Chapter 2: Exploring the Photonic Band Gap Properties of 3D Periodic Architectures – Tuning Band Gap
2.6. Comparison of Experimental Bandgap Position and Simulated Bandgap Results
To numerically investigate the electromagnetic properties of the octahedron nanolattices and see how they compare to experimental results, we employed the Finite Difference Time Domain (FDTD) solver Lumerical. The simulated geometries were determined by using SEM images of the fabricated experimental samples; the polymer refractive index of n~1.49 was determined by fitting FTIR reflectance data for TPL polymerized IP-Dip thin films using a scattering matrix approach combined with a minimization technique (described in Appendix C). We accounted for the almost infinite transverse extent of the lattices by designating Bloch boundary conditions in the x-y plane. In the z-direction normal to the plane, 6.5 unit cells were stacked on top of one another to complete the 3D model of the lattice—similar to our fabricated samples—and the whole stack was assumed to reside on an infinite slab of Si with a fixed refractive index of 3.4 over all frequencies. The entire model was then surrounded by perfect matched layers (PML) along the z-direction.
20 Figure 11. Results of simulated reflectance for effectively strained nanolattices compared to experimental data.
(a) Normalized reflection spectra under normal illumination for a simulated 45° unstrained octahedron nanolattice, and three simulated angle-varied nanolattices, each with corresponding degrees of increasing effective strain. The main peak for each simulated nanolattice corresponds to a 1st order Bragg reflection; secondary peaks that appear at longer wavelengths are caused by the higher order Bragg reflections. (b) Strain-stopband plots comparing data from the as-fabricated angle-varied lattices under corresponding effective strain, experimentally strained nanolattices using a compression cell over multiple cycles, and the simulated angle-varied lattices at normal incidence, and 25.7°
incidence.
Figure 11(a) shows the reflectance spectra for four different simulated angle-varied octahedron unit cells, 45°, 40°, 35°, and 30°, illuminated with a normal-incidence plane wave. The simulations show a clear blueshift of the reflection peak with decreasing apex angle of the octahedron unit cell, in agreement with experimental results.
These numerical results also display a linear trend between reflection peak position and effective strain, as do the experiments on the as-fabricated and in-situ strained nanolattices (see Figure 11(a)). Despite the difference in the actual position of λpeak for the fabricated and simulated angle-varied lattices, the slopes of these lines are very similar. While the simulations appear to over-predict the stopband wavelength, this discrepancy in λpeak position is in large part due to the angle of incident light used for illuminating simulated and fabricated samples; simulated samples are illuminated using plane waves incident at a single angle, and fabricated samples are under illumination from a Gaussian beam with an angular range between 16-35.5°.
Simulating nanolattices at an average illumination angle of 25.7° does shift the calculated stopband peak closer to the experimentally-measured peaks. Further discrepancies can be attributed to some non-idealities
21 in the fabricated samples, like the imperfect uniformity of beams and unit cells and buckling at the joints between unit cells.
The reflection peak observed in our experiments and simulations can be attributed to the 1st order Bragg reflection in the lattice. Bragg’s law is formulated as 2𝑑 × 𝑐𝑜𝑠𝜃 = 𝑛𝜆, where d is the vertical separation between two layers in the lattice, θ is the angle of the incident beam with the normal line, and n is the order of the Bragg reflection. A monotonic decrease in d from uniaxial strain will result in a monotonic decrease in the resonance wavelength 𝜆𝑝𝑒𝑎𝑘. This result is in agreement with the general blueshift trend observed both in the numerical and experimental data. Additional reflection peaks observed at longer wavelengths can be attributed to higher orders of the Bragg grating, and are substantially weaker than the main peak of the lattice.
As previously mentioned, the primary peak of the reflection can be attributed to the 1st order Bragg reflection in the lattice. One consequence of this fact is that, increasing the angle of incidence light, relative to the normal, will shift the resonant reflection peak to shorter wavelengths. The color map shown in Figure 12 shows the variation in reflection as a function of incident angle and wavelength for the case of a simulated, unstrained, 45° lattice. It can be clearly seen that as the angle of incidence with respect to the normal increases and the peak resonance wavelength shifts to the shorter values. For further clarification, we superimposed a dashed black line which shows the relation between the incident angle and the resonant wavelength as predicted by the Bragg condition equation. Notice that the line closely follows the resonant wavelengths obtained by numerical simulations. The discrepancies between the peak values of the colormap and the black line are due to the presence of the silicon substrate in the numerically modeled lattice. This substrate redshift the resonance following the Bragg condition, and is more substantial at larger angles of incidence. It should also be noted that the angle of incidence affects the stopband position in a nearly identical manner for the 40°, 35°, and 30° lattices.
22 Figure 12. A color map depicting the variation in reflection as a function of incident angle and wavelength for a simulated 45° nanolattice. Note that the angle of incidence here is measured with respect to the line normal to the surface. At larger angles of incidence, the peak in reflection shifts to shorter wavelengths, and the black dashed line overlaying the data shows the relation between the incident angle and the resonant wavelength given by the Bragg condition equation.