Hyperbolic metamaterials are characterized by uniaxally
anisotropic constitutive relationships with both positive and negative
components of the permittivity (or permeability) tensor.
This yields a *hyperbolic* (as opposed to spherical in conventional isotropic media)
dispersion relationship, which allows for the propagation of (otherwise evanescent)
waves with large wave vectors, resulting in a high (theoretically unbounded) photonic density of states.

Typical multilayered implementations of hyperbolic metamaterials are based the stacking of alternating subwavelength layers with negative and positive permittivities (e.g., metallic and dielectric at optical wavelengths). For this class, the effective-medium theory provides a particularly simple model in terms of a homogeneous uniaxially anisotropic permittivity tensor with components given by the Maxwell-Garnett mixing formulas. However, a series of recent papers have pointed out the limitations of this model in predicting nonlocal effects that can take place (even in the presence of deep subwavelength layers) due to the coupling of surface-plasmon polaritons propagating along the interfaces separating layers with oppositely signed permittiv- ities. This may result, for instance, in the misprediction of additional extraordinary waves, as well as of the broadband Purcell effect.

Although the effective-medium-theory model describing the local response is independent of the positional order of the layers and depends only on the permittivities of the two constituents and their filling fractions, one would intuitively expect the positional order of the layers to sensibly affect the nonlocal response.

Within this framework, in [1], we studied the nonlocal response of hyperbolic metamaterials
constituted by multilayer superlattices based on the Thue-Morse geometry.
The chosen geometry (schematized in the figure top panel) is particularly interesting since different iteration orders differ solely
in the positional order of the constituent material layers. Interestingly, we identified some configurations
for which these nonlocal effects are rather weak at the first two iteration orders
(corresponding to standard periodic multilayers) and become markedly more prominent at higher
iteration orders.
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 showed that their onset is accompanied
by a strong wave-vector dependence in the effective constitutive parameters, as well as
the appearance of
additional extraordinary waves, as shown in the figure center panels. The bottom panels
illustrate the propagation of such an additional extraordinary wave in a finite-thickness metamaterial slab.
In particular, the transverse distribution exhibits large amplitude variations and peaks at the interfaces between dielectric
and metallic layers, due to the propagation of surface-plasmon-polaritons.

Despite the inherent periodicity enforced by the unavoidable (Bloch-type) supercell terminations, such strong nonlocality is retained at any arbitrarily high-order iteration, i.e., approaching the actual aperiodic regime. To the best of our knowledge, against the many implications and applications, this represents the first evidence in connection with optical nonlocality. Besides the inherent academic interest, from the application viewpoint, this constitutes an important and technologically inexpensive additional degree of freedom in the engineering of optical nonlocality, which may also be fruitfully exploited within the recently introduced framework of nonlocal transformation optics.

More recently, we have explored some peculiar *boundary effects* that can enhance the (otherwise neglibibly weak)
nonlocality in aperiodically ordered multilayered dielectric metamaterials wity *deeply subwavelength* layers [2].

## Relevant papers

- Savoia, S., Castaldi, G., & Galdi, V. (2013). Optical nonlocality in multilayered hyperbolic metamaterials based on Thue-Morse superlattices.
*Physical Review B**87*(23), 235116.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} }

- Coppolaro, M., Castaldi, G., & Galdi, V. (2018). Aperiodic order induced enhancement of weak nonlocality in multilayered dielectric metamaterials.
*Physical Review B**98*(19), 195128.Recent studies on fully dielectric multilayered metamaterials have shown that the negligibly small nonlocal effects (spatial dispersion) typically observed in the limit of deeply subwavelength layers may be significantly enhanced by peculiar boundary effects occurring in certain critical parameter regimes. These phenomena, observed so far in periodic and randomly disordered geometries, are manifested as strong differences between the exact optical response of finite-size metamaterial samples and the prediction from conventional effective-theory-medium models based on mixing formulae. Here, with specific focus on the Thue-Morse geometry, we make a first step toward extending the studies above to the middle-ground of aperiodically ordered multilayers, lying in between perfect periodicity and disorder. We show that, also for these geometries, there exist critical parameter ranges that favor the buildup of boundary effects leading to strong enhancement of the (otherwise negligibly weak) nonlocality. However, the underlying mechanisms are fundamentally different from those observed in the periodic case, and exhibit typical footprints (e.g., fractal gaps, quasi-localized states) that are distinctive of aperiodic order. The outcomes of our study indicate that aperiodic order plays a key role in the buildup of the aforementioned boundary effects, and may also find potential applications to optical sensors, absorbers and lasers.

@article{IJ134_PRB_98_195128_2018, title = {Aperiodic order induced enhancement of weak nonlocality in multilayered dielectric metamaterials}, author = {Coppolaro, Marino and Castaldi, Giuseppe and Galdi, Vincenzo}, journal = {Physical Review B}, volume = {98}, issue = {19}, pages = {195128}, numpages = {12}, year = {2018}, month = nov, publisher = {American Physical Society}, doi = {10.1103/PhysRevB.98.195128} }