## 2 MM-AWG and Optical Coupling
For optical coupling between single-mode PLC waveguide and standard single-mode fiber with core size from 8 to 10 \(\upmu\)m, the optimal index difference in our PLC to minimize coupling loss is about 0.5%. The transmission spectrum of a fabricated MM-AWG is shown in Figure 3b. The insertion loss of the MM-AWG is as low, as less than 1 dB in each passband when multimode fiber is connected to the output port of the AWG. The 0.5-dB bandwidth is larger than 13 nm, which corresponds to channel spacing as wide as 65%. The demultiplexed multimode optical signal is not appropriate for retransmission through a single-mode optical fiber due to the loss of higher modes. Therefore, a photoreceiver in which the multimode signal is instantly converted into electric signal should be used. However, even if the spectral shape is flat at the output of the MM-AWG, poor coupling with PDs degrades the shape. Next, we need to design the optical coupling between the multimode output of the MM-AWG and the PD. The output beam from a multimode waveguide can be approximated by combining higher-order Gaussian beams. The electric field radiated from a waveguide, \(E_{m,n}(x,y,z)\), is represented as:  
\[ \begin{array}{l}E_{m,n}(x,y,z)=E_{0}\frac{\omega_{0}}{\omega(z)}\cdot H_{m} \left(\sqrt{2}\frac{x}{\omega}\right)\cdot H_{n}\left(\sqrt{2}\frac{y}{\omega }\right)\\ exp\left[-\frac{x^{2}+y^{2}}{\omega^{2}(z)}-\frac{\bar{x}\left(x^{2}+y^{2} \right)}{2R(z)}-ikz+(m+n+1)\mathcal{O}(z)\right]\end{array}  \tag{1}\]  
Figure 3: Design layout and measured spectrum of MM-AWG. The design condition is based on a silica PLC for the 1.5-μm band with an index difference by 0.75%. (**a**) Schematic of design of the AWG chip and cross-sectional view of the waveguide. (**b**) Transmission spectrum of fabricated MM-AWG. Each color line corresponds to individual output channel. Figure 2: Estimation of spectral shape. The design condition is based on a silica planar lightwave circuit (PLC) for the 1.3-μm band with index difference of 2%. (**a**) Effective index vs. waveguide width. (**b**) Calculated spectral shape of multimode-output arrayed waveguide grating (MM-AWG) dependent on the mode order \(m\). where \(E_{0}\) and \(\omega_{0}\) are the electric field and beam waist (spot size) at the waveguide edge, respectively, and \(H_{m}\) is a Hermite polynomial of \(m\) order. Here, \(\omega(z)\), \(R(z)\), and \(\phi(z)\) are defined as:  
\[\omega(z)=\sqrt{\omega_{0}\bigg{[}1+\bigg{(}\frac{\lambda z}{\pi\omega_{0}z} \bigg{)}^{2}\bigg{]}} \tag{2}\]  
\[R(z)=\bigg{[}1+\bigg{(}\frac{\pi\omega_{0}z}{\lambda z}\bigg{)}^{2}\bigg{]} \tag{3}\]  
\[\phi(z)=\tan^{-1}\bigg{(}\frac{\lambda z}{\pi\omega_{0}z}\bigg{)} \tag{4}\]  
From the above Equations, coupling efficiency between the electric field of the MM-AWG output and PD are estimated. Figure 4a shows the calculated relative responsivity of optical coupling for the highest mode. The design condition is based on a silica PLC for the 1.5-\(\upmu\)m band with the index difference of 0.75%. Here, \(z\) is the distance between the edge of the MM-AWG output and the PD surface, and \(dx\) represents displacement from the center. The diameter of the PD is assumed to be 80 \(\upmu\)m. Since the beam output from the MM-AWG creates complicated speckle patterns that can be mixed by fundamental and higher modes, a sufficiently large detection area of the PD is required to capture all the modes. We consider the following conditions: \(dx\) and \(z\) are small enough to obtain flat responsivity but large enough for alignment and tolerance to assembly error. The diameter of the PD should be large enough to provide flat responsivity but small enough to obtain high-speed operation. This plot reveals that received intensity degrades depending on both \(z\) and \(dx\).