, the device length in the proposed structure is more than 300 µm challenging its implementation for applications such as photonic integrated switch matrices. In [23], such power splitting is achieved using a shallow etch MMI requiring two-step etching, adding to the complexity of fabrication. Noticeable in [21–23] is the relatively high modal crosstalk observed in simulation. As no experimental performance was reported for these structures, one may expect the modal crosstalk to be even higher in practice. In [20], the reported measured crosstalk is -15.7 dB, an insufficient performance for practical integrated MDM systems. Indeed, to maintain a BER of 10−12 for a 10 Gb/s NRZ payload transmission, the modal crosstalk should be less than -22 dB [24]. MMIs have relatively good tolerance to fabrication non-uniformity effects [25,26]; however, non-SWG multimode MMIs require relatively large lengths. The multimode MMI used in [5] has a length greater than 300 µm with an optical bandwidth of operation limited to 40 nm. Therefore, the realization of compact and broadband multimode power splitter with low modal crosstalk requires innovative solutions in their development.
The advancements in sub-wavelength grating (SWG) integrated on-chip refractive index engineering [27–30]. These structures have been employed in a wide range of applications including fiber-to-chip coupling [31,32], wavelength multiplexing [33], polarization splitting [34–36], among other innovations. In this paper, we design and fabricate an on-chip broadband multimode power splitter based on an SWG MMI structure. The structure footprint is 5 µm × 48 µm. The excess loss and modal crosstalk are less than 0.4 dB (0.65 dB), and -32 dB (-17 dB) in simulation (experiment) for the two lowest TE modes over an optical bandwidth of 100 nm. The results demonstrate improvement in compactness, excess loss, and modal crosstalk over past published works.
2. Design and working principle of the dual-mode SWG MMI
The SWG MMI is designed for transverse electrical (TE) polarization using 220-nm-thick channel waveguides surrounded by a 2 µm buried oxide layer (BOX) and 2.2 µm oxide top cladding. The single-mode waveguide width is set to be 0.5 µm. The dual-mode waveguide width is selected to be 1 µm to support both TE0 and TE1. Figure 1(b) shows the simulated effective index as a function of waveguide width using a commercial CAD tool (Lumerical Mode solution) for the first three TE modes.
Fig. 1. (a) Cross-section of the single-mode waveguide (TE0) and dual-mode waveguide (TE1); (b) simulated effective refractive index as a function of waveguide width; (c) schematic of the proposed broadband dual-mode MMI power splitter with dimensions shown.
Figure 1(c) schematically shows the dual-mode broadband compact MMI (DBcMMI) optical power splitter, where LMMI and WMMI are the length and width of the multimode region. NMMI and NT are the number SWG blocks in the multimode region and tapers, respectively. WA is the width of the input and output tapers connecting to the multimode region (also referred to as the access waveguide width). LT is the length of the input and output tapers which are separated by the distance S. The pitch and width of the silicon blocks are noted by $\Lambda $ and a, respectively. Light at any of the two input ports with TE0 and TE1 modes is passed through the structure and split at the output ports. The MMI is working following the general interference mode principle and optimized to generate two-fold images for two input modes. The principle for an MMI with multiple input modes has been previously derived in [37]. The input transverse electric modes, TEm where $m = 1\; \textrm{or}\; 2$ representing the mode order, excite multiple modes in the MMI region. The input mode can be defined as the sum of all the excited modes.
The effective refractive indices of the SWG anisotropic metamaterial can be obtained using effective medium theory [28,38] as follows:
The design parameters WMMI, WA, and S are initially set to values corresponding to the conventional dual-mode MMI design. The SWG pitch $\mathrm{\Lambda }$ is initially set small ensuring that the design works in the sub-wavelength regime. The initial value for LMMI is calculated based on the above-presented formulas. The entire structure is finally optimized using Lumerical 3D FDTD simulations. By setting the duty SWG cycle $f = \frac{a}{\mathrm{\Lambda }}$ to be 0.5, the ordinary and extraordinary refractive indices are calculated based on Eqs. (3) and (4). These values are imported into Lumerical Mode solutions and the propagation constants of the first 2 modes (${\beta _0}(\lambda )\; \textrm{and}\; {\beta _1}(\lambda )$) in the multimode region are calculated. The length of MMI, LMMI, is initially approximate to be $3{L_\pi }/2$ according to Eq. (2) to form the first 2-fold image before performing the 3D optimizations. Figure 2 shows the simulated transmission obtained for the top input port to the bar/top output port of the MMI for four possible pitch sizes. The result shows that for a pitch of 220 nm, the structure provides the flattest response for both input modes from 1500 nm to 1600 nm. According to simulations, the TE1 transmission is more sensitive to the changes in the beat length compared to TE0. A pitch of 240 nm does not lead to a relatively constant value for the beat length from 1500 nm to 1600 nm as the TE1 transmission deteriorates. The dips in the transmission for TE1 at Λ = 240 nm is explained by partial reflections in the multimode section due to the relatively large pitch size to wavelength ratio of SWG, Λ/λ, at Λ = 240 nm. The phase-matching condition for successive back reflection is occurring modes having higher contribution in generating the TE1 2-fold image than the TE0 one.
