Optical rotation and electromagnetically induced transparency in a chiral metamaterial with C<sub>4</sub> symmetry

Wei Wu; Wei Wu; Wenbing Liu; Wenbing Liu; Zenghui Chun; Yonghong Ling; Jifei Ding; Xiaochuan Wang; Lirong Huang; Lirong Huang; Hanhui Li; Hanhui Li

doi:10.1364/OE.403421

1. Introduction

Chirality refers to the geometric property of a structure that cannot be superimposed with its mirror by simple rotation or translation [1], and the media composed of chiral molecules or particles are called chiral media. With the development of optical nanotechnology, artificial chiral materials, especially chiral metamaterial with strong electromagnetic responses, have been demonstrated for various applications [2], including ultra-thin waveplates [3], circular polarizers [4–6], asymmetric transmission [7], negative refractive index [8], nonlinearity enhancement [9], biological monitoring [10,11], etc.

When linearly polarized light propagates through chiral media, its plane of polarization will rotate, which is called optical rotation [12]. This phenomenon was first observed more than 200 years ago [13,14], after two centuries of research, relatively complete theory has been developed for the propagation of electromagnetic waves in bi-isotropic media (BI media), i.e., a kind of isotropic chiral media [15]. However, there are still some deficiencies in theoretical explanations about two-dimensional chiral materials. For example, magnetic dipole plays an important role in chiral response and keeping the symmetry of material equations [16,17], nonetheless, when Born-Kuhn model is employed to calculate the oscillation intensity in optical activity and optical rotation, magnetic dipole and magnetic field are sometimes indirectly embodied via electric displacement by the requirement of symmetry condition [18–20]. Another popular explanation is helix model. Helix is a bent wire which has straight and looped-sections. The straight-section excites electric dipole, while the looped-section induces magnetic dipole [21–24]. The helix model can well explain the optical rotation in helix structure, but it is not suitable for two-dimensional chiral metamaterials (i.e., chiral metasurfaces), because electric quadrupole is not excited in helix structure and this model does not take electric quadrupole into consideration. However, electric quadrupole will be excited in many other chiral metasurfaces, such as two-dimensional metamaterials with 4-fold rotational symmetry (C₄ symmetry), which consist of corner-stacked rods [18], enantiomeric helicoidal bilayered structures [25], Four-U-SRRs [26], conjugated gammadion resonator pairs [27], and so on. In these cases, the effect of electric quadrupole on optical rotation should be considered.

In this paper, we design and fabricate a double-layered chiral metamaterial with C₄ symmetry, which simultaneously exhibits optical rotation and electromagnetically induced transparency (EIT) phenomena. Compared with other metamaterial-based structure for optical rotation, the device we designed has high transmission because of the EIT effect. More importantly, by using analytical equivalent circuit model and Lorentz's oscillator model, we interpret the physical mechanism behind EIT and optical rotation, and find that magnetic dipole and electric quadrupole play important roles in optical rotation and keeping the symmetry of material equations.

The rest of the paper is organized as follows. In sections 2 and 3, we present device structure, EIT phenomenon and optical rotation observed in simulation and experimental results. In section 4, we interpret EIT effect by using analytical equivalent circuit model. In section 5, based on Lorentz's oscillator model, we investigate the physical mechanism behind optical rotation phenomenon and derive material equations, revealing why and how magnetic dipole and electric quadrupole play important roles in optical rotation and keeping the symmetry of material equations. In section 6, we use the relevant parameters in the derived material equations to verify EIT and optical rotation effects. Finally, conclusion is given in section 7.

2. Device structure

Figure 1 schematically shows the structure unit of the designed double-layered chiral metamaterial. The upper and lower layers are composed of copper (Cu) metallic bars, the middle layer is a FR-4 substrate. All metallic bars have the same size parameters, and they are arranged such that the metamaterial satisfies C₄ symmetry to eliminate anisotropy [18,28]. The orientation angle between the upper and lower metallic bars is θ. It is worth mentioning that this chiral metamaterial is a kind of BI medium [18,21,27,29]. As shown in Figs. 1(b)–1(d), the geometric parameters are as follows: the thicknesses of the FR-4 substrate and Cu are h₁= 0.4 mm, h₂= 0.07 mm, respectively; period p = 20 mm, the width and length of Cu bars are l₁= 2 mm, l₂= 8 mm, respectively; the distances are w₁= 4 mm and w₂= 5 mm, and orientation angle θ is 55°. Unless otherwise specified, these parameters keep unchanged throughout the paper. The conductivity of Cu is σ_Cu= 5×10⁷S/m. The relative permittivity of FR-4 is ε_FR-4= 4.3 + 0.025i. Because the period p is smaller than the operation wavelength, only zeroth-order diffraction takes place [30,31].

