Collective effects in second-harmonic generation from plasmonic oligomers
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1 Supporting Information Collective effects in second-harmonic generation from plasmonic oligomers Godofredo Bautista,, *, Christoph Dreser,,, Xiaorun Zang, Dieter P. Kern,, Martti Kauranen, and Monika Fleischer,,* Laboratory of Photonics, Tampere University of Technology, Korkeakoulunkatu 3, Tampere, Finland Institute for Applied Physics, University of Tübingen, Auf der Morgenstelle 10, Tübingen, Germany Center for Light-Matter-Interaction, Sensors and Analytics LISA +, University of Tübingen, Auf der Morgenstelle 15, Tübingen, Germany These authors contributed equally to this work. Corresponding Authors * godofredo.bautista@tut.fi * monika.fleischer@uni-tuebingen.de 1
2 Details of BEM calculations The numerical simulation for SHG is performed under the undepleted-pump approximation, so that the SHG is calculated in two separate linear optical processes. In the first linear process, we solve the light scattering problem with incident light at fundamental frequency ω. The light at the fundamental frequency ω = ω + ω results in second-order nonlinear polarization in the gold nanorod, i.e., =ϵ χ : with ϵ being the vacuum permittivity and χ being the second-order nonlinear susceptibility tensor of rank 3. 1 In the second linear process, the nonlinear polarization is regarded as the excitation for SHG, which is solved at the doubled frequency without considering any incident light. Both linear processes are solved under the BEM formulation 2,3 where only the surfaces V of the nanorod are discretized. In addition, the outgoing scattered wave into the unbounded surrounding medium V is accounted by imposing the Silver-Müller radiation conditions at the infinite boundary V in the Stratton-Chu equations. 2,4 The oligomer consists of several gold nanorods. For gold, bulk SHG is not allowed because of centrosymmetry. However, surface SHG is allowed due to symmetry breaking at the gold-dielectric interfaces. 1 Therefore, in our BEM formulation, the second-order nonlinear susceptibility χ is only defined over the surfaces of the nanorods. Specifically, we only include the χ tensor component of the surface second-order nonlinear susceptibility where n is the local surface normal. This is based on earlier findings that the surface second-order susceptibility of plasmonic metals is dominated by this tensor component. 5,6 In our simulations, the local response is integrated over multiple surfaces of the whole structure, fully describing the experiment. This approach has been proven to describe several other experiments very well. 7,8 In the calculations, the refractive index of gold was taken from literature. 9 To account for the glass substrate, the oligomers were embedded in homogenous medium (n m = 1.45). 10 We believe that the role of the interfaces, e.g., air-to-glass, is critical when the out-of-plane modes of the nanoparticles are significant and/or the symmetry of the illuminating beam is broken, e.g., under oblique incidence. 10 Since the role of out-of-plane modes of the oligomers is almost negligible, the use of an embedding homogeneous medium to account for substrate effects is justified. The resonance wavelengths of the plasmons may be somewhat red-shifted compared to glass/air. The relative shift depending on the nanorod number will be discussed later. Similar to 2