Fig. 2. Simulated transmission from the input port to the bar output port of the MMI with respect to different pitch sizes. (a) TE0 input, (b) TE1 input.
The adiabatically tapered input access waveguides are optimized next. The taper length is optimized to be long enough to ensure the transition of both input modes (TE0 and TE1) while kept compact. The taper length is optimized at LT = 8 µm leading to less than 0.1 dB insertion loss (IL) for both input modes.
We then optimized the access waveguide width WA to support both modes. The separation S should be kept as small as possible; however, it also requires to be large enough to ensure that there is no coupling between the parallel input and output waveguides. Figure 3 shows the simulation results for optimizing the WA and S parameters. The design parameters are optimized to give the lowest possible loss and imbalance for both input modes. Results show that TE1 excess loss and imbalance are more sensitive to changes in design parameters. A higher number of modes in the multimode region are required to create the 2-fold image for TE1 leading to higher sensitivity and loss. The TE1 slab to SWG waveguide mode conversion at the input and output tapers has 0.1 dB additional loss per conversion. This also contributes to a higher loss for TE1 mode.
Fig. 3. Excess loss (EL) and imbalance (Im) from simulation of the proposed design as function of wavelength for different S design values while other parameters are kept constant (Table 1); (a) TE0 input and (b) TE1 input. EL and Im from simulation of the proposed design as a function of wavelength for different WA while other parameters are kept constant (Table 1); (c) TE0 and (d) TE1 input.
Table 1. Final Optimized Values of the Proposed dBcMMI Structure
The excess loss and imbalance used to characterize the MMI for each input mode where the input power is pin = 1 mW is defined as:
Careful calibration of lithography and etching is required to meet these duty-cycle and width specifications. The recommended minimum feature size and spacing in the foundry used is 70 nm. Therefore, the SWG duty cycle design value, f, should not be less than 0.32 or more than 0.68. In simulations, we observed that $f = 0.5$ gives the lowest excess loss and imbalance for both input modes. Table 1 summarizes the final optimized values for the fabricated structure.
Figure 4 shows the simulated electric field distribution for TE0 and TE1 input modes through the structure. The input mode profiles are reproduced at the output ports at the same relative position. Figure 5 shows the excess loss, imbalance, and modal crosstalk of the designed structures for both input TE0 and TE1 modes over the 100-nm-bandwidth from 1500 nm to 1600 nm. The highest modal crosstalk in simulation is -32 dB in Fig. 5(a) likely caused by the non-perfect 2-fold imaging of the MMI. Ideally, a perfect 2-fold image of input modes is created at the output ports. However, in practice, the multimode region supports a finite number of modes which results in non-perfect 2-fold imaging. Our simulation results show that the structure excess loss starts to increase at wavelengths less than 1450 nm. The bandwidth of the structure can span cover the O-band to C-band by lowering the pitch size [39] at the cost of performance degradation in the C-band as the beat length changes in the C-band increases at lower pitch sizes. As the higher-order modes, i.e., TE1 are more sensitive to beat length variations, the TE1 performance experiences more degradation in the C-band by lowering the pitch size.
Fig. 4. Electric field propagation in the designed dBcMMI for (a) TE0 input; (b) TE1 input. Inset: shows the constructed 2-fold image at the end of the MMI section.
Fig. 5. Simulated transmission spectra of the proposed design for (a) TE0 input, (b) TE1 input modes; simulated (c) excess loss and (d) imbalance for TE0 and TE1 input modes.
