Parity-time (PT) symmetry is a fascinating concept introduced in quantum physics in order to make sense of non-Hermitian Hamiltonians. This concept, however, is rather general and pervasive, and has resonated in several research communities, triggering a surge of interest in the study of more general non-Hermitian systems.
In particular, thanks to the well-known formal analogies between quantum mechanics and (paraxial) optics, non-Hermitian concepts can be translated to electromagnetic structures by means of spatial modulation of loss and gain, which is becoming technologically viable in artificial materials and metamaterials. In fact, in metamaterial engineering, the introduction of material constituents featuring gain is a well-established strategy to overcome the inevitable presence of losses. Nonetheless, the above ideas have inspired novel, unconventional ways of mixing material constituents featuring loss and gain, so as to attain a wealth of anomalous light-matter interactions that extend well beyond the mere loss-compensation effects.
In a series of recent investigations, we have been concerned with the study of field-manipulation and wave-guiding properties of non- Hermitian metamaterials.
In particular, in [1] and [2], we proposed a complex-coordinate extension of the transformation-optics framework that allows to handle complex-valued constitutive parameters (typical of non-Hermitian metamaterials), while retaining an insightful geometrical interpretation in terms of the complexification of geometrical objects. In particular, in conjunction with the complex source-point formalism, this approach can naturally handle the manipulation of beam-like wave objects in the presence of loss and gain. Moreover, it can also be applied to the manipulation of leaky waves.
In a series of subsequent studies, we have focused on some interesting tunneling [3] and wave-guiding [4, 5] phenomena that can be induced in non-Hermitian bi-layer and tri-layer systems. More specifically, in connection with bi-layer systems (as schematized in the figure top panel), we showed that the PT-symmetry condition inherently yields unattenuated propagation along the gain-loss interface, sustained by a transverse component of the power flux (from the gain to loss region). The figure bottom panels show the field distribution, exponentially localized at the gain-loss interface. In particular, we showed that in the epsilon-near-zero regime (i.e., vanishingly small real part of the permittivities) the gain/loss level necessary to sustain the wave confinement and guiding effects can be significantly reduced.
Overall, such wave-guiding mechanism looks quite attractive in a perspective of wave-guiding on demand. For instance, one could envision the deploying of gain medium channels in a lossy background, and their selective enabling (and possibly reconfiguring) via optical pumping or other gain-inducing mechanisms.
In a related investigation [6], we applied the supersymmetry concept to the design of non-Hermitian optical couplers with higher-order mode-selection functionalities, with potential applications to mode-division multiplexing in optical links.
More recently, in collaboration with Andrea Alù’s Group (University of Texas at Austin), we have explored cylindrical lensing systems composed of pairs of metasurfaces with suitably tailored non-Hermitian and nonlocal properties in order to attain magnified imaging with reduced aberrations [7].
In collaboration with Filippo Capolino’s Group (University of California at Irvine), we have also explored the emergence of exceptional points in non-Hermitian photonic coupled chains of scatterers [8].
Our results may find potentially intriguing applicability to a variety of exotic light-matter interaction effects (e.g., unidirectional invisibility, coherent perfect absorption, lasing, negative refraction and focusing, cloaking), and may set the stage for the development of novel optical components and devices.
We extend the transformation-optics paradigm to a complex spatial coordinate domain, in order to deal with electromagnetic metamaterials characterized by balanced loss and gain, giving special emphasis to parity-time (PT) symmetric metamaterials. We apply this general theory to complex-source-point radiation and anisotropic transmission resonances, illustrating the capability and potentials of our approach in terms of systematic design, analytical modeling, and physical insights into complex-coordinate wave objects and resonant states.
@article{IJ104_PRL_110_173901_2013, title = {PT metamaterials via complex-coordinate transformation optics}, author = {Castaldi, Giuseppe and Savoia, Silvio and Galdi, Vincenzo and Al\`u, Andrea and Engheta, Nader}, journal = {Physical Review Letters}, volume = {110}, issue = {17}, pages = {173901}, numpages = {5}, year = {2013}, month = apr, publisher = {American Physical Society}, doi = {10.1103/PhysRevLett.110.173901}, url = {http://link.aps.org/doi/10.1103/PhysRevLett.110.173901}, note = {/assets/papers/IJ104_PRL_110_173901_2013_SM.pdf} }
Transformation optics (TO) is conventionally based on real-valued coordinate transformations and, therefore, cannot naturally handle metamaterials featuring gain and/or losses. Motivated by the growing interest in non-Hermitian optical scenarios featuring spatial modulation of gain and loss, and building upon our previous studies, we explore here possible extensions of the TO framework relying on complex-valued coordinate transformations. We show that such extensions can be naturally combined with well-established powerful tools and formalisms in electromagnetics and optics, based on the ‘complexification’ of spatial and spectral quantities. This enables us to deal with rather general non-Hermitian optical scenarios, while retaining the attractive characteristics of conventional (real-valued) TO in terms of physically incisive modeling and geometry-driven intuitive design. As representative examples, we illustrate the manipulation of beam-like wave-objects (modeled in terms of ‘complex source points’) as well as radiating states (leaky waves’, modeled in terms of complex-valued propagation constants). Our analytical results, validated against full-wave numerical simulations, provide useful insight into the wave propagation in non-Hermitian scenarios, and may indicate new directions in the synthesis of active optical devices and antennas.
