# Purely Gain-Coupled Distributed-Feedback Bragg Semiconductor Laser Diode Emitting at 770 nm
## 2 Materials and Methods  
The epitaxial structure of our devices is shown in Table 1. We used metal-organic chemical vapor deposition (MOCVD) to grow the epitaxial wafer. The ridge waveguides are made using i-line lithography, after which inductively coupled plasma (ICP) is used to etch them. The periodic and growth electrodes are made via i-line lithography. Figure 1a shows a schematic of the epitaxial structure of our devices. Every device uses an AlInGaAs single quantum well as the active region. The ridge width of the devices is 5.8 \(\upmu\)m, and the etching depth of the ridge is 800 nm. Figure 1b is a scanning electron microscope image of one of our devices. Figure 1c is the transverse-mode simulation diagram obtained by COMSOL Multiphysics ([[http://cn.comsol.com/](http://cn.comsol.com/)]([http://cn.comsol.com/](http://cn.comsol.com/)) (accessed on 22 December 2020), Sweden). The periodic electrical-injection windows through the surface of the devices form a periodic gain contrast in the active region, as shown in Figure 1d, and thus a gain-coupled mechanism [29, 30]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Structure** & **Thickness (\(\upmu\)m)** & **Dopant** \\ \hline GaAs & Cap & 0.1 & Carbon \\ Al(x)GaAs & P-cladding & 0.8 & Carbon \\ Al(x)GaAs & P-wave guide & 0.4 & Carbon \\ Al(x)In(y)GaAs & Quantum well & 0.0055 & Undoped \\ Al(x)GaAs & N-wave guide & 1 & Silicon \\ Al(x)GaAs & N-cladding & 1.5 & Silicon \\ GaAs & Buffer & 0.5 & Silicon \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the epitaxial wafer. The coupling coefficient \(\kappa\) of the devices can be expressed as [30]:  
\[\kappa=k_{0}\Gamma(\Delta n+i\frac{\Delta g}{4k_{0}}) \tag{1}\]  
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 our devices, \(\Delta n\) is the mode effective refractive-index change in the devices, and \(\Delta g\) represents the gain/loss change in the active region. To calculate \(\kappa\), we need to know the values of \(\Gamma\), \(\Delta n\), and \(\Delta g\). The optical confinement factor in our device calculated by COMSOL Multiphysics is \(\Gamma=0.0084\). The gain distribution of two periods in the quantum well (as shown in Figure 2a) can be obtained from PICS3D ([[https://crosslight.com.cn/wordpress/products/pics3d](https://crosslight.com.cn/wordpress/products/pics3d)]([https://crosslight.com.cn/wordpress/products/pics3d](https://crosslight.com.cn/wordpress/products/pics3d)) (accessed on 12 December 2020), Canada) by setting periodic electric injection in simulation; the result is \(\Delta g=695.8\) cm\({}^{-1}\). Therefore, the imaginary part of the coupling coefficient is \(\kappa_{\text{Im}}=1.46\) cm\({}^{-1}\). Due to the plasma dispersion effect, and the refractive index of GaAs material will decrease while the concentration of injected carrier increases [33; 34], as shown in Figure 2b. We calculate the mode effective indices \(n_{eff1}\) and \(n_{eff2}\) by COMSOL Multiphysics from the maximum and minimum index-change values, obtaining \(\Delta n=3.13\times 10^{-6}\) (\(n_{eff1}=3.446107541\) and \(n_{eff2}=3.446104411\)). Therefore, the real part of the coupling coefficient is \(\kappa_{\text{Re}}=0.00215\) cm\({}^{-1}\). As the cavity length is 1 mm in our devices, we obtain the coupling strength \(\kappa L=0.000215+0.146i\) cm\({}^{-1}\). Figure 1: (**a**) Schematic of the epitaxial structure of our device; (**b**) Scanning electron microscope image of a device; (**c**) Transverse-mode distribution map simulated by COMSOL Multiphysics; (**d**) Schematic of carrier-distribution cross section for periodic electric injection. Figure 2: (**a**) One-dimensional gain distribution in the quantum well; (**b**) One-dimensional index change in the quantum well. As can be seen, the real part is only 0.0015 times larger than the imaginary part.