Polarizabilities of complex individual dielectric or plasmonic nanostructures

When the sizes of photonic nanoparticles are much smaller than the excitation wavelength, their optical response can be efficiently described with a series of polarizability tensors. Here, we propose a universal method to extract the different components of the response tensors associated with small plasmonic or dielectric particles. We demonstrate that the optical response can be faithfully approximated, as long as the effective dipole is not induced by retardation effects, hence do not depend on the phase of the illumination. We show that the conventional approximation breaks down for a phase-driven dipolar response, such as optical magnetic resonances in dielectric nanostructures. To describe such retardation induced dipole resonances in intermediate-size dielectric nanostructures, we introduce “pseudopolarizabilities” including first-order phase effects, which we demonstrate at the example of magnetic dipole resonances in dielectric spheres and ellipsoids. Our method paves the way for fast simulations of large and inhomogeneous metasurfaces.

Figure 1

Sketch used to implement the concept of generalized electromagnetic propagator. (a) Transparent reference medium with ${\varepsilon }_1\left({\omega }_0\right)=n_1^2$ and ${\mu }_1=1$, and (b) material system of arbitrary shape, also called the source zone, embedded in the reference medium [permittivity ${\varepsilon }_s\left({\omega }_0\right)$ and permeability ${\mu }_s\left({\omega }_0\right)$].

Figure 2

Spectral variation of the imaginary part of the dipolar polarizability of a spherical gold particle of radius 5nm. (a) Nanosphere suspended in vacuum ($n_1=1$). Comparison of the analytical Clausius-Mossotti polarizability (blue) with the numerical calculation (red). (b) Numerically calculated in-plane (red) and out-of-plane (green) polarizability tensor elements for a nanosphere deposited on a silica substrate ($n_2=1.48$). Insets show sketches of the particle environment. For the numerical calculations, we discretized the spheres using 6337 identical mesh cells on a hexagonal compact grid.

Figure 3

Spectral variation of the real (e)–(h) and imaginary part (i)–(l) of the terms of the dipolar polarizability matrix for various structures represented in (a)–(d). Geometries consist in (a) a single isotropic pad of size $(50\text{nm}\times 50\text{nm}\times 25\text{nm})$, (b) an anisotropic pad of size $(50\text{nm}\times 100\text{nm}\times 25\text{nm})$, (c) an “L” shape structure included in the $xOy$ plane, (d) a “3D” shape structure with ramifications in the three directions of space. For the first three structures computations were performed with a discretization step $d=2.5$ nm while we used a step $d=2$ nm in the last case for a good convergence of the calculation. The color diagrams show the degenerate components of the polarizability tensor.

Figure 4

Dielectric nanosphere ($n=4$) of radius $r=100$ nm in vacuum, illuminated by a plane wave of linear polarization. (a) Extinction spectrum (blue line) and electric (ED) as well as magnetic dipole (MD) contributions to the extinction (orange, respectively green line), calculated as described in Ref. [28]. (b) Electric dipole extinction via the polarizability tensor ${\alpha }^{EE}$ with (solid orange line) and without the phase term (dashed orange line). The ED extinction from (a) is shown as dashed black line for comparison. (c) Magnetic dipole extinction via the polarizability tensor ${\alpha }^{HE}$ with (solid green line) and without the phase term (dashed green line). The MD extinction from (a) is shown as dashed black line for comparison. At the top, the internal electric field distribution (real parts) is shown at the ED (left) and MD resonance (right).

Figure 5

Dielectric nanospheroid ($n=4$) of long axis diameter $D_1=250$ nm (along $Ox$) and short axis diameter $D_2=120$ nm (along $Oy$ and $Oz$) placed in vacuum, illuminated by a plane wave of linear polarization (top: $s$ polarization, bottom: $p$ polarization) for varying incident angles ${\vartheta }_{\text{inc}}$. The total extinction cross section (blue solid line) is compared to the extinction induced by the effective electric and magnetic dipole moments, obtained through full-field simulations (dashed orange and green lines), to the pseudopolarizability (solid orange and green lines) as well as to the “static” polarizabilities (dotted orange and green lines). The pseudopolarizability superposition approximation is used in the three cases of oblique incidence ($22.5^{\circ }$, ${45}^{\circ }$, and $67.5^{\circ }$).