Fig. 1. Structure unit of the metamaterial. Perspective view (a), side view (b), top view (c) and bottom view (d).

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The photograph of the fabricated metamaterial is shown in Fig. 2(a), and experimental measurement is carried out in a microwave anechoic chamber shown in Fig. 2(b), in which the incoming and transmitted electromagnetic waves are measured and analyzed by a microwave network analyzer (Agilent PNA-X N5264A). While for numerical simulations, we adopt full three-dimensional finite-difference time-domain method.

Fig. 2. (a) Photograph of the fabricated metamaterial. (b) Experimental setup.

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3. EIT phenomenon and optical rotation observed in simulation and experiment

The simulation and experimental transmittance spectra shown in Fig. 3(a) are obtained when the metamaterial is under the normal incidence of y-polarized electromagnetic waves. One can see obvious EIT phenomenon peaking at 13.4 GHz for the simulation result, and 14.2 GHz for the experimental one. The 0.8 GHz (14.2 GHz - 13.4 GHz) discrepancy is mainly due to fabrication imperfection.

Fig. 3. (a) Simulated (blue line) and experimental (red line with dot markers) transmittance spectra. (b) Simulated polarization azimuth rotation angle (θ_R) and (c) ellipticity (η) of the transmitted wave as a function of frequency.

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In addition to EIT phenomenon, this metamaterial also exhibits stronger optical rotation effect around the transparency window. Figures 3(b) and 3(c) plot the simulated polarization azimuth rotation angle (θ_R) and ellipticity (η) of the transmitted wave as a function of frequency. Here, the polarization azimuth rotation angle is θ_R = 1/2[arg(T_L) - arg(T_R)] and ellipticity is η = 1/ 2 tan^-1[(|T_L|² -|T_R|²)/(|T_L|² +|T_R|²)]. T_L and T_R are the transmissions of the left-handed and right-handed circularly polarized waves, respectively [27]. As can be seen in Figs. 3(b) and 3(c), there are obvious optical rotation phenomena from 12.8 GHz to 15.3 GHz. Especially at the transmission peak 13.4 GHz, the polarization direction of the transmitted wave is rotated counterclockwise by about 33° with respect to the y-polarized incident wave, and η is nearly zero (-0.3°), meaning the transmitted wave is almost linearly polarized wave.

In what follows, we will investigate the physical mechanisms behind the EIT and optical rotation phenomena.

4. Investigation of EIT effect by analytical equivalent circuit model

First, we start with a relatively simple structure, which has only a layer of Cu bars on the upper surface of the FR4 substrate, as shown in the inset of Fig. 4(a). The corresponding transmittance spectrum under the normal incidence of y-polarized waves is plotted in Fig. 4(a). As can be seen, a transmission dip appears at 13.5 GHz, which is a resonance peak due to the collective oscillation of electrons in the metal.

Fig. 4. Transmittance spectra when there is only a layer of Cu bars (a) and when the two layers of Cu bars are arranged in parallel in the upper and lower layers (b). Current density distributions in the metallic bars at 9.2 GHz (c) and at 13.9 GHz (d), the black rectangle marks the outline of the metallic bar. Circuit diagram at 9.2 GHz (e) and at 13.9 GHz (f).

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Then, we consider a double-layer metallic structure which has two layers of Cu bars arranged in parallel (i.e. θ = 0°) on the upper and lower surfaces of the FR-4 substrate, as shown in the inset of Fig. 4(b). The corresponding transmittance spectrum for normally incident y-polarized waves is plotted in Fig. 4(b). As can be seen, two transmission dips appear, one is oscillation peak A at 9.2 GHz, while the other is oscillation peak B at 13.9 GHz, which is very near the oscillation peak 13.5 GHz of the single-layer metallic structure shown in Fig. 4(a). We further map the current density distributions in the metallic bars at 9.2 GHz in Fig. 4(c), it shows the current directions in the upper and lower metallic bars are anti-parallel to each other, and this signifies the existence of a current loop which can induce a magnetic-dipole along the x-direction at oscillating peak A. In contrast, the current directions in the upper and lower metallic layers are parallel to each other at 13.9 GHz, as shown in Fig. 4(d), it signifies the existence of electric-dipole moment along the y-direction at oscillating peak B.

By using the equivalent circuit model [32–34], we can classify oscillation peak A at 9.2 GHz as antisymmetric mode generated by magnetic response. In its equivalent circuit shown in Fig. 4(e), the black arrow represents current direction; and the upper and lower metallic bars can be equivalent to a parallel-plate capacitor C_m, while the adjacent metallic bars can be equivalent to capacitor C_e. Meanwhile, the upper and lower metallic bars can also be equivalent to two inductance coils, whose inductances are both L_m. And then the antisymmetric oscillation of free electrons in the upper and lower metallic bars is equivalent to the antiparallel currents flowing through the two inductance coils.

