Chirality is a universal feature in nature, as observed in fermion interactions and DNA helicity. Much attention has been given to chiral interactions of light, not only regarding its physical interpretation, but also focusing on intriguing phenomena in the excitation, absorption, refraction, and topological phases. Although recent progress in metamaterials has spurred artificial engineering of chirality, most approaches are founded on the same principle of the mixing of electric and magnetic responses. Here we propose non-magnetic chiral interactions of light based on low-dimensional eigensystems. Exploiting the mixing of amplifying and decaying electric modes in a complex material, the low dimensionality in polarization space having a chiral eigenstate is realized, in contrast to two-dimensional eigensystems in previous approaches. The existence of an optical-spin black hole from low-dimensional chirality is predicted, and singular interactions between chiral waves are confirmed experimentally in parity-time-symmetric metamaterials.
© 2016 Optical Society of America
Complex potentials that violate the Hermitian condition have been treated restrictively to describe non-equilibrium processes . However, since the pioneering work of Bender , it has been widely accepted that the condition of parity-time (PT) symmetry allows real eigenvalues, even in complex potentials with non-Hermitian Hamiltonians. The concept of PT symmetry has opened a pathway for handling complex potentials, overcoming traditional Hermitian restrictions, and stimulating the field of complex quantum mechanics . In contrast to Hermitian potentials, PT-symmetric potentials support regimes in which some eigenvalues are complex . Accordingly, the phase of eigenvalues can be divided into a real and a complex regime, with a border line called the exceptional point (EP) [2,4,5] marking the onset of PT symmetry breaking.
Substantial researches have focused on optical analogues of PT-symmetric dynamics [4–12]. Based on the equivalence of the Schrödinger and paraxial-wave equations, the classical simulation of complex quantum mechanics has been tested . Exotic behaviors of light have also been implemented in PT-symmetric potentials: asymmetric modal conversion [8,13,14], abnormal beams [4,12], unidirectional invisibility , and PT-symmetric resonances [6,7,10]. These phenomena originate from complex eigenstates near the EP, in relation to their skewness [8,14] and unidirectional modal conversion [4,13]. Although there has been an effort to observe PT-symmetric dynamics in polarization space  as well, previous work has focused only on the observation of PT-symmetric phase transition of eigenstates around the EP, lacking the in-depth investigation in the context of chiral “light-matter interactions”: handedness-dependent optical phenomena through the interactions with two-dimensional (2D) or three-dimensional (3D) chiral structures. For example, because the PT-symmetric metamaterial in Ref.  was considered as a meta-“surface” between the air and substrate, the studies for the evolution of propagating waves along PT-symmetric “bulk” materials, for the increase of singular dynamics at the EP and for the incidences with arbitrary polarizations, have been neglected.
In this paper, generalizing light interactions with PT-symmetric electrical materials, we investigate a new class of 2D chiral “bulk materials” with complex potentials, which possess the low-dimensional (1D) eigensystem and can be applied to exhibit the design of arbitrary singular polarization states on the Poincaré sphere. In contrast to previous approaches [17–24] for chiral optical materials, we derive chiral interactions from the mixing of amplifying and decaying electric responses, not from the mixing of electric and magnetic responses. We demonstrate that the dimensionality of the PT-symmetric chiral system is reduced to one at the EP, leading to a perfectly singular modal helix. This provides a pathway toward chiral light-matter interactions fundamentally distinct from conventional optical chirality [17–20,23,24], all of them based upon 2D eigensystems. Unique properties arising from the low dimensionality are experimentally demonstrated in high-index metamaterials. We also show the convergence of arbitrarily polarized incidences to a single chiral eigenstate without any reflections, realizing an optical-spin black hole.
2. EIGENSTATES IN PT-SYMMETRICALLY POLARIZED MATERIALS
First, we assume a homogeneous and anisotropic optical material that is PT-symmetric with respect to the axis. Because the permittivity tensor corresponds to the Hamiltonian for plane waves in polarization space (see Supplement 1, Section 1), it is necessary to define the permittivity tensor satisfying the PT-symmetry condition for the potential of the form . The permittivity tensor satisfying the necessary condition of PT symmetry  () is characterized as4] defining the EP (, Supplement 1, Section 1). The eigenstates are non-orthogonal () except for the Hermitian potentials (), which naturally derive intermodal coupling  between . Because the eigenstates have elliptical polarizations in general, and the handedness of the elliptical polarizations is left handed, the PT-symmetric potential becomes naturally chiral, favoring the left circular polarization (LCP, ) mode.
