circular-polarization-dependent mode hybridization and

Circular-polarization-dependent mode hybridizationand slow light in vertically coupled planar chiraland achiral plasmonic nanostructuresKUN JIANG,1 CHENG HE,1 XIAO-PING LIU,1 MING-HUI LU,1,* BO CUI,2 AND YAN-FENG CHEN1

1Department of Materials Science and Engineering, College of Engineering and Applied Sciences, and National Laboratory of Solid StateMicrostructures, Nanjing University, Nanjing 210093, China2Department of Electrical and Computer Engineering and Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo,Ontario N2L3G1, Canada*Corresponding author: [email protected]

Received 14 May 2015; revised 14 August 2015; accepted 17 August 2015; posted 19 August 2015 (Doc. ID 241029); published 10 September 2015

We numerically studied a square array of plasmonic nanostructures with the unit cell composed of a z-shapedchiral structure and an achiral strip pair. Circular-polarization-dependent mode hybridization between thez-shaped structure’s radiant mode and the strip pair’s subradiant mode is observed: the resonance amplitudesof two hybrid modes are different for left-handed and right-handed circularly polarized waves, and anelectromagnetically induced transparency-like transparency window is obvious for only one circularly polarizedwave. Consequently, slow light with small group velocity is only evident for the corresponding polarizationstate. © 2015 Optical Society of America

OCIS codes: (160.3918) Metamaterials; (160.1585) Chiral media; (260.3910) Metal optics.

http://dx.doi.org/10.1364/JOSAB.32.002088

1. INTRODUCTION

Metamaterials refer to one kind of artificially structured mate-rials which possess some unique properties beyond what naturalmaterials have, such as simultaneous negative permittivity andpermeability and consequently negative refraction for electro-magnetic waves [1–4]. Due to these novel properties, they havefound applications in manipulating wave propagation, amongwhich are the superlens to increase imaging resolution beyondthe conventional diffraction limit [5,6] and cloaking invisibilityto guide wave rays in the transformation media surrounding theobject [7,8]. Metallic structures have played an important rolein constructing metamaterials due to their strong electromag-netic response by surface plasmon polaritons. Considered as asystem composed of artificial atoms or molecules, namely,meta-atoms or meta-molecules, metamaterials have also beenapplied to realize some effects analogous to that in naturalmaterials. Mode hybridization through coupling building com-ponents together in one composite unit cell has been realized[9–13] and attracted much attention in the past several years, asit provides a new guide to designing metamaterials, producing avariety of novel effects. Classical analogs of some effects inatomic physics, including Fano resonance [14–19] and particu-larly electromagnetically induced transparency (EIT) [20–28],are undoubtedly among the most interesting coupling effects.The unit cell of the plasmonic structure array to mimic the EIT

effect frequently consists of an active radiant resonator directlyexcited by an external field and a passive subradiant resonatorindirectly excited by the active resonator through couplinginteraction, which in turn interacts with the active resonatorand suppresses its resonance in a narrow frequency range bydestructive interference with the external field.

At the same time, circular dichroism and optical rotationdispersion, namely, optical activity, originally existing in naturalmolecules have also been achieved by introducing chirality tothe unit cells of metamaterials [29,30], in which the specificrotation is several orders of magnitude larger than that in natu-ral chiral media. Particularly, optical activity was manifested inplanar [31] and stacked planar achiral or chiral structures[32,33], which greatly facilitates the fabrication of artificiallystructured chiral materials. Circularly polarized states, whichcorrespond to the spin states of photons, offer another typeof information carried by lights beside amplitude, phase,and frequency. The left-handed and right-handed circularly po-larized (LCP and RCP) states are degenerate in achiral media,while the degeneracy is lifted in chiral media. The lifting ofdegeneracy by introducing chirality to the achiral structureswould lead to new properties not possessed by the original me-dia, for example, the negative refraction of circularly polarizedlight [34–37]. Therefore, it is of interest to design chiral struc-tures to make some existing effects in metamaterials circularpolarization selective.

