**4.2 High order Bragg diffraction in 2-D PhC with triangular lattice**

At point (E) which satisfies the Bragg's law, Fig. 17(a) and (b) show the in-plane and vertical diffraction of the light wave diffracted in three Γ-K directions to three K' points. In the wave-vector diagram of one K' point, the light wave is diffracted to an angle tilted 30 normally from the sample surface as shown in Fig. 17(b). Therefore, the lasing behavior of K2 mode would emit at this specific angle of about 30.

At point (F), Fig. 18(a) and (b) represented the in-plane and vertical diffraction that the light wave is diffracted in two different Γ-M directions and reaches to three M' points. Fig. 18(b) shows the wave-vector diagram of one M' point where the light wave is diffracted into three independent angles tilted of about 19.47, 35.26, and 61.87 normally from the sample surface.

Fig. 17. Wave vector diagram of (a) in-plane and (b) vertical direction at point (E) (or K2 mode); *ki* and *kd* indicate incident and diffracted light wave.

Fig. 18. Wave vector diagram of (a) in-plane and (b) vertical direction at point (F) (or M3 mode); *ki* and *kd* indicate incident and diffracted light wave.

Angular-Resolved Optical Characteristics and Threshold

M Γ K

**(a)**

**λ )**

**0.5**

**0.48**

**0.46**

**Normalized Frequency (a/**

**0.44**

**0.42**

diagrams.

Gain Analysis of GaN-Based 2-D Photonics Crystal Surface Emitting Lasers 17

**(b)**

**0.5**

**λ )**

**0.48**

**0.46**

**Normalized Frequency (a/**

Fig. 19. The measured AR μ-PL diagram near the Г1 mode ((a) pumped by YVO4 pulse laser; (b) pumped by He-Cd laser), the dash lines represent the calculated photonic band diagram.

Fig. 20. The measured AR μ-PL diagram near the K2 mode ((a) pumped by YVO4 pulse laser; (b) pumped by He-Cd laser). The dash lines represent the calculated photonic band

diffracted lines can be observed and well matched to the calculated 2-D TE-like photonic band diagram, by using parameters of *r*/*a* = 0.285, *a* = 210 nm, *nb* = 2.560, *na* = 2.343, and *neff* = 2.498 for calculation shown as the dash lines in Fig. 20. In addition, the experiment results show the lasing beam emission angle of about 29 degree off from the normal along the Γ-K direction, which is exactly matched to the estimated value of about 30 degree derived in the previous section. Furthermore, we measured another PCSEL devices exhibited characteristics of M3 band edge mode along the Γ-M direction. The measured dispersion diagrams pumped by a YVO4 pulse laser and a He-Cd CW laser are shown in Fig. 21(a) and (b), respectively. The lasing peaks can be clearly seen in Fig. 21(a). The diffracted patterns can be observed in Fig. 21(b) and well matched by using parameters of *r*/*a* = 0.204, *a* = 230 nm, *nb* = 2.617, *na* = 1.767, and *neff* = 2.498. Shown as the dash lines in Fig. 21, the emission angle of lasing beam was about 59.5 degree off from the normal along the Γ-K direction, which was also quite matched to one of the estimated values of about 61.87

**0.44**

**0.42**

M Γ K
