Topological Engineering of High-Order Exceptional Points through Transformation Optics

TL;DR

The paper tackles engineering high-order exceptional points (EPs) in optical nanostructures where traditional Hamiltonian design struggles to map to experimental degrees of freedom. It introduces a transformation optics (TO) framework that links spectral singularities to physically tunable parameters, enabling non-PT-symmetric EP design and full-field mode solutions. Key contributions include a cubic resonance equation $A \epsilon_m^3 + B \epsilon_m^2 + C \epsilon_m + D = 0$ and discriminant conditions $\Delta = \Delta_1 = 0$ to realize EP$^3$, together with a conformal mapping $w' = \frac{g}{\exp(w)-1}$ that maps the solution to a nanowire geometry. The authors demonstrate EP3 in a coupled core-shell/monomer nanowire, EP2 arcs, and EP4 in a symmetric dimer, with finite-element validation showing good agreement and explicit mode distributions at degeneracy, highlighting practical routes to robust, topology-enabled photonic devices and enhanced light-matter interactions.

Abstract

Exceptional points (EPs) in non-Hermitian photonic systems have attracted considerable research interest due to their singular eigenvalue topology and associated anomalous physical phenomena. These properties enable diverse applications ranging from enhanced quantum metrology to chiral light-matter interactions. Practical implementation of high order EPs in optical platforms however remains fundamentally challenging, requiring precise multi-parameter control that often exceeds conventional design capabilities. This work presents a novel framework for engineering high order EPs through transformation optics (TO) principles, establishing a direct correspondence between mathematical singularities and physically controllable parameters. Our TO-based paradigm addresses critical limitations in conventional Hamiltonian approaches, where abstract parameter spaces lack explicit connections to experimentally accessible degrees of freedom, while simultaneously providing full-field mode solutions. In contrast to prevailing parity-time-symmetric architectures, our methodology eliminates symmetry constraints in EP design, significantly expanding the possibilities in non-Hermitian photonic engineering. The proposed technique enables unprecedented control over EP formation and evolution in nanophotonic systems, offering new pathways for developing topological optical devices with enhanced functionality and robustness.

Figures (5)

• Figure 1: TO approach to designing exceptional points. (a) Schematic diagram of planar geometry considered before (left) and plasmonic nanowire geometry consider after (right) transformation. (b) Exceptional arcs (EA) of exceptional points (EPs), defined by $\Delta = 0$ (see main text), traced in physical parameter space upon variation of the electric permittivity of gain material and interface separation $d_2$. The intersection of the two EAs corresponds to an exceptional nexus (EX), a point at which a higher order EP exists. (c) As (b) albeit for variations in host material.
• Figure 2: EP3 in core-shell/monomer coupled nanowire system. (a) Color density plot of the logarithm of the scattering cross-section of the coupled core-shell and monomer structures as a function of gain and frequency. Solid lines represent the real parts of the three eigenfrequencies (corresponding imaginary parts are shown in the inset). (b) scattering cross-sections corresponding to I-V in (a) as found using our TO approach (red lines) and finite element simulations (blue markers). (c) Potential distributions for eigenmodes at points labelled 1-9 in (a). The $+$ and $-$ markers denote a positive and negative potential respectively. Cyan lines demark the nanowire interfaces (only a small part of core-shell nanowire in vicinity of monomer is shown to enhance visibility). Panels 4-6 correspond to the EP3.
• Figure 3: EP2 in core-shell/monomer coupled nanowire system (a) As Figure \ref{['fig:EP3']}(a) albeit for system parameters lying on the EA away from the EX (see Figure \ref{['fig:schematic']}) (b) Potential distributions for 1-6 in (a), in which $+$ and $-$ denote a positive and negative potential respectively. Panels 5 and 6 correspond to the crossing of the real part of the eigenfrequecies, whereas the identical distributions in 1 and 2 signify an EP2.
• Figure 4: EP3s with different angular momenta. Potential distributions for modes with angular momenta (left) $n=2$, (middle) $n=3$ and (right) $n=4$, evaluated at an EP3 in a system analagous to that shown in Figure \ref{['fig:EP3']}.
• Figure 5: EP4 in core-shell dimer nanowire system. (a) Color density plot of the logarithm of the scattering cross-section of the coupled core-shell dimer structure as a function of $\epsilon$ and frequency. Solid and dashed lines represent the real parts of the four eigenfrequencies (corresponding imaginary parts are shown in the inset). (b) Scattering cross-sections corresponding to I-V in (a) as found using our TO approach (red lines) and finite element simulations (blue markers). (c) Potential distributions for eigenmodes at points labelled 1-9 in (a). The $+$ and $-$ markers denote a positive and negative potential respectively. Cyan lines demark the nanowire interfaces (only a small part of core-shell nanowires are shown to enhance visibility). Panel 5 corresponds to the EP3.