2088 Vol. 32, No. 10 / October 2015 / Journal of the Optical Society of America B Research Article

0740-3224/15/102088-08$15/0$15.00 © 2015 Optical Society of America

http://dx.doi.org/10.1364/JOSAB.32.002088

Circular-polarization-dependent mode hybridization hasalready been reported in stacked structures comprising twotwisted identical resonators of the same radiative loss[38,39]. Either hybrid mode discriminates against one circu-larly polarized wave (LCP or RCP) and in favor of the otherone (RCP or LCP), and an EIT-like transparency window isobvious for neither of the two circularly polarized waves. Toinvestigate the influence of chirality on the mode hybridizationbetween resonators of different radiative losses and obtain acircular-polarization-dependent EIT-like transparency window,we replace the achiral strip in the π-shaped structure [25,26]with a z-shaped chiral structure. The unit cell of this squarearray therefore comprises a z-shaped planar chiral structureand an achiral strip pair beneath it. Circular-polarization-dependent mode hybridization is observed in this coupledstructure: the resonance amplitudes of two hybrid modes aredifferent for left-handed and right-handed circularly polarizedwaves, and the EIT-like transparency window is obvious foronly one circularly polarized wave. Consequently and moreimportantly, it was demonstrated that the group velocity forthe corresponding polarization state is much smaller than forthe other one in the central region of the transparency window,which could be ascribed to the circular-polarization-dependentdispersion relation.

2. CIRCULAR-POLARIZATION-DEPENDENTMODE HYBRIDIZATION IN VERTICALLYCOUPLED PLANAR CHIRAL AND ACHIRALSTRUCTURES

As shown in Fig. 1, the unit cell of the square array consists of az-shaped structure at the top layer and a strip pair at the bottomlayer, which are both made of silver and separated by a polymerlayer. This coupled structure evolves from the π-shapedstructure [25,26] by replacing the achiral strip with the chiralz-shaped structure. For the π-shaped structure, realization ofobvious mode hybridization and an EIT-like transparency win-dow requires the polarization of the incident wave to be parallelto the long side of the strip. For this coupled structure withchirality, however, the EIT-like effect is expected to discrimi-nate against one circularly polarized state (LCP) and favor theother circularly polarized state (RCP).

The dimensions of this structure are as follows: the widthof the z-shaped structure g1 is 620 nm and that of the strip pairg2 is 480 nm; the length of the z-shaped structure l1 is 335 nmand that of the strip pair l2 is 200 nm; the line width of thez-shaped structure w1 is 100 nm and that of the strip pair w2 is120 nm; the thicknesses of the two oscillators t1 and t2 are both40 nm, and the vertical distance along the z axis between thetwo structures d is 30 nm; the orientation angle θ of the strippair with respect to the z-shaped structure is 38°; and the hori-zontal displacement of the strip pair with respect to the inclinedbar of the z-shaped structure s is 100 nm. The structure is laidon a quartz substrate, whose thickness is set as semi-infinite inthe finite simulation region, and the unit cell is arranged alongtwo perpendicular directions (x and y axes) to form a squarelattice, with the lattice constant of 800 nm.

The finite-difference time-domain (FDTD) method sup-ported by commercial software Lumerical FDTD Solutions

was employed to investigate the hybridization process in thiscoupled structure. The Drude model was adopted for thesilver, with the parameters provided by Ordal et al. [40]: theplasmon angular frequency ωp � 2π × 0.22 × 1016 rad∕s, andthe damping constant was set as ωτ � 2π × 0.13 × 1014 rad∕s,which is three times of that in the thin film considering thesevere scattering of near-free electrons at the grain boundaryof the strip structure [41]. The refractive index of the polymerlayer is set as 1.55 and the quartz substrate as 1.50. The trans-mission, reflection, and absorption spectra of the z-shapedstructure alone on the polymer layer are shown in Fig. 2(a).A broad transmission dip and reflection peak as well as a subtleabsorption peak indicate a radiant mode with a low quality fac-tor in the various polarized wave incidence cases, includingRCP, LCP, and linearly polarized (LP) waves with the polari-zation parallel to the inclined bar of the z-shaped structure.