@article{IJ118_JO_18_044027_2016, author = {Savoia, S and Castaldi, G and Galdi, V}, title = {Complex-coordinate non-Hermitian transformation optics}, journal = {Journal of Optics}, volume = {18}, number = {4}, pages = {044027}, year = {2016}, month = apr, doi = {10.1088/2040-8978/18/4/044027} }
We show that obliquely incident, transversely magnetic-polarized plane waves can be totally transmitted (with zero reflection) through epsilon-near-zero (ENZ) bilayers characterized by balanced loss and gain with parity-time (PT) symmetry. This tunneling phenomenon is mediated by the excitation of a surface wave localized at the interface separating the loss and gain regions. We determine the parameter configurations for which the phenomenon may occur and, in particular, the relationship between the incidence direction and the electrical thickness. We show that, below a critical threshold of gain and loss, there always exists a tunneling angle which, for moderately thick (wavelength-sized) structures, approaches a critical value dictated by the surface-wave phase-matching condition. We also investigate the unidirectional character of the tunneling phenomenon, as well as the possible onset of spontaneous symmetry breaking, typical of PT-symmetric systems. Our results constitute an interesting example of a PT-symmetry-induced tunneling phenomenon, and may open up intriguing venues in the applications of ENZ materials featuring loss and gain.
@article{IJ111_PRB_89_085105_2014, title = {Tunneling of obliquely incident waves through PT-symmetric epsilon-near-zero bilayers}, author = {Savoia, Silvio and Castaldi, Giuseppe and Galdi, Vincenzo and Al\`u, Andrea and Engheta, Nader}, journal = {Physical Review B}, volume = {89}, issue = {8}, pages = {085105}, numpages = {10}, year = {2014}, month = feb, publisher = {American Physical Society}, doi = {10.1103/PhysRevB.89.085105}, url = {http://link.aps.org/doi/10.1103/PhysRevB.89.085105} }
Inspired by the parity-time symmetry concept, we show that a judicious spatial modulation of gain and loss in \(ε\)-near-zero metamaterials can induce the propagation of exponentially bound interface modes characterized by zero attenuation. With specific reference to a bilayer configuration, via analytical studies and parametrization of the dispersion equation, we show that this waveguiding mechanism can be sustained in the presence of moderate gain/loss levels, and it becomes leaky (i.e., radiative) below a gain/loss threshold. Moreover, we explore a possible rod-based metamaterial implementation, based on realistic material constituents, which captures the essential features of the waveguiding mechanism, in good agreement with our theoretical predictions. Our results may open up possibilities for the design of optical devices and reconfigurable nanophotonics platforms.
@article{IJ114_PRB_91_115114_2015, title = {PT-symmetry-induced wave confinement and guiding in \(\epsilon\)-near-zero metamaterials}, author = {Savoia, Silvio and Castaldi, Giuseppe and Galdi, Vincenzo and Al\`u, Andrea and Engheta, Nader}, journal = {Physical Review B}, volume = {91}, issue = {11}, pages = {115114}, numpages = {10}, year = {2015}, month = mar, publisher = {American Physical Society}, doi = {10.1103/PhysRevB.91.115114}, url = {http://link.aps.org/doi/10.1103/PhysRevB.91.115114} }
Following up on previous studies on parity-time-symmetric gain-loss bilayers, and inspired by formal analogies with plasmonic waveguides, we study non-Hermiticity-induced wave confinement and guiding phenomena that can occur in loss-gain-loss three-layers. By revisiting previous well-established “gain-guiding” concepts, we investigate analytically and numerically the dispersion and confinement properties of guided modes that can be supported by this type of structures, by assuming realistic dispersion models and parameters for the material constituents. As key outcomes, we identify certain modes with specific polarization and symmetry that exhibit particularly desirable characteristics, in terms of quasireal propagation constant and subwavelength confinement. Moreover, we elucidate the effects of material dispersion and parameters and highlight the potential advantages by comparison with the previously studied gain-loss bilayer configurations. Our results provide additional perspectives on light control in non-Hermitian optical systems and may find potentially intriguing applicability to reconfigurable nanophotonic platforms.
