**3.3 Far field patterns (FFP) of PhC fundamental mode lasers**

The lasing area of the GaN-based 2-D PCSEL, obtained by a CCD camera, is relatively large and covers near the whole area of PhC pattern with only one dominant lasing wavelength as shown in Fig. 9. It's interesting to note that the threshold power density of GaN-based 2-D PCSEL is in the same or even better order than the threshold of the GaN-based VCSEL we have demonstrated recently8. Unlike the small emission spots observed in the GaN-based VCSELs, the large-area emission in 2-D PCSEL has great potential in applications and requires high power output operation.

Fig. 6. The angular-resolved μ-PL (AR μ-PL) system

nm with a linewidth of 0.19 nm can be observed.

requires high power output operation.

**3.3 Far field patterns (FFP) of PhC fundamental mode lasers** 

**3.2 Threshold characteristics of fundamental mode of PhC lasers** 

In the optical pumped experiments of PCSEL devices, the lasing action was clearly observed in several devices with different lasing wavelength ranging from 395 nm to 425 nm. Fig. 7 shows the output emission intensity versed the pumping energy density with the PhC lattice constant of about 254nm. In the figure, the clear threshold pumping energy shows at the threshold pumping energy density of 2.8 mJ/cm2, and a peak power density of 5.6 MW/cm2. When the laser pumping energy exceeds the threshold energy, the laser output intensity increases abruptly and linearly with the pumping energy. Fig. 8 shows the excitation energy dependent emission spectrums of 0.8 Eth, 1 Eth, 1.2 Eth, and 1.3 Eth. These spectrums clearly show the transition behavior from spontaneous emission to stimulated emission. Furthermore, above the threshold, only one dominant peak wavelength of 419.7

The lasing area of the GaN-based 2-D PCSEL, obtained by a CCD camera, is relatively large and covers near the whole area of PhC pattern with only one dominant lasing wavelength as shown in Fig. 9. It's interesting to note that the threshold power density of GaN-based 2-D PCSEL is in the same or even better order than the threshold of the GaN-based VCSEL we have demonstrated recently8. Unlike the small emission spots observed in the GaN-based VCSELs, the large-area emission in 2-D PCSEL has great potential in applications and

Fig. 7. Laser intensity as a function of pumping energy density

Fig. 8. The lasing spectrums under different pumping energy densities

Fig. 9. The lasing CCD image is at 1.3 Eth and the dash circle is the PhC nanostructure region of about 50μm

Angular-Resolved Optical Characteristics and Threshold

follows:

The *Ani*

(*N* values).

obtained as below9:

and *Tni*

normalized frequency from *k* =*ω*/*c* in Eq. (1).

**(a) (b)**

**C**

**A B**

**A B**

**C D**

**D**

Gain Analysis of GaN-Based 2-D Photonics Crystal Surface Emitting Lasers 11

The complex dielectric constant is the light amplification in GaN-based material shown as

where *ε*GaN represents the dielectric constant varied with frequency of light and *ka"* represents the amplitude gain coefcient of the material. A point source transmitted monochromatic waves are placed at the original point. The total system matrix can be

*i i ij j <sup>i</sup> n n ln l <sup>n</sup>*

incident waves, respectively. Here, according to main dipole oscillation in the GaN active region, only the transverse electric (TE) mode polarization (polarization direction perpendicular to the cylinder axis) is considered10. Eq. (2) could be simplified to an eigen value problem: MA=T. If the value of vector A / T is divergent, the laser oscillation condition would be achieved. Therefore, det(M)=0 is the complex determinant equation which is used to search for a pair of variables of threshold amplitude gain *kam"* and

*A GA T* 

1,

*j j il*

 

Fig. 12. (a) Photonic band diagram of a PhC triangular lattice with TE mode polarization calculated by PWEM near the first Γ band edges showing four different modes; (b) Normalized frequencies of lasing modes calculated by MSM for different PhC shells

According to PWEM, the first Γ band edge of photonic band diagram with the PhC triangular lattice and TE mode polarization are calculated as shown in Fig. 12(a). We can find four different band edges causing four resonant modes (A – D) since modes B and D are doubly degenerate. In Fig. 12(b), the normalized frequencies of lasing modes is calculated for different PhC shells (*N* values) by MSM., where the parameter *N* is

*N*

" <sup>2</sup> () () *GaN backgound GaN a*

> , ,

are matrixes representing expansion coefficients of scattering waves and

*<sup>c</sup> i k*

(1)

(2)

The far-field patterns (FFP) of the laser were detected by an angular-resolved optical pumped system as shown in Fig. 10. In this figure, the lasing far field profiles with different distances from the sample surface were measured. When we increased the measurement distance from the sample surface, the lasing spot sprits of four points with two axes, Г-M and Г-K directions, indicated that the lasing has strong direction and energy concentration properties in real space. Then, we re-plotted the lasing spot sizes as a function of the measurement distance as shown in Fig. 11. In the figure, it shows the divergence angle of PCSEL determined by the distance of two lasing spot axes of about 5.6 degrees. It is smaller than edge emitting laser (~100~200) and VCSEL(80).

