T hr e sh ol d pow e r ( m W )
Г1
K2 M3
Figure 5. 16 The threshold power versus normalized frequency for Г1 K2 M3 groups
The normalized frequencies as a function of r/a ratio were plotted as square points in Figure 5. 17. On the other hand, we apply the plane-wave expansion method in two-dimensions with an effective index model considering the effects of partial modal overlap of electromagnetic fields with the PhC structures to calculate the band diagram of the hexagonal PhC patterns in this structure[7]. The solid (black), dot (red), and dash (green) lines are the calculated band edge frequencies at the Г, K, and M Brillouin-zone boundaries as a function of r/a ratio, which were in accordance with the measured results.
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0.18 0.21 0.24 0.27 0.30
0.40 0.45 0.50 0.55 0.60 0.65 0.70
N or m a li ze d fr e que nc y (a/ λ)
r/a ratio
Lasing modes Simulation M Simulation Γ Simulation K
Figure 5. 17 Normalized frequency verses r/a ratios. The solid, dot, and dash lines represent the simulation results of Г, K, and M lasing groups by PWEM. The square points, inserted in the diagram, present the experiment results mapped and compared with the simulation results.
5-3 Polarization characteristics at different band edge modes
In this section, we will discuss the characteristics of GaN-based PCSELs and demonstrate the specific lasing characteristics at different band edges: Γ, K, and M points calculated by using the plane-wave expansion method. The lasing modes corresponding to the different points of Brillouin-zone boundary can be confirmed by the polarization directions of the laser emissions.
The measured polarization curves for different band edge lasers grouped into Г(circle points and solid line), K(triangle points and dot line), and M(square points and dash line) boundaries calculated by the PWEM are shown in
Polarization
Figure 5. 18(a) and the
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degree of polarization from the emission defined as (Imax-Imin)/(Imax+Imin) was somehow around 50%. The polarization angles from the emissions of devices with different normalized frequencies grouped into Γ, K or M band edge lasers were different. Since the photonic crystal lattices provide the optical feedback, which is the origin of the band edge laser operation, the direction and the polarization of the laser light will strictly follow the photonic crystal lattice vectors. The symmetric feedback directions provided by the 2-D lattice vectors could result in a relatively low degree of polarization if the measurement of the polarization is from the top of the device [8]. As a result, it should be rather difficult to distinguish the specific polarization directions in PCSELs when they are categorized as Γ, K or M band edge lasers. However, the feedback beams could not be equally diffracted by photonic crystal lattices probably due to some disorders or imperfections in the structure. This will result in some beams diffracted in specific directions having higher intensity. The ideally symmetric polarization directions will also be broken. The main polarization directions and the main diffracted laser beams, which are normal to the main polarization directions, can be drawn in a K-space map corresponding to our hexagonal PhC lattice as shown in Figure 5. 18(b). These main diffracted laser beams, shown as dash lines in Figure 5. 18(b), point exactly to the Γ(solid line), K(dot line) and M(dash line) boundaries. The distinct polarization directions provide solid evidence that the lasing actions of our photonic crystal laser originate from different band edges.
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Figure 5. 18 (a) The measured polarization curves for different band edge lasers grouped into Г(circle points and solid line), K(triangle points and dot line), and M(square points and dash line) boundaries calculated by the plane-wave expansion method. (b) The main polarization directions obtained in (a) and their corresponding diffracted laser beams, which are normal to the polarization directions in a K-space map corresponding to our hexagonal PhC lattice.
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5-4 Angular-resolved optical characteristics at different band-edge modes
We set up a rotational stage just under the sample stage to measure the light field distribution which emitted by PCSELs. In this way, the rotation center would match the sample center roughly. We have a detected arm connected the rotational stage and the fiber detector. Rotating the rotational stage, the fiber could collect PL spectrum at different angle. Therefore, we have a series of PL spectra distributed in continuous space, we named these spectra as “angle-resolved μ-PL” (AR μ-PL).