The field profiles 15 nm away above the top face of thez-shaped structure at the resonance frequency (f � 242 THz)of its radiant mode for various polarized states are shown inFig. 3. The field profiles illustrate some differences of thisradiant mode between the three polarized states that are notdistinct in the transmission, reflection, and absorption spectra.From the electric and magnetic intensity field profiles, thecharge and current distribution could be deduced. The currents

Fig. 1. Schematic of the structure and its dimensions.(a) Demonstration of the evolution from the π-shaped structure byreplacing the achiral strip with a z-shaped structure. (b) Perspectiveview of the unit cell. (c) Vertical view. (d) Side view. The red marksindicate two point probes for the magnetic field intensity componentH θ. The unit cell is arranged along x and y axes to form a squarelattice, with the lattice constant of 800 nm.

Research Article Vol. 32, No. 10 / October 2015 / Journal of the Optical Society of America B 2089

are shown by white arrows whose lengths indicate the currentamplitudes. The charges accumulated at the two corners (theleft top one and the right bottom one) of the z-shaped structureare of the same amount but different signs for each polarizedstate. In the RCP wave incidence case, the current amplitude in

the two parallel bars along the y axis is much larger than that inthe inclined bar, while in the LCP wave incidence case, thecurrent amplitudes in the two parallel bars are a little smallerthan that in the inclined bar and the amplitudes are not as largeas that in the RCP incidence. In the LP wave incidence case, thecurrent distribution and amplitudes are between that for RCPand LCP waves.

For the strip pair, there are two eigenmodes, one symmetricmode j1i � �1; 1�T and the other antisymmetric mode j2i ��1; −1�T , where the elements of the vector (1 or −1) denote thecurrents on the two strips. The symmetric mode forms a pair ofradiant electric dipoles, while the antisymmetric mode forms asubradiant magnetic dipole with a large quality factor (the con-duction currents in the strip pair and the displacement currentsbetween the ends of the strip pair form a closed current loop).Since the electric fields at the two corners of the z-shaped struc-ture are of opposite directions, not rigorously antisymmetricabout the mirror plane of the two strips as shown in Fig. 1(b)by a white dashed line, the strip laid beneath it, as shown inFig. 1, would be excited in the form of superposition modejsi � c1j1i � c2j2i through the near-field quasi-electrostaticforces. The radiant loss of this superposition mode is betweenthat of the symmetric and the antisymmetric mode, lower thanthat of the z-shaped structure, and is called the subradiantmode. It should be noted here that in the LCP and RCP waveincident cases, the incident waves contain an electric compo-nent along the long side of the strip pair, so the strip pair couldbe excited by the external field beside the excitation through thenear-field quasi-electrostatic forces.

A strip pair with the resonance frequency of the antisym-metric mode almost coinciding with that of the z-shaped struc-ture’s radiant mode was laid beneath the z-shaped structure.The transmission, reflection, and absorption spectra of thecoupled structure are shown in Fig. 2(b). Two hybrid modesappear in the original frequency region of the z-shaped struc-ture’s radiant mode, which are indicated by the two transmis-sion dips, two reflection peaks, and two absorption peaks(some absorption peaks are not distinct due to the low contrastof the absorption spectrum). In the LCP and LP wave incidencecases, the frequency of the hybrid mode at higher frequency(248 THz) and the central frequency of the transparency win-dow (244 THz) are a little lower than that for RCP wave (249and 245 THz, respectively). The spectrum line profiles arevery different for the three polarized states. In the RCP waveincidence case, the resonance amplitude of the hybrid mode atthe lower frequency is comparable with that at the higher fre-quency, so an obvious EIT-like transparency window opens. Inthe LCP and LP wave incidence cases, however, the resonanceamplitude of the hybrid mode at the lower frequency is muchsmaller than that at the higher frequency, and the transparencywindow is not distinct. The polarization-dependent modehybridization is ascribed to the differences in the z-shapedstructure’s radiant mode for the three polarized states, whichleads to different near-field coupling interactions between thetwo oscillators and consequently different superposition modesof the strip pair.

