Dispersion engineering via nonlocal transformation optics

Abstract

Transformation optics (TO) has established itself as a powerful and versatile approach to the synthesis of metamaterials with prescribed field-manipulation capabilities, via suitable spatial modulation of their constitutive properties inspired by local distortions of the spatial coordinate reference frame. From the mathematical viewpoint, this approach can be reformulated in the frequency-wavenumber reciprocal phase space so as to engineer nonlocal interactions and spatial dispersion effects, which are becoming increasingly relevant in electrodynamics and optics. Here, we present a general nonlocal-TO framework, based on complex-valued, frequency-dependent wavenumber coordinate transformations, and explore its possible applications to scenarios of interest for dispersion engineering. A key attribute of our approach, similar to conventional TO, is the separation of the conceptual design (based on intuitive geometrical considerations) from the actual metamaterial synthesis (based on a suitable approximation of analytically derived constitutive "blueprints"). To illustrate the capabilities and potential of the proposed approach, we address the engineering (from the conceptual design to the actual synthesis) of multilayered metamaterials exhibiting various exotic dispersion effects, including "one-way" (nonreciprocal) propagation, "frozen-mode" regime, and Dirac-point conical singularities. Our approach may open up new perspectives in the systematic design of metamaterials with broad field-manipulation capabilities as well as complex spatiotemporal dispersion effects, with potential applications to nonreciprocal optics, topological photonics, and "computational metamaterials."

Degenerate-band-edge engineering inspired by nonlocal transformation optics

“…For instance, we showed that (non)reciprocal effects and the appearance of additional extraordinary waves [6] are directly related to the (non-)centersymmetry and multivaluedness, respectively, of the transformations [13]. Moreover, we derived the analytical properties of classes of transformations that could induce stationary points in the dispersion diagram, and we also showed that frequency-dependent wavevector transformations enable a finer tailoring in the phase space, thereby opening up the possibility to engineer complex spatio-temporal dispersion effects such as Dirac-point conical singularities [14].…”

“…In reference [14], within the framework of nonlocal TO, we derived a class of wavevector transformations that could induce stationary points (of arbitrary order) in the dispersion relationship. In particular, we explored in detail the case m = 3 in equations (1) and (2), corresponding to the ''frozenmode'' regime.…”

“…A similar approach could be applied to deal with the DBE (m = 4) case of interest here. For brevity, we do not repeat here the analytical developments, which can be found in reference [14]. In fact, as discussed in reference [14], while providing a systematic, constructive derivation of the required material blueprints, our approach is not the only possible one, as the problem does not admit a unique solution.…”

“…For brevity, we do not repeat here the analytical developments, which can be found in reference [14]. In fact, as discussed in reference [14], while providing a systematic, constructive derivation of the required material blueprints, our approach is not the only possible one, as the problem does not admit a unique solution. Moreover, for the particular 2-D, coordinate-separable and non-magnetic structure of the arising constitutive blueprints, their interpretation is rather straightforward.…”

“…In a series of recent investigations [13,14], we have been concerned with possible extensions of the well-established transformation-optics (TO) framework [15,16] so as to systematically engineer metamaterials exhibiting desired nonlocal effects. As opposed to the conventional TO formulation, which exploits the form-invariant properties of Helmholtz or Maxwell's equations with respect to spatial (and, hence, inherently local) coordinate transformations, the basic idea underlying our approach is to apply the coordinate-transformation machinery in the wavevector-frequency phase-space, accessed via spatial Fourier transform.…”