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Polarization-dependent Fano-like Resonance and Universal Threedimensional orientation-dependent scattering trait: A Route Boosting Single Anisotropic Plasmonic Nanoparticle 3D Orientation Determination Zhihua Xu, Lin Cheng, Weijie Kong, Yuzhen Wang, and Xiaoping Zhang J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 10 Jul 2015 Downloaded from http://pubs.acs.org on July 11, 2015

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The Journal of Physical Chemistry

Polarization-Dependent Fano-Like Resonance and Universal Three-Dimensional OrientationDependent Scattering Trait: a Route Boosting Single Anisotropic Plasmonic Nanoparticle 3D Orientation Determination (old title would be replaced by “Single Anisotropic Plasmonic Nanoparticle ThreeDimensional Orientation Determination Based on Fano-Like Resonance and Universal 3D OrientationDependent Scattering Trait.”) Zhihua Xu, Lin Cheng, Weijie Kong, Yuzhen Wang and Xiaoping Zhang* School of Information Science and Engineering, Lanzhou University, Lanzhou 730030, China KEYWORDS: plasmonic Fano-like resonance · single-particle orientation determination · nanorice · polarization · gold nanorod

ACS Paragon Plus Environment

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ABSTRACT: Single-nanoparticle orientation determination plays a vital role in studying complex nanoscale motion. In the present paper, we systematically investigate the threedimensional (3D) orientation-dependent far- and near- field optical properties of single (Au core)-(dielectric shell) nanorice and gold nanorod. It shows that the scattering spectrum of the single anisotropic nanoparticle with arbitrary orientation is a linear superposition of a set of basic scattered spectra. And the scattering spectra of the single nanorice show a polarization-dependent Fano-like resonance which can be well described by a model with two coupled oscillators. Furthermore, by means of coordinate transformation, a universal analytic formula for description single nanoparticle 3D orientation- and polarization- dependent scattering intensity is proposed. Then we develop a new method to detect the 3D orientation of single nanoparticle, i.e., by fitting the scattering intensity under different incident polarization directions based on our analytic formula and the 3D orientations could be resolved explicitly. Both of the experimental and numerical simulation data can be well described by our analytical formula. Our method for single particle 3D orientation determination has high precision only with subdegree uncertainty. In addition, based on the Fano resonances caused by the efficient coupling between the dielectric elliptical shell and the gold nano-ellipsoid core, we found that the 3D orientation of single nanorice could be confirmed from either the transverse or longitudinal plasmon mode polarization-dependent scattering trait, while it is almost impossible for single gold nanorod based on the transverse plasmon mode. It’s worth noting that the transverse plasmon mode of the nanorice is mostly insensitive to the aspect ratio then it allows nanorices with different lengths to be 3D orientation sensors without altering the incident laser wavelength.

INTRODUCTION


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In recent years, plasmonic Fano-like resonances, characterized by a pronounced dip in the scattering spectra, originated from interactions between a subradiant “dark” mode and a superradiant “bright” mode in metal nanostructures have attracted much attention.1, 2 Due to their narrower spectral width and larger induced field enhancement compared to traditional plasmonic resonances,3 plasmonic Fano resonances show great potential in the applications of plasmonic nanorulers,4-6 biological sensing7-9 and nonlinear optical devices.10 And they have been observed in plasmonic oligomers,11-13 Au dolmen nanostructures,5, 14, 15 ring-disk nanocavities,16, 17 (core)(shell) nanostructures without geometrical symmetry breaking,18-21 and a single Ag nanorod with large dimensions and high-order resonant modes.22, 23 However, up to date, Fano-like resonances in single small anisotropic plasmonic nanoparticle have barely been observed. Anyway the polarization-dependent optical properties induced by the anisotropic geometry and larger scattering cross sections will be very useful in single particle orientation and rotational tracking (SPORT).24 During the last few years, single particle orientation and rotational tracking has received much attention in the field of single-molecule spectroscopy.25, 26 Due to the shape-induced anisotropic optical properties,27,

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gold nanorods (AuNRs) have been the popular choice in rotational

tracking and orientation sensing.24 An in-plane orientation sensor, which depends on the polarization-dependent scattering traits of AuNRs has been proposed by Carsten Sonnichen et al in 2005.29 The in-plane orientation determination of single AuNR based on the surface plasmon absorption has been demonstrated by Wei-Shun Chang et al in 2010.30 While these are simple to implement, the out-of-plane orientation of the AuNRs cannot be determined. Since then, image recognition has been widely applied to determine the three-dimensional (3D) orientation of single anisotropic particle.24-27, 31 In 2013 by using imaging recognition combination with the


