Abstract

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

1. INTRODUCTION

Complex potentials that violate the Hermitian condition have been treated restrictively to describe non-equilibrium processes [1]. However, since the pioneering work of Bender [2], 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 [3]. In contrast to Hermitian potentials, PT-symmetric potentials support regimes in which some eigenvalues are complex [2]. 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 [412]. Based on the equivalence of the Schrödinger and paraxial-wave equations, the classical simulation of complex quantum mechanics has been tested [11]. Exotic behaviors of light have also been implemented in PT-symmetric potentials: asymmetric modal conversion [8,13,14], abnormal beams [4,12], unidirectional invisibility [15], 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 [16] 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. [16] 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 [1724] 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 [1720,23,24], 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 (y±z)/21/2 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 V(x)=V*(x). The permittivity tensor satisfying the necessary condition of PT symmetry [2] (ε(r)=ε*(r)) is characterized as

εr=(εr0000εr0+iεi0εκ00εκ0*εr0iεi0),
where εr0, εi0, and εκ0 have real values for a non-magnetic material (for εi00, y: gain axis, z: loss axis), and r and r satisfy the mirror (or parity) symmetry with respect to (y±z)/21/2. For a plane wave propagating along the x-axis (Ey,Ez), there exist two eigenstates, each with an eigenvalue (or effective permittivity) εeig1,2=εr0±λPT and a corresponding eigenstate veig1,2=η1,2·(εκ0,iεi0±λPT)T, where η1,2=[1/(|εκ0|2+|iεi0±λPT|2)]1/2, and λPT=(εκ02εi02)1/2 is the interaction parameter [4] defining the EP (λPT=0, Supplement 1, Section 1). The eigenstates veig1,2 are non-orthogonal (veig1·veig2*0) except for the Hermitian potentials (εi0=0), which naturally derive intermodal coupling [8] between veig1,2. Because the eigenstates veig1,2=η1,2·(εκ0,iεi0±λPT)T 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, vL=(1/2)1/2·(1,i)T) mode.

Figures 1(a) and 1(b) presents the real and imaginary part of the effective permittivity εeig1,2 for PT-symmetric systems d to h of different imaginary potentials εi0. Similar to other PT-symmetric potentials [412], the variation of εi0 derives the generic square-root curve from the definition of λPT and results in the apparent phase transition of eigenvalues from the real to complex phase across the EP (point f, εi0=εκ0 for λPT=0). The normalized density of optical chirality [21,22] χ/(βr0Ue) for each eigenstate in dh is shown in Fig. 1(c) (see Appendix A for the calculation of χ, Ue=|E|2 and βr0=εr01/2·2π/Λ0), along with the corresponding profiles of the eigenpolarizations [Figs. 1(d)1(h)].

 figure: Fig. 1.

Fig. 1. Eigenvalues and spatial evolution of eigenstates in PT-symmetric chiral material. The real and imaginary parts of the effective permittivity εeig1,2 are shown in (a) and (b) with respect to εi0. (c) The density of chirality χ, normalized by the product of the electric field intensity Ue and βr0 (orange: εκ0=εr0/103>0, blue: εκ0=εr0/103<0, line: eigenstate 1, symbol: eigenstate 2). (d)–(h) Spatial evolution of eigenstates corresponding to points dh marked in (a)–(c). (d) εi0=0, (e) 0<εi0<εκ0, (f) εi0=εκ0, (g) and (h) εi0>εκ0. The red and blue arrows represent the axes of Ey (amplifying mode) and Ez(decaying mode). At the EP (f), the complex eigenstate has the singular form of a modal helix. (i) CIM and (j) CCS as functions of (εi0/εκ0). εr0=12.25 for (a)–(h), and εr0=6.5 for (j). εκ0=εr0/103>0 for (a), (b), (d)–(h), and (j). Leff=103 for (b).

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For the Hermitian case [Fig. 1(d)], the eigenstates take linear polarizations, constituting a 2D orthogonal basis set. As εi0 increases [Fig. 1(e)], the eigenstates begin to converge and take the left-handed chiral form of elliptical polarizations, with non-orthogonality (veig1·veig2*0) and increased chirality [Fig. 1(c), df]. At the EP [Fig. 1(f)], two chiral eigenstates coalesce to an LCP basis with the reduced geometric multiplicity (εeig1=εeig2). 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 f]. After the EP [Figs 1(g) and 1(h)], the 2D eigensystem is recovered with decreased chirality [Fig. 1(c), fh], as each eigenstate is saturated to a linear mode (y, amplifying; z, decaying). Note that the handedness of the chiral eigenstates can be reversed by changing the sign of εκ0 [Fig. 1(c), orange for left- and blue for right-handedness]. For completeness, the imperfect PT symmetry is also investigated (Re[εy]Re[εz] or Im[εy]Im[εz]) 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. [16], 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 vR,L=(1/2)1/2·(1,±i)T, the transfer relation between the incident Einc=(ERI,ELI)CPT and transmitted field Etrn=(ERT,ELT)CPT is written as Etrn=MPTEinc, using only structural (propagation distance d) and material (εr0, εi0, and εκ0) parameters (Supplement 1, Section 3).

A closer investigation of the transfer matrix MPT in Supplement 1, Section 3 provides a straightforward understanding of the chiral interactions in PT-symmetric potentials. First, the inequality between off-diagonal terms (|tRL|>|tLR|) 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 (tRR=tLL), the chiral response of the system is governed by the intermodal chirality CIM as

CIM=|tRLtLR|=|εκ0+εi0εκ0εi0|=|1+εi0εκ01εi0εκ0|,
which is obtained in Supplement 1, Section 3.