@article{IJ121_PRA_94_043838_2016, title = {Non-Hermiticity-induced wave confinement and guiding in loss-gain-loss three-layer systems}, author = {Savoia, Silvio and Castaldi, Giuseppe and Galdi, Vincenzo}, journal = {Physical Review A}, volume = {94}, issue = {4}, pages = {043838}, numpages = {10}, year = {2016}, month = oct, publisher = {American Physical Society}, doi = {10.1103/PhysRevA.94.043838} }
Supersymmetry has been shown to provide a systematic and effective framework for generating classes of isospectral optical structures featuring perfectly-phase-matched modes, with the exception of one (fundamental) mode which can be removed. More recently, this approach has been extended to non-Hermitian scenarios characterized by spatially-modulated distributions of optical loss and gain, in order to allow the removal of higher-order modes as well. In this paper, we apply this approach to the design of non-Hermitian optical couplers with higher-order mode-selection functionalities, with potential applications to mode-division multiplexing in optical links. In particular, we highlight the critical role of the coupling between non-Hermitian optical waveguides, which generally induces a phase transition to a complex eigenspectrum, thereby hindering the targeted mode-selection functionality. With the specific example of an optical coupler that selects the second-order mode of a given waveguide, we illustrate the aforementioned limitations and propose possible strategies to overcome them, bearing in mind the practical feasibility of the gain levels required.
@article{IJ113_SREP_5_8568_2015, author = {Principe, Maria and Castaldi, Giuseppe and Consales, Marco and Cusano, Andrea and Galdi, Vincenzo}, title = {{Supersymmetry-inspired non-Hermitian optical couplers}}, journal = {Scientific Reports}, year = {2015}, volume = {5}, pages = {8568}, month = feb, doi = {10.1038/srep08568}, note = {/assets/papers/IJ113_SREP_5_8568_2015_SM.pdf} }
We show that a cylindrical lensing system composed of two metasurfaces with suitably tailored non-Hermitian (i.e., with distributed gain and loss) and nonlocal (i.e., spatially dispersive) properties can perform magnified imaging with reduced aberrations. More specifically, we analytically derive the idealized surface-impedance values that are required for “perfect” magnification and imaging and elucidate the role and implications of non-Hermiticity and nonlocality in terms of spatial resolution and practical implementation. For a basic demonstration, we explore some proof-of-principle quasilocal and multilayered implementations and independently validate the outcomes via full-wave numerical simulations. We also show that the metasurface frequency-dispersion laws can be chosen so as to ensure unconditional stability with respect to arbitrary temporal excitations. These results, which extend previous studies on planar configurations, may open intriguing venues in the design of metastructures for field imaging and processing.
@article{IJ123_PRB_95_115114_2017, title = {Magnified imaging based on non-Hermitian nonlocal cylindrical metasurfaces}, author = {Savoia, Silvio and Valagiannopoulos, Constantinos A. and Monticone, Francesco and Castaldi, Giuseppe and Galdi, Vincenzo and Al\`u, Andrea}, journal = {Physical Review B}, volume = {95}, issue = {11}, pages = {115114}, numpages = {13}, year = {2017}, month = mar, publisher = {American Physical Society}, doi = {10.1103/PhysRevB.95.115114} }
We demonstrate the existence of exceptional points of degeneracy (EPDs) of periodic eigenstates in non-Hermitian coupled chains of dipolar scatterers. Guided modes supported by these structures can exhibit an EPD in their dispersion diagram at which two or more Bloch eigenstates coalesce, in both their eigenvectors and eigenvalues. We show the emergence of a second-order modal EPD associated with the parity-time (PT) symmetry condition, at which each particle pair in the double chain exhibits balanced gain and loss. Furthermore, we also demonstrate a fourth-order EPD occurring at the band edge. Such a degeneracy condition was previously referred to as a degenerate band edge in lossless anisotropic photonic crystals. Here, we rigorously show it under the occurrence of gain and loss balance for a discrete guiding system. We identify a more general regime of gain and loss balance showing that PT symmetry is not necessary to attain EPDs. Moreover, we investigate the degree of detuning of the EPD when the geometrical symmetry or balanced condition is broken. Furthermore, we demonstrate a realistic implementation of the EPD in a coupled chain made of pairs of plasmonic nanospheres and active core-shell nanospheres at optical frequencies. These findings open avenues toward superior light localization and transport with application to high-Q resonators utilized in sensors, filters, low-threshold switching and lasing.
@article{IJ125_PRB_95_104305_2017, title = {Exceptional points of degeneracy and PT symmetry in photonic coupled chains of scatterers}, author = {Othman, Mohamed A. K. and Galdi, Vincenzo and Capolino, Filippo}, journal = {Physical Review B}, volume = {95}, issue = {10}, pages = {104305}, numpages = {12}, year = {2017}, month = mar, publisher = {American Physical Society}, doi = {10.1103/PhysRevB.95.104305}, url = {http://link.aps.org/doi/10.1103/PhysRevB.95.104305} }