Singularities and Poincar\'{e} Indexes of Electromagnetic Multipoles

Electromagnetic multipoles have been broadly adopted as a fundamental language throughout photonics, of which general features such as radiation patterns and polarization distributions are generically known, while their singularities and topological properties have mostly slipped into oblivion. Here we map all the singularities of multipolar radiations of different orders, identify their indexes, and show explicitly the index sum over the entire momentum sphere is always $2$, consistent with the Poincar\'{e}-Hopf theorem. Upon those revealed properties, we can attribute the formation of bound states in the continuum to overlapping of multipolar singularities with open radiation channels. This insight unveils a subtle connection between indexes of multipolar singularities and topological charges of those bound states, revealing that essentially they are the same. Our work has fused two fundamental and sweeping concepts of multipoles and topologies, which can potentially bring unforeseen opportunities for many multipole related fields within and beyond photonics

Multipolar Conversion Induced Subwavelength High-Q Supermodes with Unidirectional Radiations

The two-mode coupling model with energy splitting and formation of supermodes with different life times has been pervasive in almost every discipline of physics. We revisit this fundamental model from a different perspective of multipolar expansions, and manage to reveal a hidden dimension of it, by establishing a subtle connection between two seemingly unrelated properties of Q-factors and far-field angular radiation patterns. We discover that, in both regimes of negative and positive couplings, significant Q-factor enhancement can be attributed to dramatic redistributions of radiations that originate from multipolar conversions from lower to higher orders. Relying on this connection and generalized Kerker effects of interferences among different multipoles, we synchronize both outstanding features of high-Q factor and unidirectional radiation into one subwavelength supermode. The implications of our study are not confined to optics and photonics, and can potentially shed new light on coupling between resonances of mechanical, phononic, electronic or other hybrid natures

Geometric Phase-Driven Scattering Evolutions

Conventional approaches for scattering manipulations rely on the technique of field expansions into spherical harmonics (electromagnetic multipoles), which nevertheless is non-generic (expansion coefficients depend on the position of the coordinate system's origin) and more descriptive than predictive. Here we explore this classical topic from a different perspective of controlled excitations and interferences of quasi-normal modes (QNMs) supported by the scattering system. Scattered waves are expanded into not spherical harmonics but radiations of QNMs, among which the relative amplitudes and phases are crucial factors to architect for scattering manipulations. Relying on the electromagnetic reciprocity, we provide full geometric representations based on the Poincar\'e sphere for those factors, and identify the hidden underlying geometric phases of QNMs that drive the scattering evolutions. Further synchronous exploitations of the incident polarization-dependent geometric phases and excitation amplitudes enable efficient manipulations of both scattering intensities and polarizations. Continuous geometric phase spanning $2\pi$ is directly manifest through scattering variations, even in the rather elementary configuration of an individual particle scattering waves of varying polarizations. We have essentially established a profoundly all-encompassing framework for the calculations of geometric phase in scattering systems, which will greatly broaden horizons of many disciplines not only in photonics but also in general wave physics where geometric phase is generic and ubiquitous

Multi-Mode Optical Chirality Extremizations on Incident Momentum Sphere

We study the momentum-space evolutions for chiral optical responses of multi-mode resonators scattering plane waves of varying incident directions. It was revealed, in our previous study [Phys. Rev. Lett. $\mathbf{126}$, 253901 (2021)], that for single-mode resonators the scattering optical chiralities characterized by circular dichroism ($\mathbf{CD}$) are solely decided by the third Stokes parameter distributions of the quasi-normal mode (QNM) radiations: $\mathbf{CD}=\mathbf{S}_3$. Here we extend the investigations to multi-mode resonators, and explore numerically the dependence of optical chiralities on incident directions from the perspectives of QNM radiations and their circular polarization singularities. In contrast to the single-mode regime, for multi-mode resonators it is discovered that $\mathbf{CD}$s defined in terms of extinction, scattering and absorption generally are different and cannot reach the ideal values of $\pm 1$ throughout the momentum sphere. Though the exact correspondence between $\mathbf{CD}$ and $\mathbf{S}_3$ does not hold anymore in the multi-mode regime, we demonstrate that the positions of the polarization singularities still serve as an efficient guide for identifying those incident directions where the optical chiralities can be extremized

Arbitrary Polarization-Independent Backscattering or Reflection by Rotationally-Symmetric Reciprocal Structures

We study the backward scatterings of plane waves by reciprocal scatterers and reveal that $n$-fold ($n\geq3$) rotation symmetry is sufficient to secure invariant backscattering for arbitrarily-polarized incident plane waves. It is further demonstrated that the same principle is also applicable for infinite periodic structures in terms of reflection, which simultaneously guarantees the transmission invariance if there are neither Ohmic losses nor extra diffraction channels. At the presence of losses, extra reflection symmetries (with reflection planes either parallel or perpendicular to the incident direction) can be incorporated to ensure simultaneously the invariance of transmission and absorption. The principles we have revealed are protected by fundamental laws of reciprocity and parity conservation, which are fully independent of the optical or geometric parameters of the photonic structures. The optical invariance obtained is intrinsically robust against perturbations that preserve reciprocity and the geometric symmetries, which could be widely employed for photonic applications that require stable backscatterings or reflections