Note that the intermodal chirality CIM is solely determined by the ratio of εi0 and εκ0, directly related to the degree of PT symmetry, i.e., λPT=(εκ02εi02)1/2. Accordingly, at the EP (εi0=εκ0), a one-way chiral conversion CIM 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, CIM decreases [Fig. 1(i)] due to two elliptically polarized eigenstates [Figs. 1(e) and 1(g)]. Note that this low-dimensional PT-chiral system (tRR=tLL and tRLtLR0) is fundamentally distinct from conventional chirality based on a 2D Hermitian eigensystem of circular birefringence (tRRtLL) and zero intermodal coupling (tLR=tRL=0) [18,20,22,23,25].

When the intermodal chirality CIM defines the unique origin of low-dimensional chirality in PT-symmetric material, the strength of the chiral conversion CCS [Fig. 1(j)] is determined by the competition between the intermodal transfer and the self-evolution (CCSRL=|tRL/tRR|, CCSLR=|tLR/tLL|, see Supplement 1, Section 4 for details). In agreement with the observations made for CIM, CCSRL is always larger than CCSLR, resulting in LCP-favored chiral conversion [Fig. 1(j)]. Near the EP (see Supplement 1, Section 4, CCSRL2πLeff·(εi0/εr0) and CCSLR0), the chiral conversion becomes unidirectional, and its strength is solely determined by the material and structural parameters: εi0/εr0 and effective interaction length Leff=neff·d/Λ0, where neff is the effective index of the structure. We note that while the magnitudes of the gain and loss εi0 contribute both to intermodal chirality CIM and chiral conversion CCS, the achievement of large neff, e.g., with unnaturally high-index metamaterials [26], enables a strong chiral conversion (or large CCS) in the “bulk”, overcoming the material restriction on ±εi0.

Figures 2(a) and 2(b) show the chirality of the transmitted wave for RCP and LCP incidences. While εi0/εr0 determines the regime of chiral transfers (oscillatory for εi0/εr0<1, and LCP-favored for εi0/εr01), Leff describes the effect of the interaction length of PT-symmetric chiral materials. For large Leff, 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 [1724] or the analysis of the “eigenstate” singularity in PT-symmetric metasurfaces [16]. 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 [27].

 figure: Fig. 2.

Fig. 2. Chiral dynamics within PT-symmetric optical material. The output power ratio of LCP over RCP (IL/IR=|ELT/ERT|2 in dB) is shown for the case of (a) LCP and (b) RCP incidence as a function of the imaginary permittivity (εi0/εr0) and the interaction length (Leff=εr01/2·d/Λ0). The black dotted lines in (a) and (b) represent the EPs, where the dimensionality reduces to one. (c) LCP-convergent spin black hole dynamics on the Poincaré sphere at the EP, demonstrated with randomly polarized incidences. The interaction lengths are Leff=0, 80, 160, and 240, clockwise from the upper left. The movie is shown in Visualization 1. All the results are based on the transfer matrix method. εr0=6.5, and εκ0=εr0/103 in (a) and (b), and εκ0=εr0/200 in (c).

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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 Leff=εr01/2·d/Λ0 can be considered [Fig. 3(a)]. The resonator is composed of a PT-symmetric anisotropic material at the EP (length d=834nm, same parameters as those of Fig. 2), sandwiched between two metallic mirrors (of thickness δ). Here, an S-matrix analysis [28] 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 y- and z-orthogonal basis vectors, the field inside the PT-symmetric material is expressed by the non-orthogonal basis of veig1,2, as E(x)=Eeig1+·veig1·exp(iβ1·x)+Eeig1·veig1·exp(iβ1·x)+Eeig2+·veig2·exp(iβ2·x)+Eeig2·veig2·exp(iβ2·x), including forward (+) and backward (–) components. From the continuity condition of the electric field across the boundary (E and xE), we derive the S-matrix relation of (EOy+,EOz+,EIy,EIz)T=S·(EIy+,EIz+,EOy,EOz)T. For mirrors with thicknesses of δ=40, 50, 60, or 70 nm, the obtained Q values of the resonators are 620, 1500, 3600, and 8200, respectively.

 figure: Fig. 3.

Fig. 3. Giant chiral conversion through the resonant structure. (a) Schematics of the chiral resonator for the S-matrix analysis (green: PT-symmetric anisotropic material of d=834nm, εr0=6.5 and εκ0=εi0=εr0/1000; gray: metallic mirrors, εmetal=100, Λ0=1500nm). Leff=1.4. The power ratios of the LCP over the RCP in the (b) transmitted and (c) reflected wave for different mirror thicknesses. (d) S-matrix-based spatial evolutions of waves through the resonator. Arrows denote the propagating direction of each wave.

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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 y-linear polarized incidence (EIy+=1). 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), IL/IR=20dB within Leff=1.4 at Q=8200; to compare, for the non-resonant structure, [IL/IR=0.08dB, Leff=1.4] and [IL/IR=20dB, Leff=1450]). 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 CIM 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 [29] 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 CCSRL=|tRL/tRR|2πLeff·(εi0/εr0) is directly proportional to neff=(μrεr)1/2. Considering that most of capacitive metamaterials (including the split-ring resonators in Ref. [16]) have strong diamagnetic behavior (μr1) [30], hindering the realization of high effective index, we employ an I-shaped metamaterial, which provides an ultrahigh permittivity [26] (εr1) and suppressed magnetic moments (μr1) [30], achieving strong and purely electrical light-matter interactions in the THz regime. We also note that the multilayer extension can be obtained for I-shaped structures [26], 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 [31].

 figure: Fig. 4.

Fig. 4. Chiral polar metamaterial for low-dimensional chirality. (a) Schematics of a PT-symmetric, point-wise anisotropic permittivity material for Eq. (1). (b) Lorentz model for an I-shaped patch with different material regimes. The effective anisotropic permittivity of chiral metamaterials (Supplement 1, Section 8): (c) the dielectric realization with propagating mode, and (e) the metallic realization with evanescent mode. The fabricated samples of a chiral metamaterial are shown in (d) and (f). Insets of (c) and (e) are the expanded images of the real and imaginary parts near the EP (red dotted line). The width of each polarized patch in (c) and (d) is set unequally to wy=5.5μm and wz=7.5μm, and the other structural parameters are g=1.0μm, L=34.5μm, a=20.5μm, t=100nm, and d=2μm. The arm length of each polarized patch in (e) and (f) is set unequally to ay=25μm and az=40μm, and the other structural parameters are g=1.5μm, L=50μm, w=3.0μm, t=100nm, and d=2μm. See Supplement 1, Fig. S5(a) for the definitions of the structural parameters.