To investigate the excited mode of the strip pair whencoupled with the z-shaped structure, two point probes were

Fig. 2. Comparison between the z-shaped structure alone and thecoupled structure by their transmission, reflection, and absorptionspectra. (a) Spectra for the z-shaped structure alone. (b) Spectra forthe coupled structure. The three vertical dashed lines illustrate the res-onance frequencies of the two hybrid modes and the central frequencyof the EIT-like transparency window for the RCP wave (241, 249, and245 THz, respectively). In the LCP and LP wave incidence cases, thefrequency of the hybrid mode at higher frequency (248 THz) and thecentral frequency of the transparency window (244 THz) are a littlelower than that for the RCP wave.

Fig. 3. Normalized field profiles of the z-shaped structure alone atthe resonance frequency (f � 242 THz) of its radiant mode (15 nmaway above the top face of the z-shaped structure). (a) RCP wave in-cidence case. (b) LCP wave incidence case. (c) LP wave incidence case.The red marks in the top panel illustrate the polarization of incidentwaves. The white arrows in the bottom panel illustrate the currentswith the lengths indicating the current amplitudes.


placed 15 nm away beneath the center of the bottom face ofeach strip, as shown by red marks in Figs. 1(b) and 1(c). The θcomponent of the magnetic field intensity (H θ � Hx cos θ−Hy sin θ, perpendicular to the long side of the strip pair) ofthe two probes reflects the current oscillation on the two stripsand thus the excited mode. The time evolution ofH θ was plot-ted in Fig. 4 for the resonance frequencies of the two hybridmodes and the central frequency of the transparency window inthe RCP, LCP, and LP wave incidence cases. The combinationcoefficients (c1 and c2) for the superposition mode (jsi �c1j1i � c2j2i) were obtained from the currents on the twostrips and are shown at the bottom of each panel. In the LPwave incidence case, the oscillation of the strip pair is purely

excited by the z-shaped structure through the near-field quasi-electrostatic force, so the currents on the two strips have a largephase difference Δφ (Δφ � 1.4π for f � 241 THz, Δφ �1.0π for f � 244 THz, and Δφ � 0.9π for f � 248 THz).However, as mentioned above, the oscillation of the strip pair isnot rigorously antisymmetric but a superposition of the pri-mary antisymmetric and the secondary symmetric mode, un-like that in the π-shaped structure [25,26]. The superpositionmode of the strip pair thus could not completely suppress theoscillation of the z-shaped structure’s radiant mode at the cen-tral frequency of the EIT-like transparency window (the fieldprofile of the z-shaped structure was not shown due to itscomplexity), which is therefore not as distinct as that in theπ-shaped structure. In the RCP and LCP wave incidence cases,there are also differences of varying degrees between the phasesof currents on the two strips, which indicates that the strip pairstill oscillates in the form of superposition mode, though theexternal field also excites the strip pair beside the near-fieldquasi-electrostatic force by the z-shaped structure. The ratioof c2 to c1 (c2∕c1) was shown in each panel in order to confirmthat the antisymmetric mode j2i plays an important role in theoscillation of the strip pair at the three frequencies for variouspolarized waves and that the superposition mode of the strippair is indeed subradiant.

The ratios of c2 to c1 (c2∕c1) at the three frequencies aredifferent for each polarized wave, which suggests that the modehybridization involves three modes: the z-shaped structure’s ra-diant mode and the strip pair’s two eigenmodes (j1i and j2i).The hybrid modes shown in Fig. 2(b) are the two of all thehybrid modes produced by the hybridization (other modesare not shown in this paper). The differences in the amplitudesand phase relations between the currents on the two strips forthe three polarized states show that the superposition mode ofthe strip pair is polarization dependent, which stems from thepolarization-dependent z-shaped structure’s radiant mode andconsequently the near-field coupling interactions between thetwo oscillators as mentioned above.