Scattering invariance for arbitrary polarizations protected by joint spatial-duality symmetries

We reveal how to exploit joint spatial-electromagnetic duality symmetries to obtain invariant scattering properties (including extinction, scattering, absorption) of self-dual scattering systems for incident waves of arbitrary polarizations. The electromagnetic duality ensures the helicity preservation along all scattering directions, and thus intrinsically eliminates the interferences between the two scattering channels originating from the circularly polarized components of incident waves. This absence of interference directly secures invariant scattering properties for all polarizations located on the same latitude circle of the Poincar\'{e} sphere, which are characterized by polarization ellipses of the same eccentricity and handedness. Further incorporations of mirror and/or inversion symmetries would lead to such invariance throughout the whole Poincar\'{e} sphere, guaranteeing invariant scattering properties for all polarizations. Simultaneous exploitations of composite symmetries of different natures render an extra dimension of freedom for scattering manipulations, offering new insights for both fundamental explorations and optical device engineering related to symmetry dictated light-matter interactions

Scattering and absorption invariance of nonmagnetic particles under duality transformations

We revisit the total scatterings (in terms of extinction, scattering and absorption cross sections) by arbitrary clusters of nonmagnetic particles that support optically-induced magnetic responses. Our reexamination is conducted from the perspective of the electromagnetic duality symmetry, and it is revealed that all total scattering properties are invariant under duality transformations. This secures that for self-dual particle clusters, the total scattering properties are polarization independent for any fixed incident direction; while for non-self-dual particle clusters, two scattering configurations that are connected to each other through a duality transformation would exhibit identical scattering properties. This electromagnetic duality induced invariance is irrelevant to specific particle distributions or wave incident directions, which is illustrated for both random and periodic clusters

Ideal Kerker scattering by homogeneous spheres: the role of gain or loss

We reexamine a recent work [Phys. Rev. Lett. \textbf{125}, 073205 (2020)] that investigates how the optical gain or loss (characterized by isotropic complex refractive indexes) influences the ideal Kerker scattering of exactly zero backward scattering. There it has been rigourously proved that, for non-magnetic homogeneous spheres with incident plane waves, either gain or loss prohibits such ideal Kerker scattering, provided that only electric and magnetic multipoles of a specific order are present and contributions from other multipoles can all be made precisely zero. Here we reveal that, when two multipoles of a fixed order are perfectly matched in terms of both phase and magnitude, multipoles of at least the next two orders cannot possibly be tuned to be all precisely zero or even perfectly matched, and consequently cannot directly produce ideal Kerker scattering. Moreover, we further demonstrate that, when multipoles of different orders are simultaneously taken into consideration, the loss or gain can serve as a helpful rather than harmful contributing factor, for the eliminations of backward scattering

Generalized coupled mode formalism in reciprocal waveguides with gain/loss, anisotropy or bianisotropy

In anisotropic or bianisotropic waveguides, the standard coupled mode theory fails due to the broken link between the forward and backward propagating modes, which together form the dual mode sets that are crucial in constructing couple mode equations. We generalize the coupled mode theory by treating the forward and backward propagating modes on the same footing via a generalized eigenvalue problem that is exactly equivalent to the waveguide Hamiltonian. The generalized eigenvalue problem is fully characterized by two operators, i.e., $( \bar{\bm{L}},\bar{\bm{B}})$, wherein $\bar{\bm{L}}$ is a self-adjoint differential operator, while $\bar{\bm{B}}$ is a constant antisymmetric operator. From the properties of $\bar{\bm{L}}$ and $\bar{\bm{B}}$, we establish the relation between the dual mode sets that are essential in constructing coupled mode equations in terms of forward and backward propagating modes. By perturbation, the generalized coupled mode equation can be derived in a natural way. Our generalized coupled mode formalism can be used to study the mode coupling in waveguides that may contain gain/loss, anisotropy or bianisotropy. We further illustrate how the generalized coupled theory can be used to study the modal coupling in anisotropy and bianisotropy waveguides through a few concrete examples

Extremize Optical Chiralities through Polarization Singularities

Chiral optical effects are generally quantified along some specific incident directions of exciting waves (especially for extrinsic chiralities of achiral structures) or defined as direction-independent properties by averaging the responses among all structure orientations. Though of great significance for various applications, chirality extremization (maximized or minimized) with respect to incident directions or structure orientations have not been explored, especially in a systematic manner. In this study we examine the chiral responses of open photonic structures from perspectives of quasi-normal modes and polarization singularities of their far-field radiations. The nontrivial topology of the momentum sphere secures the existence of singularity directions along which mode radiations are either circularly or linearly polarized. When plane waves are incident along those directions, the reciprocity ensures ideal maximization and minimization of optical chiralities, for corresponding mode radiations of circular and linear polarizations respectively. For directions of general elliptical polarizations, we have unveiled the subtle equality of a Stokes parameter and the circular dichroism, showing that an intrinsically chiral structure can unexpectedly exhibit no chirality at all or even chiralities of opposite handedness for different incident directions. The framework we establish can be applied to not only finite scattering bodies but also infinite periodic structures, encompassing both intrinsic and extrinsic optical chiralities. We have effectively merged two vibrant disciplines of chiral and singular optics, which can potentially trigger more optical chirality-singularity related interdisciplinary studies