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polarization-dependent scattering intensity, Ning Fang firstly resolve the 3D orientation of gold nanorods with high precision.26 But the single nanoparticle 3D orientation determination only based on the scattering trait at a broad wavelength range (i.e., not only around the longitudinal plasmon mode, but also around the transverse plasmon mode) is still a challenge.24 In this paper we systematically discuss the 3D orientation-dependent optical property of the single anisotropic plasmonic nanoparticle, i.e., single (Au core)-(dielectric shell) nanorice and gold nanorod, with certain incident polarization direction. By comparatively analyzing the nearand far- field optical properties of these anisotropic nanoparticles, we show that the scattering spectrum for both of single nanorod and nanorice with arbitrary orientation is a linear superposition of a group of basic scattered spectra. Based on the 3D orientation-dependent scattering trait and coordination transformation, we propose a simple analytical formula to determine the 3D orientation of the single anisotropic plasmonic nanoparticle. The polarizationdependent experimental result of single gold nanorod in reference26 can be well described by our simplified analytical formula. Furthermore, by utilizing numerical simulation, we demonstrate that our method can determine the 3D orientation angles range from (0°, 0°) to (180°, 180°) without angle degeneracy and there’s only subdegree uncertainty. In addition, for the (Au core)(dielectric shell) nanorice, there are two kinds of plasmonic Fano resonances in scattering spectra corresponding to the longitudinal and transverse modes, respectively. And both of the longitudinal and transverse Fano resonances can be well described utilizing a simple coupledoscillator model. Due to the efficient coupling between a broad and strong scattering background generated by the large refractive index dielectric elliptical shell and the narrow resonance of the Au nano-ellipsoid core, our studies show that the transverse plasmon mode can contribute significantly to determine the 3D orientation of single nanorice. Contrastively, single gold


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nanorod is unable to accomplish based on the transverse mode scattering trait.30, 32 Moreover the transverse plasmon mode is mostly insensitive to the aspect ratio of nanorice, the 3D orientation of nanorice with any length can be confirmed just by one laser wavelength. RESULTS AND DISCUSSION Simulation Section The 3D orientation-dependent near- and far- field optical property of single (Au core)-(dielectric shell) nanorice and gold nanorod (Figure 1a) is numerically investigated by utilizing Comsol Multiphysics 4.3a based on the finite element method (FEM). Fixed incident polarization direction and rotating the nanoparticle, we investigated the 3D orientationdependent optical response of the single anisotropic plasmonic nanostructure. By fixing the orientation of the anisotropic nanostructure and changing the incident polarization direction, we numerically demonstrated the accuracy of our analytical model of determination the single anisotropic plasmonic nanoparticle’s 3D orientation. The total scattering cross sections of the plasmonic nanoparticle are obtained by integrating the scattered power flux over a sphere surface outside it. To describe the gold metal we use a Lorentz Model33

ε (ω ) = ε ∞ −

ω p2

ω ( ω − iΓ p )

+

f1ω12 (ω12 − ω 2 ) + iωΓ 1

where Γ p , ω p , f1 , ω1 and Γ1 is the relaxation rate, plasma frequency, weighting factor, Lorentz resonance width and Lorentz oscillator damping rate, respectively. The specific values of these parameters can be found in reference,33 which are obtained by fitting the experimental data tabulated in literature.34 Given the surrounding medium is water and the refractive index is 1.33. Scattering boundary conditions and perfectly matched layers were used to properly define the calculations. The parameters of the (Au core)-(dielectric shell) nanorices and bare gold nanorods are listed in Figure 1d.


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Figure 1. (a) The schematic of the (Au core)-(dielectric shell) nanorice. The semi-major of the Au ellipsoid core is a1 and the semi-minor is b1. The semi-major of the nanorice is a2 and the semi-minor is b2. (b) The Cartesian coordinate system for the 3D orientation (β, γ) of the nanorice. The red bi-direction arrow indicates r the incident polarization. e p denotes the longitudinal direction of the nanorice. (c) Schematic of the coupled oscillators model proposed to fit the scattering spectra of a single (Au core)-(dielectric shell) nanorice. (d) Dimensions, refractive index of the dielectric shell and aspect ratio of (Au core)-(dielectric shell) nanorices and bare gold nanorods. The refractive index 1.7, 2.2 and 2.74 are corresponding to the quartz glass, ZnS and ZnSe, respectively. *For the bare gold nanorods that a1=a2 and b1=b2.

Polarization-Dependent Plasmonic Fano-Like Resonances in Single (Au Core)-(Dielectric Shell) Nanorice. In Figure 2a and 2b, we show the polarization-dependent scattering spectra for a single (Au core)-(dielectric shell) nanorice with a1=17.5, b1=12.5, a2=52, and b2=32.5 nm (called as nanorice1, the shell refractive index is n=2.2) and the corresponding hollow elliptical shell and Au nano-ellipsoid core. No matter the polarization direction along with or perpendicular to the long-axis of the hollow shell, the scattering spectra of the hollow shell show a decreasing scattering background toward the long-wavelength region. The scattering intensity of the hollow elliptical shell is much larger than that of the ellipsoid core. This scattering is ascribed to refractive index difference between the hollow elliptical shell and the surrounding medium. When the hollow elliptical shell and ellipsoid core are integrated together, both of the longitudinal and transverse scattering spectra of the nanorice1 exhibit an asymmetric line shape, characterized by a rising background in the high-frequency region and a resonance peak in the low-frequency region. The similar situation can also be found in the spherical nanoshells without geometrical symmetry breaking.19 Such an asymmetric line shape cannot be fitted by the


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conventional Lorentz model, which is a prominent feature of Fano resonance. In addition, in comparison to that of the unshelled Au nano-ellipsoid core in water, the longitudinal plasmon resonance of the nanorice1 red-shift from 560 to 690 nm as shown in Figure 2a, similarly the transverse plasmon resonance red-shift from 525 to 605 nm in Figure 2b. Obviously the dielectric elliptical shell induces the light scattering increasingly. These results will permit the transverse plasmon mode of the nanorice to determine the 3D orientation of single nanoparticle, which is almost impossible for single gold nanorod based on the scattering trait, which will be detailed later.

