Simulation of Observed Experimental Results

Chapter V: Near Perfect Anti-reflection in Lossless Dielectric Nanocone

5.4 Simulation of Observed Experimental Results

0 2 4 6 8

10 ^{Glass dbot 540nm}

0°10° 20°

30° 40°

50° 60°

500 600 700 800 900 1000 0

2 4 6 8

10 ^{dbot 650nm}

500 600 700 800 900 1000 dbot 900nm

Reflectance

Wavelengths (nm)

% Reflection

Figure 5.8: The polarization averaged reflection of a glass cover on a silicon solar cell. This is the extracted reflection from the glass interface, and not of the entire stack. Even these relatively short nanocones with aspect ratio between 0.65 and 1.08 can reduce reflection significantly out to large AOI compared to flat glass.

infrared spectrum. Unpolarized and incoherent measurements were not performed due in part to the need to isolate collection to only the patterned area of the glass and a mismatch between that instrument’s larger spot size, the shape of the diced silicon cells, and nanocone array high fidelity areas making the samples impossible to properly mount.

and red dashed lines, wheremk indicates an integer multiple of the relevant lattice vector. The free space and substrate dispersion lines are folded in the first zone for values up tok₀= 2M =1.154.

We trace the substrate light line from the zero momentumΓpoint in theMdirection, past its first fold to when it crosses back to theΓpoint at ak₀of 2M/n_{substr ate} =0.759 marked as the blue horizontal line. Here the Rayleigh condition of Equation (5.1) is satisfied and allowing the first transmitted diffraction order. Free space light at normal incidence has enough momentum to scatter into the plane of the lattice with a new component ~kk ≤ ~k0. At this k0, the experimental response starts to dip below the baseline value as diffractive light trapping is activated. Waves are trapped between the photodiode efficiently until the substrate light line crosses the free space light line, allowing for easier scattering back into free space. The shoulder (just below the cyan horizontal line) in the response is indicative of this scattering occurring in experiment with low efficiency. The low efficiency is likely due to difficulty coupling back into the previous scattering mode after undergoing reflection from the photodiode. The cyan line indicates the k0 that is equivalent to the lattice momentum of 2K/n_{substr ate}, where the first order diffraction in theK direction is allowed into the substrate. This diffraction creates a second channel for diffracted light with large~kk that is also trapped. The geometric lattice vector magnitude from 2M/n_{substr ate}and 2K/n_{substr ate}onsets is plotted and overlaid as the dark blue and cyan dashed curves up to theΓpoint withk₀= 2M. This matches the simulation of the complex cone structure well, indicating that the lattice momentum function and the band structure are in agreement. First order free space diffraction is allowed when the ambient light line returns to theΓpoint atk₀ =2M, the horizontal magenta line. Here, the first free space mode can efficiently trade momentum|2M| with either of the two substrate diffraction modes. At thisk0, light that was diffracted and subsequently reflected back to the cone array can easily scatter into oblique free- space modes without a large change in momentum. The light trapping effect seen in experiment is essentially quenched up until the second order diffractive modes, as indicated by the right hand side line spectra of aspect ratio 0.65.

Last, we look at the parameter sweep of aspect ratio for the canonical cone dis- playing the content of diffracted light in the substrate and free-space directions in

M K k 0.25

0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.25

k0

k0namb/ k0ns/2M

k0ns/K k0namb/2M kk00nnambs/mk/mk

Refl. Decrease (a.u.)0 t/d_bot=1.08 t/d_bot=0.9 t/d_bot=0.65

Figure 5.9: The FDTD computed band structure of the canonical cone array on glass.

Overlay are lines corresponding to either the ambient or substrate dispersion or a diffracted order dispersion relation. Horizontal lines at Rayleigh points are meant to guide the eye between features of the band diagram and the normalized experimental resutls plotted on the right. Experimental results from measurements with the smooth photodiode shown by green dots, orange dashes, and solid blue represent experimental arrays ofd_bot=900 nm, 650 nm, and 540 nm respectively.

Figure 5.10. The only difference between experimental cones and this parameter sweep is a top-bottom diameter ratio difference, which is seen in Figure 5.2 to have low influence on the response below 0.3, and the experimental range is between 0.1 and 0.18. The horizontal lines are carried over from Figure 5.9, marking the Rayleigh points. The vertical lines represent the experimental samples. Immedi- ately seen in these plots is that no power found in diffraction modes below their allowed momentum, consistent with the already presented results and basic diffrac- tion theory. Of immediate importance is an explanation to why the light trapping in experiment manifests with a slow onset from the first allowed substrate diffrac- tion. In Figure 5.10A) the allowed diffraction has low efficiency starting at the first order (blue dotted line), increases more quickly after the next order is allowed (cyan dotted line) leading up to the first reflected diffraction (magenta dotted line) with a diffraction strength of about 30%. These results corroborate the conclusions drawn from analysis of the band structure diagram. At the lower k_O values, it is possible to see that the lower aspect ratio cones have more diffraction strength than the larger aspect ratio samples, which is also seen in the experimental curves of Figure 5.9.

Transmitted diffraction is responsible for the majority of transmission at higher k₀.

is allowed (magenta line), experimental reflection increases rapidly, much quicker than the transmitted diffraction. This provides reasonable explanation of the similar rapid change of reflection reduction in experiments. This plot also provides clues to the differences in magnitude of the experimental reflection increase after the new diffraction order is allowed. The lowest aspect ratio of 0.65 (green dotted line) sees the greatest reflection increase for k₀ above the free space diffraction, followed by the other samples. Thek0at which the reflection begins to decrease again for each experimental curve in Figure 5.9 matches with the fraction of power that is diffracted to free space. Had it been possible to measure decreasingly small wavelengths for the array of d_bot = 540 nm, it would be expected to display light trapping into higherk₀s since it’s aspect ratio cuts through a region of high transmitted diffraction strength and between peaks in free-space diffraction strength. In fact, taking the ratio of A and B showed that an aspect ratio of 1.08 is roughly ideal for light trapping by maximizing the substrate diffraction strength while minimizing the free space diffraction strength. To target the visible to near-infrared light spectrum between 400 nm and 1000 nm, a cone withd_bot u900 nm, height of 972 nm, andd_topless than 250 nm should be ideal. Last, the periodic nature of the axial reflection modes show up in Figure 5.10B) as this mechanism competes with free space diffraction. Of note is the same periodic maxima along the aspect ratio axis appear with free space diffraction as they did with substrate diffraction, and deserve further investigation.