Figures 1(a) and 1(b) presents the real and imaginary part of the effective permittivity for PT-symmetric systems to of different imaginary potentials . Similar to other PT-symmetric potentials [4–12], the variation of derives the generic square-root curve from the definition of and results in the apparent phase transition of eigenvalues from the real to complex phase across the EP (point , for ). The normalized density of optical chirality [21,22] for each eigenstate in is shown in Fig. 1(c) (see Appendix A for the calculation of , and ), along with the corresponding profiles of the eigenpolarizations [Figs. 1(d)–1(h)].
For the Hermitian case [Fig. 1(d)], the eigenstates take linear polarizations, constituting a 2D orthogonal basis set. As increases [Fig. 1(e)], the eigenstates begin to converge and take the left-handed chiral form of elliptical polarizations, with non-orthogonality () and increased chirality [Fig. 1(c), ]. At the EP [Fig. 1(f)], two chiral eigenstates coalesce to an LCP basis with the reduced geometric multiplicity (). This low-dimensional existence of a chiral eigenstate forming a modal helix, which is distinguished from the structural helix of electric and magnetic mixing [18,23], yields perfect chirality with pure handedness [Fig. 1(c), point ]. After the EP [Figs 1(g) and 1(h)], the 2D eigensystem is recovered with decreased chirality [Fig. 1(c), ], as each eigenstate is saturated to a linear mode (, amplifying; , decaying). Note that the handedness of the chiral eigenstates can be reversed by changing the sign of [Fig. 1(c), orange for left- and blue for right-handedness]. For completeness, the imperfect PT symmetry is also investigated ( or ) with respect to the modal chirality (Supplement 1, Section 2), showing the experimental tolerance and gauge-transformed (active and passive) PT symmetry.
3. CHIRAL INTERACTIONS IN PT-SYMMETRICALLY POLARIZED MATERIALS
To examine the role of chiral eigenstates in PT-symmetric materials, here we focus on the “chiral interaction” with singularity, extending the discussion in Ref. , which focused on the eigenstate itself. First, we study the modal transfer between circular polarization (CP) modes through propagation. Utilizing the eigenstates and eigenvalues from Eq. (1) and employing the CP basis of , the transfer relation between the incident and transmitted field is written as , using only structural (propagation distance ) and material (, , and ) parameters (Supplement 1, Section 3).
A closer investigation of the transfer matrix in Supplement 1, Section 3 provides a straightforward understanding of the chiral interactions in PT-symmetric potentials. First, the inequality between off-diagonal terms () leads to an asymmetric modal conversion between the right- and left-CP (RCP and LCP) modes. Because the self-evolutions of CP modes are identical (), the chiral response of the system is governed by the intermodal chirality asSupplement 1, Section 3.
Note that the intermodal chirality is solely determined by the ratio of and , directly related to the degree of PT symmetry, i.e., . Accordingly, at the EP (), a one-way chiral conversion from the RCP to the LCP mode is achieved [Fig. 1(i)], as expected from the reduction of the eigensystem to a 1D LCP eigenstate [Fig. 1(f)]. Before and after the EP, decreases [Fig. 1(i)] due to two elliptically polarized eigenstates [Figs. 1(e) and 1(g)]. Note that this low-dimensional PT-chiral system ( and ) is fundamentally distinct from conventional chirality based on a 2D Hermitian eigensystem of circular birefringence () and zero intermodal coupling () [18,20,22,23,25].
When the intermodal chirality defines the unique origin of low-dimensional chirality in PT-symmetric material, the strength of the chiral conversion [Fig. 1(j)] is determined by the competition between the intermodal transfer and the self-evolution (, , see Supplement 1, Section 4 for details). In agreement with the observations made for , is always larger than , resulting in LCP-favored chiral conversion [Fig. 1(j)]. Near the EP (see Supplement 1, Section 4, and ), the chiral conversion becomes unidirectional, and its strength is solely determined by the material and structural parameters: and effective interaction length , where is the effective index of the structure. We note that while the magnitudes of the gain and loss contribute both to intermodal chirality and chiral conversion , the achievement of large , e.g., with unnaturally high-index metamaterials , enables a strong chiral conversion (or large ) in the “bulk”, overcoming the material restriction on .
