# An \(N\)-Port Universal Multimode Optical Router Supporting Mode-Division Multiplexing
## 2 Materials and Methods
In another case, when the microring resonator is in the OFF state, it indicates that the wavelength of the input optical signal is not equal to the resonant wavelength of the microring resonator. Instead of being coupled into the microring, the optical signal is transmitted directly to Port III. Specifically, the red and blue arrows shown in Figure 3 represent the trajectory diagram of the TE\({}_{x}\) mode transmission. Firstly, the TE\({}_{x}\) mode is input from Port I, which is demultiplexed to the fundamental mode by the mode demultiplexer. Subsequently, the fundamental mode pass through the single-mode bending and the microring resonator. When the microring resonator is in the OFF state, the fundamental mode is transmitted directly to Port III through the microring resonator for mode multiplexing, which is restored to the TE\({}_{x}\) mode. Finally, the TE\({}_{x}\) mode is output from Port III and transmitted to the next multimode optical device, as shown by the blue arrow in Figure 3. When the microring resonator is in the ON state, the fundamental mode signal is coupled into the microring and transmitted to Port IV. After that, the fundamental mode is restored to the TE\({}_{x}\) mode, which is output by Port IV of the multimode switching element and transmitted to the next multimode optical device. This process is illustrated by the red arrow in Figure 3.  
Thus, by controlling the state of the microring resonator (ON state, OFF state), the multimode switching element can control the high-order mode output from Port III or Port IV. Therefore, the multimode switching element can overcome the disadvantage that the traditional single-mode CSE can only transmit the fundamental mode. It can be used to transmit different high-order modes and control their transmission direction. The insertion loss and crosstalk noise are generated when the TE\({}_{x}\) mode passes through the multimode switching element. However, transmission loss varies with input port or output port. The source of insertion loss and crosstalk noise can be analyzed from the following aspects:  
(1) The insertion loss and crosstalk noise generated by mode-division (de)multiplexing. We can see this from the structure of multimode switching element in Figure 3, where the mode (de)multiplexer is designed and implemented based on the asymmetric directional coupler (ADC), and each mode (de)multiplexer contains \(N\)-1 ADCs, where \(N\) is the number of modes supported by the multimode switching element. Figure 4 shows a mode (de)multiplexer for TE\({}_{x}\) mode, which consists of a single-mode waveguide and a multimode taper waveguide. As shown in the brown and red arrows in Figure 4a, when the function of mode multiplexing is realized, the fundamental mode signal with the same wavelength are input from Port 1 and Port 2, respectively. When the TE\({}_{0}\) mode is input from Port 1, if the phase matching condition is satisfied, the TE\({}_{0}\) mode is converted into different high-order modes through the coupling region due to different widths of the multimode taper waveguide; hereafter, the TE\({}_{x}\) mode output from Port 3. The output power of the TE\({}_{x}\) mode from Port 3 can be given by Equation (3), where the \(L_{mxx}\) is the mode multiplexed loss coefficient of the TE\({}_{0}\) mode converted to the TE\({}_{x}\) mode. When the TE\({}_{0}\) mode is input from Port 2, it is output directly from Port 3 without mode conversion, and the corresponding output power can be calculated by Equation (4). The \(L_{mx0}\) is the loss coefficient of the TE\({}_{0}\) mode passing through the multimode taper waveguide during mode multiplexing. Utilizing the symmetrical characteristics of directional coupler, the inverse process of the multiplexing process can realize the mode demultiplexing function. As shown by the red and brown arrows in Figure 4b, TE\({}_{0}\) mode and TE\({}_{x}\) mode input from  
Port 3 are demultiplexed.