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Because the I-shaped patch supports effective permittivity following the Lorentz model [26], its spectral response [Fig. 4(b)] is divided into dielectric (Re[ε]0) and metallic (Re[ε]<0) states, both of which can be subdivided into low-loss (|Re[ε]||Im[ε]|) and high-loss (|Re[ε]||Im[ε]|) regimes. To compose the PT-symmetric polar metamaterial, we utilize the low- and high-loss regimes for y and z polarization, respectively.

Figures 4(c)4(f) show the experimental realization of PT-symmetric polar 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 εy and εz [at θ=0 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 y- and z-axis patches and the existence of spectral EP (with Re[εy]=Re[εz] and Im[εy]Im[εz], Supplement 1, Section 2). Note that our single-layered structure embedded in a subwavelength-thick polyimide (t<λ0/100) can be considered as a homogenized metamaterial [26]. To introduce the coupling εκ0, we apply an oblique alignment with a tilted angle θ [Supplement 1, Fig. S5(a)].

 figure: Fig. 5.

Fig. 5. Observation of EP in chiral polar metamaterials. (a), (b) The experimental and (c), (d) the simulated results of CIM are shown in a spectral regime for (a), (c) dielectric and (b), (d) metallic realizations. Dotted lines represent the condition of EPs in spectral and θ domains. All simulated results were obtained using COMSOL Multiphysics.

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By changing θ (the coupling εκ0) in the fabricated sample, now we measure θEP, where the one-way chiral conversion of singularity occurs with εκ0=|Im[εyεz]|. The experimentally measured intermodal chirality CIM(θ,ω) 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, θEP of singularity in (θ, ω) 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 CIM=17.4dB is observed at θEP=2.0°, and CIM=16.4dB is observed at θEP=1.6° 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 y and z 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. [16] 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 tTi) under a lossless Si-strip waveguide imposes the selective decay of the z mode, which is well separated from the y mode [Fig. 6(b), εy for low loss and εz for high loss]. The structural parameters are designed to satisfy Re[εy]=Re[εz], and the coupling εyz is achieved with the deviation Δ, which breaks the orthogonality between the y and z polarized modes. It is worth mentioning that the sign change of εκ0(=εyz) can also be controlled by the mirror offset of Δ 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 Δ for the backward (x) view, which is absent in the structural helicity [18,23].

 figure: Fig. 6.

Fig. 6. Chiral waveguides supporting the basis of modal helices. (a) Cross sections of a complex-strip waveguide using isotropic materials (graded color: silicon; purple: titanium; green: silica substrate; graded color represents the effective loss by the titanium layer). The lossless silicon (εSi=12.1) is assumed to compose the strip structure on top of the lossy titanium layer (εTi=1.66i·30.1) above a silica substrate (εSiO2=2.07), supporting both a low-loss y-polarized mode and a high-loss z-polarized mode. The complex-strip waveguide satisfies PT symmetry based on the gauge transformation (Re[εy]=Re[εz] and Im[εz]<Im[εy]<0). The effect of the loss can be controlled by changing the depth of the titanium layer. The red and blue arrows describe a corresponding point-wise anisotropic permittivity. (b) The intensity profile and the polarization (in arrows) of the eigenmodes for the structure (εyz=0. Δ=0). (c) shows the modal chirality by IL/IR as a function of Δ and tTi. (d) The absolute value of the difference between eigenvalues as a function of Δ and tTi. The intensity profile and the local chirality (IL(y,z)/IR(y,z)) at the EP are shown in (e). All results were obtained using COMSOL Multiphysics with an optical wavelength of Λ0=1500nm. L11=190nm, L12=300nm, L21=620nm, and L22=190nm.

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Figures 6(c) and 6(d) show the COMSOL-calculated modal chirality and the difference between eigenvalues as a function of the structural parameters (tTi and Δ). Because the control of the Ti layer alters the complex part of ε, the two structural parameters provide three degrees of freedom (Re[ε], Im[ε], and εκ0), resulting in a single EP in the 2D parameter space (tTi=19nm, Δ=91nm, IL/IR=21dB, and 18 dB for modes 1 and 2). The finite modal chirality (20dB) originates in the separated intensity profiles of the y and z 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 (IL/IR10dB in Δ=80100nm) 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 [16].

7. CONCLUSION

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 [27]. 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 [26] 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 [32], temporal retardation [33], and singular scattering [16], 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 [25], such as on-chip guided-wave devices for chiral lasers, amplifiers, absorbers, and switches [34], as well as complex chiral metamaterials [9] 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 [37]. 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

For the time-harmonic field of E=E0·eiωt and B=B0·eiωt, the time-varying representation of the optical chirality density [21,22] χ=[ε0εr·E(t)·(×E(t))+(1/μ0)·B(t)·(×B(t))]/2 is simplified to the time-averaged form of χ=ω·Im[E0*·B0]/2. Because E0=veig1,2·exp(iβ1,2x), where β1,2=2πεeig1,21/2/Λ0, and the magnetic field B0 is

B0=η1,2β1,2ω·[iεi0±λPTεκ0]·eiβ1,2x.

From the definition of χ and the condition of weak coupling in the PT-symmetric system (εi0εκ0εr0), the chirality density of each eigenstate before and after the EP is now expressed as

χ1,2=β1,22·εi0εκ0,
χ1,2=Re[β1,2]2·2εκ0·(εi0εi02εκ02)εκ02+(εi0εi02εκ02)2·e2Im(β1,2)x,
where exp(2·Im[β1,2]x)=|E0|2Ue represents the amplifying and decaying electric field intensities after the EP [Fig. 1(b), Ue=1 before the EP]. Herein, we adopt χ1,2/Ue to express the energy-normalized chirality of the eigenstates. Note that χ1χ2 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 I-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 CIM

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 10,0001.

