A strip waveguide-based bi-directional mode-division multiplexer is proposed. A mathematical model has been proposed to analyze the performance, and the results are simulated. The design concept of this device to (de) multiplex three modes simultaneously has been studied previously for slab waveguides, both mathematically using the perturbative mode-coupled theory and by simulation using 2D FDTD Solutions (FDTD, finite difference time domain). As slab waveguides are not suitable for extracting

more »

... abrication parameters for most silicon-oninsulator applications, we apply the concept to a more practical device that involves strip waveguides rather than slab waveguides. The effective index method (EIM) has been used to develop the mathematical model and to get approximate forms for both the profiles and coupling coefficients. The return loss of different modes is taken into consideration to fully characterize the device performance. Simple formulas for both insertion and return losses of all multiplexing modes have been derived. In addition, full vectorial 3D FDTD simulations are performed so as to validate our mathematical model. Different design parameters have been used to get numerical results of the proposed device. Our results reveal that the EIM has enough accuracy to characterize the performance of our device compared to that of the complex full vectorial simulation. In order to validate the used model, the device has been fabricated and tested. Good insertion losses and crosstalks for all modes have been obtained. same idea has been reported for TM 0 and TM 1 with short common coupling length for both modes of 15.2 μm and low IL of −0.7 dB. An eight-channel mode (de)multiplexer has been proposed in Ref. [20] using six cascaded directional couplers. The total length of the (de)multiplexer is more than 330 μm. Although the simulated IL for all channels is promising (< − 0.5 dB), the measured IL ranges from −0.2 to −3.5 dB for different modes. A (de)multiplexer has been designed in Ref. [21] that is based on asymmetric Y -junction with lengthdependent frequency response. It has been used for two modes with IL less than −1.5 dB and length more than 100 μm. In Ref. [22], a three-mode (de)multiplexer has been proposed. The maximum measured IL is about −5.7 dB at a wavelength of 1550 nm, and the worst crosstalk is −9.5 dB. The length of the (de)multiplexer for the three modes is more than 350 μm. In Ref. [25], an add/drop MDM for both TE 0 and TE 1 has been proposed, which is based on a Mach-Zehnder interferometer assisted with a periodic structure on the arms. The IL for both modes is less than 1 dB, while the length is more than 100 μm. Another add/drop MDM has been proposed in Ref. [30] using two waveguides and a Bragg grating. Y -junction-and MMI-based MDMs cannot conveniently accommodate more than two modes and require high fabrication tolerance and complex design. Tapered directional couplers-based MDMs are intrinsically limited in bandwidth. However, some studies have been done to increase the bandwidth and reduce the sensitivity to fabrication errors [23, 31] . In Ref. [29], a three-mode (de)multiplexer has been proposed based on two slab waveguides and a Bragg grating. It has been referred to as a bidirectional mode-division multiplexer (BMDM). The device concept has been verified both mathematically and with the aid of 2D FDTD simulations. However, a slab concept is not always suitable for extracting parameters for silicon-on-insulator (SOI) fabrication process. In this paper, we use strip waveguides to provide practical insights to the performance of the BMDM assisted with 3D FDTD simulation. We develop a mathematical model to study the performance of the BMDM with strip waveguides. The rigorous analysis makes it possible to design the device at any wavelength, for any modes, and for any waveguide dimensions by using only the final equations. Specifically, the mode profiles for all involved modes in a strip waveguide are written using the effective index method (EIM). The EIM has proven to be in good agreement with full vectorial methods, yet is much simpler [32] [33] [34] [35] [36] . In Ref. [32], the implementation of EIM has been proven to be applicable to a wide range of guiding structures (strip, buried, and diffused guides) and has been tested against analytical solutions and against other methods' results to prove good agreement. Using this method, a 2D problem is transformed into 1D problems (slab) that are easy to express. The first slab is in the horizontal direction and has a thickness as the height of the strip waveguide. The second slab is in the vertical direction and has a height as the width of the strip waveguide. The vertical slab uses modes resulted from solving the horizontal slab. In addition, we drive approximated expressions for different coupling coefficients in the strip waveguide using perturbative coupled-mode theory. The return loss (RL) in the input waveguide due to the existence of Bragg grating is taken into account in our analysis to fully characterize the