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Analytical chiroptics of the 2D and 3D nanoantennas Atefeh Fazel Najafabadi, and Tavakol Pakizeh ACS Photonics, Just Accepted Manuscript • Publication Date (Web): 16 May 2017 Downloaded from http://pubs.acs.org on May 21, 2017
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Analytical chiroptics of the 2D and 3D nanoantennas Atefe Fazel Najafabadi and Tavakol Pakizeh* Faculty of Electrical Engineering, K. N. Toosi University of Technology, Tehran 1631714191, Iran
KEYWORDS: Chiral plasmonics, chiroptics, 2D/3D nanoantennas, circular dichroism. Abstract: The optical interaction of the circularly-polarized light with structurally chiral optical antennas earns the chiral nanophotonics and the antenna-enhanced chiral molecular sensing. Here we develop a simple analytical framework, connecting the far-field chiroptical transmission (circular dichroism) of the feasible 2D and 3D nanoantennas with the electromagnetic polarizability of their elements. We report that the real and imaginary parts of the polarizability determine the line-shape of the circular dichroism spectra, while the optical coupling of the comprising elements controls the spectral width and the absolute amplitude. On the example of the coupled-nanorods antennas, as a fundamental nanoplasmonic structure, we visualize the effect of the structural parameters variation on properly addressing and controlling the observed chiroptical effects.
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Optical activity serves as a biomolecular structural fingerprint and a source of diverse nanophotonic applications, making fundamentally attractive its control in the optical antennas at the nanoscale.1-7 It is usually associated with the optical chirality (OC) and the circular dichroism (CD) / optical rotatory dispersion (ORD), related to the CD via the Kramers-Kronig (KK) relation.8-10 The OC is the near-field phenomenon, and the CD/ORD are characterized in the farfield. Nanometallic optical antennas boost the optical activity of the chemical and biomolecular chiral substances in the near-field, creating efficient enantiospecific optical sensors.3,4,11-15 Chiral nanoantennas are successfully employed in manipulating the phase and the amplitude of the electromagnetic fields aiming at perfect lensing, designing the negative refractive index materials and in the dynamic nanorheological sensors, to name the few.5,16-19 A rich library of designs for the chiroptical nanoantennas is created to date, including helical,1,5,12 G-shaped,20 U-shaped,21-22 and various dimers23-26 nanoantennas. Among these, a nanorod dimer is an archetypical nanoantenna, addressed theoretically as a model system for the emergence of plasmon chiroptics.23 It experimentally displays a strong and tunable CD,23-25,27 providing the building blocks for the ultra-thin chiroptical metasurfaces and monolithic photodetectors that discriminate the right- and left-circularly polarized (RCP and LCP) light,28 and evidencing orders of magnitude enhancement of the molecular chiroptical response.11 Here we take this exemplary chiroptical nanorod dimer antenna and build a simple quantitative analytical model that predicts the CD of its 2D and 3D configurations. Specifically, the model clarifies the relationship between the absorption, interaction, and the retardation terms of the nanodimer elements that are crucial for the efficient generation of the chiroptical effects. Although the coupling strength can be increased exponentially by making the elements closer, 2 ACS Paragon Plus Environment
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the spatial phase retardation between the incident electric field at the element’s position, as the key for the emergence of efficient chiroptics in this system, effectively decreases. The model further tackles the fact that a 2D antenna, while lacking the retardation terms and thus not displaying any chiral forward scattering, might produce the chiral scattering (and the differential absorption) of light in off-normal (other directions rather than forward and backward scattering) directions. We examine the CD spectrum of the 2D and 3D nanorods dimer antenna, defining the CD as a differential extinction (i.e., combined absorption and scattering) of the RCP and LCP light and considering all the geometrical parameters such as nanorods dimensions and interparticle spacing. We start with a 2D configuration of two Au nanorods (L=100 nm, h=20 nm, Figure 1a), addressing it with the coupled-dipoles approximation (CDA)29 enabled by the nanoantenna subwavelength dimensions and the finite-integration technique.30 The simulations are conducted with isolated dimers and the simulation volume is set to the λ0/4 (a quarter of the resonant wavelength of a single nanorod). For high accuracy we use tetrahedral meshes with cell size in the nanoantenna region set to within 1-6 nm. The perfectly matched layer (PML) boundary condition with the reflection level < 0.0001 is applied to model the free space. Although the nanoantenna is achiral and does not produce the CD in transmission31 (CD of extinction (CDext) is zero), the scattering cross sections (Csca) are not identical for the LCP and the RCP light. Figure 1b gives the optical scattering efficiency, i.e. Qsca=Csca/A, where A is the geometrical area of the nanorod. There, the spectral positions of the nanoantenna’s in-phase (at 740 nm) and out-of-phase (at 830 nm) resonant modes are denoted. Also, the induced electric fields for the hybridized modes are depicted in the inset of Figure 1b. In this regard, the angular patterns of the Csca at these resonances' wavelengths, plotted along the propagation direction (i.e., 3 ACS Paragon Plus Environment
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in the plane, perpendicular to the nanoantenna/sample) are depicted in Figure 1c and d. The scattering polar plots visualize that while the RCP and LCP scatterings are identical in the forward and backward directions (0˚ and 180˚), they substantially deviate from each other at other angles, most remarkably in the nanoantenna/sample plane at both resonances. Basically, this chirality feature arises from the 2D handedness dependent of the structure; although, lack of vertical phase retardation between the elements causes similar scattering patterns in the ±z directions, which results in CDext=0.31 However, CD of scattering (CDsca) and CD of absorption (CDabs) are not necessarily zero. Thus to keep the net circular dichroism constant, CDabs is essentially the mirror image of CDsca.
