Directional Toroidal Dipoles Driven by Oblique Poloidal and Loop

Sep 27, 2018 - The toroidal dipole is an exquisite and untraditional electromagnetic mode, categorizing separately from the well-known conventional mu...
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Cite This: J. Phys. Chem. C 2018, 122, 24304−24308

Directional Toroidal Dipoles Driven by Oblique Poloidal and Loop Current Flows in Plasmonic Meta-Atoms Arash Ahmadivand* and Burak Gerislioglu

J. Phys. Chem. C 2018.122:24304-24308. Downloaded from pubs.acs.org by UNIV OF SUNDERLAND on 10/26/18. For personal use only.

Department of Physics & Astronomy, Rice University, 6100 Main Street, Houston, Texas 77005, United States ABSTRACT: The toroidal dipole is an exquisite and untraditional electromagnetic mode, categorizing separately from the well-known conventional multipole group which can be robustly localized and squeezed in an extremely tiny spot. In optical subwavelength systems, such a dynamic mode, which can be particularly distinguished as a nonradiating charge-current configuration, manifests in a head-to-tail fashion. Here, going beyond conventional toroidal dipoles, we explained the possibility of the excitation of directional toroidal dipoles in plasmonic meta-atoms driven by controlling the oblique magnetic field and poloidal fluxes. This was done by utilizing an antisymmetric multipixel unit cell with strong toroidal response across the near-infrared band. We showed that the unique structural properties of the proposed asymmetric nanostructure lead to the formation of multiple directional toroidal modes with opposite oblique magnetic fields and reverse poloidal current flows. We envision that this understanding paves novel approaches for the practical purposes of the multiresonant toroidal meta-atoms for advanced nanophotonic applications from modulation to beam steering.

1. INTRODUCTION Toroidization or toroidal polarization has been acknowledged as the description for condensed matter principle based on toroidal multipoles.1,2 Theoretically, the residues from an electric octupole and a magnetic quadrupole that can be combined to form an exquisite nonradiating closed-loop charge-current configuration, known as an axial toroidal dipole moment (T).1,2 Discovered, revealed, and described for the first time by Zel’dovich,3 toroidal multipoles have been studied in the context of solid-state physics,4 atomic and nuclear physics,5 and condensed matters.6 In terms of the latter approach, time reversal (t → −t) and space inversion (r → −r) symmetries are the fundamentals of the toroidization principle for both vortexlike and polar configurations.2−4 Such violations in the classical electromagnetic symmetries give rise to the formation of a strongly squeezed vortex of a closed-loop arrangement (toroidal dipole) with an ultraweak far-field radiation signature, also known as “ghost resonances”.4,7,8 The major production of this interference is a nonzero contribution to the scattered far-field radiation pattern, which can be described particularly by the d summation of induced charge (Q ∝ dr (r ·jl )), transversal (M ∝ |r × J|), and radial (T ∝ |r·J|) currents as follows Es =

4πk 2 ∑Q ψ + c l ,m l ,m l ,m

∑ Ml ,mϕl ,m + ∑ Tl ,mψl ,m l ,m

l ,m

ent properties similar to the dark-side of plasmons, the toroidal dipole mode has received copious interests recently for several practical applications because of facilitating negligible radiative losses due to suppression of the radiative electric dipole.4,9,10 On the other hand, the hybrid behavior, narrow line shape, and ultrahigh sensitivity of toroidal dipole-resonant metamaterials to the environmental perturbations have led to emerging of several optical and biological-sensing devices with unique spectral properties.11−14 Very recently, several innovative methods have been utilized to increase the tunability of toroidal meta-atoms and metamolecules to enhance the functionality of developed devices,15 such as integration of multipixel planar plasmonic toroidal meta-atoms and phase-changing materials to design hybrid and fast modulators.16,17 The excitation of tunable and controllable toroidal modes helps to develop advanced functional plasmonic and nanophotonic devices with high performances. The excitation of a dynamic toroidal dipole accompanies with the formation of a head-to-tail arrangement of the magnetic field and currents flowing around the surface of a torus. The direction of both current loop and poloidal current strongly relies on the direction of the magnetic fields in the adjacent pixels of a given plasmonic unit cell; however, the direction of the induced currents cannot be effectively controlled and functionalized. In addition, both magnetic field and poloidal currents are the significant and distinct evidences for the excitation of toroidal

(1)

where k is the wavenumber, c is the speed of light in a vacuum, and Q, M, and T correspond to the electric, magnetic, and toroidal multipole components, respectively. Possessing inher© 2018 American Chemical Society

