# Symmetry Criterion and Far-Field Control of Photonic-Crystal Surface-Emitting Lasers
## 3 Results and Discussion
### The Impact of Period Number  
Figure 5 shows the dependence of divergence angle on the size of the PhC cavity in the PhC slab and PCSEL. The periods of PhC in both structures are set as 277 nm. The deformed triple-lattice structure described in Figure 3 is adopted here. After fitting the data, the following inversely proportional relationship was obtained for both curves:  
\[\theta=\frac{m\lambda}{L}, \tag{1}\]  
where \(\theta\) is the divergence angle expressed in radians, \(m\) is the coefficient mainly determined by the in-plane lattice structure whose value is about 1.87 in Figure 5, \(\lambda\) is the resonant wavelength in free space, and \(L\) is the size of PhC cavity. Note that the difference of the resonance wavelength corresponding to the two structures leads to two fitting curves that do not overlap. Based on this relationship, it can further control the shape of FFPs to obtain different divergence angles in x and y directions by adjusting the period number of PhC in these two directions, respectively. This relationship in Equation (1) can be explained through Fraunhofer diffraction theory in the following way: each cell of photonic crystal can be regarded as an independent light emitting unit, and the far field is a result of the coherent superposition of light waves from all units. Since the resonance occurs at the \(\Gamma_{2}\) point, we assume all emitting units have the same initial phase, so the complex electrical field at the far field is:  
\[\widetilde{E}_{\text{far}}(\alpha,\beta)=\widetilde{E}_{0}(\alpha,\beta)\sum \limits_{m=1}^{N_{1}}\sum\limits_{n=1}^{N_{2}}\exp[i\frac{2\pi}{\lambda}((m-1) a\cos\alpha+(n-1)a\cos\beta)], \tag{2}\]  
where \(\widetilde{E}_{0}(\alpha,\beta)\) is the complex amplitude distribution contributed by a single emitting unit, and we assume its value is the same for all emitting units for simplicity, \(N_{1}\) and \(N_{2}\) are the period number along the x and y directions, \(a\) is the period, and \(\alpha\) and \(\beta\) are the direction angle of emitting rays along the x and y directions. After simplification, the following formula can be obtained:  
\[\widetilde{E}_{\text{far}}(\alpha,\beta)=\widetilde{E}_{0}(\alpha,\beta)\exp( i\phi)(\frac{\sin N_{1}\delta_{1}}{\delta_{1}})(\frac{\sin N_{2}\delta_{2}}{ \delta_{2}}), \tag{3}\]  
where:  
\[\phi=(N_{1}-1)\delta_{1}+(N_{2}-1)\delta_{2}, \tag{4}\]  
\[\delta_{1}=\frac{\pi}{\lambda}a\cos\alpha,\;\delta_{2}=\frac{\pi}{\lambda}a \cos\beta. \tag{5}\]  
Figure 5: Divergence angle versus the size of PhC cavity for PhC slab and PCSEL, respectively. The air filling factor and hole depth are kept as 20% and 160 nm. Through Equation (3), we deduce that the nearest 1/e times amplitude from the central occurs at directions where:  
\[\sin c(N_{1}\delta_{1})\equiv\frac{\sin(N_{1}\delta_{1})}{N_{1}\delta_{1}}=\frac{ 1}{N_{1}\mathrm{e}},\ \sin c(N_{2}\delta_{2})\equiv\frac{\sin(N_{2}\delta_{2})}{N_{2}\delta_{2}}=\frac{ 1}{N_{2}\mathrm{e}}. \tag{6}\]  
Their values approaches zero when \(N_{1}\) and \(N_{2}\) are large. Then,  
\[N_{1}\delta_{1}=\pi,\ N_{2}\delta_{2}=\pi. \tag{7}\]  
Further, assuming their corresponding angles from normal direction, \(\theta_{\mathrm{x}}\) and \(\theta_{\mathrm{y}}\), are small, we obtain:  
\[\theta_{\mathrm{x}}\approx\sin\theta_{\mathrm{x}}=\cos\alpha=\pm\frac{\lambda }{N_{1}a}=\pm\frac{\lambda}{L_{1}},\ \theta_{\mathrm{y}}\approx\sin\theta_{\mathrm{y}}=\cos\beta=\pm\frac{\lambda}{N _{2}a}=\pm\frac{\lambda}{L_{2}} \tag{8}\]  
where \(L_{1}\) and \(L_{2}\) are the size of PhC cavity along the x and y directions. Equation (8) is similar to Equation (1) in expression, and implies that coefficient m in Equation (1) should have a value of around 2. In many applications, a very small beam divergence is necessary. For example, when applied for light detection and ranging and free-space optical communication, laser beams with a divergence angle less than 0.1\({}^{\circ}\) or even smaller are needed.