Topological Darkness and Goos-Hänchen Effect in Transdimensional Plasmonic Film Systems

It has been known that the reflection of a linearly polarized optical beam of finite transverse extent incident on a plane surface does not exactly follow the Snell’s law of geometrical optics [1]. Instead, the reflected beam experiences slight lateral in-plane displacement and angular deflection in the plane of incidence—the phenomenon commonly referred to as the Goos-Hänchen (GH) effect. Originating from the spatial dispersion of reflection/transmission coefficients due to the finite transverse size of the beam (and so nonlocal in its nature), the GH effect occurs for reflected/refracted light in realistic optical systems. The effect attracts much attention these days [2,3] due to the new generation of materials being available, such as transdimensional (TD) materials which can enhance nonlocal subwalength light propagation to offer new opportunities for quantum optics, quantum nanophotonics and biosensing application development. Plasmonic TD materials are ultrathin metal films of precisely controlled thickness [4]. They offer high tailorability of their optoelectronic properties not only by altering their chemical and/or electronic composition but also by merely varying their thickness (number of monolayers) [5-7]. They provide a new regime—transdimensional, in between 3D and 2D—turning into 2D as the film thickness tends to zero, whereby strong vertical quantum confinement makes the linear electromagnetic (EM) response of the TD film nonlocal and the degree of nonlocality can be controlled by the film thickness. The properties of the TD plasmonic films can be understood in terms of the confinement-induced nonlocal EM response theory built on the Keldysh-Rytova (KR) electron interaction potential [8,9]. By now the theory is verified experimentally in a variety of settings [5-7]. Here, we focus on the confinement-induced material nonlocality and use the KR theory to study the GH effect in TD plasmonic materials. We show that there are topologically protected singularities for the nonlocal reflection coefficient of the system which lead to giant lateral and angular GH shifts in the millimeter and milliradian range, respectively, to greatly exceed those of microscale reported for beams of finite transverse extent with no material-induced nonlocality [2–7]. Such singularities appear in TD materials with broken in-plane reflection symmetry where due to the confinement-induced nonlocality the eigenmode degeneracy is lifted to create the topological darkness points in the visible range [10], not existing in standard local Drude materials. With transdimensional TiN used as an example, our analysis reveals lateral and angular GH shifts ∼0.4 mm and ∼40 mrad, respectively, for the He-Ne laser beam reflected from a 40 nm thick film.

Supported by US ARO-W911NF2310206 (I.V.B.) and in part by NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP).

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