**4. Bragg diffraction mechanism**

According to Bragg diffraction theory, the first order Bragg diffraction with 2-D PhC triangular lattice will be introduced in Section 4.1. The high order diffraction mechanism will be shown in Section 4.2 together with K2 and M3 PhC modes.

### **4.1 First order Bragg diffraction in 2-D PhC triangular lattice6,13**

Fig. 14(a) shows a photonic band diagram with PhC triangular lattice. Among the points (A), (B), (C), (D), (E), and (F) in band diagram, each of them presents different lasing modes, including Γ1, K2, M1, Γ2, K2, and M2, which can control the light propagated in different lasing wavelength and band-edge region. A schematic diagram of the PhC nanostructure in reciprocal space transferred from real space are shown in Fig. 14(b). The parameter of a is

represented as the number of cylinder layers in the Γ-M direction. The dashed lines of Fig. 12 represent different resonant modes of A, B, C and D at the Γ band edge. It can be observed that the resonant mode frequencies calculated by MSM will approach to band edge frequencies calculated by PWEM when the shell number increases. Therefore, we could obtain more accurate results when the layer number goes beyond 20. Because of the shapes of photonic band diagrams, the blue-shifted or red-shifted trends of normalized

**10 20 30 40 50 60**

**Hole Filling Factor(%)**

Fig. 13. Threshold amplitude gain of four modes as a function of the hole filling factor. The inset shows the lasing mode at Γ point in the PhC plane using Bragg diffraction scheme10.

Fig. 13 shows the threshold amplitude gain of modes A-D as a function of the hole filling factor calculated by MSM. The confinement factor and effective refractive index are 0.865 and 2.482 for guided modes in the calculation, respectively. Hence, real parts of *ε*GaN and *ε*Hole are 7.487 and 3.065 for the GaN material and PhC air holes11,12. In the figure, the mode A and B have the lowest threshold gain for hole filling factors of about 35% and 30%; besides, mode C and D have the lowest threshold gain for hole filling factors of about 10% and 15%. This result shows that the proper hole filling factor can control the PhC mode

According to Bragg diffraction theory, the first order Bragg diffraction with 2-D PhC triangular lattice will be introduced in Section 4.1. The high order diffraction mechanism

Fig. 14(a) shows a photonic band diagram with PhC triangular lattice. Among the points (A), (B), (C), (D), (E), and (F) in band diagram, each of them presents different lasing modes, including Γ1, K2, M1, Γ2, K2, and M2, which can control the light propagated in different lasing wavelength and band-edge region. A schematic diagram of the PhC nanostructure in reciprocal space transferred from real space are shown in Fig. 14(b). The parameter of a is

will be shown in Section 4.2 together with K2 and M3 PhC modes.

**4.1 First order Bragg diffraction in 2-D PhC triangular lattice6,13**

frequencies are increased with the shell numbers in Fig. 12(b).

 **Mode(A) Mode(B) Mode(C) Mode(D)**

**0.00**

**0.01**

**0.02**

**Threshold Amplitude Gain (Ka")**

**4. Bragg diffraction mechanism** 

selection.

**0.03**

the PhC lattice constant. The *K1* and *K2* are the Bragg vectors with the same magnitude, |*K*|=2π/a0. Considering the TE modes in the 2-D PhC nanostructure, the diffracted light wave from the PhC structure must satisfy the Bragg's law and energy conservation:

$$k\_d = k\_i + q\_1 K\_1 + q\_2 K\_2 \, \quad \quad \quad q\_{1,2} = 0, \; \pm 1, \; \pm 2, \; \dots \tag{3}$$

$$
\alpha\_d = \alpha\_i \tag{4}
$$

where *kd* is a xy-plane wave vector of diffracted light wave; *ki* is a xy-plane wave vector of incident light wave; *q*1,2 is order of coupling; ω*d* is the frequency of diffracted light wave, and ω*i* is the frequency of incident light wave. Eq. (3) represents the momentum conservation, and Eq. (4) represents the energy conservation. When both equations are satisfied, the lasing behavior would be observed.

Fig. 14. (a) The band diagram of PhC with triangular lattice; (b) The schematic diagram of PhC with triangular lattice in reciprocal space.

In the calculation, the PhC band-edge lasing behavior would occur at specific points on the Brillouin-zone boundary, including Γ, M, and K which would split and cross. At these PhC lasing band-edge modes, waves propagating in different directions would be coupled and increase the density of state (DOS). Each of these band-edge modes exhibits different types of wave coupling routes. For example, only the coupling at point (C) involves two waves, propagating in the forward and backward directions as shown in Fig. 15(c). In different structures, all of them show similar coupling mechanism but different lasing behaviors. However, they can be divided into six equivalent Γ-M directions. It means that the cavity can exist independently in three different directions to form three independent lasers. Point (B) has an unique coupling characteristic as shown in Fig. 15(b). It forms the triangular shape resonance cavity propagating in three different directions while comparing with the conventional DFB lasers. On the other hand, the point (B) can also be six Γ-K directions in the structure shown two different lasing cavities in different Γ-K directions coexisted independently. In Fig. 15(a) point (A), the coupling waves in in-plane contain six directions of 0°, 60°, 120°, -60°, -120°, and 180°. According to the first order Bragg diffraction theory, the coupled light can emit perpendicular from the sample surface as shown in Fig. 16. Therefore, the PhC devices can function as surface emitting lasers.

Angular-Resolved Optical Characteristics and Threshold

K2 mode would emit at this specific angle of about 30.

**(a) (b)**

mode); *ki* and *kd* indicate incident and diffracted light wave.

mode); *ki* and *kd* indicate incident and diffracted light wave.

surface.

**(a)**

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

Gain Analysis of GaN-Based 2-D Photonics Crystal Surface Emitting Lasers 15

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

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

**X**

**X**

**<sup>Z</sup> <sup>Y</sup>**

**<sup>Z</sup> <sup>Y</sup>**

**(b)**

Fig. 17. Wave vector diagram of (a) in-plane and (b) vertical direction at point (E) (or K2

Fig. 18. Wave vector diagram of (a) in-plane and (b) vertical direction at point (F) (or M3

Fig. 15. Wave vector diagram at points (A), (B), (C) in Fig. 13(a); *ki* and *kd* indicate the incident and diffracted light wave.

Fig. 16. The wave vector diagram at point (A) in vertical direction.

Fig. 15. Wave vector diagram at points (A), (B), (C) in Fig. 13(a); *ki* and *kd* indicate the

Fig. 16. The wave vector diagram at point (A) in vertical direction.

**X**

incident and diffracted light wave.

**Y**