**Increase the measurement distance from the sample surface**

Fig. 10. The far field pattern with different distance from the sample surface collected by objective lens

Fig. 11. The divergence angle between the two axes.

#### **3.4 Threshold gain analysis by multiple scattering method (MSM)**

This section would introduce the multiple scattering method (MSM) shown below:

The simulation structure is composed of finite two-dimensional PhCs nanostructure with triangular-lattice patterns and parallel cylinders placed in a uniform GaN-based material.

The far-field patterns (FFP) of the laser were detected by an angular-resolved optical pumped system as shown in Fig. 10. In this figure, the lasing far field profiles with different distances from the sample surface were measured. When we increased the measurement distance from the sample surface, the lasing spot sprits of four points with two axes, Г-M and Г-K directions, indicated that the lasing has strong direction and energy concentration properties in real space. Then, we re-plotted the lasing spot sizes as a function of the measurement distance as shown in Fig. 11. In the figure, it shows the divergence angle of PCSEL determined by the distance of two lasing spot axes of about 5.6 degrees. It is smaller

**Increase the measurement distance from the sample surface** Fig. 10. The far field pattern with different distance from the sample surface collected by

**-0.5 0.0 0.5 1.0 1.5 2.0**

**Distance(mm)**

than edge emitting laser (~100~200) and VCSEL(80).

**-4**

Fig. 11. The divergence angle between the two axes.

**3.4 Threshold gain analysis by multiple scattering method (MSM)** 

This section would introduce the multiple scattering method (MSM) shown below:

The simulation structure is composed of finite two-dimensional PhCs nanostructure with triangular-lattice patterns and parallel cylinders placed in a uniform GaN-based material.

**-2**

**0**

**Radius (cm)**

**2**

**4**

 **K K M M**

objective lens

The complex dielectric constant is the light amplification in GaN-based material shown as follows:

$$
\varepsilon\_{background}(\alpha) = \varepsilon\_{GaN}(\alpha) - i \frac{2\varepsilon\sqrt{\varepsilon\_{GaN}}}{\alpha} k\_a^"\tag{1}
$$

where *ε*GaN represents the dielectric constant varied with frequency of light and *ka"* represents the amplitude gain coefcient of the material. A point source transmitted monochromatic waves are placed at the original point. The total system matrix can be obtained as below9:

$$
\Gamma\_n^i A\_n^i - \sum\_{j=1, j \neq i}^N \sum\_{l=-\infty}^\infty G\_{l,n}^{i,j} A\_l^j = T\_n^i \tag{2}
$$

The *Ani* and *Tni* are matrixes representing expansion coefficients of scattering waves and incident waves, respectively. Here, according to main dipole oscillation in the GaN active region, only the transverse electric (TE) mode polarization (polarization direction perpendicular to the cylinder axis) is considered10. Eq. (2) could be simplified to an eigen value problem: MA=T. If the value of vector A / T is divergent, the laser oscillation condition would be achieved. Therefore, det(M)=0 is the complex determinant equation which is used to search for a pair of variables of threshold amplitude gain *kam"* and normalized frequency from *k* =*ω*/*c* in Eq. (1).

Fig. 12. (a) Photonic band diagram of a PhC triangular lattice with TE mode polarization calculated by PWEM near the first Γ band edges showing four different modes; (b) Normalized frequencies of lasing modes calculated by MSM for different PhC shells (*N* values).

According to PWEM, the first Γ band edge of photonic band diagram with the PhC triangular lattice and TE mode polarization are calculated as shown in Fig. 12(a). We can find four different band edges causing four resonant modes (A – D) since modes B and D are doubly degenerate. In Fig. 12(b), the normalized frequencies of lasing modes is calculated for different PhC shells (*N* values) by MSM., where the parameter *N* is

Angular-Resolved Optical Characteristics and Threshold

behavior would be observed.

PhC with triangular lattice in reciprocal space.

Therefore, the PhC devices can function as surface emitting lasers.

Gain Analysis of GaN-Based 2-D Photonics Crystal Surface Emitting Lasers 13

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

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

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

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.

11 22 1,2 , 0, 1, 2, ... *d i k k qK qK q* (3)

*K1*

*K2*

**Γ**

*d i* (4)

wave from the PhC structure must satisfy the Bragg's law and energy conservation:

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

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 selection.