Received: August 23, 2018 Revised: September 20, 2018 Published: September 27, 2018 24304

DOI: 10.1021/acs.jpcc.8b08184 J. Phys. Chem. C 2018, 122, 24304−24308

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The Journal of Physical Chemistry C features; thus having control on the direction, quality, and squeezing of these components leads to having functional and directional toroidal resonances. As a leading work, Bellan et al.18 have analytically shown that the variations in the vorticity model of flow can be driven by induced closed-loop and poloidal surface currents using magnetohydrodynamic model equations. This method successfully explained the influence of the axisymmetric current flux surfaces on the axially directed vorticities in plasma systems. Although the excitation of multiple toroidal modes have been reported very recently,19 the influence of the magnetically induced loop current and poloidal surface fluxes on the directivity of toroidal resonances in metamaterials has not been investigated yet. In this communication, we study on the spectral response of a multiresonant toroidal plasmonic meta-atom to support directional toroidal resonances depending on the oppositely spinning (clockwise and counterclockwise rotation) oblique poloidal and loop currents across the near-infrared (NIR) region. Using wellengineered, asymmetric multipixel aluminum (Al) nanoresonators in a single unit cell, we demonstrated the physical mechanism behind the excitation of directional toroidal dipoles through the formation of opposite magnetic fields and corresponding current fluxes. The surface current density and optical binding force calculations strongly validate our claim for the excitation of directional toroidal mode. Moreover, we show that the exquisite design of the metastructure leads to approximately θ ≈ 22° tilting in the polarization angle of the transmitted beam through the resonator and steers the angle of the incidence.

Figure 1. (a) Schematic for the design of the proposed Al meta-atom. (b) Corresponding geometrical sizes for the proposed structure. The corresponding geometrical sizes are specified in the picture for all parameters. (c) Normalized multipole scattering intensity from the scatterer for toroidal (T), electric (p), and magnetic (m) components. (d) Normalized transmission and reflection patterns radiated from the metamaterial. (e,f) E-field maps for the toroidal dipole resonances in the meta-atom in logarithmic scale. The thickness of the unit cell is assumed to be 80 nm.

2. NUMERICAL ANALYSIS To define the exact geometries, analyze the spectral response, and understand the behavior of the proposed asymmetric toroidal nanostructure, we utilized finite-difference time-domain method (using the software and solver package Lumerical 2018). In the corresponding three-dimensional simulations, the spatial grid sizes were set to Δxyz = 1 nm in all axes and the time step was set to dt ≈ 0.0199 fs to establish the Courant stability factor close to ∼0.99.17 The incident pulse was assumed to be a plane wave with the bandwidth tuned at the NIR. The workplace was encompassed by periodic boundaries to resemble a metasurface structure with infinite number of unit cells in arrays along both x- and y-directions, whereas the normal beam propagation direction (z-axis) is bounded with 32 perfectly matched layers.

ization of the incident beam is transverse (φ = 90°) or polarized in y-coordination (the corresponding trajectory is presented in the picture). The spectra are obtained based on the radiation transmitted and reflected by the planar arrangements of infinite number of scatterers, defined by the approach provided by Marinov et al.22 for toroidal metamolecules. The plotted curves reveal the excitation of two distinguished resonances located at λ ≈ 975 and ∼1450 nm, correlating with the toroidal dipole moments (T). According to the toroidization concept of the optical physics, the projected toroidal radiation field (Es) includes the signature of both electric and magnetic components as described below23,24 ÅÄÅ ÑÉÑ 2 μ c2 l o ̂ × ÅÅÅÅm − k m(1)ÑÑÑÑ − + Es = 0 2 o ik p ik R m ÅÅ ÑÑ o 10 2Δ o ÅÇ ÑÖ n É ÅÄÅ Ñ 2 ÑÑ k 2Å 2 (e) (1) − k ÅÅÅÅT − T ÑÑÑÑ + k [Q ·R̂ ] ÅÅÇ ÑÑÖ 10

3. RESULTS AND DISCUSSION Figure 1a shows the schematic of the proposed meta-atom located on a quartz substrate with the permittivity of ε = 2.1 (not to scale). To model the unit cell in the current analyses, we used the empirically measured and defined Al dielectric function derived using ellipsometric data by assuming the presence of a thin diametric (∼2 nm) alumina (Al2O3) layer as a coverage coating the structure.20 This was done based on mixing the tabulated values of Al and Al2O3 layers using Bruggeman approach, reported by Palik.21 The exact geometrical sizes are reported in Figure 1b, obtained by judiciously selecting and simulating various parameters for each resonator, and these values are assumed to be fixed along the work. The simulated normalized multipole decomposition intensity profile for the electromagnetic response of the proposed plasmonic nanostructure is plotted in Figure 1c. The polar-

