Universal framework for anisotropic particles with resonance laws and splitting

TL;DR

The paper develops a universal full-wave framework for anisotropic nanoparticles, deriving closed-form eigenmodes and resonance conditions for uniaxial and biaxial spheres and extending to ellipsoids. It reveals axial-eigenpermittivity sum rules that cause resonance splitting and, in uniaxial cases, degeneracy that enables superposition of modes, while also providing analytic Q factors that connect anisotropy to mode localization and energy loss. The theory is validated against extensive full-wave simulations for hBN and α-MoO3 particles, aligning with experimental observations and enabling tunable multispectral responses and directional emission. By unifying anisotropic nanostructure behavior across optics, thermal transport, and magnetism, this framework enables new photonic devices and sensing modalities with environment- and geometry-tunable spectral features. These insights pave the way for multispectral biomarkers, directional emitters, and novel applications in nanophotonics and related fields.

Abstract

Nanophotonics enables unprecedented control over light-matter interactions, yet conventional isotropic materials limit the spectral range and mode response in subwavelength structures. Anisotropic nanoparticles -- ubiquitous in natural and engineered systems -- offer new degrees of freedom that couple geometry and material properties, unlocking previously inaccessible spectral regions. Here, we establish a universal full-wave framework describing the eigenmodes and resonance conditions of uniaxial and biaxial nanoparticles. Closed-form solutions reveal axial-permittivity sum rules and material-anisotropy-induced symmetry breaking, manifesting as resonance splitting and novel radiation patterns. Generalizing the theory to ellipsoids provides geometric tunability of the multispectral response, while analytic predictions of quality factors elucidate how anisotropy governs mode localization and energy loss. Full-wave simulations of h-BN and $α$-MoO3 nanoparticles, together with our recently reported experimental observations, confirm the theory. This framework unifies the understanding of anisotropic nanostructures across optics, thermal transport, and magnetism, enabling a new generation of photonic devices with tunable multispectral response and directional emission.

Figures (8)

• Figure 1: Scope of applicability of the proposed framework
• Figure 2: Angular distributions of the quasi-electrostatic potential eigenstates for isotropic, uniaxial, and biaxial spheres. The modes of the isotropic sphere are directed along the field/dipole excitation direction bergman1978dielectricfarhi2017eigenstate whereas the modes of the anisotropic spheres are aligned with the crystal axes and multispectral (colors correspond to frequencies). The high-order modes of anisotropic particles exhibit resonance splitting, and degeneracy for uniaxial spheres, leading to superimposed modes.
• Figure 3: Spectrum and eigenmodes of a subwavelength uniaxial sphere. (a) A uniaxial sphere and the permittivity of bulk hBN. (b) The scattering spectrum and $|\mathbf{E}|^2$ of the dipole modes on an hBN sphere surface. Here we see scattering peaks at approximately $820\, \mathrm{(1/cm)}$ (red line) and $1540 \,(\mathrm{1/cm})$ (blue line) associated with dipole modes in $z$ and $x,$ respectively. The crystal axes [100],[010],[001] are directed along the $x,y,z$ axes, respectively. (c) The scattering spectrum and $|\mathbf{E}|^2$ of the second-order modes, where the $\psi=xz$ and $\psi=yz$ modes and the uniaxial and $\psi=xy$ modes are both doubly degenerate in frequency and there is a resonance splitting compared to the bulk material. (d) Scattering spectrum and $|\mathbf{E}|^2$ of the third-order mode, where $\psi=xyz$ is excited at two $\omega$s. Moreover, there are geometric-anisotropic $\omega$ shifts compared to the bulk for all modes. Interestingly, high-order resonances have lower $\omega\mathrm{s}$ compared to the dipole modes, in contrast to isotropic spheres. (e) $|\mathbf{E}|$ of the radiation patterns of four resonances in the $xy$ plane for a dipole source with $\mathbf{r}_0\propto\mathbf{p}\propto(1,1,1)$ resulting in the excitation of the modes $\psi=x^2-y^2,\,\,\psi=xz,\,\,\psi=x^{3}-3y^{2}x,\,\,\,\psi=(x^2-y^2)z.$
• Figure 4: Spectrum and eigenmodes of a subwavelength biaxial particle. (a) A biaxial sphere with the bulk permittivities of $\alpha-\mathrm{MoO_3}$. The scattering spectra and $|\mathbf{E}|^2$ of the eigenmodes on a $\alpha-\mathrm{MoO_3}$ sphere surface for the dipole (b), second-order (c), and third-order (d) modes. Note that there are geometric-anisotropic frequency shifts for all modes and resonance splitting for the second-order modes compared to the bulk. (e) $|\mathbf{E}|$ in the radiation patterns of two representative modes of $\psi=xz,\,\,\psi=xyz$ in the $xy$ plane.
• Figure 5: Scattering spectrum, peak widths, and fields near resonance for a uniaxial hBN sphere of a radius $a=11.5\,\mathrm{nm}$ excited by an oscillating dipole located at $\mathbf{r}_0=(1,0,1)\cdot22\,\mathrm{nm}$ with a dipole moment of $\mathbf{P}=(1,0,1)\,\mathrm{A\cdot m}.$. Top: $\int |\mathbf{E}|^2 da$ over the particle envelope calculated in COMSOL and compared to the peaks predicted analytically using $1/|\mathrm{resonance\,condition}|$ with excellent agreement. The circles are the peak widths obtained analytically from $1/|\mathrm{resonance\,condition}(\omega)|^2=\mathrm{max}/2$ compared to the ones calculated from the simulations, with very good agreement. (a),(b),(c),(d),(e) $|\mathbf{E}|$ for the resonance conditions: $\epsilon_{1x}=-2,$$\epsilon_{1x}=-1.5,$$\epsilon_{1x}+\epsilon_{1z}=-3,$$\epsilon_{1x}=-4/3,$$2\epsilon_{1x}+\epsilon_{1x}=-4$ calculated at $f=46.301,\,$$46.619,$$\,44.53,\,$$\,46.725,$$45.401\mathrm{(THz)},$ respectively, using COMSOL compared to the mode fields calculated analytically with very good agreement. Bottom: $\int |\mathbf{E}|^2 da$ over the particle envelope calculated in COMSOL for a dipole located at $\mathbf{r}_0=(1,0,0)31\mathrm{nm}$ (same dipole distance as the previous case) with a dipole moment of $\mathbf{P}=(1,0,0)\,\mathrm{A\cdot m}.$ As predicted the $m=\pm(l-1)$ modes disappeared from the spectrum.