Funding

National Research Foundation of Korea (NRF) (2014M3A6B3063708, K20815000003, 2016R1A6A3A04009723).

Acknowledgment

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.

REFERENCES

1. N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77, 570–573 (1996). [CrossRef]  

2. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998). [CrossRef]  

3. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002). [CrossRef]  

4. S. Yu, D. R. Mason, X. Piao, and N. Park, “Phase-dependent reversible nonreciprocity in complex metamolecules,” Phys. Rev. B 87, 125143 (2013). [CrossRef]  

5. L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760–1767 (2014). [CrossRef]  

6. B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014). [CrossRef]  

7. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014). [CrossRef]  

8. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010). [CrossRef]  

9. R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113, 023903 (2014). [CrossRef]  

10. B. Peng, Ş. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014). [CrossRef]  

11. S. Deffner and A. Saxena, “Jarzynski equality in PT-symmetric quantum mechanics,” Phys. Rev. Lett. 114, 150601 (2015). [CrossRef]  

12. K. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008). [CrossRef]  

13. S. Yu, X. Piao, K. Yoo, J. Shin, and N. Park, “One-way optical modal transition based on causality in momentum space,” Opt. Express 23, 24997–25008 (2015). [CrossRef]  

14. S. Yu, X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A 86, 031802 (2012). [CrossRef]  

15. X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38, 2821–2824 (2013). [CrossRef]  

16. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014). [CrossRef]  

17. V. Fedotov, P. Mladyonov, S. Prosvirnin, A. Rogacheva, Y. Chen, and N. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97, 167401 (2006). [CrossRef]  

18. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009). [CrossRef]  

19. Z. Li, M. Gokkavas, and E. Ozbay, “Manipulation of asymmetric transmission in planar chiral nanostructures by anisotropic loss,” Adv. Opt. Mater. 1, 482–488 (2013). [CrossRef]  

20. J. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004). [CrossRef]  

21. Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010). [CrossRef]  

22. Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332, 333–336 (2011). [CrossRef]  

23. M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three‐dimensional bi‐chiral photonic crystals,” Adv. Mater. 21, 4680–4682 (2009). [CrossRef]  

24. N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012). [CrossRef]  

25. S. Chen, D. Katsis, A. Schmid, J. Mastrangelo, T. Tsutsui, and T. Blanton, “Circularly polarized light generated by photoexcitation of luminophores in glassy liquid-crystal films,” Nature 397, 506–508 (1999). [CrossRef]  

26. M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–373 (2011). [CrossRef]  

27. I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech, 1994).

28. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002). [CrossRef]  

29. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009). [CrossRef]  

30. J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009). [CrossRef]  

31. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013). [CrossRef]  

32. A. Pors, M. G. Nielsen, G. Della Valle, M. Willatzen, O. Albrektsen, and S. I. Bozhevolnyi, “Plasmonic metamaterial wave retarders in reflection by orthogonally oriented detuned electrical dipoles,” Opt. Lett. 36, 1626–1628 (2011). [CrossRef]  

33. S.-C. Jiang, X. Xiong, P. Sarriugarte, S.-W. Jiang, X.-B. Yin, Y. Wang, R.-W. Peng, D. Wu, R. Hillenbrand, and X. Zhang, “Tuning the polarization state of light via time retardation with a microstructured surface,” Phys. Rev. B 88, 161104 (2013). [CrossRef]  

34. X. Piao, S. Yu, J. Hong, and N. Park, “Spectral separation of optical spin based on antisymmetric Fano resonances,” Sci. Rep. 5, 16585 (2015). [CrossRef]  

35. S. Yu, X. Piao, J. Hong, and N. Park, “Bloch-like waves in random-walk potentials based on supersymmetry,” Nat. Commun. 6, 8269 (2015). [CrossRef]  

36. M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87, 043819 (2013). [CrossRef]  

37. S. Yu, X. Piao, J. Hong, and N. Park, “Metadisorder for designer light in random-walk systems,” arXiv:1510.05518 (2015).

