Evolution and global charge conservation for polarization singularities emerging from non-Hermitian degeneracies

Core concepts in singular optics, especially the polarization singularities, have rapidly penetrated the surging fields of topological and non-Hermitian photonics. For open photonic structures with non-Hermitian degeneracies in particular, polarization singularities would inevitably encounter another sweeping concept of Berry phase. Several investigations have discussed, in an inexplicit way, connections between both concepts, hinting at that nonzero topological charges for far-field polarizations on a loop are inextricably linked to its nontrivial Berry phase when degeneracies are enclosed. In this work, we reexamine the seminal photonic crystal slab that supports the fundamental two-level non-Hermitian degeneracies. Regardless of the invariance of nontrivial Berry phase (concerning near-field Bloch modes defined on the momentum torus) for different loops enclosing both degeneracies, we demonstrate that the associated far polarization fields (defined on the momentum sphere) exhibit topologically inequivalent patterns that are characterized by variant topological charges, including even the trivial scenario of zero charge. Moreover, the charge carried by the Fermi arc actually is not well defined, which could be different on opposite bands. It is further revealed that for both bands, the seemingly complex evolutions of polarizations are bounded by the global charge conservation, with extra points of circular polarizations playing indispensable roles. This indicates that although not directly associated with any local charges, the invariant Berry phase is directly linked to the globally conserved charge, physical principles underlying which have all been further clarified by a two-level Hamiltonian with an extra chirality term. Our work can potentially trigger extra explorations beyond photonics connecting Berry phase and singularities.

Pioneered by Pancharatnam, Berry, Nye, and others (1–10), Berry phase and singularities have become embedded languages all across different branches of photonics. Optical Berry phase is largely manifested through either polarization evolving Pancharatnam–Berry phase or the spin-redirection Bortolotti–Rytov–Vladimirskii–Berry phase (2, 4, 5, 11–15); while optical singularities are widely observed as singularities of intensity (caustics) (6), phase (vortices) (7), or polarization (8–10). As singularities for complex vectorial waves, polarization singularities are skeletons of electromagnetic waves and are vitally important for understanding various interference effects underlying many applications (16–20).There is a superficial similarity between the aforementioned two concepts: Both the topological charge of polarization field [Hopf index of line field (21, 22)] and Berry phase are defined on a closed circuit. Despite this, it is quite unfortunate that almost no definite connections have been established between them in optics. This is fully understandable: Berry phase is defined on the Pancharatnam connection (parallel transport) that decides the phase contrast between neighboring states on the loop (3, 4); while the polarization charge reflects accumulated orientation rotations of polarization ellipses, which has no direct relevance to the overall phase of each state. This explains why in pioneering works where both concepts were present (23–27), their interplay was rarely elaborated on.Spurred by studies into bound states in the continuum, polarization singularities have gained enormous renewed interest in open periodic photonic structures, manifested in different morphologies with both fundamental and higher-order half-integer charges (28–50). Simultaneously, the significance of Berry phase has been further reinforced in surging fields of topological and non-Hermitian photonics (1, 23, 26, 51–56). In open periodic structures involving band degeneracies, Berry phase and polarization singularity would inevitably meet, which sparks the influential work on non-Hermitian degeneracy (36) and several other following studies (40, 43, 45) discussing both concepts simultaneously. Although not claimed explicitly, those works hint that nontrivial Berry phase produces nonzero polarization charges.Aiming to bridge Berry phase and polarization singularity, we reexamine the seminal photonic crystal slab (PCS) that supports elementary two-level non-Hermitian degeneracies. It is revealed that with an invariant nontrivial π Berry phase, the corresponding polarization fields on different isofrequency contours enclosing both non-Hermitian degenerate points (or equivalently exceptional points [EPs]) (26) exhibit diverse patterns characterized by different polarization charges, even including the trivial zero charge. It is further revealed that the charge carried by the Fermi arc is actually not well defined, which could be different on opposite bands. We also discover that such complexity of field evolutions is constrained by global charge conservation for both bands, with extra points of circular polarizations (C points) playing pivotal roles. This reveals the explicit connection between globally conserved charge and the invariant Berry phase, underlying which the physical mechanisms have been further clarified by a two-level Hamiltonian with an extra chirality term (25). We show that such an unexpected connection is generically manifest in various structures, despite the fact that Berry phase and polarization charge actually characterize different entities of near-field Bloch modes and their projected far polarization fields, respectively: Bloch modes are defined on the momentum torus and can be folded into the irreducible Brillouin zone; while polarization fields are defined on the momentum sphere, due to the involvement of out-of-plane wave vectors along which there is no periodicity. Our study can spur further investigations in other subjects beyond photonics to explore conceptual interconnectedness, where both the concepts of Berry phase and singularities are present.