Figure 6

Polarizability of a gold sphere (radius $r=50$ nm) calculated using a nonretarded (blue line) and a retarded (green and red lines) Green's tensor for the description of the substrate. The blue and green curves correspond to the ${\alpha }_{xx}$ component of the electric-electric polarizability of the nanosphere lying on a dielectric substrate ($n_2=1.45$, illustrated in the top left inset). In the case of the red dashed curve an additional 50 nm thick gold layer is inserted between silica substrate and gold sphere (see sketch in top right inset). In both cases, the top medium is air ($n_1=1$).

Figure 7

Extinction efficiency spectra computed for a gold sphere of diameter $D=100$ nm using Mie theory (blue curve), the full GDM simulation (orange curve) and the effective polarizability approximation (green curve). The sphere is placed in vacuum and illuminated by a linear polarized plane wave.

Figure 8

An $s$-polarized (${\mathbf{E}}_0\parallel Y$) plane wave is incident from below at and oblique angle of ${45}^{\circ }$ in the $XZ$ plane. The mesoscale structures are made from a constant index dielectric with $n=4$, placed in vacuum. The figure shows the extinction spectrum calculated via full GDM simulation (solid lines) and obtained from the pseudopolarizability model (dashed lines) for (a) a “full” nanocuboid and (b) the same cuboid but with on of the top corners removed.

Figure 9

Near-field intensity distribution, calculated 100 nm above the top-surface of a plasmonic nanostructure for (a)–(d) a small gold sphere and (e)–(h) an “L”-shaped gold nanoparticle. (b) and (f) show the near-field calculated using the dipolar effective polarizability approximation. (c) and (g) are calculated based on the full GDM simulation. (d) and (h) show the relative error of the dipole approximation with respect to the full simulation. (b), (c), (f), and (g) show the normalized near-field intensity ${\left|\mathbf{E}\right|}^2/{\left|{\mathbf{E}}_0\right|}^2$. The incident plane wave (normal incidence) is polarized along $X$ at a wavelength of ${\lambda }_0=520$ nm for the sphere and ${\lambda }_0=690$ nm for the “L” shaped structure. All colorplots show an area of $800\times 800$ ${\mathrm{nm}}^2$.

Figure 10

Coupling of five gold nanorods of dimensions $100\times 50\times 50$ ${\mathrm{nm}}^3$ ($X\times Y\times Z$), aligned along the $OY$ axis. As illustrated in (a), the rods are separated by a variable distance $D$ (center-to-center) and are illuminated by a plane wave of linear polarization along $Y$. [(b)–(e)] For different distances $D$ between the nanorods, comparison of top: scattering spectra and bottom: near-field intensity ${\left|\mathbf{E}\right|}^2/{\left|{\mathbf{E}}_0\right|}^2$ in a plane parallel to $XY$ at a height $Z=50$ nm above the rod's top surface. We compare the full GDM simulation (blue lines, leftmost color maps) with the effective polarizability model, in which case we either include optical interactions between the dipoles via a self-consistent coupling scheme (orange lines, center color maps) or we assume that the dipoles are optically isolated, and only interference effects occur (no coupling, corresponding to the Born approximation, red lines). The shown area in the near-field intensity maps is $3D/2\times 3D$, the wavelength of the illumination is 550 nm. Note that in (b) the near-field maps are not on the same color scale.

Figure 11

Extinction (solid lines) and scattering (dashed lines) cross sections of a dielectric sphere (constant and real refractive index $n=4$) with radius $r=100$ nm, placed in vacuum and illuminated by a linear polarized plane wave. The sharp resonance around 525 nm can be attributed to a quadrupole mode.