References

  • View by:
  • |
  • |
  • |

  1. N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77, 570–573 (1996).
    [Crossref]
  2. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  3. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
    [Crossref]
  4. S. Yu, D. R. Mason, X. Piao, and N. Park, “Phase-dependent reversible nonreciprocity in complex metamolecules,” Phys. Rev. B 87, 125143 (2013).
    [Crossref]
  5. L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760–1767 (2014).
    [Crossref]
  6. B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
    [Crossref]
  7. H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
    [Crossref]
  8. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [Crossref]
  9. R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113, 023903 (2014).
    [Crossref]
  10. B. Peng, Ş. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
    [Crossref]
  11. S. Deffner and A. Saxena, “Jarzynski equality in PT-symmetric quantum mechanics,” Phys. Rev. Lett. 114, 150601 (2015).
    [Crossref]
  12. K. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [Crossref]
  13. S. Yu, X. Piao, K. Yoo, J. Shin, and N. Park, “One-way optical modal transition based on causality in momentum space,” Opt. Express 23, 24997–25008 (2015).
    [Crossref]
  14. S. Yu, X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A 86, 031802 (2012).
    [Crossref]
  15. X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38, 2821–2824 (2013).
    [Crossref]
  16. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
    [Crossref]
  17. V. Fedotov, P. Mladyonov, S. Prosvirnin, A. Rogacheva, Y. Chen, and N. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97, 167401 (2006).
    [Crossref]
  18. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
    [Crossref]
  19. Z. Li, M. Gokkavas, and E. Ozbay, “Manipulation of asymmetric transmission in planar chiral nanostructures by anisotropic loss,” Adv. Opt. Mater. 1, 482–488 (2013).
    [Crossref]
  20. J. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
    [Crossref]
  21. Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
    [Crossref]
  22. Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332, 333–336 (2011).
    [Crossref]
  23. M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three‐dimensional bi‐chiral photonic crystals,” Adv. Mater. 21, 4680–4682 (2009).
    [Crossref]
  24. N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
    [Crossref]
  25. S. Chen, D. Katsis, A. Schmid, J. Mastrangelo, T. Tsutsui, and T. Blanton, “Circularly polarized light generated by photoexcitation of luminophores in glassy liquid-crystal films,” Nature 397, 506–508 (1999).
    [Crossref]
  26. M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–373 (2011).
    [Crossref]
  27. I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech, 1994).
  28. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
    [Crossref]
  29. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
    [Crossref]
  30. J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009).
    [Crossref]
  31. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
    [Crossref]
  32. A. Pors, M. G. Nielsen, G. Della Valle, M. Willatzen, O. Albrektsen, and S. I. Bozhevolnyi, “Plasmonic metamaterial wave retarders in reflection by orthogonally oriented detuned electrical dipoles,” Opt. Lett. 36, 1626–1628 (2011).
    [Crossref]
  33. S.-C. Jiang, X. Xiong, P. Sarriugarte, S.-W. Jiang, X.-B. Yin, Y. Wang, R.-W. Peng, D. Wu, R. Hillenbrand, and X. Zhang, “Tuning the polarization state of light via time retardation with a microstructured surface,” Phys. Rev. B 88, 161104 (2013).
    [Crossref]
  34. X. Piao, S. Yu, J. Hong, and N. Park, “Spectral separation of optical spin based on antisymmetric Fano resonances,” Sci. Rep. 5, 16585 (2015).
    [Crossref]
  35. S. Yu, X. Piao, J. Hong, and N. Park, “Bloch-like waves in random-walk potentials based on supersymmetry,” Nat. Commun. 6, 8269 (2015).
    [Crossref]
  36. M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87, 043819 (2013).
    [Crossref]
  37. S. Yu, X. Piao, J. Hong, and N. Park, “Metadisorder for designer light in random-walk systems,” arXiv:1510.05518 (2015).

2015 (4)

S. Deffner and A. Saxena, “Jarzynski equality in PT-symmetric quantum mechanics,” Phys. Rev. Lett. 114, 150601 (2015).
[Crossref]

S. Yu, X. Piao, K. Yoo, J. Shin, and N. Park, “One-way optical modal transition based on causality in momentum space,” Opt. Express 23, 24997–25008 (2015).
[Crossref]

X. Piao, S. Yu, J. Hong, and N. Park, “Spectral separation of optical spin based on antisymmetric Fano resonances,” Sci. Rep. 5, 16585 (2015).
[Crossref]

S. Yu, X. Piao, J. Hong, and N. Park, “Bloch-like waves in random-walk potentials based on supersymmetry,” Nat. Commun. 6, 8269 (2015).
[Crossref]

2014 (6)

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113, 023903 (2014).
[Crossref]

B. Peng, Ş. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760–1767 (2014).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

2013 (6)

S. Yu, D. R. Mason, X. Piao, and N. Park, “Phase-dependent reversible nonreciprocity in complex metamolecules,” Phys. Rev. B 87, 125143 (2013).
[Crossref]

X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38, 2821–2824 (2013).
[Crossref]

Z. Li, M. Gokkavas, and E. Ozbay, “Manipulation of asymmetric transmission in planar chiral nanostructures by anisotropic loss,” Adv. Opt. Mater. 1, 482–488 (2013).
[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
[Crossref]

S.-C. Jiang, X. Xiong, P. Sarriugarte, S.-W. Jiang, X.-B. Yin, Y. Wang, R.-W. Peng, D. Wu, R. Hillenbrand, and X. Zhang, “Tuning the polarization state of light via time retardation with a microstructured surface,” Phys. Rev. B 88, 161104 (2013).
[Crossref]

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87, 043819 (2013).
[Crossref]

2012 (2)

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

S. Yu, X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A 86, 031802 (2012).
[Crossref]

2011 (3)

Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332, 333–336 (2011).
[Crossref]

A. Pors, M. G. Nielsen, G. Della Valle, M. Willatzen, O. Albrektsen, and S. I. Bozhevolnyi, “Plasmonic metamaterial wave retarders in reflection by orthogonally oriented detuned electrical dipoles,” Opt. Lett. 36, 1626–1628 (2011).
[Crossref]

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–373 (2011).
[Crossref]

2010 (2)

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

2009 (4)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009).
[Crossref]

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three‐dimensional bi‐chiral photonic crystals,” Adv. Mater. 21, 4680–4682 (2009).
[Crossref]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

2008 (1)

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

2006 (1)

V. Fedotov, P. Mladyonov, S. Prosvirnin, A. Rogacheva, Y. Chen, and N. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97, 167401 (2006).
[Crossref]

2004 (1)

J. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[Crossref]

2002 (2)

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

1999 (1)

S. Chen, D. Katsis, A. Schmid, J. Mastrangelo, T. Tsutsui, and T. Blanton, “Circularly polarized light generated by photoexcitation of luminophores in glassy liquid-crystal films,” Nature 397, 506–508 (1999).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

1996 (1)

N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77, 570–573 (1996).
[Crossref]

Aieta, F.

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Albrektsen, O.

Alù, A.