Figure 2. The normalized scattering spectra of nanorice1 and the corresponding Au ellipsoid core and hollow dielectric elliptical shell under the orientation (a) β = 90°, γ = 0° and (b) β = 90°, γ = 90° calculated according FEM (solid line) and fitting of the FEM scattering spectra of the nanorice1 with the oscillator model (dotted line). (c-f) Real part of the y-component of the electric field (Ey) and the charge distribution for the plasmon modes at the selected wavelengths shown in figure (a) under the orientation β = 90°, γ = 0°. The red area and blue area means the forward and reverse y-axis direction, respectively. Thus, some red areas and blue areas are both marked as positive. (g-j) The norm electric field |E| distribution on x-y plane corresponding to the same


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cases as shown in figures (c-f). The parameters of nanorice1 are shown in Figure 1d. The surrounding media is water.

For boosting the understanding of the fundamental origin of the Fano resonance in (Au core)(dielectric shell) nanorice, the charge distribution analysis of the plasmon modes at specific wavelengths for nanorice1 are exhibited in Figure 2c-2f. A simple way to recognize the sign of the charges is to compute the real part of the relevant component of the electric field, because the direction of electric field is related to the sign of the charges.3, 35 Under light excitation in the appropriate wavelength range, both of the polarization charge oscillation in the dielectric elliptical shell and the free electron oscillation in the gold nano-ellipsoid core can be induced by the free-space electromagnetic field.19 Based on the Coulombic interaction, the polarization charges of the elliptical shell can interact with the free electrons of the nano-ellipsoid core. As shown in Figure 2d, the free electron oscillation of the gold ellipsoid core and the polarization charge oscillation of the dielectric elliptical shell are out-of-phase, causing a small total dipole moment of sub-radiant nature. And there is a π phase difference between the charge distributions of the core-shell interfaces in Figure 2(c) and 2(f). Due to the π phase jump across the narrow plasmon resonance, the coupling changes from constructive to destructive.19, 36 The destructive coupling reduces the whole oscillation strength because of the partial cancellation between the polarization charges and the free electron oscillations. Thus, a non-Lorentz asymmetric scattering spectrum is produced. Figure 2g-2j also shows the electric field distributions at the corresponding wavelengths under the incident polarization along with the longitudinal-axis of the nanorice. As expect, strong near field enhancements are observed on the longitudinal ends of the nanorice. Also the near field distribution is sensitive to the excitation frequency. Similar situation of the charges and electric field intensity distribution can also be found on the


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nanorice1 with the incident polarization direction along with its transverse-axis at different wavelengths (Figure S1 in Supporting Information). To provide better insights into the (Au core)-(dielectric shell) nanorice Fano resonance phenomena, we also analyse it utilizing a coupled oscillator model consisting of two interacting oscillators as shown in Figure 1c. The plasmon resonance of the Au nano-ellipsoid core is modeled as the interacting oscillator of frequency ω2 and that of the broad scattering background of the dielectric elliptical shell is modeled as ω1. The two oscillators interact together with a massless spring. In contrast to the traditional plasmonic Fano systems37 only driven by one periodic harmonic force, the two oscillators in our (Au core)-(dielectric shell) nanorice are both driven by G1 = g1e−iωt and G2 = g2e−iωt , respectively. The motion equations of the two oscillators can be written as && x1 + γ 1 x&1 + ω12 x1 −ν122 x2 = g1e−iωt

(1)

&& x2 + γ 2 x&2 + ω22 x2 −ν122 x1 = g2 e−iωt

(2)

where parameter ν12 describes the interacting strength of the oscillators. x1 and x2 are the displacements from equilibrium position of the oscillators 1 and 2, respectively. The friction coefficient γi is used to account for the energy dissipation of the oscillator i. The solutions for Equation (1) and (2) are also harmonic: x1 = k1e− iωt , x2 = k2 e−iωt , and the corresponding amplitudes solved by maple are given as k1 =

g2ν 122 + g1 (ω22 − ω 2 − iγ 2ω ) (ω12 − ω 2 − iγ 1ω )(ω22 − ω 2 − iγ 2ω ) −ν 124

(3)

k2 =

g1ν 122 + g2 (ω12 − ω 2 − iγ 1ω ) (ω12 − ω 2 − iγ 1ω )(ω22 − ω 2 − iγ 2ω ) − ν124

(4)

The scattering spectrum of the (Au core)-(dielectric shell) nanorice system can then be confirmed by Sca (ω ) ∝ && x1 + && x2

2

(5)


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Both of the longitudinal (Figure 2a) and transverse (Figure 2b) scattering spectra of the nanorice1 obtained from FEM can be well fitted by those calculated from the oscillator model. The parameters used in the oscillator model are shown in Table S1 (Supporting Information).

