Multipole analysis of substrate-supported dielectric nanoresonator metasurfaces via the $T$-matrix method

Substrates, and layered media in general, are ubiquitous and affect the properties of any object in their vicinity. However, their influence is, in an arbitrary framework, challenging to quantify analytically, especially for large arrays which additionally escape explicit numerical treatment due to the computational burden. In this work, we utilize a versatile $T$-matrix-based framework to generalize the coupled multipole model toward arbitrarily high multipole orders and substrate-supported arrays. We then employ it to study substrate-supported random/amorphous arrays of high index dielectric nanoparticles which are of wide interest due to relatively low losses and a highly tunable optical response, making them promising elements for nanophotonic devices. We discuss how multipole coupling rules evolve in the presence of a substrate in amorphous arrays for three interaction mechanisms: direct coupling between particles, substrate-mediated interparticle coupling, and substrate-mediated self-coupling. We show how the interplay of array density, distance from the substrate, and the latter's refractive index determine the optical response of an array. As an example, we use this framework to analyze refractometric sensing with substrate-supported arrays and demonstrate that the substrate plays a crucial role in determining the array sensitivity.

Figure 1

(a) Small-scale schematic representation of substrate-supported random arrays of nanodisks (c-Si, $n_{\mathrm{sub}}=2$, diameter $D=160$ nm, thickness $H=120$ nm) for comparison between FDTD and $T$-matrix calculations with (b) obtained extinction spectra showing very good agreement. (c) Comparison of $T$-matrix calculations of explicit substrate-supported arrays of 1751 nanodisks with refractive index 4 [other parameters from (b)] with proposed model shows good agreement. Coupling in the random arrays with different density induces significant changes of the peak positions, widths, and amplitudes.

Figure 2

(a) Coupling matrices for (left) direct coupling between particles, (center) substrate-mediated interparticle coupling, and (right) substrate-mediated self-coupling for CC $=3.5D,\lambda =700$ nm assuming $D=160$ nm and $H=160$ nm. (b) Real and imaginary parts of the dipole matrix elements at $\lambda =700$ nm as a function of center-to-center distance for direct coupling (direct), substrate-mediated coupling between the same dipole type (refl) and cross coupling between electric and magnetic dipoles (refl-c), and self-coupling also between the same dipoles (self-) and cross coupling (self-c). (c) The magnitude of the dipole matrix element as function of wavelength for CC $=3.5D$ shows that the self-coupling is the dominant factor modifying the optical properties, however, jointly the remaining contributions can be, depending on the relative phases, of similar magnitude.

Figure 3

Comparison between the optical response of amorphous arrays of c-Si cylinders with $D=150$ nm and $H=225$ nm for selected center-to-center distances: (a) in air, (b) on a substrate with $n_{\mathrm{sub}}=1.45$, and (c) in an index-matched medium of 1.28. (d)–(f) Multipole decomposition (D, dipole; Q, quadrupole; O, octupole) of extinction spectra for the three cases for CC = 5. Note the substrate-mediated magnetoelectric coupling around 600 and 725 nm, which is absent in homogeneous environments.

Figure 4

Coupling dependence vs distance from substrate and substrate index. (a) Real and (b) imaginary parts of the dipole-dipole coupling term $W$ (neglecting magnetoelectric cross coupling) for an amorphous array on a substrate ($n_{\mathrm{sub}}=2$) at $\lambda =500$ nm as function of array-substrate distance $z$ and minimum center-to-center $l_{cc}$ distance. The distance $z=0$ corresponds to an array of disks of height equal to 100 nm placed on a substrate. Both real and imaginary parts of total interparticle coupling shows distinct dependence on $l_{cc}$ for each array-substrate distance. (c) The self-coupling term $W_r^{S,S}$. likewise affected. (d) $W_r^{S,S}$ increases significantly with $n_{\mathrm{sub}}$, while interparticle coupling is weakly dependent on $n_{\mathrm{sub}}$ and only for very dense arrays with small $l_{cc}$ is larger than $W_r^{S,S}$.

Figure 5

Extinction cross-section spectra for silicon nanodisks with 100-nm diameter and 100-nm height placed on a substrate with $n_{\mathrm{sub}}=2$ for selected center-to-center distances and (a) $z=0$ nm, (b) $z=90$ nm, and (c) $z=240$ nm. The modification of the resonances is determined by the array density, which is inversely proportional to CC, and the array-substrate separation distance. (d) Example of magnetic resonance dependence on the disk's distance from the substrate $z$ and $l_{cc}$, which results from an interplay of various coupling terms. White marks the resonance wavelength of a single disk in free space.

Figure 6

Bulk sensing ($n_d\in \langle 1.33,1.53\rangle$) characteristics of substrate-supported dielectric nanodisks. (a) Sensitivity of a nanodisk in a homogeneous environment (dashed line) is very small, but when the disk is placed on a substrate (solid with circles) with $n_{\mathrm{sub}}\in \langle 1.1,2\rangle$ it varies significantly with $n_{\mathrm{sub}}$. (b) The average sensitivity (over the whole $n_d$ range) of substrate-supported random arrays of disks (solid lines) varies with the center-to-center distance and the mean single-particle value (dashed lines) around which it oscillates depends on $n_{\mathrm{sub}}$. (c) The local-index ($\partial \lambda /\partial n_d$) sensitivity for substrate $n_{\mathrm{sub}}=1.45$ is nonlinear and there is considerable spread of its value once near-field coupling effects have decayed for the center-to-center distance above $\sim 2.5D$.

Figure 7

Pair correlation function for 2D amorphous arrays are obtained from fitting numerical data.