Accurate Modeling of Dark-Field Scattering Spectra

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Rudolf Wilkerson
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1 Accurate Modeling of Dark-Field Scattering Spectra of Plasmonic Nanostructures (Supporting Information) Liyong Jiang 1,2,, Tingting Yin 2,, Zhaogang Dong 3,, Mingyi Liao 4, Shawn J. Tan 3, Xiao Ming Goh 3, David Allioux 4, Hailong Hu 4, Xiangyin Li 1, Joel K. W. Yang 3,5, *, and Zexiang Shen 2,4, * 1 Nanophotonic Laboratory, Department of Physics, Nanjing University of Science and Technology, Nanjing 2194, China 2 Centre for Disruptive Photonic Technologies, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore Institute of Materials Research and Engineering, A*STAR (Agency for Science, Technology and Research), 3 Research Link, Singapore School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore Singapore University of Technology and Design, 8 Somapah Road, Singapore *Correspondence and requests for materials should be addressed to Z.X.S. ( zexiang@ntu.edu.sg) and J. K. W. Y. ( joel_yang@sutd.edu.sg). 1

2 1. The characteristic of the outer and inner Gaussian beams in modeling Figure S1. (a,b) Electric field distribution of the outer and inner Gaussian beams in the x-y plane for constructing the dark-field (DF) light source. The electric field distributions of the outer and inner Gaussian beams at z = can be written as E e e and ( )/ outer A x y w i t E e e. Here, A is the magnitude and w is the beam waist at z=. ( )/ ( ) inner A x y w i t These two Gaussian beams are having the same w, but a phase difference of π. The spot size of outer and inner Gaussian beams is 24 nm and 1952 nm respectively. (c) Electric field distribution of the superposed Gaussian beams in the x-y plane. In the overlapping region, the two Gaussian beams will experience a destructive interference, so as to create an annular-like DF light source (see Movies S1 and S2). Although the boundary truncation error may bring some slight deviations, the electric field distribution in (c) shows that such truncation error is still acceptable for the purpose of simulating the annular-like DF illumination. 2

3 2. Demonstration of the overall blue-shifted response using the conventional approach with varied incident angles Figure S2. (a) Simulated dark-field scattering spectrum (DFSS) of the gold nanodisk with a diameter of 2 nm by using the conventional plane-wave illumination approach with incident angles varying from to 5 with a step of 1. For either p- or s- polarization, the fundamental peak is gradually blue-shifted with the increasing incident angle. (b) Simulated DFSS of different gold nanodisks in Figure 2a by using the conventional approach with increasing incident angles from 32 to 5 with a step of 3. The colored curves are the multi-peak Lorentzian curve fitting results for nanodisks e-g. 3

4 From Figure S3b, we can also observe an overall blue shift of the whole spectra for those relatively large nanodisks. In conclusion, the overall blue shift is due to the varied incident angles at the boundary of gold nanodisks. Our presented DF simulation approach can well reproduce the overall blue shift behavior of experimental spectra because it can automatically reproduce the varied oblique incidence angle at the boundary of gold nanodisks. Our simulation also reproduces the experiment results more accurately than the conventional plane-wave approach with increasing incident angles. 4

5 3. Main factors influencing the DFSS Simulation. Figure S3. (a-c) Simulated DFSS for gold nanodisk with a diameter of 2 nm when different substrates, focal plane s positions, and NA outer -NA inner combinations were considered. The black solid line in each sub figure is the origin curve used in Figure 2d. The following factors can affect the accuracy of DFSS simulation, i.e., the substrate effects, the position of focal plane, and the NA outer and NA inner of DF lens. Firstly, the substrate effect should be fully considered during simulation. In our simulation work, the substrate consists of a 1 nm thick SiO 2 layer on Si. Figure S3a shows the simulated DFSS for the case of NA col =.9 when different substrates were used. It is clear that the missing of silicon substrate will bring significant discrepancies. The use of highly reflective silicon substrate is help to enhance the backward scattering of plasmonic structures, especially at large oblique incidence condition. Meanwhile, a 2 nm thick Si is good enough to reach a consistent response with the bulk one. When 1 nm Cr adhesion was included in the simulation, a slight damping in intensity is observed. Secondly, since the white light source used in experiment has a long focal length, it is important 5

6 to choose a suitable position of focal plane in simulation. For instance, the effective focal plane of DF light source (NA outer =.99, NA inner =.985) is varied from z = -51 nm (λ = 58 nm) to z = - 13 nm (λ = 4 nm). As shown in Figure S3b, the simulated DFSS at different focal planes show obvious changes in curve profile and intensity. Thirdly, in DFSS simulation, it is also important to set suitable values of NA outer and NA inner. As shown in Figure S3c, when a larger or smaller set of NA outer and NA inner was used, the spectra show some changes in curve profile and intensity. In our simulations, the substrate effects have been fully taken into accounted. There are still some discrepancies between simulation and experiment and they are likely due to the uncertain position of focal plane as well as the NA outer and NA inner of the DF lens which are not exactly given in experiment. 6

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Large Area Plasmonic Color Palettes with Expanded Gamut Using Colloidal Self-Assembly Liancheng Wang 1, Ray J. H. Ng 1, Saman Safari Dinachali 1, Mahsa Jalali 1, Ye Yu 1, and Joel K.W. Yang 1,2 1 Engineering