3 the experimental conditions (fundamental excitation wavelength of 1060 nm, NA = 0.8), the signal was collected point-wise in reflection over a scanning area of 3 µm 3 µm where the focal plane is taken to be situated at the plane that cuts across the oligomer. The far-field scanning optical images for the oligomers were calculated at the second-harmonic frequency. To model an idealized version of the experimental oligomers, the dimensions of the meshes were deduced from the SEM measurements. We individually performed the SHG calculations for all oligomers consisting of 1 to 16 nanorods for the two different nanorod arrangements. BEM was also used to calculate the corresponding extinction cross sections and associated near-fields of the oligomers under different polarizations. For each linear extinction spectrum, the wavelength was scanned from 400 nm to 1700 nm with a step of 20 nm. The calculated near-field maps cover a 250 nm 400 nm area which is located at the transverse plane 20 nm above the mirror plane. Here, the real parts of the instantaneous radial E r and azimuthal E φ electric field components are plotted in polar coordinates. The radial vector points outward from the center of the oligomer. Similarly, the azimuthal vector is defined along the counterclockwise direction. In these maps, the red and blue colors represent field components that are 180 out of phase. Supporting References (1) Boyd, R. W. Nonlinear Optics; Academic Press, (2) Mäkitalo, J.; Suuriniemi, S.; Kauranen, M. Opt. Express 2011, 19 (23), (3) Butet, J.; Gallinet, B.; Thyagarajan, K.; Martin, O. J. F. J. Opt. Soc. Am. B 2013, 30 (11), (4) Stratton, J. A.; Chu, L. J. Phys. Rev. 1939, 56 (1), 99. (5) Krause, D.; Teplin, C. W.; Rogers, C. T. J. Appl. Phys. 2004, 96 (7), (6) Wang, F. X.; Rodríguez, F. J.; Albers, W. M.; Ahorinta, R.; Sipe, J. E.; Kauranen, M. Phys. Rev. B 2009, 80 (23), (7) Canfield, B. K.; Husu, H.; Laukkanen, J.; Bai, B.; Kuittinen, M.; Turunen, J.; Kauranen, M. Nano Lett. 2007, 7 (5), (8) Butet, J.; Thyagarajan, K.; Martin, O. J. F. Nano Lett. 2013, 13 (4), (9) Johnson, P. B.; Christy, R. W. Phys. Rev. B 1972, 6, (10) Knight, M. W.; Wu, Y.; Lassiter, J. B.; Nordlander, P.; Halas, N. J. Nano Lett. 2009, 9 (5),
4 (11) Bautista, G.; Kauranen, M. ACS Photonics 2016, 3 (8), (12) Sancho-Parramon, J.; Bosch, S. ACS Nano 2012, 6 (9), (13) Biss, D. P.; Brown, T. G. Opt. Lett. 2003, 28 (11),
5 Figure S1 Numerical simulation of the scattering spectra of individual gold nanorods (thickness t of 20 nm, width w of 40 nm, lengths l of 150 nm, 165 nm and 180 nm) on top of ITO-coated glass. 5
6 Figure S2 Verification of SHG signals from oligomer samples. (a) Nonlinear signals at 531 ± 11 nm (Semrock FF01-531/22-25) were detected from four identical radial oligomers consisting of 8 nanorods (l = 145 ± 5 nm, w = 40 ± 5 nm, t = 20 ± 5 nm, R = 472 ± 5 nm, top row) only when the incident beam (excitation wavelength of 1060 nm, average input power of 1 mw, radial polarization) is pulsed (140 fs). When the same laser is operated in continuous wave (CW) mode, the detected signals from the same oligomers are close to the background signal level. When pulsed laser excitation is used, the nonlinear signals from the array of oligomers with increasing rod numbers (l = 165 ± 5 nm, w = 46 ± 5 nm, t = 20 ± 5 nm, R = 485 ± 5 nm, bottom row) are found to vanish when detected with a notch filter around 530 ± 20 nm before the photomultiplier tube, i.e., when the SHG is excluded. The background signals are very low, i.e., about three orders of magnitude smaller, and attributed to nonlinear luminescence. All SHG images are plotted using the same color scale. Scale bar = 1 µm (top row), 2 µm (bottom row). (b) Quadratic power dependence of the detected nonlinear signal from an azimuthal oligomer (8 nanorods, l = 145 ± 5 nm, w = 40 ± 5 nm, t = 20 ± 5 nm, R = 472 ± 5 nm) at 531 ± 11 nm using pulsed excitation (wavelength of 1060 nm, azimuthal polarization). The data points are averaged over four identical azimuthal oligomers. Only the data when the oligomer is symmetrically illuminated are included. Altogether, these results prove that the detected nonlinear emission from the oligomer is dominated by SHG. 6
