## 2 Device Structure, Design and Fabrication Methods  
Figure 1a depicts the schematic of the proposed LC-DFB laser array in which laterally coupled gratings are along both sides of the ridge waveguide to provide both lateral optical confinement and longitudinal feedback. A 2-\(\upmu\)m-thick SiO\({}_{2}\) dielectric layer supports the planar electrodes and passivates the side wall of ridge waveguides. A p-type electrode pad with a diameter of 100 \(\upmu\)m introduces a pad capacitance \(C\) of 0.18 pF. The intrinsic modulation speed of DMLs is limited by the relaxation oscillation frequency (\(f_{r}\)). The \(f_{r}\) is given by [10]. \[f_{r}=\sqrt{\left(\frac{\Gamma v_{g}a\eta_{i}}{qV}\right)}\frac{\sqrt{\left(I-I _{th}\right)}}{2\pi}=D\sqrt{\left(I-I_{th}\right)} \tag{1}\]  
where \(\Gamma\) is the active region optical confinement factor, \(v_{g}\) is the group velocity, \(a=\partial g/\partial N\)[11] is the differential gain, \(\eta_{i}\) is the internal quantum efficiency, and \(I\) and \(I_{th}\) are the bias current and the threshold current, respectively. \(q\) is the elementary charge, and \(V\) is the active volume. The \(D\) factor \(D=1/2\pi\sqrt{\left(\Gamma v_{g}a\eta_{i}\right)/\left(qV\right)}\) describes the slope of \(f_{r}\) on the square root of the bias current above the threshold [12]. Strained multi-quantum wells (MQWs) were designed for high speed modulations which considered the improving of the differential gain \(a\)[13]. The epitaxial structure is the same as Li, A.K., which consists of five periods of 5-nm-thick InGaAs quantum-well and 8.5-nm-thick InGaAs barrier [14]. The optical confinement factor of the active region \(\Gamma\) is also an important parameter for \(f_{r}\). The relationship of the coupling coefficient \(\kappa\) and \(\Gamma\) versus the ridge width are calculated by the finite element analysis (FEA) method as shown in Figure 1b. The optical confinement factor \(\Gamma\) increases with the width of the waveguide, yet a narrow waveguide is needed for an LC-DFB to increase the grating coupling coefficient \(\kappa\). A narrow waveguide would result in more power leakage out of the waveguides, thus decreasing \(\Gamma\). However, it enhances the optical field in the grating region and increases \(\kappa\). The strength of the grating's feedback is described by the normalized coupling coefficient (\(\kappa L\)). The cavity length is inverse to \(\kappa\) when \(\kappa L\) is a constant. The \(f_{r}\) is proportional to the \(\kappa\) and \(\Gamma\) by Equation (1). The ridge width is designed as 1.2 \(\upmu\)m considering a compromise between \(\kappa\) and \(\Gamma\). Figure 1c shows the variation of \(\kappa\) versus duty cycle \(\gamma\) assuming a ridge width of 1.2 \(\upmu\)m and a grating order of 3. A duty cycle of 0.9 is designed to obtain a higher \(\kappa\) taking the etching broadening effect into account. To fabricate the deep lateral gratings, the pattern was formed by electron beam lithography (EBL) on 400-nm-thick HSQ photo-resist. Deep lateral gratings were formed by  
Figure 1: (**a**) Schematic of the structure of the laterally coupled distributed feedback (LC-DFB) laser array with deep and planar lateral gratings. (**b**) \(\kappa\) (blue line) and \(\Gamma\) (red line) versus the ridge width with the third order gratings and a duty factor of 0.5. (**c**) \(\kappa\) as a function of the duty cycle \(\gamma\) on third order gratings with a ridge width of 1.2 \(\upmu\)m.  
inductively coupled plasma (ICP) etching with CH\({}_{4}\)/H\({}_{2}\)/Ar gas mixture. Finally, diluted hydrochloric acid solution was adopted to smooth the side wall of lateral gratings, whose microscope image captured by scanning electron microscope (SEM) is shown in Figure 2a.