Figures 2(a) and 2(b) show the chirality of the transmitted wave for RCP and LCP incidences. While determines the regime of chiral transfers (oscillatory for , and LCP-favored for ), describes the effect of the interaction length of PT-symmetric chiral materials. For large , strong LCP chirality from the one-way chiral conversion is apparent near the EP, which emphasizes the role of the singularity. This chiral singularity forms the optical-spin black hole [the south pole of the Poincaré sphere, Fig. 2(c)] where all the states of polarization (SOP) converge (Supplement 1, Section 5 for details, and Visualization 1 for the spin black hole behavior at the EP), differentiating the analysis based on low-dimensional chirality from conventional chirality [17–24] or the analysis of the “eigenstate” singularity in PT-symmetric metasurfaces . Because the eigenstate of the polarization singularity can be designed systematically [e.g., see Supplement 1, Section 6 for a low-dimensional eigenstate of linear polarization (LP)], the polarization black hole can be achieved in the entire polarization space of the Poincaré sphere by combining the imaginary potentials for the linear and circular basis sets. We also reveal another unique property of low-dimensional chirality in Supplement 1, Section 7, chirality reversal for LP incidences (see Visualization 2 for LP incidence), which is impossible for the optical activity in conventional chiral materials .
4. GIANT CHIRAL CONVERSION IN THE RESONANT STRUCTURE
To achieve a large chiral conversion within a compact footprint, a resonant structure for the effective increase of the interaction length can be considered [Fig. 3(a)]. The resonator is composed of a PT-symmetric anisotropic material at the EP (length , same parameters as those of Fig. 2), sandwiched between two metallic mirrors (of thickness ). Here, an S-matrix analysis  is utilized to calculate the frequency-dependent transmission, reflection, and field distributions inside the resonator. In detail, while the fields of the background air and mirrors are expressed as the linear combinations of the - and -orthogonal basis vectors, the field inside the PT-symmetric material is expressed by the non-orthogonal basis of , as , including forward (+) and backward (–) components. From the continuity condition of the electric field across the boundary ( and ), we derive the S-matrix relation of . For mirrors with thicknesses of , 50, 60, or 70 nm, the obtained values of the resonators are 620, 1500, 3600, and 8200, respectively.
Figures 3(b) and 3(c) show the S-matrix calculated power ratio of the LCP over the RCP of a transmitted and reflected wave for the forward -linear polarized incidence (). Also shown in Fig. 3(d) are the S-matrix-calculated wave evolutions at the on-resonance condition of the 3/2 wavelength. Enhanced by the chiral standing wave in the resonator, a giant LCP-favored chirality of the transmitted wave is observed (Fig. 3(b), within at ; to compare, for the non-resonant structure, [, ] and [, ]). It is also notable that, in contrast to the non-resonant structure where the reflection is absent, perfectly chiral reflection (RCP only) results from the backward-propagating RCP waves inside the resonator [Fig. 3(c)].
5. LOW-DIMENSIONAL CHIRAL METAMATERIAL
We now investigate the experimental realization of the PT-symmetric chiral medium, focusing on the observation of the one-way chiral conversion as a clear and direct evidence of the low dimensionality. To achieve the complex anisotropic permittivity of Eq. (1) with isotropic media, we employ a platform of metamaterials, which enables the use of local material parameters from the subwavelength structure and the implementation of gauge-transformed  PT-symmetry in a passive manner (Supplement 1, Section 2) through the designer permittivity.
Figure 4 shows the realization of low-dimensional chiral metamaterials, achieved by transplanting the ideal point-wise anisotropic permittivity [Fig. 4(a)] into the subwavelength structure (see Appendices B and C for the fabrication and THz measurement). We emphasize that the chiral conversion is directly proportional to . Considering that most of capacitive metamaterials (including the split-ring resonators in Ref. ) have strong diamagnetic behavior () , hindering the realization of high effective index, we employ an -shaped metamaterial, which provides an ultrahigh permittivity  () and suppressed magnetic moments () , achieving strong and purely electrical light-matter interactions in the THz regime. We also note that the multilayer extension can be obtained for -shaped structures , and the operation condition of the high effective index and purely electrical response can be satisfied in the optical regime as well, for example, by utilizing hyperbolic metamaterials .
