[Paper Review] Magnetic and electric Purcell factor control through geometry optimization of high index dielectric nanostructures

Yoann Brûlé, Peter R. Wiecha+3 more|arXiv (Cornell University)|Jan 5, 2022

Plasmonic and Surface Plasmon Research62 references30 citations

TL;DR

This paper presents a geometry-optimized design of high-index silicon dielectric nanoantennas using differential evolution and Green's tensor methods to enhance or suppress the magnetic and electric Purcell factors for europium ion emitters. It achieves up to a 2,000-fold enhancement in the magnetic Purcell factor through engineered Mie resonances and circular grating structures, with physical insights from modal and deterministic analysis.

ABSTRACT

We design planar silicon antennas for controlling the emission rate of magnetic or electric dipolar emitters. Evolutionary algorithms coupled to the Green Dyadic Method lead to different optimized geometries which depend on the nature and orientation of the dipoles. We discuss the physical origin of the obtained configurations thanks to modal analysis but also emphasize the role of nanoscale design of the LDOS. We complete our study using finite element method and demonstrate an enhancement up to 2000 of the magnetic Purcell factor in europium ions. Our work brings together random optimizations to explore geometric parameters without constraint, a first order deterministic approach to understand the optimized designs and a modal analysis which clarifies the physical origin of the exaltation of the magnetic Purcell effect.

Motivation & Objective

• To design planar silicon nanoantennas that control the emission rate of magnetic and electric dipole emitters using geometry optimization.
• To overcome the inherently weak magnetic dipole interaction in optics by enhancing the magnetic local density of states (LDOS).
• To explore the physical origin of Purcell enhancement through modal and deterministic analysis beyond black-box optimization.
• To validate optimized geometries using finite element method simulations for realistic cylindrical shapes.
• To demonstrate both enhancement and inhibition of Purcell factors for magnetic and electric dipoles through tailored nanostructure design.

Proposed method

• Employed differential evolution (DE) algorithm to perform global, constraint-free optimization of silicon nanopillar geometries in a 1.68×1.68 µm² area.
• Used the Green Dyadic Method (GDM) with pyGDM to compute decay rates and Purcell factors in multi-material nanostructures with 20 nm mesh resolution.
• Applied modal analysis and deterministic design principles from Mignuzzi *et al.* to interpret the physical origin of optimized structures.
• Performed finite element method (FEM) simulations to validate results on rounded, cylindrical geometries suggested by DE optimization.
• Defined fitness functions to maximize or minimize the magnetic (Γ⊥_m/Γ₀, Γ∥_m/Γ₀) and electric (Γ⊥_e/Γ₀, Γ∥_e/Γ₀) Purcell factors at λ = 590 nm and 610 nm.
• Derived analytical expressions for local magnetic and electric LDOS contributions using reciprocity theorems and mixed Green's tensors (GHE, GHH).

Experimental results

Research questions

• RQ1 Can evolutionary algorithms effectively optimize high-index dielectric nanostructures to enhance the magnetic Purcell factor for rare-earth ions?
• RQ2 What are the physical mechanisms underlying the optimized geometries that maximize magnetic dipole emission?
• RQ3 How do the optimized structures differ for in-plane versus out-of-plane magnetic dipoles, and what role does symmetry play?
• RQ4 To what extent can finite element method simulations validate the results from the Green Dyadic Method for realistic, rounded nanostructures?
• RQ5 Can both enhancement and suppression of Purcell factors be achieved through geometric design for both electric and magnetic dipole transitions?

Key findings

• The differential evolution algorithm successfully converged on a consistent circular grating structure around a Si core for both in-plane and out-of-plane magnetic dipole configurations.
• A maximum magnetic Purcell factor of up to 2,000 was achieved for out-of-plane magnetic dipole transitions in Eu3+ ions at 590 nm using optimized Si nanoantennas.
• For in-plane magnetic dipoles, the optimized Purcell factor reached approximately 120, with structures showing two symmetric lobes perpendicular to the dipole orientation.
• Electric Purcell factor enhancements were limited to Γ⊥_e/Γ₀ ≈ 2 for out-of-plane dipoles, achieved with a circular grating but without a central core.
• Modal analysis confirmed that the enhancement arises from excitation of Mie resonances, particularly magnetic dipole modes, in the optimized dielectric structures.
• The finite element method simulations validated the GDM results and confirmed the robustness of the optimized geometries for realistic cylindrical nanopillars.