Pumping Energy Density(mJ/cm2)

Figure 5. 8 The spontaneous emission coupling factor of GaN-based 2-D PCSEL

In general, the cavity quality factor is defined as Quality factor of the VCSEL structure

Q λ

= λ

∆ , where λ is the wavelength emitted from cavity and ∆λ is the FWHM of the emission peak. In our experiment, we cannot obtain the Q value of our PCSEL structure because our PCSEL devices operate at the band-edge modes of the dispersion curve. Therefore, the devices have more loss paths. According to the following equation, the value can be estimated but have some inaccuracies because the theoretical estimation doesn’t consider the fabrication problems and non-uniform of InGaN material in the MQWs region. First, the spontaneous emission coupling factor is estimated of about 5*10^-3 and then we can calculate the Purcell factor of about 5*10^-3 as shown in Eq. (5. 1), where Fp is Purcell factor. In Eq. (5. 2), n is the GaN refractive index of about 2.5 and VC is the optical volume of laser emission. Here, the PhC fabrication area with a PhC lattice constant of

110

about 234nm is a 50-μm circle in diameter. Then, Vc has a volume of about 47 μm³ and lasing wavelength is about 402 nm. Finally, we can calculate the quality factor (Q) about 743.

Figure 5. 9 The calculation schematic diagram and the plane-view SEM image

Characteristic Temperature

Figure 5. 10 shows the seminatural-logarithm plots of the dependence of the threshold pumping energy (In (E_th) on the operation temperature (T₀). The threshold energy gradually increased as the operation rose from 100 K to 300 K. The relationship between the threshold energy and the operation temperature could be characterized by the equation: E_th=E_o*exp (T/T_o), where T_o is the characteristic temperature and E_o is a constant. Therefore, we obtain a characteristic temperature of about 148 K by linear fitting of the experiment data, which is close to the value reported for GaN-based edge emitting lasers ^[6]

111

100 150 200 250 300 5.6

6.0 6.4 6.8

7.2

Linear fit of Experiment data

ln (E th )

Temperature (K)

Figure 5. 10 Temperature dependence of the lasing threshold pumping energy (Eth).

The far-field patterns (FFP) of the laser were detected by an angular-resolved optical pumped system as shown in

Far field pattern (FFP)

Figure 5. 11. In this figure, the lasing far field profiles with different distances from the sample surface were measured. When we increased the measurement distance, the lasing spot sprits into four points with two axes, Г-M and Г-K directions, indicated 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 Figure 5. 11. From the figure, it shows the divergence angle of PCSEL determined by the distance of two lasing spot axes of about 5.6degree which is smaller than edge emitting laser (~10⁰~20⁰) and VCSEL(8⁰).

112

Increase the measurement distance from the sample surface

Figure 5. 11 The far field pattern with different distance from the sample surface collected by objective lens

-0.5 0.0 0.5 1.0 1.5 2.0

-4 -2 0 2 4

K K M M

R a di us ( c m )

Distance(mm)

Figure 5. 12 The divergence angle between the two axes.

5-2 Threshold power characteristics with different coupling coefficients

In our experiment, we had measured fundamental mode and other high order modes has different threshold pumping powers. In order to understand the physical mechanism, we applied the couple-wave model in our PCSEL devices and discussed the operation principle in section 4.2 for triangular PhC lattice. In the following section, we will calculate the coupling coefficient at Г1, K2 M3 band- edge modes and figure out the relation between the calculated coupling coefficient and the experimental threshold

113

pumping power.

The design for lasing action at Г1 device which parameters are described as follows:

Г1 numerical results

r/a=0.25, a=180nm, nb=2.65, na=1.87, neff=2.495

put these parameters in R-soft software and plot the dispersion curve for TE-like mode as shown in Figure 5. 13

Г1

Figure 5. 13 Dispersion curve for TE like mode for Г1 case

For the band-edge Г1, there are four cavity mode frequencies, two are degenerated.

T The cavity mode frequency can be obtained via simple transform of the normalized frequency which derived from R-soft. The normalized frequency values from lower to higher are 0.448, 0.460, 0.4704, and 0.4930. Once the cavity mode frequency at the individual band-dges can be obtained, we can derive the coupling coefficients κ1, κ2, and κ3 from Eq. (4. 42) and its values are 17480 cm^-1, 11240 cm^-1 and 11248 cm^-1, respectively. Therefore, for Г1, the lasing oscillation forms a hexagonal cavity which provided the major significant contribution to support the lasing action.

The design for lasing action at K2 device which parameters are described as follows:

K2 numerical results

114

r/a=0.26, a=220nm, nb=2.63, na=2.0, neff=2.495

put these parameters in R-soft software and plot the dispersion curve for TE-like mode as shown in Figure 5. 14.

K2

Figure 5. 14 Dispersion curve for TE like mode for K2 case

For the band-edge K2, there are two cavity mode frequencies, one is degenerated.

The cavity mode frequency can be obtained via simple transform of the normalized frequency which derived from R-soft. The normalized frequency values from lower to higher are 0.5326 and 0.5413. Once the cavity mode frequency at the individual band- edges can be obtained, we can derive the coupling coefficients κ from Eq. (4. 46) is 4089 cm^-1. Therefore, for K2, the lasing oscillation forms a triangular cavity which provided energy to support the lasing action.

The design for lasing action at M3 device which parameters are described as follows:

M3 numerical results

r/a=0.266, a=247nm, nb=2.64, na=2.01, neff=2.495

put these parameters in R-soft software and plot the dispersion curve for TE-like mode as shown in Figure 5. 15

115

M3

Figure 5. 15 Dispersion curve for TE like mode for M3 case

For the band-edge M3 mode, there are four cavity mode frequencies. The cavity mode frequency can be obtained via simple transform of the normalized frequency which derived from R-soft. The normalized frequency values from lower to higher are 0.60898, 0.60943, 0.61409, and 0.62335. Once the cavity mode frequency at the individual band-edges can be obtained, we can derive the coupling coefficients κ1, κ2, and κ3 from Eq. (4. 50) and its values are 1241 cm^-1,1356 cm^-1 and 2683 cm^-1, respectively. Therefore, we know the lasing oscillation back and forth provided the major significant contribution to support the lasing action.

The threshold gain is determined by two factors, one is the gain medium and the other is coupling coefficient. It is expected that the lasing action occurs at Г1 band edge should have the lowest threshold gain due to the largest coupling coefficient. Therefore, we analyze the threshold gain of PCSELs with its r/a, ranges from 0.25 to 0.26, as a function of normalized frequency as shown in Figure 5. 16. It is obvious to see the Г1 indeed has the lowest threshold gain and M3 has highest threshold gain which is corresponding to our expectation. In the future, for the electrical pump PCSELs fabrication, one can follow this rule and design for Г1 group to achieve lasing action.

116

0.4 0.5 0.6 0.7

0.2 0.3 0.4 0.5 0.6