The phase difference between the two ports in an MMI should ideally be 90 degrees over the wavelength range from 1500 nm to 1600 nm. Figure 6 presents the simulated relative phase between the two output ports. The simulated relative phase of two MMI outputs are 90° ± 0.9° and 90° ± 2.8° for TE0 and TE1, respectively, over the wavelength range from 1500 nm to 1600 nm.
As the horizontal modes in a standard 220-nm-thick waveguides share a common beat length, the proposed multimode power splitter may scale up to split higher order TE modes (e.g., TE3 and TE4) by following the design procedure explained in Section 2. Higher-order mode MMIs require a wider multimode section. Thus, they require longer lengths and have larger footprints [40]. Figure 7 presents the multimode length in SWG-based and conventional MMIs for few-mode MMIs. As shown, the use of SWG structures for higher-order mode MMIs leads to reduced structure length.
3. Fabrication and characterization of the dual-mode broadband compact MMI
the dBcMMI is fabrication through Applied Nanotools Inc in Alberta, Canada. The silicon device layer is patterned using a 100 KeV electron-beam lithography (EBL) followed by an inductively coupled plasma-induced reactive ion etching (ICP-RIE) process. A 2.2-µm-thick SiO2 cladding is deposited by plasma-enhanced chemical vapor deposition (PECVD). The design is characterized using continuous wave (CW) optical input. The polarization-controlled input light from a tunable C-band laser source (Yenista Tunics T100R) is swept from 1500 nm to 1600 nm. The light from the polarization controller (PC) is coupled into the chip using a grating coupler (GC). To realize a multimode input signal, we used an on-chip mode (de)multiplexers before and after the structure as shown in Fig. 8. The light coming from the GC is multiplexed into TE0 and TE1 modes in a multimode waveguide and is passed through the dBcMMI design and then demultiplexed into two TE0 modes before coupling out to fiber array for measurement. The output light from the GC is measured using an optical power meter (ILX Lightwave FPM-8200). A single-mode waveguide connected to two single-mode surface grating couplers is fabricated on the same chip to normalize the loss of the design. The optical micrograph image of the fabricated structures is shown in Fig. 8.
Fig. 8. Optical microscope image of the fabricated structure showing the 2-input mode multiplexer, power splitter, and the mode demultiplexers.
3.1 Optical characterization and modal crosstalk
A test structure consisting of an ADC-based 2-mode multiplexer cascaded with a demultiplexer is first characterized by a CW optical input. The worst IL of the mode multiplexer is 0.34 dB and 1.04 dB for TE0 and TE1 input modes from 1500 nm to 1600 nm wavelength operating range, and the highest crosstalk is −19.66 dB.
The normalized optical transmissions of the dBcMMI as a function of wavelength from 1500 nm to 1600 nm are presented in Figs. 9(a) and 9(b) for TE0 and TE1 input modes. The measured excess loss of the dBcMMI for TE0 and TE1 input modes is 0.28 dB and 0.65 dB for TE0 and TE1 input modes respectively excluding the loss of the mode (de)multiplexers before and after the DBcMMI. The modal crosstalk of the dBcMMI in the experiment is -17 dB. The crosstalk is defined as the transmission of the input TE mode m to the output TE mode n, where m = 1, 2 represents the input mode order, n = 1, 2 represents the output mode order at the output ports. To quantify the crosstalk, we measured the optical power of mode n at one output port (${P_{{n_{\textrm{out}}}}}$) while the input is excited with the other mode m (${P_{{m_{\textrm{in}}}}};\;n \ne m$). The crosstalk is measured at the output of the entire MDM link system which includes the mode (de)multiplexers. As such, the modal crosstalk measured is the contribution of the modal crosstalk in the mode multiplexer and demultiplexer, along with the modal crosstalk from the MMI proposed. The definition used to report modal crosstalk is the following:
To reduce modal crosstalk, further optimization of the mode (de)multiplexers is necessary. Further, digital signal processing techniques (DSP) may be considered to compensate for the crosstalk between the different mode channels using multiple inputs and multiple-output (MIMO) techniques [41].
Fig. 9. Measured normalized transmission as a function of wavelength for (a) TE0 input; (b) TE1 input.
Recently, several mode-insensitive power splitters have been demonstrated [19–23]. The comparison is presented in Table 2. As shown, our characterized design has a more compact footprint, lower loss, and crosstalk, and a broader bandwidth compared to these works.
