## 2 Materials and Methods  
The device consists of two parts: the front part is the DFB laser, and the rear part is the tapered waveguide as shown in Figure 1a. In the DFB laser section, the period of the grating was 6.0 \(\upmu\)m, and the ridge waveguide (W = 4.0 \(\upmu\)m, L = 1.0 mm) was in the center of the grating. Electrode windows (R = 1.5 \(\upmu\)m) were patterned on the grating to form periodic electrical injection channels, and there was no current injection anywhere except at the electrode windows. The topography of the surface grating and periodic electrodes windows can be seen by scanning electron microscopy (SEM) as shown in Figure 1b. A tapered waveguide (L = 1.5 mm) with a 6\({}^{\circ}\) taper angle increased the output power of the device. The width of the tapered waveguide port was 150 \(\upmu\)m. The insulating channel between the two parts is 250 nm to ensure that the two parts are independently connected to electricity without affecting each other. Ti/Pt/Au metal layers are deposited as p-and n-electrodes by magnetron sputtering. The lift-off technique is used to define the p-electrode shape. Thermal annealing is then applied to improve metal contact and lower electrical resistance. Then the lasers are obtained after wafer cleaving, and two cleaved facets are coated with silicon oxide layer for anti-reflection and protection. The current that defines the access front part is I\({}_{\text{DFB}}\), and the rear part is I\({}_{\text{Tap}}\). The device (W = 500 \(\upmu\)m, L = 2.5 mm) is soldered to a heat sink with the p-side facing down to form an ohmic contact mounted on a thermoelectric cooling (TEC) controlled stage for temperature tuning. No complex process technology is used in the entire process, which makes the fabrication simple and low-cost. The surface gratings, ridge waveguides, taper waveguides, and periodic electrodes were all fabricated by i-line lithography, after which inductively coupled plasma (ICP) is used to etch them. The surface grating was fabricated by inductively coupled plasma shallow etching, which helped to reduce scattering loss. The periodic electrical injection creates periodic gain differences in the active area, as can be seen in Figure 1c. Considering that surface gratings cause simultaneous changes in the real and imaginary parts of the refractive index, we have carefully designed them to avoid too many refractive index coupling effects to prevent higher-order scattering, which can cause additional optical losses and increase the threshold. The coupling coefficient \(\kappa\) is expressed as:  
\[\kappa=k_{0}\Gamma\Delta n\frac{\sin\left(\frac{l\pi}{\Lambda}L_{g}\right)}{l \pi}+i\Gamma^{\prime}\frac{\Delta g}{4} \tag{1}\]  
where \(k_{0}=2\pi/\lambda_{0}\) is the vacuum wave number for the vacuum wavelength \(\lambda_{0}\). \(\Gamma\) is the optical confinement factor of the gating in our device. \(\Delta n\) is the refractive index change in the waveguide. \(Lg\) represents the grating's groove width. \(\Lambda\) is the period for the \(l\) order grating (\(\Lambda=l\pi/n_{eff}\)) and \(\Gamma^{\prime}\) is the optical confinement factor of the active region. \(\Delta g\) represents the gain/loss change in the waveguide. By analyzing the coupling coefficient \(\kappa\), the influence of the refractive index coupling on the device is reduced by decreasing \(\Delta n\) and \(\Gamma\) in the real part, and increasing \(\Delta g\) in the imaginary part makes the gain coupling occupy the main modulation position in practical work. It is known by calculation that the imaginary part is 4 orders of magnitude higher than the real part. This means that the strength of the refractive index coupling is negligible, and the device is mainly modulated by the gain coupling. The epitaxial structure of the device was fabricated by metal-organic chemical vapor deposition (MOCVD), as shown in Figure 1d. The active region of the device was made of AlGaInAs material, which reduced the linewidth of the device.