Figure 3. (a) The normalized scattering spectra of three different shell thickness nanorices (i.e., nanorice0, nanorice1 and nanorice2) with the same shell refractive index of 2.2 under the orientations β = 90°, γ = 0° and β = 90°, γ = 90°. (b) Calculated dependence of the scattering spectra on the nanorices with different aspect ratio: nanorice1 a2/b2=1.6, nanorice5 a2/b2=2.4. The shell refractive index is the same of 2.2. (c) Calculated dependence of the scattering spectra on the nanorices with different shell refractive index: nanorice1 n=2.2, nanorice3 n=2.74 and nanorice4 n=1.7. Nanorice1, nanorice3 and nanorice4 have the same dimensions. The parameters of nanorices are shown in Figure 1d. The surrounding media is water.

Furthermore, the influences of the dielectric shell thickness (Figure 3a), the aspect ratio (Figure 3b) and shell refractive index n (Figure 3c) of the (Au core)-(dielectric shell) nanorice on the Fano resonance have been studied by using FEM and the oscillator model comparatively. All of the longitudinal and transverse scattering spectra of the different type (Au core)-(dielectric shell) nanorices exhibit an asymmetric line shape that is characterized by a rising background in


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the short-wavelength region and a resonance peak in the long-wavelength region. And all of them can be well fitted by the oscillator model (Figure S2 in Supporting Information) with the parameters referred from Table S1 (Supporting Information). Figure 3a shows the scattering spectra of the nanorices with different shell thickness, i.e., nanorice0, nanorice1 and nanorice2. As increasing the dielectric shell thickness under the constant a1, b1 and a2/b2=1.6, the longitudinal resonance wavelength is gradually red-shifted from 675 to 705 nm and the scattering intensity is increased. These results are ascribed to the energy transfer from the shell to core as the shell thickness is increased. In Figure 3b, similar to the single bare gold nanorod, the longitudinal plasmon resonance wavelength shows a distinct red-shifting with increasing the aspect ratio of the nanorice and the transverse plasmon mode is almost insensitive to various aspect ratios. However, the scattering intensity of transverse plasmon mode in the (Au core)(dielectric shell) nanorices become more distinctly than that of the gold nanorod. Therefore, there’s great potential that the 3D orientation of individual nanorice with different length can be determined just by the scattering intensity distribution at one laser wavelength. Figure 3c presents the dependence of the scattering spectrum on the refractive index of nanorice shell with the same dimension (a1=17.5, b1=12.5, a2=52, and b2=32.5 nm), due to the decreasing of the dielectric shell’s refractive index, both of the longitudinal and transverse plasmon peak exhibits a distinct blue-shifting and the scattering intensity gives a decreasing trend. Universal Three-Dimensional Orientation-Dependent Optical Property of the Single Anisotropic Plasmonic Nanoparticle. For understanding the universal 3D orientationdependent optical response of the anisotropic plasmonic nanoparticle, we systematically investigate the 3D orientation-dependent far-field scattering spectra and near-field response of the single (Au core)-(dielectric shell) nanorice and bare gold nanorod. Figure 4a shows the


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scattering spectra of the nanorice1 (aspect ratio a2/b2=1.6) with different orientations under the polarization direction along x-axis (Figure 1b). Within the wavelength range 450-850 nm, there are two distinct plasmon resonances when (β=90°, γ=0°) and (β=90°, γ=90°) which are corresponding to the longitudinal plasmon mode (λ2=690 nm) and transverse plasmon mode (λ1=605 nm) of the nanorice1, respectively. The corresponding electric field distributions at the resonance wavelengths are shown in Figure 5. The scattering spectra of the nanorice are highly

Figure 4. The comparison of the orientation-dependent normalized scattering spectra of the (Au core)(dielectric shell) nanorice and the bare gold nanorod under the incident polarization along with the x-axis (Figure 1b). (a) and (b) corresponding to the nanorice1 and nanorice5, respectively, with the same shell refractive index of different aspect ratios. (c) and (d) corresponding to the bare gold nanorod1 and nanorod3, respectively. Nanorice1 (nanorice5) and nanorod1 (nanorod3) has the same dimension, respectively. The parameters of nanorices and bare gold nanorods refer to Figure 1d. The surrounding media is water.

sensitive to the nanorice’s 3D orientations. Both of the scattering intensity at the resonance wavelengths λ1 and λ2 are decreasing with the longitudinal and transverse ends of the nanorice far away from the incident polarization direction. This indicates that we can utilize the