To further understand the mechanism underlying themode hybridization and engineer the EIT-like transparencywindow, we study the dependence of the transparency windowon the intrinsic loss of the metallic structure and the horizontaldisplacement s of the strip pair with respect to the inclined barof the z-shaped structure. The damping constant ωτ in theDrude model characterizes the intrinsic loss of the metallicstructure and was, respectively, set as one, three, and five timesthat provided by Ordal et al. [40], with the other parameters thesame as mentioned above. As shown in Fig. 5, with the decreaseof the metal loss, the contrast of the spectral line for backwardscattering (reflection) is increased, and the transparency win-dow becomes more obvious, with its width unchanged. This isin agreement with the model for two coupled oscillators [26],in which the transparency window becomes more distinct asthe damping constants of the two oscillators are both reduced.Thus, the intrinsic loss of the structure is a critical parameter forengineering the transparency window and might also be re-duced and actively tuned by incorporating gain materials [42].

The horizontal displacement s was varied from 0 to 150 nm,with a step of 50 nm, and the other parameters remained the

Fig. 4. Characterization of the mode of the strip pair in this coupledstructure by the time evolution of normalized magnetic field intensitycomponent H θ (H θ � Hx cos θ −Hy sin θ, perpendicular to thelong side of the strip pair) at three important frequencies. (a) Forthe hybrid mode at lower frequency. (b) For the hybrid mode at higherfrequency. (c) For the central frequency of the EIT-like transparencywindow. P1 and P2 indicate two point probes 15 nm away beneaththe center of the bottom faces of the two strips, as shown by redmarks in the H θ field profile at the right bottom of this figure andin Figs. 1(b) and 1(c). The H θ field profile at the right bottom(15 nm away beneath the bottom faces of the strip pair) shows the H θ

field profile at the moment of ωt � 0.12π for f � 245 THz in theRCP incidence case (marked by a vertical dashed line in the corre-sponding panel). The white arrows illustrate the currents on the twostrips with the lengths indicating the current amplitudes. c1 and c2given in each panel are the combination coefficients for the superpo-sition mode jsi � c1j1i � c2j2i, where j1i and j2i are the symmetricand antisymmetric eigenmode of the strip pair.


same as that mentioned above. For s � 0 nm, the transparencywindow nearly disappears, and the spectra are similar to thoseof the z-shaped structure alone. It suggests that the couplingbetween the two oscillators in this case is weak and that thetransparency window in the nonzero s value cases indeed stemsfrom the coupling of the two oscillators. For s set as nonzerovalues, the evolution of the transparency window is not as regu-lar as that in the π-shaped structure [25,26] (the spectra are notshown here). It is because that for different s values, the ratio ofc2 to c1 (c2∕c1) for the superposition mode of the strip pairwere different. In the π-shaped structure case, variation ofthe horizontal displacement s only affects the coupling strength,leaving the antisymmetric mode of the strip pair unchanged.Still, s is another important parameter for tailoring the trans-parency window, and a series of values of smaller step could beattempted to obtain a transparency window of desired spectralline shape.

It should be noted here that this structure does not have afour-fold rotational symmetry axis along the normal direction;in the circularly polarized wave incidence case, the transmittedand reflected waves, according to the Jones matrix theory [43],are no longer rigorously circularly polarized but elliptically po-larized. We have attempted to arrange the unit cell to create afour-fold rotational symmetry axis by rotating the neighboringunit cell by 90°, with four unit cells constituting a compositeunit cell, but the EIT-like transparency window is not obviousdue to the far-field interference. Although this coupled struc-ture does not support the propagation of rigorous RCP or LCPwave as its polarization eigenstate due to the broken four-foldrotational symmetry, the transmitted waves are dominated byonly one circularly polarized component, the one the same as

that of the incident wave, as will be proved in the followingsection. Therefore, RCP and LCP waves could be approxi-mately considered as the two polarization eigenstates of thiscoupled structure.