In a similar way, Fig. 4(f) shows the equivalent circuit for oscillation peak B at 13.9 GHz, which is classified as symmetric mode excited by electric response. The symmetric oscillation of free electrons in the upper and lower metallic bars is equivalent to the parallel currents flowing through the two inductance coils.

According to the analytical equivalent circuit model, inductance L_m, capacitance C_m and oscillation frequency f_m for the magnetic response can be deduced as [32]:

(1)$${L_m} = \frac{{\mu {h_1}{l_2}}}{{2{l_1}}}$$

(2)$${C_m} = {c_1}\frac{{{\varepsilon _r}{\varepsilon _0}{l_1}{l_2}}}{{{h_1}}}$$

(3)$${f_m} = \frac{1}{{2\pi \sqrt {{L_m}({C_m} + {C_e})} }} \approx \frac{1}{{2\pi \sqrt {{L_m}{C_m}} }} = \frac{1}{{2\pi \sqrt {{c_1}{\varepsilon _r}/2} }}\frac{c}{{{l_2}}}$$

in which μ is permeability, approximately equal to the permeability μ₀ in vacuum. ε₀ is dielectric constant in vacuum, c = 1/(μ₀ε₀)^1/2 is the speed of light in vacuum, ε_r is relative dielectric constant. l₁ and l₂ are the width and length of the metallic bar, respectively; h₁ is the thicknesses of the FR-4 substrate. c₁ is the efficiency factor of capacitor. Because charge density is not uniformly distributed in the metallic bars, capacitance C_m is reduced and then the efficiency factor c₁ of the capacitor is introduced. Besides, C_m is also strongly dependent on the orientation angle θ between the upper and lower metallic layers because the effective area of the parallel-plate capacitor will change when twisting the two metallic-bar layers. For example, the effective area reaches the maximal value of l₁l₂ when θ=0°.

For the electric response mode, its oscillation frequency is f_e = 1/[2π(L_eC_e)^1/2] [32]. As illustrated in Fig. 4(f), current does not pass through capacitance C_m. As a consequence, electric response frequency f_e will keep relatively stable when C_m is changed. This is in sharp contrast to magnetic response frequency f_m, which will increase when C_m is decreased, as expressed in Eq. (3).

As we know, when two resonant modes are at the same frequency and with proper phase difference, strong coupling effect will take place and thus induce EIT phenomenon. For the double-layer metallic structure shown in Fig. 4(b), no EIT effect is generated because the magnetic and electric response modes have unequal resonant frequency, with f_m ( = 9.2 GHz) much smaller than f_e ( = 13.9 GHz).

In order to induce EIT phenomenon, one should first make f_m nearly equal to f_e. Noting that f_m will increase with a reduced C_m and remembering that capacitance C_m is heavily dependent on the orientation angle θ, we investigate the dependence of f_e and f_m on θ for a double-layer structure, which has two pairs of Cu bars within a structural unit, as shown in Fig. 5(a). Its transmittance spectra for different values of θ is given in Fig. 5(b), in which the varying trend of f_e and f_m is marked by dashed lines. It should be mentioned that, for the purpose of clarity, the transmittance (T) spectra are vertically displaced by T = 1 each time when θ is increased. With the increase of θ, f_m is significantly increased as expected, because the increase in θ will decrease the effective area of the capacitor C_m; in contrast, f_e is not much sensitive to θ. As a consequence, f_m finally equals f_e when θ is increased to 55°. However, even though f_e = f_m, EIT effect is hardly observable in this case, as show in Fig. 5(b), this is because the electric and magnetic resonance modes do not satisfy the required phase relationship for EIT effect [35–38], i.e., their radiating transmission electromagnetic fields are not in-phase. Only when their phase difference is integer multiples of 2π may the superposition of the two transmission waves result in constructive interference and thus a maximal transmittance [39–41].

Fig. 5. (a) A double-layer structure with two pairs of metallic bars in the structural unit, the orientation angle is θ. (b) Transmittance spectra for different angles, the blue and red dashed lines are the guide to the eye for f_e and f_m, respectively. (c) The C₄ symmetrical metamaterial structure. (d) Transmittance spectra of the C₄ symmetric structure when θ is 20° and 55°. For clarity, spectra are vertically displaced by T = 1 each time when θ is increased.

