For this project the commercial software package COMSOL has been used to study both the electromagnetic field and the heat transfer in our system. COMSOL relies on finite element calculations to simulate a system. Finite element calculations, our COMSOL structure and the results of these simulations will be described in this section.
2.2.1 Finite element calculations using COMSOL
Because analytical solutions are generally hard to find and often hard to solve with numerical methods, most numerical methods rely on finite element calculations to find an approximate value close to the exact solution. The numerical calculation will converge onto the exact solution until a certain tolerance condition has been met.
In finite element calculation both the time and the spatial coordinates are discretized. In practice this means space is divided into points, which should obey certain conditions.[33] Time differentials are also discretized after which they can be numerically integrated using standard integration methods as Runge-Kutta.
The heat transfer calculations were done using the heat transfer module. The laser was simulated by a rectangle source input, as is shown in Figure 15. The boundaries at the edge of the system were chosen to act as a heat sink and were chosen at approximately the position where the undercut would end, while the air holes were chosen to be thermally insulating. These simulations were done in 2-D due to computational limits.
In COMSOL the electromagnetic solutions for the photonic crystal are found by solving the Maxwell’s equations using a finite element calculation. For this purpose the RF module has been used, which is described in more detail in [34]. The boundaries are chosen to be absorbing, to ensure a finite system without reflections from the edges. The field was excited using a point source. All these simulations have also been done in 2-D because of computational limits.
2.2.2 Heat transfer simulations
The heat transfer simulations were performed to investigate the effects of a laser spot as a local heating source on the sample temperature and more specifically if it would be possible to heat and thus shift the Noda and the FP cavity independently. For the calculations the following heat transfer coefficients were used[35]:
The optical absorption is considered to be the same for both GaAs and InP. Although this is not completely true, taking into account the uncertainties in the reflectance due to the large numerical aperture of our objective, the difference in absorption can be neglected.
The results of the heat transfer simulations are shown in Figure 15.
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Figure 15: Calculated temperature distribution using the heat transfer simulations performed for our structure for GaAs (a) and InP (b). The laser spot was placed at the rectangle and the absorbed laser power was the same (100μW) in both cases. The Noda cavity is on the left side of the structure.
It can be seen from Figure 15 that both materials have a similar temperature distribution but the maximum temperature reached is severely different. The higher maximum temperature for InGaAsP is caused by the much lower thermal conduction, which requires a higher temperature difference to reach thermal equilibrium. The temperature distribution is mainly dictated by the boundaries, which are, apart from a scaling factor, similar for both materials. This is because the boundaries have been set as a heat sink so the position of the maximum and minimum temperature are forced upon the system
It can also be seen that although the Noda cavity is still heated somewhat the FP cavity is heated much more and it should thus be possible to tune the two cavities nearly independently. In the actual experiments, due to practical limitations, the laser spot will be much closer to the Noda cavity, causing a larger portion of the heat shifting the Noda cavity. This shows that the position of the second spot is crucial for performing the right thermo-optic experiments.
2.2.3 Electromagnetic simulations
The electromagnetic simulations served several purposes; testing the structure and its ideal parameters, comparing spectra found in experiments with simulations and to get the mode distributions. For the electromagnetic simulations a wavelength dependent effective refractive index library has been used for the electrodynamic material properties.
Figure 16 shows the wavelength spectra found in 2-D simulations for the Noda cavity and the FP cavity and corresponding mode distributions in the combined structure. In these simulations both the excitation and collection spot are a point and thus the spectrum from only that particular cavity can be studied. In practice however, both spot sizes are larger than the Noda cavity. This causes for the modes from the barrier to become visible on the spectra. Therefore these two spectra have been combined for the spectra of the Noda cavity.
0 10 20 30 40 50 60 70 80 90 100 0 1 2 3 4 5 6 7 8 9 10
(a)
(b)
ΔT:
ΔT:
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1304 1308 1312 1316 1320
Noda mode Barrier modes
|E y|2 (a.u.)
|E y|2 (a.u.)
Wavelength (nm)
1366 1368 1370 1372 1374 1376
|E y|2 (a.u)
Wavelength(nm)
From Figure 16 it can be seen that the Noda mode, combined with the barrier mode, shows one distinct peak, which is attributed to the Noda cavity and a series of Fabry-Pérot like peaks from the barrier. This barrier acts as an FP cavity due to reflections at the edge of the crystal.
The FP mode has two different resonances, the even and the odd modes, at higher energies. This is also displayed in Figure 9 in the previous chapter. Both these resonances show clear slow-light behaviour where the modes are squeezed together and get an increased Q-factor. This is in agreement with the band calculations of next section. In Figure 16 only the even modes have been plotted.
The resonant wavelength of the FP cavity was now changed by decreasing the refractive index in the most distant half of the cavity. In the meantime the factor of the Noda was determined. The Q-factor was much higher than the expected value due to the fact that these were 2-D simulations and thus there were no losses in the third dimension, which in practice would be the main loss channel.
However for the purpose of this project 2-D simulations would suffice since the goal is to change the in-plane losses. The ΔQ/Q will only be lower in practice because of the lower Q-factor for the Noda cavity.
Figure 17 shows the resulting Q-factor which was acquired from these simulations. Figure 17(a) shows the Q change in the weak coupling regime. The weak coupling regime in this case was achieved by reducing the barrier at the end of the FP cavity to be the same size as the barrier between Noda and FP. Since the main loss channel in this case is the far end of the FP cavity, as apparent by the high Q of the uncoupled Noda, this ensured that the losses and the coupling were comparable. This should be the ideal Q-tuning region, which is also apparent in the high dQ/Q displayed. Figure 17(b) shows the Q change in case of strong coupling, where the barrier at the other end of the FP cavity was increased and the losses thus decreased.
Figure 16: Spectra of the Noda mode and Barrier mode (a) and the Fabry-Pérot mode (b) as acquired from 2-D COMSOL simulations. The FP spectrum shows the FP modes (green arrows) close to the band edge. The high Q of the modes can be explained by the fact that the simulations were done in 2-D and there were thus no vertical losses.
(a) (b)
24 A resemblance between Figure 17 and Figure 11 can also be seen. Especially the smaller Q-tuning region in the strong coupling region is visible in the COMSOL simulations as well. At the lowest point, 2 peaks could be observed which shows the mode anti-crossing and crossing of the Q-factor of the highest Q mode from one mode to the other. This indicates that the effect observed in the COMSOL simulations is really a coupling effect and strengthens the validity of both theoretical descriptions.