Spatial dispersion, i.e., the nonlocal character of the electromagnetic constitutive relationships, is typically regarded as a negligible effect for most natural media. However, there is currently a growing interest in its study, in view of its critical relevance in the homogenized (effective-medium) modeling of many artificial electromagnetic materials of practical interest, as well as in a variety of related effects including artificial magnetism, wave splitting into multiple beams, beam tailoring, and ultrafast nonlinear optical response. If, for most metamaterials, spatial dispersion is seen as a nuisance, counterproductive for practical applications, its proper tailoring and engineering may add novel degrees of freedom in the wave interaction with complex materials.
In [1], in collaboration with Nader Engheta (University of Pennsylvania) and Andrea Alù (University of Texas at Austin), we have extended the transformation-optics approach by enabling the manipulation and control of nonlocal light-matter interactions. Instead of spatially-changing refractive properties, we proposed a framework of “transformation media” that can tailor the wave interaction in the reciprocal space of spatial wave-numbers, so as to produce desired nonlocal effects.
Subsequently, in [2] and [3], we further generalized this approach in the frequency-wavenumber reciprocal phase space, so as to engineer a broad variety of nonlocal interactions and spatial dispersion effects, including including “one-way” (nonreciprocal) propagation, “frozen-mode” and “degenerate-band-edge” regimes, and Dirac-point conical singularities
For instance, the figure top panel illustrates a desired dispersion effect, namely, a degenerate-band-edge condition at a prescribed frequency. Such exotic dispersion effect (manifested by the appearance of a higher-order stationary point in the dispersion relationship) is eliciting a growing attention in view of its potential relevance to diverse applications including slow light, solid-state lasers, quantum-cascade lasers, sensors, optical delay lines, traveling-wave tubes, distributed solid-state amplifiers, and switches.
The figure bottom panel shows the corresponding field distribution in a unit cell of a multilayered metamaterial synthesized with our approach. As it can be observed, the field is transmitted (with small reflection) in the metamaterial half-space, where it gets converted into an extended mode with growing amplitude. At a distance of about five wavelengths from the vacuum-metamaterial interface, the amplitude reaches a maximum value that is over a factor 1000 larger than the incident one, and then it starts decreasing.
A key attribute of our approach, similar to conventional transformation optics, 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”.
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.”
More recently, in a study [4] in collaboration with Carlo Rizza (University of l’Aquila) and Alessandro Ciattoni (CNR-SPIN), we have explored a novel class of structures, based moderate-permittivity inclusions, for which nonlocal effects are enhanced. Among other things, these results may significantly boost the practical applicability of nonlocal transformation optics.
Moreover, in a recent study [5] in collaboration with Andrea Alù, we have explored some peculiar boundary effects that can enhance the (otherwise neglibibly weak) nonlocality in multilayered dielectric metamaterials wity deeply subwavelength layers.
In a different, but related, study [6] we have investigated the emergence of aperiodic-order-induced nonlocal effects in multilayered metamaterials.
We show that the powerful framework of transformation optics may be exploited for engineering the nonlocal response of artificial electromagnetic materials. Relying on the form-invariant properties of coordinate-transformed Maxwell’s equations in the spectral domain, we derive the general constitutive “blueprints” of transformation media yielding prescribed nonlocal field-manipulation effects and provide a physically incisive and powerful geometrical interpretation in terms of deformation of the equifrequency contours. In order to illustrate the potentials of our approach, we present an example of application to a wave-splitting refraction scenario, which may be implemented via a simple class of artificial materials. Our results provide a systematic and versatile framework which may open intriguing venues in dispersion engineering of artificial materials.
@article{IJ98_PRL_108_063902_2012, title = {Nonlocal transformation optics}, author = {Castaldi, Giuseppe and Galdi, Vincenzo and Al\`u, Andrea and Engheta, Nader}, journal = {Physical Review Letters}, volume = {108}, issue = {6}, pages = {063902}, numpages = {5}, year = {2012}, month = feb, publisher = {American Physical Society}, doi = {10.1103/PhysRevLett.108.063902}, url = {http://link.aps.org/doi/10.1103/PhysRevLett.108.063902}, note = {/assets/papers/IJ98_PRL_108_063902_2012_SM.pdf} }
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.”
@article{IJ117_Optica_3_179_2016, author = {Moccia, Massimo and Castaldi, Giuseppe and Galdi, Vincenzo and Al\`{u}, Andrea and Engheta, Nader}, journal = {Optica}, keywords = {Dispersion; Effective medium theory ; Metamaterials}, number = {2}, pages = {179--188}, publisher = {OSA}, title = {Dispersion engineering via nonlocal transformation optics}, volume = {3}, month = feb, year = {2016}, url = {http://www.osapublishing.org/optica/abstract.cfm?URI=optica-3-2-179}, doi = {10.1364/OPTICA.3.000179}, note = {/assets/papers/IJ117_Optica_3_179_2016_SM.pdf} }
We address the engineering of degenerate-band-edge effects in nonlocal metamaterials. Our approach, inspired by nonlocal-transformation-optics concepts, is based on the approximation of analytically-derived nonlocal constitutive “blueprints”. We illustrate the synthesis procedure, and present and validate a possible implementation based on multilayered metamaterials featuring anisotropic constituents. We also elucidate the physical mechanisms underlying our approach and proposed configuration, and highlight the substantial differences with respect to other examples available in the topical literature.