3. CIRCULAR-POLARIZATION-DEPENDENTSLOW LIGHT

Given that LCP and RCP waves could be approximately viewedas the two polarization eigenstates of this coupled structure, thetransmission, reflection, and absorption spectra for LCP andRCP waves shown in Fig. 2 could be considered as the spectrafor the two polarization eigenstates. Since the EIT-like trans-parency window is more obvious for the RCP wave than for theLCP wave, according to effective medium theory [44], a steepnormal dispersion of the real part of the refractive index aroundthe center of the transparency window, and consequently a slowgroup velocity originally existing in coherent media with EITeffect [45,46], is expected to be smaller for a RCP pulse than fora LCP pulse.

LCP and RCP waves should have their own dispersion re-lations kL∕R � ωnL∕R∕c0, respectively. The different groupvelocities of LCP and RCP pulses result from their own refrac-tive index dispersions nL∕R�ω� in this coupled structure,

vL∕Rg ��dkL∕Rdω

�−1

� c0

�nL∕R � ω

dnL∕Rdω

�−1

� c0nL∕Rg

; (1)

where vg denotes the group velocity, k denotes the real part ofthe wave vector, n denotes the real part of refractive index, c0 isthe velocity of light in vacuum, ng � n� ωdn∕dω denotes thegroup index, and the superscript or subscript L∕R denotes theLCP or RCP pulse.

Viewed as an effective medium, the refractive index nL∕Rcould be retrieved by the phase delays of transmitted wavesaccording to the relation nL∕R�ω� � �ΦL∕R∕2π�∕�l∕λ0�, whereΦL∕R is the phase delay by this structure, l represents itsthickness, and λ0 denotes the wavelength in vacuum corre-sponding to the angular frequency ω. It is noted here thatΦL∕R�ω� � arg�E2

L:∕R�ω�∕E1L:∕R�ω��, where E2

L:∕R�ω� is thecomplex amplitude of the transmitted wave through thiscoupled structure and E1

L:∕R�ω� is that of the transmitted wavethrough the substrate as reference; arg denotes a function forobtaining the phase of a complex number. In the LCP/RCPwave incidence case, the transmitted waves contain a smallamount of RCP/LCP waves, so the transmitted LCP/RCPwaves should be extracted by basic vector transformation,shown by Eq. (3) in the following text. Thus, the group indexnL∕Rg could be obtained by its relation with the refractive indexnL∕R . As shown in Fig. 6, the refractive index nL∕R and group

index nL∕Rg were retrieved for three values of damping constantωτ. The refractive index indeed possesses a normal dispersionregion between two abnormal dispersion regions for both LCPand RCP waves. The group index dispersion nL∕Rg �ω� suggeststhat a region of positive values corresponds to the normaldispersion region of the refractive index nL∕R . The nRg reachesits maximum at the central region of the transparency window(f � 244 THz) marked by inclined lines, while the nLg gets its

Fig. 5. Effect of the intrinsic loss of the metallic structure on themode hybridization and EIT-like transparency window. The dampingconstant ωτ in the Drude model characterizes the intrinsic loss.(a) LCP wave incidence case. (b) RCP wave incidence case.


peak near the hybrid mode at the higher frequency(f � 248 THz). This stems from the circular-polarization-dependent mode hybridization that the resonance amplitudesof the two hybrid modes are comparable for RCP waves, and bycontrast, the resonance amplitude of the hybrid mode at thehigher frequency is much larger than that at the lower fre-quency for LCP waves. The transmission coefficients in thecentral region of the window should be larger than that nearthe hybrid mode at the higher frequency. It is expected thatin the marked region, a RCP pulse should obtain a longer timedelay than a LCP pulse and both retain relatively large ampli-tudes. Figure 6 also indicates that as the metal loss is reduced,the slope of the normal dispersion of nL∕R is increased and the

nL∕Rg consequently becomes larger. It results from the fact thatthe transparency window becomes more obvious with the de-crease of the metal loss.