−

k2 ̂ ik3 (T) ̂ [Q ·R ] − ik3[(O(e)·R̂ )R̂ ] R × [Q(m)·R̂ ] − 2 3 | o ik3 − [(O(m)·R̂ )R̂ ] o × exp( −ikR ) } o o 180 ~ (2)

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DOI: 10.1021/acs.jpcc.8b08184 J. Phys. Chem. C 2018, 122, 24304−24308

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The Journal of Physical Chemistry C

Figure 2. (a) Schematics of the formation of toroidal mode in the structure and the rotation of the poloidal current are shown in the donut shape field. (b,c) Top-view vectorial surface current density maps for the directional toroidal modes. (d,e) Cross-sectional xz-plane vectorial H-field maps for the excitation of toroidal field across the structure at two different wavelengths.

where μ0 is the magnetic permeability of free space, A is the area of the unit cell, r is the perpendicular distance of the monitor from the metasurface, and r̂ is the unit vector perpendicular to the unit cell array. In addition, m(1) and T(1) denote the meansquare radii of magnetic and toroidal dipoles, respectively, which are considered for the unit cells with finite geometries. Additionally, the other contributed terms to the emitted field are electric (p), magnetic (m), and toroidal dipoles (T), electric (Q(e)), magnetic (Q(m)), and toroidal (Q(T)) quadrupoles, and electric (O(e)), magnetic (O(m)), and toroidal (O(T)) octupoles. It should be underlined that we neglected the influence of the multipolar toroidal modes (quadrupolar, octupolar, etc.) in our analysis because of having an ultraweak far-field scattering signature; hence, the dominant multipoles are shown merely in the profile. Obviously, at both resonant wavelengths (λ ≈ 975 and 1450 nm), the toroidal modes become dominant because of the suppression of all classical electromagnetic multipoles. The noteworthy point here is the noticeable amplitude of the quadrupolar electric and dipolar magnetic modes, which originated from the normalization of the profile. Figure 1d validates the consistency between the calculated multipole scattering intensity and transmission/reflection amplitude spectra for the proposed metasurface. Accordingly, the optically induced toroidal dipoles act as leading modes consistent with our studies demonstrated in Figure 1c. This plane also contains the transmission curve for the longitudinal polarization (φ = 0°), which attests the strong polarization dependency of the unit cell to the polarization perturbations. Figure 1e,f illustrates the electric-field (E-field) maps for the excitation of toroidal dipoles under transverse polarization excitation in a logarithmic scale. The remarkable point here is the confinement and intensification of the excited fields at specific points of the capacitive openings between the central and proximal resonators at different wavelengths. Clearly, for the toroidal dipole at λ ≈ 975 nm, the field concentration is in the central part of the unit cell, and conversely, for the toroidal moment at λ ≈ 1450 nm, the fields are localized at the tips of the outermost of the meta-atom. This effect reveals the formation of opposite toroidal dipoles, and the reason will be explained in the following subsection. As discussed earlier, the induced surface currents and magnetic field direction in the proximal resonators in a

multipixel meta-atom have fundamental and substantial impacts on the formation and direction of toroidal resonances. Figure 2a exhibits the artistic illustrations for the formation zone and rotation of the toroidal field, squeezed between the proximal and central resonators. This graphic also allows us to understand the formation of the oblique closed-loop arrangement of the magnetic field, loop current, and poloidal current fluxes around the central block and rotation of poloidal current in tilted fashion around the surface of the torus. In Figure 2b,c, we computed and plotted the vectorial surface current density (J) maps for both toroidal resonances at different wavelengths to validate the excitation of directional toroidal resonances enforced via opposite oblique current fluxes. These planes clearly show the required mismatch between the induced magnetic fields in each structure and also illustrate how the direction of the toroidal dipole inversely changes because of the variations in the direction of the poloidal current flow. The direction of the incident normal plane-wave is shown in the inset of these planes. Here, the direction of the induced current for each toroidal dipolar mode is demonstrated by arrows. Technically, for the mode at λ ≈ 975 nm, the surface current flow direction in peripheral resonators converges to the center of the unit cell, consistent with Figure 1e. For the other mode at λ ≈ 1450 nm, the surface current direction on the proximal half-rings is in the direction to the outermost part of unit cell in both sides, consistent with Figure 1f. This can be easily understood by plotting the cross-sectional (xz-plane) magnetic-field (H-field) vectorial boards. We depicted the direction of the opposite surface currents at different wavelengths of the transmission dips Figure 2d,e. For the toroidal dipole at λ ≈ 975 nm, we monitored a counterclockwise rotation of magnetically induced current, and conversely, for the toroidal mode around λ ≈ 1450 nm, the poloidal current spins in the clockwise path (see Figure 2a). This phenomenon allows for the formation of directional toroidal dipolar moments in a single unit cell. Further analysis over the surface current density feature also enables us to compare the strength and quality of each induced dipolar mode. Figure 2f,g evaluates the surface current across the structure at the toroidal resonance wavelengths in yz-plane (cross-sectional view). Obviously, the induced current density at higher energies (λ ≈ 975 nm) is approximately 2 times stronger than the toroidal 24306