Because the -shaped patch supports effective permittivity following the Lorentz model , its spectral response [Fig. 4(b)] is divided into dielectric () and metallic () states, both of which can be subdivided into low-loss () and high-loss () regimes. To compose the PT-symmetric polar metamaterial, we utilize the low- and high-loss regimes for and polarization, respectively.
Figures 4(c)–4(f) show the experimental realization of PT-symmetric metamaterials using a dielectric [Figs. 4(c) and 4(d)] and metallic [Figs. 4(e) and 4(f)] state (see Supplement 1, Section 8). Figures 4(c) and 4(e) show anisotropic permittivity and [at in Supplement 1, Fig. S5(a)] for dielectric and metallic realizations [Figs. 4(d) and 4(f), respectively]. These figures show the spectral overlap between low- and high-loss regimes for the - and -axis patches and the existence of spectral EP (with and , Supplement 1, Section 2). Note that our single-layered structure embedded in a subwavelength-thick polyimide () can be considered as a homogenized metamaterial . To introduce the coupling , we apply an oblique alignment with a tilted angle [Supplement 1, Fig. S5(a)].
By changing (the coupling ) in the fabricated sample, now we measure , where the one-way chiral conversion of singularity occurs with . The experimentally measured intermodal chirality is shown each for dielectric [Fig. 5(a)] and metallic [Fig. 5(b)] state realization and is in good agreement with the COMSOL simulation [Figs. 5(c) and 5(d), respectively]. In both samples, of singularity in (, ) space, satisfying the one-way chiral conversion for the sensitive EP, are observed (crossed points of black dotted lines). For the dielectric state metamaterial with propagating waves inside, the large is observed at , and is observed at in the design in the metallic state with evanescent waves inside the metamaterial. Despite the different propagating features of the two regimes, the EP design derives the chiral interaction of light in terms of “one-way chiral conversion” for both regimes, confirming the role of low-dimensionality with a singular chiral eigenstate. Note that the separated and local modes that are highly concentrated within the gaps are well-converted to a single plane-wave-like beam due to the deep-subwavelength scale of the structure, conserving the pure spin angular momentum of light without additional orbital angular momentum.
6. LOW-DIMENSIONAL MODAL HELIX IN GUIDED-WAVE STRUCTURES
While Ref.  has been focused on the singular “scattering” from PT-symmetric meta-“surface” which can be analyzed through the scattering matrix, the impact of the “eigenstates” defining the system Hamiltonian is critical for propagating waves along the “bulk.” Extending the discussion to the guided wave and the optical frequency, we propose a modal helix in an optical waveguide platform utilizing isotropic materials. The point-wise permittivity [Fig. 4(a)] is transplanted into the complex-strip waveguide as a passive form [Fig. 6(a)], where the lossy Ti layer (gray region, thickness ) under a lossless Si-strip waveguide imposes the selective decay of the mode, which is well separated from the mode [Fig. 6(b), for low loss and for high loss]. The structural parameters are designed to satisfy , and the coupling is achieved with the deviation , which breaks the orthogonality between the and polarized modes. It is worth mentioning that the sign change of can also be controlled by the mirror offset of for the deterministic control of the handedness [Fig. 1(c)]. Therefore, the chirality of the proposed modal helix has directionality from the sign reversal of for the backward () view, which is absent in the structural helicity [18,23].
Figures 6(c) and 6(d) show the COMSOL-calculated modal chirality and the difference between eigenvalues as a function of the structural parameters ( and ). Because the control of the Ti layer alters the complex part of , the two structural parameters provide three degrees of freedom (, , and ), resulting in a single EP in the 2D parameter space (, , , and 18 dB for modes 1 and 2). The finite modal chirality () originates in the separated intensity profiles of the and modes [Fig. 6(b)], resulting in the non-uniform local chirality [82 dB maximum, Fig. 6(e)]. Therefore, the chiral guided wave includes the additional orbital angular momentum from the varying wavefront, in contrast to the case of the subwavelength structure (Fig. 4), which supports a plane wave. With an experimentally accessible geometry ( in ) and the coalescence of eigenmodes [Fig. 6(d)], the complex-strip waveguide will be an ideal building block for chiral guided-wave devices and for the utilization of active materials such as GaInAsP. It is also worth mentioning that our approach based on the guided-wave platform can derive the low-dimensional chirality through the simple spatial displacement of the lossy dielectric waveguide, without the use of molecular designs for low-loss and high-loss components .
