Pancharatnam-Berry phase for Voigt waves and Voigt surface waves

Geometric phase is a unifying concept that spans both classical and quantum physics. Pancharatnam--Berry (PB) phase is an optical manifestation of geometric phase which provides a measure of the dissimilarity of two electromagnetic plane waves of the same frequency, based on the evolution of their polarization states. Currently PB phase is the focus of intense research activity, especially involving anisotropic and/or non-homogeneous materials in which its effects are most striking. Applications of PB phase in wavefront tailoring, polarization-dependent lenses, holograms and waveguiding are being pursued. These developments have been fuelled by advances in engineered materials, including metamaterials and metasurfaces. Voigt waves propagate in bulk materials. Voigt surface waves propagate at the planar interfaces of materials. Both are singular forms of optical plane-wave propagation that arise when the matrix governing their propagation fails to exhibit Hermitian symmetry and has repeated eigenvalues. In contrast, non-singular propagation is characterized by Hermitian symmetry and distinct eigenvalues. A distinguishing feature of Voigt waves and Voigt surface waves is that both are unusually localized. Voigt waves decay in the direction of propagation in a manner that depends on the product of the propagation distance and its exponential (in contrast to non-singular plane waves whose decay is purely exponential). Voigt surface waves decay in the direction perpendicular to the planar interface in a manner that depends on the product of the distance from the interface and its exponential (in contrast to non-singular surface waves whose decay is purely exponential). This unusual localization may be exploited for applications in optical sensing and communications. Material anisotropy is essential for the non-Hermitian conditions to be satisfied for both Voigt wave and Voigt surface wave propagation. Crucially, Voigt wave propagation and Voigt surface wave propagation are both polarization dependent. Indeed, materials supporting Voigt wave propagation can discriminate between left- and right-circularly polarized light. To date research on PB phase has concentrated exclusively on non-singular optical propagation -- the issue of PB phase for singular optical propagation has not been considered. In particular, an open question is as yet unanswered: What PB phase is associated with Voigt waves and Voigt surface waves? The proposed research will address this matter by calculations of PB phase for plane waves transmitted through slabs of anisotropic materials. The transfer-matrix method, which is a powerful mathematical formalism that has been developed to determine the reflection and transmission characteristics of slabs of complex materials, will be applied to calculate the PB phase. Since this method can be adapted to accommodate singular optical propagation for the most general types of linear materials, it is ideally suited to the proposed research. Specifically, the PB phase will be calculated for the transmitted wave relative to the incident wave for: (i) Voigt-wave propagation within an anisotropic slab; and (ii) Voigt-surface-wave propagation at the planar interface of an anisotropic slab. The PB phase will be related to the symmetries and constitutive parameters of the supporting anisotropic materials. By choosing the supporting anisotropic materials to be engineered materials, such as homogenized composite materials, the constitutive parameters of the supporting anisotropic materials may be varied to tune the PB phase as required. Therefore, not only will our understanding of PB phase for singular optical propagation be elucidated, but a means of tailoring the PB phase to suit the requirements of particular applications will be developed.