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orientation-dependent scattering trait to confirm the orientation of the nanorice. And the scattering spectrum induced by arbitrary orientation is always fluctuating between the longitudinal scattering spectrum and the transverse scattering spectrum. Similar situation also can be found in the nanorice5 (Figure 4b) with the aspect ratio a2/b2=2.4, bare gold nanorod1 (Figure 4c) and bare gold nanorod3 (Figure 4d). Note that nanorice1 and nanorod1 (nanorice5 and nanorod3) has the same dimensions. In addition, from Figure 4 we discover that the longitudinal resonance wavelength shows a distinct red-shifting with increasing the aspect ratio both of the (Au core)-(dielectric shell) nanorice and bare gold nanorod, but the transverse plasmon modes are almost insensitive to various aspect ratio. For single (Au core)-(dielectric shell) nanorice, no matter the aspect ratio is 1.6 or 2.4, the scattering intensity at the transverse plasmon mode either shows a distinct orientation-dependent trait. This is caused by the efficient coupling between the dielectric elliptical shell with a large refractive index and the Au nanoellipsoid core. However, this phenomenon cannot be found in single bare gold nanorod in Figure 4c and 4d. Such a prominent merit of single (Au core)-(dielectric shell) nanorice can be fully exploited to determine the 3D orientation of the nanorice with different length. The further specific comparison of the (Au core)-(dielectric shell) nanorice’s and the bare gold nanorod’s 3D orientation-dependent scattering intensity at the transverse plasmon mode can be found in Figure 6. Obviously, the flat scattering intensity distribution in Figure 6f implies that the gold nanorod cannot tell the orientation at all based on the transverse plasmon mode. To provide better insights into the 3D orientation-dependent optical property of the single anisotropic plasmonic nanoparticles, we also investigate its orientation-dependent near-field traits. Figure 5 shows the distribution of the electric field at the resonance wavelengths with different 3D orientations of the nanorice1. As expected, the electric field distribution of the


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nanorice is highly sensitive to the 3D orientations. Corresponding to the far-field orientationdependent characteristics, both of the electric field intensity distributions at the longitudinal and transverse ends of the nanorice are decreasing with the orientation far away from the incident polarization direction at the longitudinal and transverse resonance wavelength, respectively. At the same time, we analyzed the electric field distribution at different wavelengths with certain

Figure 5. Electric field intensity contours obtained from the FEM calculations of a single (Au core)-(dielectric shell) nanorice1 at different wavelengths excited by different orientations (β, γ).

orientations. As shown in Figure 5, when the incident polarization direction along with (β=90°, γ=0°) or perpendicular to (β=90°, γ=90°) the long axis of the nanorice at different wavelengths, the distributions of the electric field at the longitudinal or transverse ends of the nanorice are in dominant, respectively. Particularly at the resonance wavelength λ1=605nm and λ2=690nm, the


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electric field distribution is separately focused on the transverse and longitudinal ends of the nanorice. It is almost insensitive to the variation of the nanorice’s orientations. While under the orientation (β=90°, γ=45°), the electric field distributions are observed to rotate away from the ends of the nanorice at the off-resonance wavelengths (550 and 750 nm). This is attributed to the phase difference between the transverse and longitudinal charge oscillations under the offresonance excitation.38 On the basis of the analysis of the near- and far- field orientation-dependent optical properties of the single anisotropic plasmonic nanoparticle, we discovered a quantitative relationship between the 3D orientation of the anisotropic plasmonic nanoparticle and the scattering spectra, i.e., the scattering spectra of the nanoparticle excited by arbitrary 3D orientation (β, γ) is a linear superposition of a set of basic scattered spectra. It can be written as Equation 6. Sca ( β , γ , λ ) = Scax ( λ ) cos 2 γ sin 2 β + Scaz ( λ ) cos 2 β + Scay ( λ ) sin 2 γ sin 2 β

(6)

where Sca is the scattering spectrum of the anisotropic plasmonic nanoparticle, Scax(λ) = Sca(ߚ=90°, ߛ=0°,λ), Scay(λ) = Sca(ߚ=90°, ߛ=90°,λ)，and Scaz(λ) = Sca(ߚ=0°, ߛ=0°,λ). To the best of our knowledge, it’s the first time we put forward an analytic formula to define the inherent relationship of the scattering spectra with arbitrary orientations of the single anisotropic nanoparticle. With an anisotropic electromagnetic dipole approximation of the nanoparticle, the theoretical derivation process of Equation 6 is shown in Supporting Information. Figure 6a and 6b show obviously the scattering spectra of the nanorice with different aspect ratio a2/b2=1.6 (nanorice1) and 2.4 (nanorice5) obtained from FEM and data-fitting by equation (6) accord with each other very well, respectively. According equation (6), we plot the curved surfaces of the normalized scattering intensity of nanorice under the 3D orientation angles change from (0º, 0º) to (180º, 180º) at the transverse resonance wavelength of the nanorice1, nanorice5 and nanorod1,


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respectively. Evidently, as shown in Figure 6d-6f, the curved surfaces and the solid points overlap with each other perfectly. As expected, the scattering intensity of the bare gold nanorod at the transverse plasmon mode is insensitive to the 3D orientation angles. However, the scattering intensity of the nanorices at the transverse plasmon resonance wavelength is