We proposed and investigated a new class of optical chiral interactions based on complex potentials, without any bi-anisotropic mixing of electric and magnetic dipoles . Based on the mixing of amplifying and decaying responses and on the presence of the low-dimensional eigenstate at the EP, exotic chiral behaviors of one-way CP convergence are observed, which cannot be observed in conventional chiral or gyrotropic materials. The reduced dimensionality also enables reflectionless CP generation following the dynamics of optical-spin black holes, which is impossible in conventional approaches based on Hermitian elements. We also emphasize that our result, supporting the saturation of linear polarizations to a single circular polarization (Supplement 1, Section 7), is distinct from conventional chirality, which maintains optical activity for linear polarizations. We demonstrated the physics of low-dimensional chirality by using ultrahigh index polar metamaterials  with two design strategies: utilizing propagating and evanescent waves. The results prove the existence of EP and the one-way chiral conversion in the spectral regime, and the manipulation of the angular property is achieved. As an application, a chiral waveguide using isotropic materials is also proposed, which can be achieved by transplanting the point-wise anisotropic permittivity of an ideal complex potential. Compared to previous results for polarization tunability based on detuned dipoles , temporal retardation , and singular scattering , our study focuses on designing target eigenstates in a chiral form, also revealing the exotic phenomenon of optical-spin black hole.
Our new findings of low-dimensional chirality will pave a route toward active chiral devices , such as on-chip guided-wave devices for chiral lasers, amplifiers, absorbers, and switches , as well as complex chiral metamaterials  and topological phases. From the evident correlation between complex potentials and optical chirality, we can also imagine the eigenstate with non-trivial optical spins in “non-PT-symmetric” complex potentials by utilizing the supersymmetry technique [35,36] or the inverse design of an eigenstate in disordered media . Based on the general framework of non-Hermitian physics, we note that our work can be further extended using a different polarization basis set (Supplement 1, Section 6) to enable SOP collection for arbitrary designer polarization.
APPENDIX A: DENSITY OF OPTICAL CHIRALITY FOR COMPLEX EIGENSTATES
From the definition of and the condition of weak coupling in the PT-symmetric system (), the chirality density of each eigenstate before and after the EP is now expressed as1(b), before the EP]. Herein, we adopt to express the energy-normalized chirality of the eigenstates. Note that from the condition of weak coupling [line and symbol of Fig. 1(c)].
APPENDIX B: FABRICATION PROCESS OF THZ CHIRAL POLAR METAMATERIALS
Serving as a flexible and vertically symmetric environment of a metamaterial, a polyimide solution (PI-2610, HD MicroSystems) was spin coated (1 μm) onto a bare Si substrate and converted into a fully aromatic and insoluble polyimide (baked at 180°C for 30 min and cured at 350°C). A negative photoresist (AZnLOF2035, AZ Electronic Materials) was spin-coated and patterned using photolithography. Then, Au (100 nm) was evaporated on a Cr (10 nm) adhesion layer and patterned as crossed -shaped array structures via the lift-off process. Repeating the polyimide coating and curing (1 μm), single-layered metamaterials were fabricated by peeling off the metamaterial layers from the substrate.
APPENDIX C: THZ-TDS SYSTEM FOR THE MEASUREMENT OF INTERMODAL CHIRALITY
To generate a broadband THz source, a Ti:sapphire femtosecond oscillator was used (Mai-Tai, Spectra-physics, 80 MHz repetition rate, 100 fs pulse width, 800 nm central wavelength, and 1 W average power). The pulsed laser beam was focused onto a GaAs terahertz emitter (Tera-SED, Gigaoptics). The emitted THz wave was then focused onto the samples using a 2 mm spot diameter. The propagating THz radiation was detected through electro-optical sampling using a nonlinear ZnTe crystal. The THz-TDS system has a usable bandwidth of 0.1–2.6 THz and a signal-to-noise ratio greater than .
National Research Foundation of Korea (NRF) (2014M3A6B3063708, K20815000003, 2016R1A6A3A04009723).
We thank J. Hong for reading the manuscript and providing useful feedback. This work was supported by NRF through the Global Frontier Program (GFP) and the Global Research Laboratory (GRL) Program, and the Brain Korea 21 Plus Project in 2015, which are all funded by the Ministry of Science, ICT & Future Planning of the Korean government. S. Yu was also supported by the Basic Science Research Program through the NRF, funded by the Ministry of Education of the Korean Government.
See Supplement 1 for supporting content.
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