The above predictions by refractive index dispersion nL∕R�ω�and group index nL∕Rg could be confirmed by simulations in timedomain. LCP and RCP pulses, both with a bandwidth of about4 THz around the central frequency of 244 THz, coincidingwith the central region of the transparency window markedby inclined lines in Fig. 6 were simulated to inject into the sub-strate (as reference) and this coupled structure (including thesubstrate). For the transmitted waves, frequency spectra oftwo orthogonal electric field component, �E x�ω�; E y�ω��T ,could be obtained by applying a Fourier transformation tothe time signal of the transmitted pulses [the time factor ofthe harmonic component is defined as exp�−iωt��,

�E x�ω�E y�ω�

�� 1

2π

Z∞

0

�Ex�t�Ey�t�

�exp�iωt�dt: (2)

If each frequency component �E x�ω�; E y�ω��T contains againboth LCP and RCP components, the two orthogonal circularpolarized components EL�ω� and ER�ω� could be separatedby basic vector transformation, from the linearly polarized setof �1; 0�Tx and �0; 1�Ty to the circularly polarized set of�1; −i�TL and �1; i�TR (the transmitted wave propagates alongthe negative direction of the z axis),

EL∕R�ω� �1

2�E x�ω� � iE y�ω��: (3)

Then the time signal of the separate LCP and RCP pulses couldbe obtained by applying an inverse Fourier transformation to thefrequency spectra of the LCP and RCP pulses given by Eqs. (2)and (3),

�Ex�t�Ey�t�

�L∕R

�Z

∞

−∞EL∕R�ω�

�1�i

�exp�−iωt�dω; (4)

where �Ex�t�; Ey�t��TL∕R represents the time signal of the sepa-rate LCP and RCP pulses. It is worthwhile to note that�Ex�t�; Ey�t��TL∕R differs from �Ex�t�; Ey�t��T in that the latterone represents the time signal of the total transmitted pulse withLCP and RCP pulses mixed together.

The frequency spectra of the transmitted LCP and RCPpulses from the substrate jEL∕R�ω�j, obtained by Eqs. (2)and (3), and the time signals of the transmitted LCP and RCPpulse, �Ex�t�; Ey�t��TL∕R , obtained by Eq. (4), are shown in thetop panel of Figs. 7(a) and 7(b), with the left top panel for theLCP pulse incidence case and the right top panel for the RCPpulse incidence case. The transmitted pulse is also rigorouslythe LCP or RCP pulse, whose frequency spectra are the same,because the substrate are achiral, maintaining the polarizationand having the same transmission coefficient for any normallyincident polarized wave. The envelopes of Ex�t� and Ey�t� arealmost the same owing to the narrow bandwidth, so only theEx�t� signal is given here.

The frequency spectra of the transmitted LCP and RCPpulses from the coupled structure, jEL∕R�ω�j, are obtained byEqs. (2) and (3), and the time signals of the transmitted LCPand RCP pulses, �Ex�t�; Ey�t��TL∕R , obtained by Eq. (4) areshown in the bottom panel of Figs. 7(a) and 7(b). In theLCP pulse incidence case, as shown in the bottom panel ofFig. 7(a), the transmitted pulse is dominated by a LCP pulse,with a small RCP pulse scattered by the coupled structure;similarly, in the RCP pulse incident case, as shown in the bot-tom panel of Fig. 7(b), the transmitted wave is dominated by aRCP pulse, with a small scattered LCP pulse. In fact, the timesignals for the LCP and RCP pulses were mixed together in thesimulation and were separated by Eqs. (2)–(4). The convertedRCP or LCP pulse of small amplitude stems from the fact thatthe polarization eigenstates of this coupled structure are notrigorously LCP and RCP and could be neglected comparedwith the primary LCP or RCP pulse. It thus confirms the claimat the end of the above section that the transmitted waves aredominated by one circularly polarized component of the samehandedness as that of the incident wave, and the RCP and LCP

Fig. 6. Effective refractive index nL∕R and group index nL∕Rg re-trieved from the phase delays of transmitted waves and effect ofthe intrinsic loss of metallic structure on them. The damping constantωτ in the Drude model characterizes the intrinsic loss. (a) LCP waveincidence case. (b) RCP wave incidence case. The region filled withinclined lines illustrates the frequency band where the RCP pulse isexpected to be slower than the LCP pulse and the transmitted pulsescould retain relatively large amplitudes.


waves could be approximately considered as the two polariza-tion eigenstates of this coupled structure.