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113, 023903 (2014).
[Crossref]

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B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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S. Yu, D. R. Mason, X. Piao, and N. Park, “Phase-dependent reversible nonreciprocity in complex metamolecules,” Phys. Rev. B 87, 125143 (2013).
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H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
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S. Yu, X. Piao, K. Yoo, J. Shin, and N. Park, “One-way optical modal transition based on causality in momentum space,” Opt. Express 23, 24997–25008 (2015).
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X. Piao, S. Yu, J. Hong, and N. Park, “Spectral separation of optical spin based on antisymmetric Fano resonances,” Sci. Rep. 5, 16585 (2015).
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S. Yu, X. Piao, J. Hong, and N. Park, “Bloch-like waves in random-walk potentials based on supersymmetry,” Nat. Commun. 6, 8269 (2015).
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S. Yu, D. R. Mason, X. Piao, and N. Park, “Phase-dependent reversible nonreciprocity in complex metamolecules,” Phys. Rev. B 87, 125143 (2013).
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X. Piao, S. Yu, J. Hong, and N. Park, “Spectral separation of optical spin based on antisymmetric Fano resonances,” Sci. Rep. 5, 16585 (2015).
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S. Yu, X. Piao, J. Hong, and N. Park, “Bloch-like waves in random-walk potentials based on supersymmetry,” Nat. Commun. 6, 8269 (2015).
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[Crossref]

S. Yu, X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A 86, 031802 (2012).
[Crossref]

S. Yu, X. Piao, J. Hong, and N. Park, “Metadisorder for designer light in random-walk systems,” arXiv:1510.05518 (2015).

Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
[Crossref]

Pors, A.

Prosvirnin, S.

V. Fedotov, P. Mladyonov, S. Prosvirnin, A. Rogacheva, Y. Chen, and N. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97, 167401 (2006).
[Crossref]

Rill, M. S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three‐dimensional bi‐chiral photonic crystals,” Adv. Mater. 21, 4680–4682 (2009).
[Crossref]

Rogacheva, A.

V. Fedotov, P. Mladyonov, S. Prosvirnin, A. Rogacheva, Y. Chen, and N. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97, 167401 (2006).
[Crossref]

Rotter, S.

B. Peng, Ş. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Saile, V.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Sarriugarte, P.

S.-C. Jiang, X. Xiong, P. Sarriugarte, S.-W. Jiang, X.-B. Yin, Y. Wang, R.-W. Peng, D. Wu, R. Hillenbrand, and X. Zhang, “Tuning the polarization state of light via time retardation with a microstructured surface,” Phys. Rev. B 88, 161104 (2013).
[Crossref]

Saxena, A.

S. Deffner and A. Saxena, “Jarzynski equality in PT-symmetric quantum mechanics,” Phys. Rev. Lett. 114, 150601 (2015).
[Crossref]

Schmid, A.

S. Chen, D. Katsis, A. Schmid, J. Mastrangelo, T. Tsutsui, and T. Blanton, “Circularly polarized light generated by photoexcitation of luminophores in glassy liquid-crystal films,” Nature 397, 506–508 (1999).
[Crossref]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Shen, J.-T.

J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009).
[Crossref]

Shin, J.

S. Yu, X. Piao, K. Yoo, J. Shin, and N. Park, “One-way optical modal transition based on causality in momentum space,” Opt. Express 23, 24997–25008 (2015).
[Crossref]

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–373 (2011).
[Crossref]

J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009).
[Crossref]

Sihvola, A.

I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech, 1994).

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Sounas, D. L.

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113, 023903 (2014).
[Crossref]

Tang, Y.

Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332, 333–336 (2011).
[Crossref]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref]

Thiel, M.

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three‐dimensional bi‐chiral photonic crystals,” Adv. Mater. 21, 4680–4682 (2009).
[Crossref]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Tikhodeev, S. G.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Tretyakov, S.

I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech, 1994).

Tsutsui, T.

S. Chen, D. Katsis, A. Schmid, J. Mastrangelo, T. Tsutsui, and T. Blanton, “Circularly polarized light generated by photoexcitation of luminophores in glassy liquid-crystal films,” Nature 397, 506–508 (1999).
[Crossref]

Viitanen, A.

I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech, 1994).

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

von Freymann, G.

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three‐dimensional bi‐chiral photonic crystals,” Adv. Mater. 21, 4680–4682 (2009).
[Crossref]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

Wang, Y.

L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760–1767 (2014).
[Crossref]

S.-C. Jiang, X. Xiong, P. Sarriugarte, S.-W. Jiang, X.-B. Yin, Y. Wang, R.-W. Peng, D. Wu, R. Hillenbrand, and X. Zhang, “Tuning the polarization state of light via time retardation with a microstructured surface,” Phys. Rev. B 88, 161104 (2013).
[Crossref]

Wegener, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three‐dimensional bi‐chiral photonic crystals,” Adv. Mater. 21, 4680–4682 (2009).
[Crossref]

Willatzen, M.

Wu, D.

S.-C. Jiang, X. Xiong, P. Sarriugarte, S.-W. Jiang, X.-B. Yin, Y. Wang, R.-W. Peng, D. Wu, R. Hillenbrand, and X. Zhang, “Tuning the polarization state of light via time retardation with a microstructured surface,” Phys. Rev. B 88, 161104 (2013).
[Crossref]

Xiong, X.

S.-C. Jiang, X. Xiong, P. Sarriugarte, S.-W. Jiang, X.-B. Yin, Y. Wang, R.-W. Peng, D. Wu, R. Hillenbrand, and X. Zhang, “Tuning the polarization state of light via time retardation with a microstructured surface,” Phys. Rev. B 88, 161104 (2013).
[Crossref]

Xu, N.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

Yablonskii, A. L.

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

Yang, L.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

B. Peng, Ş. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

Yang, S.

Yilmaz, H.

B. Peng, Ş. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

Yin, X.

Yin, X.-B.

S.-C. Jiang, X. Xiong, P. Sarriugarte, S.-W. Jiang, X.-B. Yin, Y. Wang, R.-W. Peng, D. Wu, R. Hillenbrand, and X. Zhang, “Tuning the polarization state of light via time retardation with a microstructured surface,” Phys. Rev. B 88, 161104 (2013).
[Crossref]

Yoo, K.

Yu, N.

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

Yu, S.