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In order to induce obvious EIT effect, we further add another two pairs of Cu bars to the double-layer structure shown in Fig. 5(a), thus obtaining a C₄ symmetrical metamaterial structure as shown in Fig. 5(c), whose transmittance spectra at θ = 20° and 55° are shown in Fig. 5(d). When θ = 20°, f_e and f_m are 13.5 GHz and 10.3 GHz, respectively, matching well with the corresponding values shown in Fig. 5(b). This is to say, after adding another two pairs of metallic bars, f_e and f_m both keep nearly unchanged. This conclusion also holds for other values of θ (not shown here). And when θ = 55°, obvious EIT phenomenon takes place around 13.4 GHz, as shown in Fig. 5(d), this is because electric and magnetic response modes not only have the same frequency, but also meet the required phase condition for EIT effect.

To briefly conclude, we in this section reveal how EIT phenomenon is generated by analyzing electric and magnetic resonance modes and their coupling effects. In the next section, we will continue to investigate the physical mechanism behind optical rotation effect and reveal how magnetic dipole and electric quadrupole play their roles in the material equations of BI media.

5. Interpretation of optical rotation by Lorentz's coupled oscillator model

As we know, bi-isotropic media (BI media) are chiral media, whose material equations are [15]

(4)$${\textbf D} = \mathrm{\varepsilon }{\textbf E} + \left( {\chi - i\kappa \sqrt {{\mu_0}{\varepsilon_0}} } \right){\textbf H}$$

(5)$${\textbf B} = \mu {\textbf H} + \left( {\chi + i\kappa \sqrt {{\mu_0}{\varepsilon_0}} } \right){\textbf E}$$

in which E is electric field, B is magnetic flux density, D is electric displacement, H is magnetic field, ε and μ are the dielectric constant and magnetic permeability of the media, χ is Tellegen parameter, κ is chirality parameter. Material Eqs. (4) and (5) are symmetric, both containing χ and κ. When χ = 0, the BI medium is reciprocity, it is called as Pasteur medium [15].

As we mentioned above, our designed chiral metamaterial is a BI medium, which obeys material Eqs. (4)–(5) and exhibits optical rotation effect. However, it is not much clear why the chiral metamaterial has such material equations. In the following we will explain how our designed structure satisfies the material equations of BI media.

Here we view the double-layer metallic bars as a whole, namely, a metallic bar-pair. The orientation angle between the upper and lower metallic bars is θ and the layer-to-layer separation distance is d. Under the normal incidence of a polarized wave, the oscillation of free electrons in the metallic bars generates electric current, which excites two orthogonal oscillation modes, i.e., mode 1 and mode 2, as indicated in Fig. 6. The black arrow indicates the direction of the current, and the current density is set as J.

Fig. 6. Oscillation modes in the two-layer metallic bar-pair. (a) mode 1; (b) mode 2. The black arrow indicates the direction of the current. The “+” and “-” signs represent positive and negative charges, respectively.

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For mode 1, the y-direction components of the current in the two-layer metallic bars are parallel to each other; while for mode 2, they are antiparallel to each other. The combination of these two modes with different amplitudes and phases can represent all oscillation modes. By analyzing the two modes, we can easily identify the existence of magnetic dipole. For example, in mode 1, the antiparallel x-components of current in the two-layer metallic bars form magnetic dipole moment m, while the parallel y-components of current in the two-layer metallic bars form electric dipole moment p. Similar conclusion holds for mode 2.

In what follows, we will first derive current density J by using Lorentz's oscillator model, and further quantitatively analyze how the current produces polarization P and magnetization M.

5.1 Derivation of current density J

To drive material equations, one should first calculate current density J, then calculate polarization P and magnetization M. Here we calculate current density J by applying Lorentz's coupled oscillator model, in which charge oscillations in the metallic bars are analogy to harmonic oscillators [18,19,42,43]. The center of the metallic bar-pair is set at r₀(0,0,z₀), and the oscillation intensity of the two harmonic oscillators in the upper and lower metallic layer are u₁ and u₂, respectively. Then, we have

(6)$$\left( {\frac{{{d^2}}}{{d{t^2}}} + \gamma \frac{d}{{dt}} + \omega_0^2} \right){u_1} + \xi {u_2} = \frac{q}{m}(\sin \frac{\theta }{2}{E_x} + \cos \frac{\theta }{2}{E_y}){e^{ - i\omega t - i{k_0}\left( {{z_0} + \frac{d}{2}} \right)}}. $$

(7)$$\left( {\frac{{{d^2}}}{{d{t^2}}} + \gamma \frac{d}{{dt}} + \omega_0^2} \right){u_2} + \xi {u_1} = \frac{q}{m}( - \sin \frac{\theta }{2}{E_x} + \cos \frac{\theta }{2}{E_y}){e^{ - i\omega t - i{k_0}\left( {{z_0} - \frac{d}{2}} \right)}}$$

in which ω₀ is the resonance frequency of harmonic oscillator, γ is the damping coefficient, q and m are the charge and effective mass of the bound electrons, respectively. ξ is the coupling coefficient between the upper and lower harmonic oscillators, and E_xexp(-iωt-ik₀z₀) and E_yexp(-iωt-ik₀z₀) are the components of the incident electric field along the x and y directions, respectively. By setting φ = -k₀d and using exp(iφ/2) ≈ 1+iφ/2, we convert the oscillation intensity of the upper and lower resonators to that of mode 1 and mode 2 shown in Fig. 6, then we obtain [20]:

