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These plasmonic arrays are ideal candidates for a variety of applications such as enhanced absorption with large angular tolerance for solar cells, surface-enhanced Raman scattering substrates, and broadband plasmonic enhancement.

The optical response of plasmonic arrays of NPs can be explained in terms of the plasmonic response of the NPs (

This phenomenon is of particular interest when the critical grating constant also coincides with the localized surface plasmon wavelength of the NP. The result is the excitation of a very sharp, so-called “

A key obstacle in analyzing the optical properties of QC arrays of plasmonic NPs and QC morphologies in general is the lack of accurate is the lack of analytical tools to accurately and efficiently model them. In periodic lattices, the solution domain of an infinite structure is reduced to a single unit cell by exploiting the translational symmetry. QC by definition lack translational symmetry and hence cannot be modeled accurately using periodic boundary conditions. Currently, the primary analytical method for QCs is the so-called super-cell approach. The method essentially takes a large segment of the structure and applies periodic boundary conditions to. Analyzing plasmonic structures using traditional full-wave finite-difference and finite-element techniques requires considerable computational resources even for relatively simple configurations.

For this reason, there has been recent interest in developing more efficient analytical tools which offer solutions several orders of magnitude faster than is possible with finite-difference and finite-element methods.

In theory any physically realizable incident wave can be expanded in terms of VSWFs. However in the original paper and code that was subsequently developed an incident plane wave was assumed. Using an incident plane wave greatly simplifies the problem on two fronts. First, a plane wave has a well-known simple expansion in terms of VSWFs. Secondly it can be shown that for an incident plane wave, the expansion coefficients at the displaced systems differ from the primary expansion coefficients only by a constant phase term and thus does not require the application of addition theorems for VSWFs. However since a plane wave has an infinite beamwidth, it is not possible to define reflection and transmission coefficients in the usual sense when considering the analysis of finite-size arrays.

Optical properties of aperiodic Au NP using the GMT method where first studied in Ref. 2 where aperiodic arrays of cylindrical Au NPs with a height of 30 nm and a diameter of 200 nm were fabricated and dark-field scattering spectroscopy was employed to characterize the scattering properties of the aperiodic arrays. For the electrodynamic calculations, the GMT method was utilized to calculate the scattering efficiencies of finite-size arrays composed of approximately 100 Au nano-spheres with radii of 100 nm. Here two important points are noted regarding the results reported in Ref. 2:

- It was argued that since the main aim of the study was to reveal the role of array morphology, simulations based on spherical particles could be used to explain the scattering features of cylindrical NP arrays.
- The second issue had to do with the incident field used in the GMT simulations. The authors used the original code developed by Xu based on an assumed plane wave excitation. However since a plane wave has an infinite beamwidth, it is not possible to define reflection and transmission coefficients in the usual sense when considering the analysis of finite-size arrays.

As it can be seen from the previous slide, there is very little resemblance between the simulated and measured results. These are mostly due to two issues that were mentioned in slide 6. We resolved both of these issues and have obtained

- First, we were able to fabricate spherical NP arrays with great accuracy.
- We implemented a modified GMT approach introduced in Ref. 3 for an incident wave excitation with a finite beamwidth. The incident beam was obtained by placing a circular aperture in front of a plane wave to obtain a beamwidth smaller than the array dimensions thereby avoiding diffraction by spheres at the edges of the lattice. Such a setup is very similar to realistic experimental conditions. Using the far-field expressions for scattered fields, generalized transmission and reflection coefficients were then defined based on total far-field energy fluxes.

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