7 Figure S3 Experimentally acquired intensity images of the CVBs used in the nonlinear microscopy experiments. The schematic diagrams of the CVBs and their corresponding transverse electric field orientations (solid white arrows) are shown in the leftmost column. The intensity images were captured using a camera and a polarization analyzer placed before the microscope objective. The orientation angles (solid black arrows) of the polarization analyzer were set to 0, 45, 90 and 135, with respect to the x axis. 7
8 Figure S4 Calculated SHG scanning microscopy image (NA = 0.8, excitation wavelength of 1060 nm) of a radial oligomer (l = 165 nm, w = 46 nm, t = 20 nm, R = 485 nm) using a radial CVB. The defects are simulated by removing some of the nanorods in the surface mesh that corresponds to the radial oligomer. The orientations of the nanorods in the oligomer are marked. 8
9 Figure S5 (a,b) Experimental SHG scanning microscopy images of the array of (a) radial and (b) azimuthal oligomers (l = 165 ± 5 nm, w = 46 ± 5 nm, t = 20 ± 5 nm, R = 485 ± 5 nm) with increasing numbers of nanorods using linear polarization (along y). The orientations of the oligomers are the same as in Figure 1. The maximum SHG intensities are shown. Scale bar: 1 µm. (c,d) Calculated SHG scanning microscopy images (NA = 0.8, excitation wavelength of 1060 nm) of the (c) radial and (d) azimuthal oligomers (l = 165 nm, w = 46 nm, t = 20, R = 485 nm) with increasing numbers of nanorods using linear polarization (along y). The orientations of the oligomers (see marks) and beams are the same as in Figure 1. The SHG maps for each 9
10 oligomer are simulated separately and then stitched together in the post-processing. The maximum SHG intensities are shown. Scale bar: 1 µm. 10
11 Figure S6 Calculated near-field distributions around a section of the (a) radial and (b) azimuthal oligomers (l = 165 nm, w = 46 nm, t = 20 nm, R = 485 nm) in the transverse plane, i.e., 20 nm above the mirror plane, at the second-harmonic wavelength of 530 nm for different numbers of nanorods (columns) under excitation with the symmetrically illuminating and matching CVB. Each figure shows the real part of the (a) radial E r and (b) azimuthal E φ component of the total electric field on the same color scale. 11
12 Figure S7 Calculated near-field distributions around a section of the (a) radial and (b) azimuthal oligomers (l = 165 nm, w = 46 nm, t = 20 nm, R = 485 nm) in the transverse plane, i.e., 20 nm above the mirror plane, at the fundamental wavelength of 1060 nm for different numbers of nanorods (columns) under excitation with the symmetrically illuminating and matching CVB. Each figure shows the real part of the (a) radial E r and (b) azimuthal E φ component of the total electric field on the same color scale. 12
13 Figure S8 Near-field distributions around a section of the radial oligomer with 16 nanorods (l = 165 nm, w = 46 nm, t = 20 nm, R = 485 nm) in the transverse plane, i.e., 20 nm above the mirror plane, (a,c) at the fundamental wavelength of 1060 nm and (b,d) at the second-harmonic wavelength of 530 nm for (a,b) radial and (c,d) azimuthal CVBs under symmetric illumination. In a,b (c,d), the real part of the azimuthal E φ (radial E r ) component of the total electric field is shown. 13
14 Figure S9 Near-field distributions around a section of the azimuthal oligomer with 16 nanorods (l = 165 nm, w = 46 nm, t = 20 nm, R = 485 nm) in the transverse plane, i.e., 20 nm above the mirror plane, (a,c) at the fundamental wavelength of 1060 nm and (b,d) at the second-harmonic wavelength of 530 nm for (a,b) azimuthal and (c,d) radial CVBs under symmetric illumination. In a,b (c,d), the real part of the radial E r (azimuthal E φ ) component of the total electric field is shown. 14
15 Figure S10 Calculated extinction spectra of azimuthal oligomers (l = 165 nm, w = 46 nm, t = 20 nm, R = 485 nm) using focused azimuthal CVB under symmetric illumination. As shown in the schematic diagrams, a nanorod is removed one at a time starting from the 16-nanorod case while the gap size is fixed. 15
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