Figure 6. Fitting the normalized orientation-dependent scattering spectra of the nanorice1 (a), nanorice5 (b), nanorod1 (c), nanorod2 (k) and nanorod3 (l) by Equation (6). The solid points are obtained from FEM by frequency sweeping. (d), (e) and (f) are the 3D orientation-dependent scattering intensity of the nanorice1, nanorice5 and nanorod1 around the transverse resonance wavelength, respectively. (h), (i) and (j) are the 3D orientation-dependent scattering intensity of the nanorice1, nanorice5 and nanorod1 around the longitudinal


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resonance wavelength, respectively. The curved surfaces are plotted according equation (6). Nanorice1 (nanorice5) and nanorod1 (nanorod3) has the same dimension, respectively. The surrounding media is water.

highly sensitive to the altering of the 3D orientations. As shown in Figure 6h-6j the scattering intensity of both the nanorices and nanorods at the longitudinal plasmon mode are highly sensitive to the 3D orientations changing. In addition, the single gold nanorod with different dimensions also obey the rules described by equation (6) as shown in Figure 6c (nanorod1), 6k (nanorod2) and 6l (nanorod3), respectively.

Figure 7. (a) Schematic geometry for determining the single anisotropic nanoparticle’s 3D orientation (β, γ) r r r ( ex , e y , e z )

respect to the coordinate system . (b) and (d) Dependence of the normalized scattering intensity on the excitation polarization angles (θ, φ) at the longitudinal (690 nm) and transverse (605 nm) resonance


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wavelength for the (Au core)-(dielectric shell) nanorice1 with different orientations (β, γ), respectively. The solid points are obtained from the FEM calculations. The curve surfaces are fitting by the orientation determination analytical formula Equation (10). (c) Fitting the normalized polarization-dependent experimental data of single gold nanorod with the orientation (β=23.4°, γ=135.9°) from literature26 panel 4 by our simplified analytic model Equation (11). Coincided with the excitation condition in the literature, the incident polarization angle α in literature is corresponding to the angle (90°-θ) with φ=90° and exchanging the incident direction of k and H.

The Implementation of Single Anisotropic Plasmonic Nanoparticle 3D Orientation Determination. Based on the above analysis, the polarization-dependent scattering intensity allows us to confirm the 3D orientation of the single anisotropic plasmonic nanoparticle. The simulated experimental excitation geometry is presented in Figure 7a. (β, γ) is the 3D orientation of the anisotropic plasmonic nanoparticle with respect to the Cartesian coordinate system (erx , ery , erz ) . r ep

is the unit vector of the longitudinal direction of the nanorice. Thus, we have  cos γ sin β  r r r r   e p = ( ex ey ez ) ×  sin γ sin β   cos β   

(7)

The unit vector of the propagation direction (k) and electric field (E) of the incident light is erk and r eE

, respectively. So, the unit vector of the magnetic field (H) is erH

r r = ek × eE

. The dashed single- and

solid double- arrowed red line indicates the propagation direction and polarization direction of the incident light, respectively. (θ, φ) is the 3D polarization angle of the incident electromagnetic field with respect to the Cartesian coordinate system (erx , ery , erz ) . Therefore, the relationship between Cartesian coordinate system (er

E

r  eE r  erH  ek 

r r r r r , eH , ek ) and (ex , ey , ez ) can

r   cos ϕ sin θ sin ϕ sin θ cosθ  ex     r  cos ϕ 0  ey   =  − sin ϕ    r    − cos ϕ cosθ − sin ϕ cosθ sin θ  ez 

be described as

(8)

So, the unit vector of the longitudinal direction of the anisotropic plasmonic nanoparticle

r ep

can

be written as r r r r e p = ( eE eH ek ) ×  cos γ sin β cos ϕ sin θ + sin γ sin β sin ϕ sin θ + cos β cos θ    − cos γ sin β sin ϕ + sin γ sin β cos ϕ    − cos γ sin β cos ϕ cosθ − sin γ sin β sin ϕ cosθ + cos β sin θ   

(9)


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Since the scattering spectrum of the anisotropic plasmonic nanoparticle with arbitrary orientation is a linear superposition of a set of basic scattered spectra, the relationship between the incident polarization direction and the scattering intensity can be described as S (λ ) = Scay(λ ) ( cos γ sin β sin ϕ − sin γ sin β cos ϕ )

2

+Scax ( λ )( cos γ sin β cos ϕ sin θ + sin γ sin β sin ϕ sin θ + cos β cosθ ) (10) 2

+Scaz ( λ )( cos γ sin β cos ϕ cosθ + sin γ sin β sin ϕ cos θ − cos β sin θ )

2

where (β, γ) is the 3D orientation angle of the anisotropic plasmonic nanoparticle, and (θ, φ) is the excitation polarization angle. The polarization-dependent scattering data determined by FEM for the (Au core)-(dielectric shell) nanorice1 with different 3D orientation angles shown in Figure 7b and 7d can be well fitted by the analytical Equation 10 at the longitudinal and transverse plasmon mode, respectively. More importantly, our simplified analytic model can describe the polarization-dependent experimental scattering data of single gold nanorod in literature26 as well. Coincided with the excitation condition in reference,26 the incident polarization angle α in literature is corresponding to the angle (90°-θ) in our simulation excitation system (Figure 7a) with φ=90° and exchanging the incident direction of k and H. Therefore, analytic model Equation 10 can be simplified as S (λ ) = Scax ( λ )( sin γ sin β cos α + cos β sin α )