@article{IJ120_EPJAM_3_2_2016, author = {Moccia, Massimo and Castaldi, Giuseppe and Galdi, Vincenzo}, title = {Degenerate-band-edge engineering inspired by nonlocal transformation optics}, doi = {10.1051/epjam/2016003}, journal = {EPJ Applied Metamaterials}, year = {2016}, volume = {3}, pages = {2}, month = jul }
We investigate a class of multilayered metamaterials characterized by moderate-permittivity inclusions and low average permittivity. Via first-principles calculations, we show that in such a scenario, first- and second-order spatial dispersions may exhibit a dramatic and nonresonant enhancement, and may become comparable with the local response. Their interplay gives access to a wealth of dispersion regimes encompassing additional extraordinary waves and topological phase transitions. In particular, we identify a configuration featuring bound and disconnected isofrequency contours. Since they do not rely on high-permittivity inclusions, our proposed metamaterials may constitute an attractive and technologically viable platform for engineering nonlocal effects in the optical range.
@article{IJ128_PRB_96_081113_2017, title = {Enhancement and interplay of first- and second-order spatial dispersion in metamaterials with moderate-permittivity inclusions}, author = {Rizza, Carlo and Galdi, Vincenzo and Ciattoni, Alessandro}, journal = {Physical Review B}, volume = {96}, issue = {8}, pages = {081113}, numpages = {5}, year = {2017}, month = aug, publisher = {American Physical Society}, doi = {10.1103/PhysRevB.96.081113}, url = {https://link.aps.org/doi/10.1103/PhysRevB.96.081113} }
Nonlocal (spatial-dispersion) effects in multilayered metamaterials composed of periodic stacks of alternating, deeply subwavelength dielectric layers are known to be negligibly weak. Counterintuitively, under certain critical conditions, weak nonlocality may build up strong boundary effects that are not captured by conventional (local) effective-medium models based on simple mixing formulas. Here we show that this phenomenon can be fruitfully studied and understood in terms of error propagation in the iterated maps of the trace and antitrace of the optical transfer matrix of the multilayer. Our approach effectively parameterizes these peculiar effects via remarkably simple and insightful closed-form expressions, which enable direct identification of the critical parameters and regimes. We also show how these boundary effects can be captured by suitable nonlocal corrections.
@article{IJ132_PRAppl_10_034060_2018, title = {Boundary effects of weak nonlocality in multilayered dielectric metamaterials}, author = {Castaldi, Giuseppe and Al\`u, Andrea and Galdi, Vincenzo}, journal = {Physical Review Applied}, volume = {10}, issue = {3}, pages = {034060}, numpages = {13}, year = {2018}, month = sep, publisher = {American Physical Society}, doi = {10.1103/PhysRevApplied.10.034060}, url = {https://link.aps.org/doi/10.1103/PhysRevApplied.10.034060} }
We show that hyperbolic electromagnetic metamaterials, implemented as multilayers based on two material constituents arranged according to the Thue-Morse aperiodic sequence, may exhibit strong nonlocal effects, manifested as the appearance of additional extraordinary waves which are not predicted by standard effective-medium-theory (local) models. From the mathematical viewpoint, these effects can be associated with stationary points of the transfer-matrix trace and can be effectively parametrized via the trace-map formalism. We show that their onset is accompanied by a strong wave-vector dependence in the effective constitutive parameters. Despite the inherent periodicity enforced by the unavoidable (Bloch-type) supercell terminations, we show that such strong nonlocality is retained at any arbitrarily high-order iteration, i.e., approaching the actual aperiodic regime. Moreover, for certain parameter configurations, at a given wavelength and for two given material layers, these effects may be significantly less prominent when the same layers are arranged in a standard periodic fashion. Our findings indicate that the (aperiodic) positional order of the layers constitutes an effective and technologically inexpensive additional degree of freedom in the engineering of optical nonlocality.
@article{IJ105_PRB_87_235116_2013, title = {Optical nonlocality in multilayered hyperbolic metamaterials based on Thue-Morse superlattices}, author = {Savoia, Silvio and Castaldi, Giuseppe and Galdi, Vincenzo}, journal = {Physical Review B}, volume = {87}, issue = {23}, pages = {235116}, numpages = {10}, year = {2013}, month = jun, publisher = {American Physical Society}, doi = {10.1103/PhysRevB.87.235116}, url = {http://link.aps.org/doi/10.1103/PhysRevB.87.235116} }