Table 2. Performance Comparison of Several Multimode Power Splitters
3.2 Sensitivity of the design and analysis of the crosstalk
We further investigated the origin of modal crosstalk in the dBcMMI using 3D simulation tools. The results show that the crosstalk induced by the dBcMMI is sensitive to the length of the multimode region (LMMI). Figure 10 shows the simulation results of the transmission of the design with respect to the different number of SWG blocks (NMMI) in the multimode section which relates to the length of the multimode region (LMMI). From the results, we see that a small change in the number of SWG periods in the multimode region from 146 to 150 increases the modal crosstalk by 10 dB. Although the modal crosstalk is still low, the change is important as the modal crosstalk is the key factor in the multimode systems to maintain a low BER. Due to the improper length of MMI (LMMI), we have non-perfect 2-fold imaging at the outputs. The non-perfect field exits as higher-order modes, adding to the crosstalk.
Fig. 10. Simulated transmission response of the dBcMMI for the different number of SWG blocks in the multimode section.
Another source of crosstalk that we investigated is the buried air gaps in the cladding of SiP structures. In the fabrication process, the SiO2 cladding may not be able to fill the gap between the fine features of the SWG during the deposition reported in [42,43]. This is illustrated in Fig. 11(a). In such cases, the performance of the SWG structure is affected and altered from the design values. The voids are represented by the cross-section of their volume (x × z) in Figs. 11(a) and 11(b). We evaluated the voids in the cladding of the chips fabricated through the ANT center. Figure 11(c) presents the FIB-HIM evaluation of the voids. The measured cross-section area for the voids is fewer than 500 nm2.
Fig. 11. (a) A sub-wavelength grating structure where L is the length, G is the gap between the blocks, and Λ is the period of the grating; Inset: voids in the SiO2 cladding measured via helium ion microscope (HIM) images of the voids in SiO2 cladding; (b) side view of the structure showing modeled 500 nm2 upper cladding voids. (c) HIM-SEM images from ANT showing worst-case voids with an area less than 500 nm2. Simulation results of the impact of ANT voids on the transmission of the proposed design for: (d) TE0 input; (e) TE1 input.
We investigate the transmission of the dBcMMI considering 500 nm2 air voids in the cladding of the structure as presented in Fig. 11(b). Simulation results in Figs. 11(d) and 11(e) show that such small voids do not have a significant impact on the excess loss of the design. However, the modal crosstalk increases by 10 dB even with such small voids.
4. Conclusion
In this work, we designed and characterized a dual-mode broadband 2 × 2 power splitter using SWG structures. The wavelength dependence of the beat length is inhibited by dispersion engineering in the SWG structures. The footprint of the designed structure is as small as 5 µm × 48 µm (including the adiabatic taper-mode converters) which is small compared to other multimode power splitters. The measurement shows low excess loss less than 1 dB and low modal crosstalk less than -17 dB for both input modes (TE0 and TE1) over a 100-nm-bandwidth which is a significant improvement over past published works. By further optimizing the dBcMMI length and mode (de)multiplexers, the modal crosstalk can be improved. We believe such a device could find its applications in high-performance WDM-compatible multimode circuits.
Funding
Canada Research Chairs Program (CSC); the Natural Sciences and Engineering Research Council of Canada (NSERC); the China Scholarship Council (CSC) program.
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 authors upon reasonable request.
References
1. “Cisco global cloud index: Forecast and methodology, 2016–2021,” Accessed: November 19, 2018. https://www.cisco.com/c/en/us/solutions/collateral/service-provider/global-cloud-index-gci/white-paper-c11-738085.html.
2. D. A. Miller, “Attojoule optoelectronics for low-energy information processing and communications,” J. Lightwave Technol. 35, 346–396 (2017). [CrossRef]
3. Z. Hu, S. Shao, Z. Xiao, X. Zhu, Y. Wu, J. Feng, and J. Guo, “100 Gb/s PSM-4 silicon photonics transceiver for intra-datacenter on a 200-mm Wafer,” in Asia Communications and Photonics Conference, (Optical Society of America, S3D. 3 (2020).