2

+ Scay ( λ )( sin γ sin β sin α − cos β cos α ) + Scaz (λ ) ( cos γ sin β )

2

(11)

2

As shown in Figure 7c, the normalized polarization-dependent experimental data of single Au nanorod with 3D orientation angle (β=23.4°, γ=135.9°) in literature26 panel 4 can be well fitted by our simplified analytic model (Equation 11). The fitting parameters (Scax(λ), Scay(λ),and Scaz(λ)) of the Equation 11 are obtained from FEM calculations with the same conditions setting in the literature.26 These results prove that our analytic model (Equation 10) for determining the 3D orientation of single anisotropic plasmonic nanoparticle is correct.


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Furthermore, we have demonstrated quantively that our method for single nanorice 3D orientation angles determination presents high accuracy no matter at the longitudinal or transverse plasmon resonance wavelength. However, this cannot be accomplished by single gold nanorod based on the scattering trait at the transverse plasmon mode. Assumed the 3D orientation angles of the nanorice1 in simulation model are (β, γ) = (45°, 45°), (15°, 75°), (45°, 135°), (135°, 45°), (135°, 135°), (90°, 45°), respectively. Here we list three conditions to illustrate our method. Condition 1 represents that the set of normalized basic spectra are certain at the longitudinal plasmon mode (690 nm), i.e., ScaxL=1, ScayL=0.122, and ScazL=0.118 which are obtained by numerical simulation at λ=690 nm. Condition 2 and condition 3 means that the set of basic spectra are unknown at the longitudinal (690 nm) and transverse (605 nm) plasmon mode, respectively, which will make us to determine the 3D orientation angles of single-particle more simply. For comparison, under condition 1 and condition 2 the corresponding values determined by utilizing the Equation (10) to fit the polarization-dependent simulation scattering intensity at the longitudinal plasmon mode (690 nm) are (β1, γ1) = (45.2°, 45.3°), (15.3°, 76.6°), (45.1°, 134.9°), (134.8°, 44.8°), (134.7°, 134.9°), (89.6°, 45.2°) and (β2, γ2) = (45.0°, 44.9°), (15.0°, 75.5°), (44.9°, 134.9°), (135.2°, 44.9°), (134.7°, 134.4°), (89.8°, 45.0°), respectively. At the same time, under condition 3 the corresponding angles determined by utilizing the Equation (10) to fit the polarization-dependent simulation scattering intensity at the transverse plasmon mode (605 nm) are (β3, γ3) = (45.0°, 44.9°), (15.0°, 75.7°), (45.0°, 134.9°), (135.3°, 44.5°), (135.7°, 133.7°), (89.9°, 44.9°). The further details can be seen in Table 1. Among condition 1, 2 and 3, the 3D orientation angles setting in simulation model and determined by our method are nearly equal only with deviation less than 1°，i.e., our method has a high precision just with subdegree uncertainty. And from Figure 7b and 7d, the polarization-


20

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dependent scattering intensity distributions of the nanorice with orientations (β, γ) = (45°, 45°), (45°, 135°), (135°, 45°), and (135°, 135°) are quite different with each other. These indicate that the method we proposed can be utilized to determine the arbitrary 3D orientation angle from (0°, 0°) to (180°, 180°) without angle degeneracy. Table 1 The comparison of the 3D orientation angles setting in simulation (β,γ) and obtained from our method by fitting the simulation data (in figure 7b and 7d solid points) under Condition 1 (β1,γ1), Condition 2 (β2,γ2) and Condition 3 (β3,γ3). Condition 1 represents the set of normalized basic spectra are certain, i.e., ScaxL=1, ScayL=0.122, and ScazL=0.118 which are obtained by numerical simulation at λ=690 nm. Condition 2 and condition 3 means that the set of basic spectra are unknown at the longitudinal (690 nm) and transverse (605 nm) plasmon mode, respectively. This will let us determine the 3D orientation angles more easily. Scax2, Scay2 and Scaz2 are the set of normalized basic spectra obtained by fitting the simulation data with Equation 10 under condition 2 (Longitudinal mode). Scax3, Scay3 and Scaz3 are corresponding to the condition 3 (Transverse mode). These data are nearly same with the ScaxL=1, ScayL=0.122, and ScazL=0.118 (Longitudinal mode) and ScaxT=0.124, ScayT=1, and ScazT=0.978 (Transverse mode), respectively. The fitting algorithm is a nonlinear least-squares fit (R2>0.995) among condition 1, condition 2 and condition 3. Setting in simulation

Condition 1

Condition 2

Condition 3

(β, γ)

(β1,γ1)

(β2,γ2)

Scax2

Scay2

Scaz2

(β3,γ3)

Scax3

Scay3

Scaz3

(45°,45°)

(45.2°,45.3°)