(8)$${u^{\prime}_1} = \frac{1}{2}({u_1} + {u_2})$$

(9)$${u^{\prime}_2} = \frac{1}{2}({u_1} - {u_2})$$

where u₁′, u₂′ are the oscillation intensity of mode 1 and mode 2, which can be deduced by combining Eqs. (6)–(9) and written as

(10)$${u^{\prime}_1} = ({F_ + } + {F_ - })(\cos \frac{\theta }{2}{E_y} + i\frac{\varphi }{2}\sin \frac{\theta }{2}{E_x}){e^{ - i\omega t - i{k_0}{z_0}}}$$

(11)$${u^{\prime}_2} = ({F_ + } - {F_ - })(\sin \frac{\theta }{2}{E_x} + i\frac{\varphi }{2}\cos \frac{\theta }{2}{E_y}){e^{ - i\omega t - i{k_0}{z_0}}}$$

where F₊ and F₋ are

(12)$${F_ + } = \frac{q}{m}\frac{{\omega _0^2 - i\gamma \omega - {\omega ^2}}}{{{{(\omega _0^2 - i\gamma \omega - {\omega ^2})}^2} - {\xi ^2}}}$$

(13)$${F_ - } ={-} \frac{q}{m}\frac{\xi }{{{{(\omega _0^2 - i\gamma \omega - {\omega ^2})}^2} - {\xi ^2}}}$$

The average current density of mode 1 and mode 2 are J₁ and J₂, respectively, which satisfy [19]

(14)$${J_j} ={-} i\omega \frac{q}{{{V_m}}}{u^{\prime}_j}\quad (j\,=\,1,2)$$

in which V_m is the volume of a single metallic bar.

After obtaining J₁ and J₂, we can further calculate polarization P and magnetization M.

5.2 Calculation of polarization P and magnetization M

For convenience, we filter out exp(-iωt-ik₀z₀) and obtain dipole moment by comparing the actual radiated field with that of metallic bar-pair [24]. In mode 1, the far-zone vector potential A of the wave propagating in the z-direction can be deduced and written as

(15)$${\textbf A}({\textbf r} )= \frac{{{\mu _0}}}{{4\pi }}\smallint \frac{{{\textbf J}({{\textbf r^{\prime}}} ){e^{ik|{{\textbf r} - {\textbf r^{\prime}}} |}}}}{{|{{\textbf r} - {\textbf r^{\prime}}} |}}{d^3}r^{\prime}$$

After approximate calculation and derivation processes (see Appendix 1), we get

(16)$${\textbf A}({\textbf r} )= \frac{{{N_0}{V_0}{\mu _0}{e^{ik({R - {z_0}} )}}}}{{4\pi R}}\left( {\frac{{\partial {{\textbf P}_{\textbf 1}}}}{{\partial t}} + ik{{\hat{{\boldsymbol e}}}_{\boldsymbol R}} \times {{\textbf M}_{\textbf 1}}} \right)$$

Polarization P₁ and magnetization M₁ for mode 1 are deduced as

(17)$${{\textbf P}_{\textbf 1}} = 2i\frac{1}{\omega }\cos \frac{\theta }{2}{J_1}\frac{{{V_m}}}{{{V_0}}}\hat{{\boldsymbol y}}$$

(18)$${{\textbf M}_{\textbf 1}} = d\sin \frac{\theta }{2}{J_1}\frac{{{V_m}}}{{{V_0}}}\hat{{\boldsymbol y}}$$

where r = r(0,0,R), ${\hat{{\boldsymbol e}}_{\boldsymbol R}} = ({\textbf r} - {{\textbf r}_{\textbf 0}})/|{{\textbf r} - {{\textbf r}_{\textbf 0}}} |= \hat{{\boldsymbol z}}$, $\hat{{\boldsymbol x}}$ and ŷ are the unit vectors along the x-axis and y-axis, respectively. The detailed calculation and derivation processes for getting Eqs. (16)–(18) are given in Appendix 1.

