Analytical model of resonant electromagnetic dipole-quadrupole coupling in nanoparticle arrays

An analytical model for investigations of multipole coupling effects in the finite and infinite nanoparticle arrays supporting electromagnetic resonances is presented and discussed. This model considers the contributions of both electric and magnetic modes excited in the nanoparticles, including electric and magnetic dipoles and electric and magnetic quadrupoles. The magnetic quadrupole propagator (Green's tensor) that describes the electromagnetic field generated by a point magnetic quadrupole source in all wave zones is derived. As an example, we apply the developed model to study infinite two-dimensional rectangular periodic arrays of spherical silicon nanoparticles supporting the dipole and quadrupole resonant responses. The correctness and accuracy of the analytical model are confirmed by the agreement of its results with the results of full-wave numerical simulations. Using the developed model, we show the electromagnetic coupling between electric dipole and magnetic quadrupole moments as well as between magnetic dipole and electric quadrupole moments even for the case of an infinite rectangular periodic array of spherical nanoparticles. The strong suppression of the dipole or quadrupole moment due to the coupling effects is demonstrated and discussed for spherical nanoparticle arrays. The analytical expressions for the reflection and transmission coefficients written with the effective dipole and quadrupole polarizabilities are derived for normal light incidences and zero-order diffraction. The derived expressions are applied to explain the lattice anapole (invisibility) states when the incident light is transmitted unperturbed through the silicon nanoparticle array. The important role of dipole and quadrupole excitations in scattering compensation resulting in the lattice anapole effect is explicitly demonstrated. The presented approach can be used for designing metasurfaces and further utilizing them in developing ultrathin functional optical elements.

Figure 1

Schematic of the rectangular periodic nanoparticle array under consideration.

Figure 2

Analytical (denoted “with coup.”) and numerical calculations (denoted “numeric”) of the intensity reflection and transmission coefficients for the infinite periodic array of silicon spheres. The lattice periods are $D_x=D_y=300$ nm, and the silicon particles have radius $R=125$ nm.

Figure 3

Multipole resonances in the individual sphere and its periodic array. The lattice periods are $D_x=D_y=300$ nm, and silicon particles have $R=125$ nm. Absolute values of (a) ED and MQ and (b) MD and EQ multipole effective polarizabilities. The effective polarizabilities are calculated with Eqs. (30) and (31) for the single sphere, Eqs. (41) for the sphere in the array without cross-multipole coupling, and Eqs. (42, 43, 44, 45) taking into account coupling. Effective polarizabilities of MQ and EQ are normalized to $k_0^2/4$ and $k_0^2/12$, respectively, following the coefficients in Eqs. (61) and (62), and all electric dipole and quadrupole polarizabilities are divided on ${\varepsilon }_0$.

Figure 4

Effective polarizabilities of (a),(c) ED and (b),(d) MQ. Panels (a) and (b) were calculated without accounting for cross-multipole coupling between ED and MQ, and panels (c) and (d) were calculated with the cross-multipole coupling. The lattice periods are $D_x=D_y$, and silicon particles have $R=125$ nm. The results are shown in a logarithmic scale. Note that the plotted quantity is ${log}_{10}$ of the absolute value of the effective multipole polarizability and takes into account $k_0^2/4$ for MQ. The dotted lines labeled RA mark the position of Rayleigh anomaly. Here all electric dipole and quadrupole polarizabilities are divided on ${\varepsilon }_0$.

Figure 5

Intensity reflection coefficient from the array of particles with the lattice periods $D_x=D_y$ and silicon particles $R=125$ nm. (a) Calculations are performed using only single-particle polarizability Eqs. (31), without taking into account the interaction of particles in the lattice. (b) Calculations are performed with polarizabilities defined through Eqs. (41) and accounting only for the interactions of multipoles of the same kind in the lattice. (c) Calculations with cross-multipole coupling of the multipoles defined by Eqs. (42, 43, 44, 45). Panels (a) and (b) include regions with unphysical results with the reflection coefficient values greater than 1. The dotted lines labeled RA mark the position of the Rayleigh anomaly.

Figure 6

Cross-multipole coupling of ED and MQ in the lattice. The lattice periods are $D_x=D_y=270$ nm, and silicon particles have $R=125$ nm. (a) Absolute values of ED and MQ effective polarizabilities without and with cross-multipole coupling. Effective polarizability of MQ is normalized to $k_0^2/4$ following the coefficients in Eqs. (61) and (62). Blue and light blue circles correspond to the minimum of effective MQ and ED polarizability accounting for cross-multipole coupling, respectively. Blue and light blue squares correspond to the same spectral point as circles and indicate $|{\alpha }_p^{\text{eff}}|=|{\alpha }_p^{\text{eff,coup}}|$ and $|{\alpha }_M^{\text{eff}}|=|{\alpha }_M^{\text{eff,coup}}|$, respectively. (b) Inverse MQ polarizability and cross-multipole coupling sum. (c) Inverse ED polarizability and cross-multipole coupling sum. In (b) and (c), blue and light blue circles correspond to the same spectral points as the circles in (a). Here all electric dipole and quadrupole polarizabilities are divided on ${\varepsilon }_0$.

Figure 7

Multipole resonances and reflection and transmission through the periodic array of spheres. The lattice periods are $D_x=530$ nm and $D_y=410$ nm, and silicon particles have $R=125$ nm. (a) Absolute values of each separate multipole term in the brackets of Eqs. (61) and (62) for the reflection and transmission coefficients (the MQ and EQ effective polarizabilities are normalized to $k_0^2/4$ and $k_0^2/12$, respectively, and ED and EQ polarizabilities are divided on ${\varepsilon }_0$). (b) Corresponding phases of the multipole terms in Eqs. (61) and (62). (c) Intensity reflection and transmission coefficients. (d) Phase of field reflection and transmission coefficients (61) and (62). At the wavelength 610–630 nm, the contributions of multipole moments in the reflection and transmission coefficients have comparable values; the phases of ED and EQ terms are almost equal to each other, and the same holds for MD and MQ. At this wavelength range, reflection is near zero, transmission is close to 1 and its phase does not change, indicating a lattice anapole state.

Figure 8

Total $E_x$ field distribution (incident and scattered waves) in numerical simulations of (a),(c) an empty domain and (b),(d) a domain with a sphere in the array with $D_x=530$ nm and $D_y=410$ nm. Simulations are performed at the wavelength (a),(b) $\lambda =629$ nm (which corresponds to the transmission phase $\varphi =0$ and the lattice anapole state) and (c),(d) $\lambda =580$ nm. Light incidence is from the left side. A cross section is taken in the $xz$ plane.