(45.0°,44.9°)

0.996

0.124

0.112

(45.0°,44.9°)

0.124

0.996

0.990

(15°,75°)

(15.3°,76.6°)

(15.0°,75.5°)

0.996

0.132

0.117

(15.0°,75.7°)

0.125

0.995

0.979

(45°,135°)

(45.1°,134.9°)

(44.9°,134.9°)

0.998

0.112

0.114

(45.0°,134.9°)

0.126

0.997

0.985

(135°,45°)

(134.8°,44.8°)

(135.2°,44.9°)

0.991

0.124

0.106

(135.3°,44.5°)

0.123

0.999

0.975

(135°,135°)

(134.7°,134.9°)

(134.7°,134.4°)

0.997

0.132

0.119

(135.7°,133.7°)

0.123

0.977

0.978

(90°,45°)

(89.6°,45.2°)

(89.8°,45.0°)

0.996

0.123

0.117

(88.9°,44.9°)

0.124

0.999

0.987

Finally, let us address the subsequent experimental realization of our method for single-particle 3D orientation determination. Firstly, the fabrication of the (Au core)-(dielectric shell) nanorices can refer to the method for the synthesis of Au-CdS core-shell nanorods.39 Secondly, we have two suggestions about recording the polarization-dependent scattering intensity. Suggestion 1: Keeping the propagation and polarization direction of the incident light and the sample unchanged to set up an original coordinate system. Then only rotate the sample to equivalently


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Page 22 of 26

realize the changing of the incident light and record the sample rotation angles respect to the original coordinate system and the corresponding scattering intensity. This can be achieved by total internal reflection scattering microscopy.26 Suggestion 2: Keeping the sample fixed, change the propagation and polarization direction of the incident light directly and record the changing directions and the corresponding scattering intensity. This can be realized by a polarizationresolved scattering technique which has been reported.4 CONCLUSIONS In conclusion, single (Au core)-(dielectric shell) nanorice presents inherent Fano resonances and anisotropic induced polarization-dependent scattering trait. We have discovered the inherent relationship of the anisotropic plasmonic nanoparticle’s (i.e., (Au core)-(dielectric shell) nanorice and bare gold nanorod) scattering spectra with different orientations, which can be defined by a trigonometric equation. Furthermore, we proposed an analytic model for determining 3D orientation of single anisotropic nanoparticle from (0°, 0°) to (180°, 180°) without angle degeneracy. Both of the experimental and numerical simulation data can be well fitted by our analytical model. The precision of the 3D orientation angles determined by our method is only with subdegree uncertainty. In addition, due to the efficient coupling between the dielectric elliptical shell and the gold nano-ellipsoid core, we show that the 3D orientation of single nanorice can be determined from either the transverse or longitudinal plasmon mode polarization-dependent scattering trait. Our orientation determination method is insensitive to the aspect ratio of nanorice, which can relax the limitation on the processing technology for nanoparticles. Our work will be useful for single particle orientation and rotational tracking.


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SUPPORTING INFORMATION. The charge distribution for the plasmon modes at the selected wavelengths of the nanorice1 under the orientation β = 90°, γ = 90°. Fano fitting of the FEM scattering spectra of the other kinds of individual (Au core)-(dielectric shell) nanorice under the orientation β = 90°, γ = 90° and β = 90°, γ = 0°. Parameters determined by fitting the calculated scattering spectra of the (Au core)-(dielectric shell) nanorice with the coupled oscillator model and the theoretical derivation process of Equation 6. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author *Address correspondence to [email protected]. Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. ACKNOWLEDGMENT This work was supported by the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2014-45, lzujbky-2014-48, lzujbky-2014-236 and lzujbky-2015-k7. REFERENCES 1. Rahmani, M.; Luk'yanchuk, B.; Hong, M. H., Fano Resonance in Novel Plasmonic Nanostructures. Laser Photon. Rev. 2013, 7 (3), 329-349. 2. Fan, J. A.; Wu, C. H.; Bao, K.; Bao, J. M.; Bardhan, R.; Halas, N. J.; Manoharan, V. N.; Nordlander, P.; Shvets, G.; Capasso, F., Self-Assembled Plasmonic Nanoparticle Clusters. Science 2010, 328 (5982), 1135-1138. 3. Lovera, A.; Gallinet, B.; Nordlander, P.; Martin, O. J. F., Mechanisms of Fano Resonances in Coupled Plasmonic Systems. ACS Nano 2013, 7 (5), 4527-4536. 4. Shafiei, F.; Wu, C. H.; Wu, Y. W.; Khanikaev, A. B.; Putzke, P.; Singh, A.; Li, X. Q.; Shvets, G., Plasmonic Nano-Protractor Based on Polarization Spectro-Tomography. Nat. Photonics 2013, 7 (5), 367-372.


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Abstract Graphic In combination with the polarization-dependent plasmonic Fano resonances and shape induced orientation-dependent scattering trait for realization single-particle three-dimensional (3D) orientation determination at both of the transverse and longitudinal plasmon mode.


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Single Anisotropic Plasmonic Nanoparticle Three ... - ACS Publications

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