In the same way, the polarization P₂ and magnetization M₂ for mode 2 can be obtained by

(19)$${{\textbf P}_{\textbf 2}} = 2i\frac{1}{\omega }\sin \frac{\theta }{2}{J_2}\frac{{{V_m}}}{{{V_0}}}\hat{{\boldsymbol x}}$$

(20)$${{\textbf M}_{\textbf 2}} ={-} d\cos \frac{\theta }{2}{J_2}\frac{{{V_m}}}{{{V_0}}}\hat{{\boldsymbol x}}$$

By now, P₁, P₂, M₁, M₂, namely, the polarization and magnetization of mode 1 and mode 2, are all obtained for the double-layer structure shown in Fig. 5(a), which has only two pairs of metallic bars in a structural unit. Rotating the two metallic bar-pairs by 90°, one can obtain another two metallic bar-pairs. Then inserting them into the structural unit, one can obtain the C₄ symmetric metamaterial, which has 4 pairs of metallic bars in one period, as shown in Fig. 5(c).

When the metallic bars are rotated by 90°, the polarization P₁^′, P₂^′ and magnetization M₁^′, M₂^′ of mode 1 and mode 2 can be obtained by using coordinate transformation (see Appendix 2). Ignoring the influence of metallic bars on each other, the total polarization P of the C₄ symmetric metamaterial is

(21)$$\begin{array}{l} {\textbf P} = \mathbf{P}=\mathbf{P}_{1}+\mathbf{P}_{2}+\mathbf{P}_{1}^{\prime}+\mathbf{P}_{2}^{\prime} = \frac{q}{{{V_0}}}[2({F_ + } + {F_ - }\cos \theta ){E_x} + ikd{F_ - }\sin \theta {E_y}]\hat{{\boldsymbol x}}\\ + \frac{q}{{{V_0}}}[2({F_ + } + {F_ - }\cos \theta ){E_y} - ikd{F_ - }\sin \theta {E_x}]\hat{{\boldsymbol y}} \end{array}$$

And the total magnetization M is

(22)$$\begin{array}{l} \textrm{M} = \mathrm{M}_{1}+\mathrm{M}_{2}+\mathrm{M}_{1}^{\prime}+\mathbf{M}_{2}^{\prime} = \omega d\frac{q}{{{V_0}}}[ - i{F_ - }\sin \theta {E_x} + k\frac{d}{2}({F_ + } - {F_ - }\cos \theta ){E_y}]\hat{{\boldsymbol x}}\\ + \omega d\frac{q}{{{V_0}}}[ - i{F_ - }\sin \theta {E_y} - k\frac{d}{2}({F_ + } - {F_ - }\cos \theta ){E_x}]\hat{{\boldsymbol y}} \end{array}$$

The detailed derivation processes for the transformation matrix, P₁^′, P₂^′, M₁^′, M₂^′, P and M are given in Appendix 2.

According to B(r,t) = B(r)exp(-iωt), we get ∇×E = -∂B/∂t = iωB. Noting that the plane wave is normally incident on the metamaterial along the negative z-axis, we have E_z = 0 and B_z = 0. Then, we have iωB_x = -∂E_y/∂z = ikE_y, iωB_y = ∂E_x/∂z = -ikE_x. With these, we can rewrite Eqs. (21)–(22) as [22]

(23)$${P_j} = \frac{{2q}}{{{V_0}}}({F_ + } + {F_ - }\cos \theta ){E_j} + i\frac{q}{{{V_0}}}\omega d{F_ - }\sin \theta {B_j}$$

(24)$${M_j} = \frac{q}{{2{V_0}}}{\omega ^2}{d^2}({F_ + } - {F_ - }\cos \theta ){B_j} - i\frac{q}{{{V_0}}}\omega d{F_ - }\sin \theta {E_j}$$

in which j = x, y.

5.3 Material equations and the role of magnetic dipole moment in material equations

One can readily note the coefficients of the second terms on the right side of Eqs. (23)–(24) are of equal magnitude (q/V₀ωdF_-sinθ) but with opposite signs, because polarization and magnetization strictly satisfy the specific relationship as given in Eqs. (17) and (18), and this relationship will lead to the symmetry of material equations. By substituting polarization P and magnetization M into D = ε₁E + P and B = μ₀H+μ₀M (here ε₁ is the dielectric constant of the environment), we get the following material equations

(25)$${D_j} = \varepsilon {E_j} - i\kappa \sqrt {{\mu _0}{\varepsilon _0}} {H_j}$$

(26)$${B_j} = \mu {H_j} + i\kappa \sqrt {{\mu _0}{\varepsilon _0}} {E_j}$$

where j = x, y, and

(27)$$\varepsilon = {\varepsilon _1} + 2\frac{q}{{{V_0}}}({F_ + } + {F_ - }\cos \theta ) + \frac{{{{(\frac{q}{{{V_0}}}\omega d{F_ - }\sin \theta )}^2}}}{{\frac{1}{{{\mu _0}}} - \frac{q}{{{V_0}}}\frac{{{\omega ^2}{d^2}}}{2}({F_ + } - {F_ - }cos\theta )}}$$