4. L.-W. Luo, N. Ophir, C. P. Chen, L. H. Gabrielli, C. B. Poitras, K. Bergman, and M. Lipson, “WDM-compatible mode-division multiplexing on a silicon chip,” Nat. Commun. 5, 1–7 (2014). [CrossRef]
5. G. Zhang, H. R. Mojaver, A. Das, and O. Liboiron-Ladouceur, “Mode insensitive switch for on-chip interconnect mode division multiplexing systems,” Opt. Lett. 45, 811–814 (2020). [CrossRef]
6. M. Bohn, P. Magill, M. Hochberg, D. Scordo, A. Novack, and M. Streshinsky, “Next-generation silicon photonic interconnect solutions,” in Optical Fiber Communication Conference, Optical Society of America, M3J. 3(2019).
7. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013). [CrossRef]
8. T.-H. Pan and S.-Y. Tseng, “Short and robust silicon mode (de) multiplexers using shortcuts to adiabaticity,” Opt. Express 23, 10405–10412 (2015). [CrossRef]
9. Y. Ding, J. Xu, F. Da Ros, B. Huang, H. Ou, and C. Peucheret, “On-chip two-mode division multiplexing using tapered directional coupler-based mode multiplexer and demultiplexer,” Opt. Express 21, 10376–10382 (2013). [CrossRef]
10. D. Dai, C. Li, S. Wang, H. Wu, Y. Shi, Z. Wu, S. Gao, T. Dai, H. Yu, and H. K. Tsang, “10-channel Mode (de) multiplexer with dual polarizations,” Laser Photonics Rev. 12, 1700109 (2018). [CrossRef]
11. H. Xiao, X. Han, and Y. Tian, “Reconfigurable two-mode multiplexer based on three-waveguide-coupling structure,” in 17th International Conference on Optical Communications and Networks (ICOCN2018), International Society for Optics and Photonics, 2019, 110483B (2019).
12. L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012). [CrossRef]
13. I. Papakonstantinou, K. Wang, D. R. Selviah, and F. A. Fernández, “Transition, radiation and propagation loss in polymer multimode waveguide bends,” Opt. Express 15, 669–679 (2007). [CrossRef]
14. Y. Wang and D. Dai, “Multimode silicon photonic waveguide corner-bend,” Opt. Express 28, 9062–9071 (2020). [CrossRef]
15. W. Chang and M. Zhang, “Silicon-based multimode waveguide crossings,” J. Phys. Photonics 2, 022002 (2020). [CrossRef]
16. R. B. Priti and O. Liboiron-Ladouceur, “Reconfigurable and scalable multimode silicon photonics switch for energy-efficient mode-division-multiplexing systems,” J. Lightwave Technol. 37, 3851–3860 (2019). [CrossRef]
17. Y. Xiong, R. B. Priti, and O. Liboiron-Ladouceur, “High-speed two-mode switch for mode-division multiplexing optical networks,” Optica 4, 1098–1102 (2017). [CrossRef]
18. R. B. Priti and O. Liboiron-Ladouceur, “A broadband rearrangable non-blocking MZI-based thermo-optic O-band switch in silicon-on-insulator,” in Photonics in Switching, (Optical Society of America, 2017), PM4D. 2.
19. H. Xu and Y. Shi, “Ultra-broadband dual-mode 3 dB power splitter based on a Y-junction assisted with mode converters,” Opt. Lett. 41, 5047–5050 (2016). [CrossRef]
20. Y. Luo, Y. Yu, M. Ye, C. Sun, and X. Zhang, “Integrated dual-mode 3 dB power coupler based on tapered directional coupler,” Sci. Rep. 6, 23516 (2016). [CrossRef]
21. L. Han, B. P.-P. Kuo, N. Alic, and S. Radic, “Ultra-broadband multimode 3 dB optical power splitter using an adiabatic coupler and a Y-branch,” Opt. Express 26, 14800–14809 (2018). [CrossRef]
22. H. Xu, D. Dai, and Y. Shi, “Ultra-broadband on-chip multimode power splitter with an arbitrary splitting ratio,” OSA Continuum 3, 1212–1221 (2020). [CrossRef]
23. C. Ye, M. Zhang, Y. Shi, and D. Dai, “Broadband dual-mode 2 × 2 3 dB multimode interference couplers with a shallowly etched multimode section,” Appl. Opt. 59, 7308–7312 (2020). [CrossRef]
24. C. Williams, G. Zhang, R. Priti, G. Cowan, and O. Liboiron-Ladouceur, “Modal crosstalk in Silicon photonic multimode interconnects,” Opt. Express 27, 27712–27725 (2019). [CrossRef]