Figure 1. (a) Schematic view of the 2D/ planar nanorods dimer antenna and the definition of the geometrical parameters. (b) The scattering spectrum of a 2D dimer and the amplitude of the electric field distribution (|E|) at the in-phase (740 nm) and out-of-phase (830 nm) modes. The scattering patterns at the hybridized plasmonic modes when the dimer is normally excited with
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the RCP (blue solid-curve) and LCP (red solid-curve) plane waves at resonance: (c) λ=740 nm and (d) λ=830 nm. The Cscas are calculated by computing the power radiated by the two oscillating dipoles32 in order to obtain the CDsca/CDabs of the 2D nanoantenna. The CDsca can be expressed as (see the Supporting Information Note 1):
CDsca ∝ k4 [| PxR |2 −| PyL |2 ]
(1)
where PxR is the vector polarizability of the induced dipole moment along the x axis for the RCP light and PyL is the vector polarizability of the y-dipole for the LCP light. Adding the effective polarizability of nanorods ( α ), eq 1 takes the form of: CDsca ∝ k 4
| α |2 Im{ A12α } |1 − ( A12α )2 |2
(2)
where A12 is the cross coupling term between the x- and y-induced dipoles moments and Im{.} stands for the imaginary part. The eq 2 expresses a general formula for the differential scattering of the RCP and the LCP (CDsca) by the 2D nanoplasmonic dimer antenna. This conveniently allows us to tune the CDsca without need to rigorous numerical or full-wave computations. Importantly, since the coupling between the plasmonic nanoresonators managed through the near-field coupling, term A12 is mainly real and one can conclude that absorption has the fundamental role in determination of CDsca/CDabs response of the 2D dimer nanoantenna. This statement is verified by recent experimental findings for a planar-chiral metasurface.33 Additionally, it is shown that absorption is dominant over scattering for in-plane nanoparticle oligomers.31 Hence, based on the eqs 1and 2, lossless planar dimers (Im{α}=0), the T-shaped arrangement (A12=0), and L-shaped structures with the same effective arms result in CDsca/CDabs
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=0. Moreover, whereas the Im{α} forms a bell shape function with positive values, CDsca preserves its sign along the entire spectrum. In order to have a closer look at the chiroptical properties of the 2D plasmonic nanoantenna, we physically model each nanorod with a prolate nanospheroid (a, b=c), where 2a is the major diameter and 2b and 2c (b=c) are the minor diameters; and the modified electrostatic polarizability of a prolate spheroid34-35 are employed (see Supporting Information Note 2). Figures 2a and b reveal the effect of the interparticle spacing g in vertical (y) direction and of the horizontal offset x0 along x-direction. In fact, these adjust the amplitude and the bandwidth of the observed scattering CDsca spectrum. As seen in Figure 2a by increasing the offset x0 from 0 to 40 nm, two hybridized plasmonic modes of the CDsca spectrum, spectrally overlap, bandwidth decreases, and optical CDsca reaches its maximum value at the localized surface plasmon resonance (LSPR) wavelength of a single nanorod (λ0= λLSPR). The scattering CDsca takes the bell shape form of Im{α}. However, according to the inset, by increasing x0 to 50 nm, its intensity rapidly declines and at x0=50 nm the system becomes entirely symmetric (T-shape dimer), the dipoles are decoupled (A12=0), and as the eq 2 predicts CDsca vanishes everywhere. Though, the numerical result is not absolute zero. In the theoretical modeling,29 the nanorods are modeled with the point dipoles, which is strictly 2D, although in reality finite spatial extension of nanorods along the light propagation direction introduce phase retardation which is different for RCP and LCP. The results seen in Figure 2b show that by increasing g, the absolute value of CDsca gradually decreases because of decreasing the electromagnetic coupling. The offset x0 = 40 nm is considered for all cases here. The inset represents |CDsca| versus g at λLSPR. The corresponding numerical results shown in Figures 2c and d agree well with that predicted from the electromagnetic modeling modeling. For the sake of competence, the results of optical CDs
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for different value of x0 are shown in Figure S1. In addition, the induced dipole moments and electric fields near each nanorod are investigated, as shown in Figure S2.
Figure 2. Scattering CDsca spectra for a 2D plasmonic nanoantenna. The effect of the offset x0 are shown in the left column theoretically (a) and numerically (c). The gap distance is g=10 nm in all cases. The |CDsca| at λLSPR versus x0 are shown as the inset. |CDsca| spectrum for g=20 and 40 nm are shown theoretically and numerically at (b) and (d), respectively (x0= 40 nm). The insets show the |CDsca| at λLSPR versus g. The schematic views of the structure from the top are plotted as the inset in (a) and (b).
Then we switch to a 3D plasmonic dimer nanoantenna which there is a vertical spacing d between the nanorods. In this case, due to spatial phase retardation between the nanoplasmonic elements, in contrast to the 2D case, the optical extinction CD is significant and highly depends on the vertical spacing d. Therefore, in this arrangement we explore the optical CD, sum of CDsca and CDabs. The associated max.{CD} spectrum versus different vertical distances is 7 ACS Paragon Plus Environment
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calculated and shown in Figure 3a. Interestingly, the resulting curve is similar to the absolute value of a damped sine curve, although it deviates at shorter distances (d