(28)$$\mu = \frac{1}{{\frac{1}{{{\mu _0}}} - \frac{q}{{{V_0}}}\frac{{{\omega ^2}{d^2}}}{2}({F_ + } - {F_ - }cos\theta )}}$$

(29)$$\kappa ={-} \frac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }}\frac{{\frac{q}{{{V_0}}}\omega d{F_ - }\sin \theta }}{{\frac{1}{{{\mu _0}}} - \frac{q}{{{V_0}}}\frac{{{\omega ^2}{d^2}}}{2}({F_ + } - {F_ - }cos\theta )}}$$

Material Eqs. (25)–(26) both contain coefficient κ and exhibit symmetrical relationship, which are similar to Pasteur medium's material equations [15]. Therefore, optical rotation will occur when a linearly polarized wave propagates through the designed metamaterial.

From above, we can conclude that, it is by taking magnetization M into consideration that we can obtain the above material Eqs. (25) and (26), which explicitly include magnetic field and hence display perfect symmetric feature. By contrast, some literatures do not analyze the effect of magnetic dipole moment m (magnetization M = m/unit volume), therefore the derived material equations do not explicitly address magnetic field [18–20]. This implies that magnetic dipole moment m plays an indispensable role in keeping the symmetry of material equations. Apart from m, later on we will find that electric quadrupole also plays an important role in optical rotation.

5.4 Role of electric quadrupole in optical rotation

Taking mode 1 of the designed C₄ symmetrical chiral metamaterial for example, on one hand, its magnetization M₁ can be calculated by using formula (18), i.e. M₁ = sin(θ/2) dJ₁V_m/V₀ŷ. On the other hand, we can also calculate it according to the strict definition of magnetization [44], that is,

(30)$${\textbf M}({{\textbf r},t} )= \frac{\textrm{1}}{\textrm{2}}\oint {\textbf J}({{\textbf r},t} )\times d{\boldsymbol l}$$

Substituting current density J₁ for mode 1(see Eq. (14)) into Eq. (30), we get the magnetization for mode 1,

(31)$$\mathbf{M}_{1}^{\prime \prime} = \frac{1}{2}d\sin \frac{\theta }{2}{J_1}\frac{{{V_m}}}{{{V_0}}}\hat{{\boldsymbol y}}$$

Comparing formula (31) with (18), we surprisingly find that M₁^″ is not equal to M₁ as expected, but M₁ = 2M₁^″. How does this contradiction happen? According to the definition of magnetization, we think M₁^″ should be more credible. However, if we replace M₁ with M₁^″ in Eq. (22), then the resulting material equations will lose symmetry. What is wrong with it?

Later on, we will understand that this is due to the existence of electric quadrupole moment. It actually plays an important role in optical rotation and hence in material equations, however, we didn’t notice this because it is not explicitly included in the formula for far-zone vector potential A, i.e., Eq. (16).

By using the strict definition for electric dipole moment p, magnetic dipole moment m^″ and electric quadrupole moment Q, which are

(32)$${\textbf p} = \smallint {\boldsymbol x^{\prime}}\rho ({{\boldsymbol x^{\prime}}} ){d^3}x^{\prime}$$

(33)$${\textbf m^{\prime\prime}} = \frac{1}{2}\smallint {\boldsymbol x^{\prime}} \times {\textbf J}({{\boldsymbol x^{\prime}}} ){d^3}x^{\prime}$$

(34)$${Q_{ij}} = \smallint {x^{\prime}_i}{x^{\prime}_j}\rho ({{\boldsymbol x^{\prime}}} ){d^3}x^{\prime}$$

we obtain p, m^″ and Q for mode 1,

(35)$${\textbf p} = 2{N_0}\textrm{cos}\frac{\theta }{2}{l_2}q^{\prime}\hat{{\boldsymbol y}}$$

(36)$${\textbf m^{\prime\prime}} = \frac{1}{2}{N_0}\textrm{sin}\frac{\theta }{2}d{l_2}I\hat{{\boldsymbol y}}$$

(37)$${\textbf Q} = \left[ {\begin{array}{ccc} 0&{{Q_{12}}}&{{Q_{31}}}\\ {{Q_{12}}}&0&0\\ {{Q_{31}}}&0&0 \end{array}} \right]$$

where q′ is the accumulated charge at the end of a metallic bar, and ∂q′/∂t = I with I being the current in the metallic bar. Q₃₁ is the electric quadrupole moment in the x-z plane, Q₁₂ is the electric quadrupole moment in the x-y plane.