S. Yu, X. Piao, K. Yoo, J. Shin, and N. Park, “One-way optical modal transition based on causality in momentum space,” Opt. Express 23, 24997–25008 (2015).
[Crossref]

X. Piao, S. Yu, J. Hong, and N. Park, “Spectral separation of optical spin based on antisymmetric Fano resonances,” Sci. Rep. 5, 16585 (2015).
[Crossref]

S. Yu, X. Piao, J. Hong, and N. Park, “Bloch-like waves in random-walk potentials based on supersymmetry,” Nat. Commun. 6, 8269 (2015).
[Crossref]

S. Yu, D. R. Mason, X. Piao, and N. Park, “Phase-dependent reversible nonreciprocity in complex metamolecules,” Phys. Rev. B 87, 125143 (2013).
[Crossref]

S. Yu, X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A 86, 031802 (2012).
[Crossref]

S. Yu, X. Piao, J. Hong, and N. Park, “Metadisorder for designer light in random-walk systems,” arXiv:1510.05518 (2015).

Zhang, P.

Zhang, S.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

Zhang, W.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

Zhang, X.

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

L. Feng, X. Zhu, S. Yang, H. Zhu, P. Zhang, X. Yin, Y. Wang, and X. Zhang, “Demonstration of a large-scale optical exceptional point structure,” Opt. Express 22, 1760–1767 (2014).
[Crossref]

X. Zhu, L. Feng, P. Zhang, X. Yin, and X. Zhang, “One-way invisible cloak using parity-time symmetric transformation optics,” Opt. Lett. 38, 2821–2824 (2013).
[Crossref]

S.-C. Jiang, X. Xiong, P. Sarriugarte, S.-W. Jiang, X.-B. Yin, Y. Wang, R.-W. Peng, D. Wu, R. Hillenbrand, and X. Zhang, “Tuning the polarization state of light via time retardation with a microstructured surface,” Phys. Rev. B 88, 161104 (2013).
[Crossref]

Zheludev, N.

V. Fedotov, P. Mladyonov, S. Prosvirnin, A. Rogacheva, Y. Chen, and N. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97, 167401 (2006).
[Crossref]

Zhu, H.

Zhu, X.

Adv. Mater. (1)

M. Thiel, M. S. Rill, G. von Freymann, and M. Wegener, “Three‐dimensional bi‐chiral photonic crystals,” Adv. Mater. 21, 4680–4682 (2009).
[Crossref]

Adv. Opt. Mater. (1)

Z. Li, M. Gokkavas, and E. Ozbay, “Manipulation of asymmetric transmission in planar chiral nanostructures by anisotropic loss,” Adv. Opt. Mater. 1, 482–488 (2013).
[Crossref]

Nano Lett. (1)

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12, 6328–6333 (2012).
[Crossref]

Nat. Commun. (1)

S. Yu, X. Piao, J. Hong, and N. Park, “Bloch-like waves in random-walk potentials based on supersymmetry,” Nat. Commun. 6, 8269 (2015).
[Crossref]

Nat. Photonics (1)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
[Crossref]

Nat. Phys. (2)

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Nature (2)

S. Chen, D. Katsis, A. Schmid, J. Mastrangelo, T. Tsutsui, and T. Blanton, “Circularly polarized light generated by photoexcitation of luminophores in glassy liquid-crystal films,” Nature 397, 506–508 (1999).
[Crossref]

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–373 (2011).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (2)

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87, 043819 (2013).
[Crossref]

S. Yu, X. Piao, D. R. Mason, S. In, and N. Park, “Spatiospectral separation of exceptional points in PT-symmetric optical potentials,” Phys. Rev. A 86, 031802 (2012).
[Crossref]

Phys. Rev. B (3)

S. Yu, D. R. Mason, X. Piao, and N. Park, “Phase-dependent reversible nonreciprocity in complex metamolecules,” Phys. Rev. B 87, 125143 (2013).
[Crossref]

S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66, 045102 (2002).
[Crossref]

S.-C. Jiang, X. Xiong, P. Sarriugarte, S.-W. Jiang, X.-B. Yin, Y. Wang, R.-W. Peng, D. Wu, R. Hillenbrand, and X. Zhang, “Tuning the polarization state of light via time retardation with a microstructured surface,” Phys. Rev. B 88, 161104 (2013).
[Crossref]

Phys. Rev. Lett. (11)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

J. Shin, J.-T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth,” Phys. Rev. Lett. 102, 093903 (2009).
[Crossref]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
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N. Hatano and D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics,” Phys. Rev. Lett. 77, 570–573 (1996).
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C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
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C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

R. Fleury, D. L. Sounas, and A. Alù, “Negative refraction and planar focusing based on parity-time symmetric metasurfaces,” Phys. Rev. Lett. 113, 023903 (2014).
[Crossref]

S. Deffner and A. Saxena, “Jarzynski equality in PT-symmetric quantum mechanics,” Phys. Rev. Lett. 114, 150601 (2015).
[Crossref]

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113, 093901 (2014).
[Crossref]

V. Fedotov, P. Mladyonov, S. Prosvirnin, A. Rogacheva, Y. Chen, and N. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97, 167401 (2006).
[Crossref]

Sci. Rep. (1)

X. Piao, S. Yu, J. Hong, and N. Park, “Spectral separation of optical spin based on antisymmetric Fano resonances,” Sci. Rep. 5, 16585 (2015).
[Crossref]

Science (5)

Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332, 333–336 (2011).
[Crossref]

J. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[Crossref]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325, 1513–1515 (2009).
[Crossref]

B. Peng, Ş. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. Bender, F. Nori, and L. Yang, “Loss-induced suppression and revival of lasing,” Science 346, 328–332 (2014).
[Crossref]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346, 975–978 (2014).
[Crossref]

Other (2)

I. V. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech, 1994).

S. Yu, X. Piao, J. Hong, and N. Park, “Metadisorder for designer light in random-walk systems,” arXiv:1510.05518 (2015).

Supplementary Material (3)

NameDescription
» Supplement 1: PDF (4112 KB)      Supplementary material for “Low-dimensional optical chirality in complex potentials”
» Visualization 1: MP4 (3460 KB)      Spin black hole behavior at the EP.
» Visualization 2: MP4 (1516 KB)      Linear polarization (LP) incidence.