The time delay of the RCP pulse (Δt ≈ 39 fs) is muchlonger than that for the LCP pulse (Δt ≈ 10 fs), confirmingthe predictions by refractive index dispersion nL∕R�ω� andgroup index nL∕Rg . Besides, both the transmitted LCP pulse(LCP pulse incidence) and the RCP pulse (RCP pulse inci-dence) maintain relatively large amplitudes and that of theRCP pulse is a little larger than LCP pulse. It is because thefrequency band of the pulses coincides with the central regionof the transparency window and that the transmission coeffi-cients of RCP waves are larger than LCP waves. It is noted herethat both the transmitted pulse signals from the substrate andthat from the coupled structure were acquired 7 μm away fromthe structure, where the fields are homogeneous for frequenciesno higher than 248 THz, which covers most of the frequencycomponents of the pulses.

As shown in Fig. 7(c), as the metal loss is reduced, the timedelays of the pulses are increased, which is in agreement withthe variation of the group index nL∕Rg with the loss shown inFig. 6. As mentioned in Section 2 of this paper, if the metalloss could be compensated by incorporating a gain medium,

the group velocity would be further slowed down and activelytuned by optical or electrical pumping [42].

4. CONCLUSIONS

Circular-polarization-dependent mode hybridization betweenresonators of different radiative losses was numerically studiedin a square array of a plasmonic nanostructure array whose unitcell is composed of a chiral z-shaped structure providing theradiant mode and an achiral strip pair providing the subradiantmode. The EIT-like transparency window is more obvious forRCP waves than for LCP waves. The circular-polarization-dependent EIT effect was first realized through the destructiveinterference of a correlated exciton pair of opposite spins inGaAs quantum wells performed at 10 K, in which the probeand coupling beams are required to have opposite circularpolarization [47]. The artificially structured material proposedin this paper provides another route to the circular-polarization-dependent EIT-like effect at room temperature, with only onebeam involved. It thus expands the region of metamaterials,which explore novel properties natural materials do not possess.Circular-polarization-dependent slow group velocity is anotherinteresting concept, which means that the steep normaldispersion of the refractive index (dominated by electric permit-tivity in this structure) discriminates against one circular polar-ized wave and is in favor of the other one in a specific frequencyregion, and it is different from that in achiral media. Thecircular-polarization-dependent slow group velocity might findsome applications in enhancing circular polarization selectivelight–matter interactions.

Funding. National Basic Research Program of China(2012CB921503, 2013CB632702); National Natural ScienceFoundation of China (NSFC) (1134006, 11474158); NaturalScience Foundation of Jiangsu Province (Jiangsu ProvincialNatural Science Foundation) (BK20140019); PriorityAcademic Program Development of Jiangsu Higher EducationInstitutions (PAPD).

Acknowledgment. The authors are grateful to Prof.Wei-Hua Zhang (Department of Quantum Electronics andOptical Engineering, College of Engineering and AppliedSciences, Nanjing University, China) for advice on the manu-script and to Li-Yang Zheng (LAUM, CNRS, Université duMaine, Le Mans, France) for advice on modifying the z-shapedstructure.

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Fig. 7. Demonstration of the time delays of far-field transmittedpulses. (a) LCP pulse incidence case. (b) RCP pulse incidence case.(c) Effect of metal loss on the time delays. The insets in (a) and (b) givethe frequency spectra of corresponding pulses. The black curves illus-trate LCP pulses and red curves illustrate RCP pulses. The dampingconstant ωτ in the Drude model characterizes the intrinsic loss. In thesimulations shown by (a) and (b), ωτ � 2π × 13 THz.


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circular-polarization-dependent mode hybridization and

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