Chapter VI: Photonic Structures Towards Perfect Reflection in Luminescent
7.3 FDTD Calculations
Figure 7.3: Folded band diagrams, calculated in 3D, of a high performing PC slab waveg- uides made from lattices of rods with index 2.2 on substrate of 1.42. The insets shows a profile view of the structure in the top left, and a schematic of the irreproducible Brillouin zone vectors in the center bottom. Bands below the substrate cone are guided laterally in the photonic crystal. The Stoke’s for CdS/CdSe QDs is also depicted
area. These modes are TIR modes, and can work in cooperation with the photonic crystal slab to guide PL in the lateral direction. Diffraction from the lattice imparts a large enough lateral momentum for PL to enter into these TIR modes should they not originally coupled to the PC waveguide modes. Free-space incident light is absorbed above the blue light cone, and a frequency down-shift due to the Stoke’s-shift allows a free-space photon to become a trapped photon as the frequency crosses the light cone, allowing PL into a new high magnitude~kkvector, as depicted. It is also worth mentioning that, in the absence of the discrete bands, the same mechanism of TIR PL trapping is enabled by this crossing from the light cone to the trapped modes of the substrate cone. This is actually the mechanism of traditional LSCs. The forbidden cone represents modes that are unavailable because they have momentum greater than allowed by the high index material. These diagrams helped to narrow the range of exploration in the computationally expensive FDTD simulations to PCs with promising waveguide modes as shown.
minimized any coherent interactions with boundary reflections. Two groups of 7 Poynting vector monitors were arranged in rectangular prisms, excluding a field monitor on the face that is adjacent to the PC slab such that all power flux leaving the area defined by the slab is recorded right before it exits the simulation via the absorbing boundaries. This allows the free-space waves to evolve into fields suitable for analytic far-field projection calculations used to determine the spherical power distribution at a radius of 30 µm into either half-space from the dipole emitter.
This distance is the thickness of waveguides used in previous LSC experiments (Chapter 6). Finally, a set of 4 Poynting vector monitors surround the edges of the slab to record the power that exits in-plane with the PC waveguide. The monitors account for 100% of the simulation radiation power input.
In these first steps, the structure symmetry was used to reduce the computational cell to only the first quadrant. This requires that the dipole source be placed directly at the xy origin to satisfy the field symmetry requirements in addition to the structure symmetry. It is noted that this limits coupling to targeted symmetric modes. Simulation results for dipoles oriented in the x, y, z give significantly different results due to the polarizations of the trapped modes, specifically, the z- oriented dipole responses dominated the average results. This dominance is the result of strong coupling and correspondingly high Pf.
Figure 7.4A) shows the behavior of a Z-oriented dipole embedded in the center of the hexagonal lattice of Figure 7.3 across the entire range of trapped~kk at the edge of the Γ in the direction of K. The stacked color spectrum shows the portions of the normalized power that travel into 4 different directions. Red indicates loss to free-space, the two shades of blue represent emission into the substrate, and finally the yellow is emission in the plane of the PC slab waveguide. PL into the substrate is subdivided by whether or not it enters at angles outside the TIR escape cone. Light outside the escape cone will be trapped at the opposite substrate interface. Plotted on the right axis is the Pf in green. Finally, the bright orange qualitatively indicates the PL of CdS/CdSe QDs when weighted by the Pf spectrum. To be quantitatively rigorous, a future detailed-balance photon accounting needs to be completed. The black dashed line indicates the wavelength for which Figure 7.4B)-D) correspond, and is the peak of combined in-plane and TIR PL containment (the sum of the yellow and light blue areas).
The yellow segment of Figure 7.4 A) shows that near a certain frequency, the portion of the light that is coupled into directions with large ~kk increases drastically; this
A) B)
D) C)
Figure 7.4: Results from a Z-oriented dipole centered in the hexagonal PC slab waveguide from Figure 7.3 A) The spectrum is parsed into the possible directions power can exit the simulation from its source in the center. The Pf spectrum and the weighted PL emission peak of the QDs previously presented in Chapter 6 are overlaid. The PL spectrum is not a rigorously quantitative representation. B) A polar plot of the emission into the substrate, where the white dashed line denotes the TIR angle outside of which PL is trapped. C) and D) show the log of electric field magnitude in XY and XZ cross-sections at the wavelength indicated by the black dashed line of A). The field is concentrated into the rods in both planes. In D) the high-angled emission into the substrate is clearly visible.
frequency is about 0.36, which from Figure 7.3A) it is seen to be the frequency where the first TM band begins to flatten as it approaches the Brillouin zone edge.
A remarkably large Pf spike occurs as predicted by theory for this frequency. At a slightly larger frequency, the fraction of in-plane flux peaks and slowly begins to decline again towards higher frequencies due to the continuum of available ~ks creating weaker preferential coupling. At lower frequencies the in-plane flux falls precipitously because there are no modes other than the linearly dispersive modes of the substrate. The fraction of totally internally reflected PL falls monotonically to the nominal value of a planar slab, 74%. Figure 7.4 B) shows the angle dependent distribution of power into the glass substrate, with the angle of TIR boundary given by the white dashed circle. The majority of the power is seen at angles greater than 44.8 deg, indicating successful coupling to large wave vectors, ~kk. The peak value of maximum transmission for this dipole orientation, location, and geometry is 98.8%, far greater that the TIR value of 74%. This demonstrates the promise of this coupled dipole approach to directional PL emission.
Figure 7.4 C) and D) show log-scale plots of the electric field magnitude in the XY plane and the XZ plane. Figure 7.4 C) shows that the field energy takes on the periodicity of a combination of the lowest order modes at the peak trapping frequency. Figure 7.4 D) further confirms that the energy is confined to the PC waveguide with some fraction being emitted at high angles into the substrate. Some loss at high angles into the positive Z direction is seen. This is seen as power directly above the slab. The power to the left and right of the center is likely evanescently bound to the waveguide.
This is only an example of a very specific favorable situation, in which the emitter was purposefully located at a high symmetry point and oriented properly to excite TM modes. The same simulation was completed with dipoles in the X and Y orientations and averaged to obtain an incoherent emitter representation. The result is a weighted average of the emission pathways, weighted by the spectrally dependent Pf. The maximum fraction trapped for an incoherent emitter was found to be a respectably high 96.4%. In this instance, the Z-orientation dominated the average because of its high Pf, while the emission is suppressed from X and Y oriented dipoles. Pf was between 1 to as low as 0.18 for both X and Y.
The relative Z position of the emitter was also symmetrically optimized. A sen- sitivity to Z-placement is shown in Figure 7.5 for the weighted average maximum trapping fraction. Surprisingly, the simulated PL is not strongly scattered into free-
Figure 7.5: Smoothed maximum of orientation averaged trapped fraction and Pf from dipole placements along the thickness of the PC waveguide from Figure 7.3A). The Z position of 0.5 corresponds to the middle and position of peak value.
space or into the substrate escape cone even when an emitter was precisely at the Z limits of the PC waveguide. Weighted average Pf also has a dependence on the Z position, indicating that the majority of the in-plane trapped light comes from the center, and the extremities are less influenced by the PC structure, approaching homogeneous values of Pf 1.