Data normalization
After measurements, we tran sfo rmed th e AR μ-PL spectrum to obtain the guided modes dispersion relation (reduced frequency u=Λ/λ0 as y-axis versus in-plane wave vector, k//, as x-axis), by using the relation k//= k0*sinθ. In addition, each wavelength, IPL(õ), is normalized relative to its integrated intensity[9]. The normalized AR μ-PL diagram reveals the clear dispersion relation of guided modes and figures out the detail information about the relative excitation and out-coupling efficiency.
Figure 5. 19 AR μ-PL diagram
shows the measured dispersion diagram at Г1 mode pumped by the YVO4 pulse laser and pumped by the He-Cd laser. The dash lines represent the simulated photonic band diagram around Г1 mode. Since YVO4 pulse laser has a higher pumping energy intensity, which can overpass the threshold of the stimulated emission provided by the PhC in-plane resonance routes to observe the lasing phenomenon from the devices in Figure 5. 19(a), it can be clearly seen that the PC laser shows the vertical emission near the normal direction to the sample surface. However, except for the lasing peaks, the diffracted lines in this figure cannot be observed clearly due to high intensity of laser peaks. Alternatively, we used a CW He-Cd laser which has a constant average power but a lower pumping intensity to collect diffracted emissions from our PCSEL structures. So the diffracted pattern can be more clearly revealed in the measured dispersion diagram shown in Figure 5. 19(b). It should be noted that the transverse upward curving lines (indicated by black arrows in Figure 5. 19(b)) are resulted from the Fabry-Perot effect provided by the vertical device structure of the p-/n-GaN layers and modulated by the
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interference of the DBR layers. Besides, the obvious diffraction lines can be observed with narrow line widths in the measured dispersion diagram, which are resulted from the in-plane PhC diffraction. In order to explain the observed diffraction patterns caused by a PhC, particular for the guided modes in devices, the electric field propagating in the PhC structure could be described as a Bloch mode: E(r) = ΣG EG×exp [i(k// + G)•r], where EG is the electric field component corresponding to harmonic reciprocal lattice vector G, and k// is the in-plane wave vector of the Bloch mode. In our PhC structure, the reciprocal lattice in K space is a 2-D hexagonal lattice rotated by 30° with respect to the direct lattice in real space and reciprocal lattice vectors can be written as: G = q1K1+ q2K2, where q1 and q2 are integers, and K1 and K2 are the two reciprocal lattice basis vectors.
Harmonics of the Bloch mode are extracted if their in-plane wave vectors are within the light cone: |k// + G| < k0, where k0is defined as 2π/a.
In Figure 5. 19(b), we can observe several groups with different slopes of diffraction lines in the dispersion diagram. The diffraction lines with different slopes represent different dispersion modes, which can be well matched to calculated dispersion curves shown as dashed lines. The parallel diffraction lines with the same slope represent different guide modes in the in-plane direction. Since the lasing peak will occur near the Г1 band edge, by comparing between Figure 5. 19(a) and Figure 5. 19(b), the lasing actually occurs at the third guided mode. It’s interesting to note that instead of lasing at the fundamental guided mode, the third guided mode may benefit from the lower scattering loss resulted from the rough interfaces at the PhC hole boundaries in this PCSEL device. On the other hand, though several band edges would appear in the calculated dispersion curves as shown in Figure 5. 19(a), only one dominant lasing peak was observed at the second lowest band edge mode and one small peak was observed at the lowest band edge mode.
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(a) (b)
M Γ K 0.5
0.48
0.46
0.44
0.42
Normalized Frequency (a/λ)
M Γ K 0.5
0.48
0.46
0.44
0.42
Normalized Frequency (a/λ)
Figure 5. 19 The measured AR-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 dispersion curves.