25. S. Darmawan, S.-Y. Lee, C.-W. Lee, and M.-K. Chin, “A rigorous comparative analysis of directional couplers and multimode interferometers based on ridge waveguides,” IEEE J. Sel. Top. Quantum Electron. 11, 466–475 (2005). [CrossRef]
26. M. Rajarajan, B. Rahman, and K. Grattan, “A rigorous comparison of the performance of directional couplers with multimode interference devices,” J. Lightwave Technol. 17, 243–248 (1999). [CrossRef]
27. P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D.-X. Xu, A. Densmore, A. Delâge, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35, 2526–2528 (2010). [CrossRef]
28. R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, and Í. Molina-Fernández, “Waveguide sub-wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9, 25–49 (2015). [CrossRef]
29. P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560, 565–572 (2018). [CrossRef]
30. I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nat. Photonics 11, 274–284 (2017). [CrossRef]
31. Y. Wang, L. Xu, A. Kumar, Y. D’mello, D. Patel, Z. Xing, R. Li, M. G. Saber, E. El-Fiky, and D. V. Plant, “Compact single-etched sub-wavelength grating couplers for O-band application,” Opt. Express 25, 30582–30590 (2017). [CrossRef]
32. Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22, 20652–20662 (2014). [CrossRef]
33. J. Chen, “A broadband wavelength demultiplexer assisted by SWG-based directional couplers,” Optik 202, 163602 (2020). [CrossRef]
34. H. Xu, D. Dai, and Y. Shi, “Ultra-broadband and ultra-compact on-chip silicon polarization beam splitter by using hetero-anisotropic metamaterials,” Laser Photonics Rev. 13, 1800349 (2019). [CrossRef]
35. C.-C. Huang, C.-X. Wu, and C.-C. Huang, “Numerical design of a high-performance polarization beam splitter assisted by composite subwavelength gratings,” Opt. Express 28, 14908–14921 (2020). [CrossRef]
36. L. Xu, Y. Wang, A. Kumar, D. Patel, E. El-Fiky, Z. Xing, R. Li, and D. V. Plant, “Polarization beam splitter based on MMI coupler with SWG birefringence engineering on SOI,” IEEE Photonics Technol. Lett. 30, 403–406 (2018). [CrossRef]
37. Y. Zhang, M. A. Al-Mumin, H. Liu, C. Xu, L. Zhang, P. L. LiKamWa, and G. Li, “An Integrated Few-Mode Power Splitter Based on Multimode Interference,” J. Lightwave Technol. 37, 3000–3008 (2019). [CrossRef]
38. R. Halir, A. Ortega-Moñux, D. Benedikovic, G. Z. Mashanovich, J. G. Wangüemert-Pérez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, “Subwavelength-grating metamaterial structures for silicon photonic devices,” Proc. IEEE 106, 2144–2157 (2018). [CrossRef]
39. A. Maese-Novo, R. Halir, S. Romero-García, D. Pérez-Galacho, L. Zavargo-Peche, A. Ortega-Moñux, I. Molina-Fernández, J. Wangüemert-Pérez, and P. Cheben, “Wavelength independent multimode interference coupler,” Opt. Express 21, 7033–7040 (2013). [CrossRef]
40. L. B. Soldano and E. C. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995). [CrossRef]
41. S. O. Arik, J. M. Kahn, and K.-P. Ho, “MIMO signal processing for mode-division multiplexing: An overview of channel models and signal processing architectures,” IEEE Signal Process. Mag. 31, 25–34 (2014). [CrossRef]
42. H. Shiran, H. R. Mojaver, J. Bachman, C. Jin, and O. Liboiron-Ladouceur, “Impact of SiO2 cladding voids in SiPh building blocks,” in 2020 IEEE Photonics Conference (IPC), (2020).
43. R. Halir, P. Cheben, J. M. Luque-González, J. D. Sarmiento-Merenguel, J. H. Schmid, G. Wangüemert-Pérez, D. X. Xu, S. Wang, A. Ortega-Moñux, and Í. Molina-Fernández, “Ultra-broadband nanophotonic beamsplitter using an anisotropic sub-wavelength metamaterial,” Laser & Photonics Reviews 10, 1039–1046 (2016). [CrossRef]