DOI: 10.1021/acs.jpcc.8b08184 J. Phys. Chem. C 2018, 122, 24304−24308

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The Journal of Physical Chemistry C mode at low energies (λ ≈ 1450 nm). This difference between the induced current density correlates with the strength of the magnetic moment in the toroidal mode position. On the other hand, we also computed the dephasing time for each induced resonances using the method developed based on the full width at half-maximum (fwhm) computations as below24−27 2ℏ (3) fwhm where h̵ is the reduced Planck’s constant. Thus, the dephasing times for the toroidal dipoles can be defined as approximately ∼8.5 and 3.2 fs for the dipolar modes at 975 and 1450 nm, respectively. This shows longer lifetime of the toroidal mode at the shorter wavelengths due to having a narrower resonance line shape. We also estimated the optical binding force (F) for the toroidal plasmonic scatterer to show the dominant enforcement arising from the poloidal current flux at the critical wavelengths (λ ≈ 975 nm and λ ≈ 1450 nm) to determine the direction of the induced toroidal resonant moments. To this end, we employed the surface integration F = ∮ ⟨T⃡ ⟩·n̂ ds using the timeS averaged Maxwell stress tensor (T⃡ )26 t=

(

Figure 4. Polarization ellipse for the oblique toroidal field angle. The inset is the far-field radiation pattern for the toroidal meta-atom.

)

metasurface. This steering in the polarization angle of the transmitted electromagnetic beam can be utilized for light bending and steering applications, enhanced by toroidal modes. In addition, the strong sensitivity of the unit cell can be utilized to develop fast telecommunication switches based on toroidal mode.9,28,29

1 I⃡ ⟨T⃡ ⟩ = Re(εε0 EE* + μμ0 HH* − (εε0|E|2 + μμ0 |E|2 )) 2 2 (4)

In the equations above, n is the normalized vector on the surface of a given meta-atom. Figure 3 demonstrates the

4. CONCLUSIONS In this work, we have numerically analyzed and explained the possibility of the excitation of polarization-dependent directional toroidal dipoles in multipixel plasmonic meta-atoms across the NIR region. Our studies have shown that the optically driven oblique loop fluxes have direct influence on the excitation of directional toroidal modes. Taking advantage of the vectorial magnetic fields and surface current density profiles, we validated the formation of the directional toroidal fields. This claim was also verified by calculating the optical binding force at the resonant wavelengths. Finally, we calculated the polarization angle variations due to the oblique shape of the spinning toroidal fields across the multipixel unit cell. We envision that this work opens new avenues to the practical applications of the multiresonant toroidal meta-atoms for advanced nanophotonic applications from sensing to beam steering.

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AUTHOR INFORMATION

Corresponding Author

Figure 3. Spectrum of the binding optical forces for the plasmonic toroidal meta-atom.

*E-mail: [email protected]. ORCID

computed binding force for a plasmonic unit cell, confirming the significant forces arising at the resonant wavelengths from the magnetic moments to push and define the direction of the toroidal moment. Ultimately, we analyzed the specific feature of the asymmetric design and variations in the incident polarization angle via oblique signature of the toroidal field and its corresponding spinning symmetry. Figure 4 shows the polarization ellipse projected from the planar plasmonic toroidal metasurface. Here, the oblique angle of the polarization of the transmitted beam from the metasurface is calculated around θ ≈ 22° (p-polarized axis) and corresponds to the normalized incident beam (spolarized axis). The inset is the far-field radiation pattern to validate the angle of the transmitted field polarization from the

Arash Ahmadivand: 0000-0003-0808-8775 Burak Gerislioglu: 0000-0001-8575-7990 Notes

The authors declare no competing financial interest.

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REFERENCES

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DOI: 10.1021/acs.jpcc.8b08184 J. Phys. Chem. C 2018, 122, 24304−24308