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Figures (6)

Fig. 1.
Fig. 1. Eigenvalues and spatial evolution of eigenstates in PT-symmetric chiral material. The real and imaginary parts of the effective permittivity εeig1,2 are shown in (a) and (b) with respect to εi0. (c) The density of chirality χ, normalized by the product of the electric field intensity Ue and βr0 (orange: εκ0=εr0/103>0, blue: εκ0=εr0/103<0, line: eigenstate 1, symbol: eigenstate 2). (d)–(h) Spatial evolution of eigenstates corresponding to points dh marked in (a)–(c). (d) εi0=0, (e) 0<εi0<εκ0, (f) εi0=εκ0, (g) and (h) εi0>εκ0. The red and blue arrows represent the axes of Ey (amplifying mode) and Ez(decaying mode). At the EP (f), the complex eigenstate has the singular form of a modal helix. (i) CIM and (j) CCS as functions of (εi0/εκ0). εr0=12.25 for (a)–(h), and εr0=6.5 for (j). εκ0=εr0/103>0 for (a), (b), (d)–(h), and (j). Leff=103 for (b).
Fig. 2.
Fig. 2. Chiral dynamics within PT-symmetric optical material. The output power ratio of LCP over RCP (IL/IR=|ELT/ERT|2 in dB) is shown for the case of (a) LCP and (b) RCP incidence as a function of the imaginary permittivity (εi0/εr0) and the interaction length (Leff=εr01/2·d/Λ0). The black dotted lines in (a) and (b) represent the EPs, where the dimensionality reduces to one. (c) LCP-convergent spin black hole dynamics on the Poincaré sphere at the EP, demonstrated with randomly polarized incidences. The interaction lengths are Leff=0, 80, 160, and 240, clockwise from the upper left. The movie is shown in Visualization 1. All the results are based on the transfer matrix method. εr0=6.5, and εκ0=εr0/103 in (a) and (b), and εκ0=εr0/200 in (c).
Fig. 3.
Fig. 3. Giant chiral conversion through the resonant structure. (a) Schematics of the chiral resonator for the S-matrix analysis (green: PT-symmetric anisotropic material of d=834nm, εr0=6.5 and εκ0=εi0=εr0/1000; gray: metallic mirrors, εmetal=100, Λ0=1500nm). Leff=1.4. The power ratios of the LCP over the RCP in the (b) transmitted and (c) reflected wave for different mirror thicknesses. (d) S-matrix-based spatial evolutions of waves through the resonator. Arrows denote the propagating direction of each wave.
Fig. 4.
Fig. 4. Chiral polar metamaterial for low-dimensional chirality. (a) Schematics of a PT-symmetric, point-wise anisotropic permittivity material for Eq. (1). (b) Lorentz model for an I-shaped patch with different material regimes. The effective anisotropic permittivity of chiral metamaterials (Supplement 1, Section 8): (c) the dielectric realization with propagating mode, and (e) the metallic realization with evanescent mode. The fabricated samples of a chiral metamaterial are shown in (d) and (f). Insets of (c) and (e) are the expanded images of the real and imaginary parts near the EP (red dotted line). The width of each polarized patch in (c) and (d) is set unequally to wy=5.5μm and wz=7.5μm, and the other structural parameters are g=1.0μm, L=34.5μm, a=20.5μm, t=100nm, and d=2μm. The arm length of each polarized patch in (e) and (f) is set unequally to ay=25μm and az=40μm, and the other structural parameters are g=1.5μm, L=50μm, w=3.0μm, t=100nm, and d=2μm. See Supplement 1, Fig. S5(a) for the definitions of the structural parameters.
Fig. 5.
Fig. 5. Observation of EP in chiral polar metamaterials. (a), (b) The experimental and (c), (d) the simulated results of CIM are shown in a spectral regime for (a), (c) dielectric and (b), (d) metallic realizations. Dotted lines represent the condition of EPs in spectral and θ domains. All simulated results were obtained using COMSOL Multiphysics.
Fig. 6.
Fig. 6. Chiral waveguides supporting the basis of modal helices. (a) Cross sections of a complex-strip waveguide using isotropic materials (graded color: silicon; purple: titanium; green: silica substrate; graded color represents the effective loss by the titanium layer). The lossless silicon (εSi=12.1) is assumed to compose the strip structure on top of the lossy titanium layer (εTi=1.66i·30.1) above a silica substrate (εSiO2=2.07), supporting both a low-loss y-polarized mode and a high-loss z-polarized mode. The complex-strip waveguide satisfies PT symmetry based on the gauge transformation (Re[εy]=Re[εz] and Im[εz]<Im[εy]<0). The effect of the loss can be controlled by changing the depth of the titanium layer. The red and blue arrows describe a corresponding point-wise anisotropic permittivity. (b) The intensity profile and the polarization (in arrows) of the eigenmodes for the structure (εyz=0. Δ=0). (c) shows the modal chirality by IL/IR as a function of Δ and tTi. (d) The absolute value of the difference between eigenvalues as a function of Δ and tTi. The intensity profile and the local chirality (IL(y,z)/IR(y,z)) at the EP are shown in (e). All results were obtained using COMSOL Multiphysics with an optical wavelength of Λ0=1500nm. L11=190nm, L12=300nm, L21=620nm, and L22=190nm.

Equations (5)

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εr=(εr0000εr0+iεi0εκ00εκ0*εr0iεi0),
CIM=|tRLtLR|=|εκ0+εi0εκ0εi0|=|1+εi0εκ01εi0εκ0|,
B0=η1,2β1,2ω·[iεi0±λPTεκ0]·eiβ1,2x.
χ1,2=β1,22·εi0εκ0,
χ1,2=Re[β1,2]2·2εκ0·(εi0εi02εκ02)εκ02+(εi0εi02εκ02)2·e2Im(β1,2)x,

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