If we enlarge the calculated dispersion curves near the Г1 band edge as shown in Figure 5. 20(a), the different Г1 band edges will correspond to different field distribution patterns in one unit cell. The magnetic field in one unit cell for each Г1 mode was simulated and shown in Figure 5. 20(b). The orange dash circle indicates the PhC air hole and blue and green areas correspond to positive and negative magnetic fields perpendicular to the plane. Since the PhC air holes were etched through the MQW region in this PCSEL device, the optical gain did not exist in the PhC air holes. As a result, we can only observe the two lowest Г1 band edge modes, for which the magnetic fields cover larger gain regions in the PCSEL structures in comparison to the two higher order Г1 band edge modes, which the magnetic fields locate mostly in air hole region with no optical gain at all, as shown in Figure 5. 20(b).
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Figure 5. 20(a) The enlarged dispersion curves near the Г1 band edge.
Labels (I) to (IV) correspond to four different Г1 band edge modes. (b) (I)-(IV) are the magnetic field distributions in one unit cell for the band edge modes labeled in (a). Blue and green areas represent positive and negative magnetic fields perpendicular to the plane. dash circles indicate the PhC air holes.
Figure 5. 21 show the measured AR-PL diagrams of another PCSEL device with different PC structure near the K2 modes along the Γ-K direction. Fig. 7(a) shows lasing peaks in the ARPL diagram by using YVO4 pulse laser pumping. In addition, the AR-PL diagram in Figure 5. 21(b) was obtained by using CW He-Cd laser pumping. Except for the upward curving lines, the diffracted lines can be observed and well matched to the calculated 2-D TE-like photonic band diagram shown as the dash lines in Figure 5. 21, by using parameters of r/a = 0.285, a = 210 nm, nb = 2.560, na = 2.343, and neff = 2.498 for calculation. In addition, the emission angle of lasing beam was about 29degree off from the normal along the Γ-K direction, which was quite matched to the estimated value (30
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degree) derived in the previous section. Finally, we measured another PCSEL device exhibited characteristics of M3 band edge mode along the Γ-M direction. The measured dispersion diagrams pumped by YVO4 pulse laser and by He-Cd laser are shown in Figure 5. 22(a) and (b), respectively. The lasing peaks can be clearly seen in Figure 5.
22(a). The diffracted patterns can be observed in Figure 5. 22(b) and well matched to the calculated 2-D TE-like photonic band diagram shown as the dash lines in Figs. 8, by using parameters of r/a = 0.204, a = 230 nm, nb = 2.617, na = 1.767, and neff = 2.498. 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 (61.87degree) derived in the previous section. The reason only one emission angle was obtained could be due to that we only measured the AR-PL diagram alone one Γ-M direction.
Figure 5. 21 The measured AR-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 dispersion curves.
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Figure 5. 22 The measured AR-PL diagram near the M3 mode ((a) pumped by YVO4 pulse laser; (b) pumped by He-Cd laser). The dash lines represent the calculated photonic band dispersion curves.
From Figure 5. 19(a), Figure 5. 21(a), and Figure 5. 22(a), each of PhC band-edge modes exhibited specific emission angle by different type of wave coupling mechanism.
Figure 5. 23 shows the divergence angles of Γ1, K2, and M3 band-edge modes on the normal plane from the sample surface despite the measurements were along different directions. The lasing emission angles are (0˚, 29˚, 59.5˚) and the divergence angles of laser beams are (1.2˚, 2.5˚, 2.2˚) for (Γ1, K2, M3) band edge modes, respectively. It should be noted that the measured emission angles might have some offset values (about 1˚ to 2˚) due to the alignment difficulties in the AR-PL system. However, from the above observation of our PCSEL devices, not only the higher band edge modes were determined but their characteristics can be properly matched to the Bragg diffraction mechanism in 2-D PhC structure.
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-40 -20 0 20 40 60
0.0 0.2 0.4 0.6 0.8 1.0 1.2
N or m a lz e d Int e ns it y ( a rb. uni t)
Angle (degree)
Γ1 K2 M3
0 degree 29 degree 59.5 degree
1.2o
2.5o
2.2o
Figure 5. 23 The emission angles and divergence angles of Γ1, K2, and M3 band-edge modes on the normal plane from the sample surface.
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