## 2 Methods  
### Coherent Beam Combining  
Dammann grating (DG) is a binary phase grating which can produce an equal space and intensity beam point array [26; 27]. The transmission function of a DG is  
\[T_{DG}(x,y)=\exp[j\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}\Delta\phi( x-mT_{x},\,y-nT_{y})] \tag{1}\]  
where \(T_{x}\) and \(T_{y}\) are the periods of DG in \(x\) and \(y\) directions, respectively, \(\Delta\phi(x,y)\) is the phase of DG in one period. When a Gaussian beam \(E(x,y)\) illuminates the DG, a Gaussian laser array with same intensity can be generated. The electric field \(E(x,y)\) of the Gaussian beam is given as  
\[E(x,y)=\exp\left(-\frac{x^{2}+y^{2}}{\omega_{0}^{2}}\right) \tag{2}\]  
where \(\omega_{0}\) is the waist radius. The optical field at the back focal plane of the lens L in Figure 1a can be expressed as  
\[U(f_{x},f_{y})=\widetilde{E}(f_{x},f_{y})\otimes\widetilde{T}_{DG}(f_{x},f_{y}) \tag{3}\]  
where \(\widetilde{E}(f_{x},f_{y})\) and \(\widetilde{T}_{DG}(f_{x},f_{y})\) are the Fourier transformation of \(E(x,y)\) and \(T_{DG}(x,y)\), respectively, and the symbol \(\otimes\) represents convolution. Suppose the DG can generate \(M\times N\) equally bright diffraction orders and other high orders \(H(f_{x},f_{y})\) with little energy,  
\[\widetilde{T}_{DG}(f_{x},f_{y})=\sum_{m}^{M}\sum_{n}^{N}\delta(f_{x}-\frac{m}{ T_{x}},f_{y}-\frac{n}{T_{y}})C_{mn}e^{i\phi_{mn}}+H(f_{x},f_{y}), \tag{4}\]  
where \(\delta(x,y)\) is Dirac function, \(C_{mn}\) and \(\phi_{mn}\) (DP) are the amplitude and phase of the diffraction order \((m,n)\). By optimizing the parameters of the DG, \(H(f_{x},f_{y})\) can be very small or even close to 0, that is, the diffraction efficiency is close to 100%. Then, Equation (3) can be simplified as  
\[U(f_{x},f_{y})=\sum_{m}^{M}\sum_{n}^{N}\widetilde{E}(f_{x}-\frac{m}{T_{x}},f_{ y}-\frac{n}{T_{y}})C_{mn}e^{i\phi_{mn}} \tag{5}\]  
When using CDG to combine the beam as shown in Figure 1b, we assume that the array beam is a perfect coherent laser. At the same time, the phase plate (PP) used in the beam combining system is the complex conjugation of the phase \(\phi_{mn}\) in Equation (4), expressed by \(\phi_{mn}^{*}\). The phase distribution of the CDG is the complex conjugate of the DG, expressed by \(T_{DG}^{*}(x,y)\). The laser array field distribution after the PP in Figure 1b is  
\[E_{1}\big{(}f_{x},f_{y}\big{)}=e_{s}\big{(}f_{x},f_{y}\big{)}\otimes\sum_{m}^{ M}\sum_{n}^{N}\delta\bigg{(}f_{x}-\frac{m}{T_{x}}\bigg{)}\delta\bigg{(}f_{y}- \frac{n}{T_{y}}\bigg{)}e^{i\phi_{mn}^{*}}. \tag{6}\]  
where \(\mathrm{e}_{s}(f_{x},f_{y})\) is a function of the incident laser. Because the relationship between the front focal plane and the back focal plane of the lens can be transformed into Fourier transform calculation, we simplify the optical field at the back focal plane of the lens L in Figure 1b as \(\widetilde{E}_{1}(x,y)\propto\Im\big{\{}E_{1}\big{(}f_{x},f_{y}\big{)}\big{\}}\), where \(\Im\) is the Fourier transformation. By combining Equations (5) and (6), \(\widetilde{E}_{1}(x,y)\) can be rewritten as  
\[\widetilde{E}_{1}(x,y)\propto\frac{\Im\big{\{}e_{s}\big{(}f_{x},f_{y}\big{)} \big{\}}}{C_{mn}}T_{DG}(-x,-y) \tag{7}\]  
Therefore, the expression of optical field behind the CDG is  
\[E_{2}(x,y)=\widetilde{E}_{1}(x,y)T_{DG}^{*}(-x,-y)\propto\frac{\Im\big{\{}e_ {s}\big{(}f_{x},f_{y}\big{)}\big{\}}}{C_{mn}}. \tag{8}\]  
From the results of Equation (8), it can be seen that the CDG can effectively eliminate the phase change during beam combination, so as to obtain a uniform optical field distribution, and a better single beam spot can be obtained in the far-field observation. In our work, we can generate 2 \(\times\) 2 array beam by specially designing the structure of DG. The simulation of generating array beams is shown in Figure 2a,b. The simulation of the 2 \(\times\) 2 CDG used for beam combining is shown in Figure 2c-e.