# Symmetry Criterion and Far-Field Control of Photonic-Crystal Surface-Emitting Lasers
## 3 Results and Discussion
### The Impact of In-Plane PhC Structure  
In this part, we adjust the in-plane PhC structure to study its influence on FFP while keeping the other parameters unchanged. The air filling factor, period number, and hole depth are set as 20%, 31, and 160 nm, respectively. The air filling factor is defined as the ratio between the area of air holes and the area of the unit cell. The corresponding FFPs of these structures are obtained by the aforementioned method and will be analyzed from the perspective of symmetry later.  
From previous investigations on photonic bands, it was shown that the modulation of light was significantly affected by the optical confinement in the PhC layer [29]. In order to maximize the influence of in-plane PhC structure on the far field, and avoid interference from other complicated factors, we use the PhC slab as the calculation model to obtain the results in Figure 3. In this case, both sides of the PhC layer are air cladded unlike the epi-structure of PCSEL model in Figure 1a. Thereby, the optical confinement in the PhC layer is maximized and so is the coupling coefficient.  
In Figure 3, we show the FFP and NFP of PhC slabs with various in-plane structures, including a circle, an isosceles triangle, an isosceles right triangle, and other six structures. They are grouped into two categories, i.e., single-lattice and multiple-lattice structures. The latter can be regarded as several single lattices nested within each other. Since the electric field components determine the final vertical radiation, we calculate their distribution in a single unit cell at the center of slabs. Among single-lattice structures, when the hole-shape of a unit cell is a circle, its electric field distribution exhibits all the same symmetries as the square lattice structure, including rotational symmetry. This leads to negative interference and eventually results in an annular FFP, as mentioned in Figure 2. When its shape changes to an isosceles right triangle, the FFP displays a circular single-lobed spot, which is usually important for a diode laser. This is because the electric field loses the rotational symmetry and then the negative interference effect degrades. When the shape is an isosceles triangle, symmetrical about the x-axis, the FFP displays an elliptical spot with a larger divergence angle in the y-direction than in the x-direction.  
Further, we study the FFP corresponding to multiple-lattice PhC structures. This type of PhC structure is more suitable for controlling the in-plane optical loss and vertical radiation factor by adjusting the space between nested lattices or adjusting the shape of holes in each lattice. Thus, some key performances of PCSELs based on them would be  
Figure 3: The schematic of various in-plane PhC structures and their corresponding electric field, FFP, and NFP. The period number of PhC is 31.  
characteristic as FFPs, as shown in Figure 3. Thus, their characteristics are not listed independently in Table 1.  
In fabrication, holes with special shapes, such as triangles, are more difficult to process than circular shapes. In this way, multi-lattice structures with several circular holes reduce fabrication difficulties compared to single-lattice designs, while achieving the same goal by tuning the hole positions. However, multiple-lattice structures tend to have smaller hole diameters, which is a drawback in the exposure and etching processes, but may be an advantage in the process of retaining air holes during regrowth.