(38)$${Q_{31}} = 3{N_0}\textrm{sin}\frac{\theta }{2}d{l_2}q^{\prime}$$

(39)$${Q_{12}} = \frac{3}{2}{N_0}\textrm{sin}\theta l_2^2q^{\prime}$$

The far-zone vector potential A associated with p, m^″, Q are [44],

(40)$${\textbf A}({\textbf r} ){\; } = \frac{{{\mu _0}{e^{ik({R - {z_0}} )}}}}{{4\pi R}}\left[ {\frac{{\partial {\textbf p}}}{{\partial t}} + ik\left( {{{\hat{{\boldsymbol e}}}_{\boldsymbol R}} \times {\textbf m^{\prime\prime}} - \frac{1}{6}{{\hat{{\boldsymbol e}}}_{\boldsymbol R}} \cdot \frac{{\partial {\textbf Q}}}{{\partial t}}} \right)} \right]$$

Now, electric quadrupole moment Q obviously appears in the expression of far-zone vector potential A.

Using Eqs. (38)–(39) and noting that ${\hat{{\boldsymbol e}}_{\boldsymbol R}}\textrm{ = }\hat{{\boldsymbol z}}$, ${\hat{{\boldsymbol e}}_{\boldsymbol R}} \cdot \partial {Q_{12}}/\partial t{\hat{\textbf x}\hat{y}} = 0$, ${\hat{{\boldsymbol e}}_{\boldsymbol R}} \cdot \partial {Q_{31}}/\partial t\hat{{\boldsymbol z}}\hat{{\textbf x}} = \partial {Q_{31}}/\partial t\hat{{\textbf x}}$, we get

(41)$$\frac{1}{6}{\hat{{\boldsymbol e}}_{\boldsymbol R}} \cdot \frac{{\partial {\textbf Q}}}{{\partial t}} = \frac{1}{2}\textrm{sin}\frac{\theta }{2}d{l_2}I\hat{{\boldsymbol x}}$$

Setting equivalent magnetic dipole moment m_Q= 1/2sin(θ/2)dl₂Iŷ and noting that ${\hat{{\boldsymbol e}}_{\boldsymbol R}} \times \hat{{\textbf y}} ={-} \hat{{\boldsymbol x}}$, then we get

(42)$${\textbf A}({\textbf r} ){\; } = \frac{{{\mu _0}{e^{ik({R - {z_0}} )}}}}{{4\pi R}}\left[ {\frac{{\partial {\textbf p}}}{{\partial t}} + ik({{{\hat{{\boldsymbol e}}}_{\boldsymbol R}} \times {\textbf m^{\prime\prime}} + {{\hat{{\boldsymbol e}}}_{\boldsymbol R}} \times {{\textbf m}_{\textbf Q}}} )} \right]$$

By now, the electric quadrupole Q is equivalent to the magnetic dipole moment m_Q, they have equal contributions to the far-field radiation.

Noting that l₂I = J₁V_m and using M = m/unit volume, then

(43)$$\mathbf{A}(\mathbf{r})=\frac{N_{0} V_{0} \mu_{0} e^{i k\left(R-z_{0}\right)}}{4 \pi R}\left[\frac{\partial \mathbf{P}_{1}}{\partial t}+i k \hat{e}_{R} \times\left(\mathbf{M}_{1}^{\prime \prime}+\mathbf{M}_{\mathrm{Q}}\right)\right]$$

in which

(44)$${{\textbf M}_{\textbf Q}} = \frac{1}{2}d\sin \frac{\theta }{2}{J_1}\frac{{{V_m}}}{{{V_0}}}\hat{{\boldsymbol y}}$$

Looking back to the expressions for M₁ and M₁^″, namely, Eqs. (18) and (31), we readily find that M₁^″ = M_Q, and

(45)$${{\textbf M}_{\textbf 1}} = \mathbf{M}_{1}^{\prime \prime} + {{\textbf M}_{\textbf Q}}$$

That is

(46)$${{\textbf M}_{\textbf 1}} = 2\mathbf{M}_{1}^{\prime \prime}$$

Then, we finally obtain the far-zone vector potential A for mode 1

(47)$${\textbf A}({\textbf r} )= \frac{{{N_0}{V_0}{\mu _0}{e^{ik({R - {z_0}} )}}}}{{4\pi R}}\left( {\frac{{\partial {{\textbf P}_{\textbf 1}}}}{{\partial t}} + ik{{\hat{{\boldsymbol e}}}_{\boldsymbol R}} \times {{\textbf M}_1}} \right)$$

Now, we can find that Eq. (47) is exactly the same as Eq. (16), and M₁ indeed equals 2M₁^″ because M₁ involves not only magnetization M₁^″ (the contribution from magnetic dipole moment m) but also equivalent magnetization M_Q (the contribution from electric quadrupole Q). Only under this prerequisite can the deduced material equations still keep symmetry.

To sum up, in this section, we use Lorentz's oscillator model to derive the material equations similar to those of Pasteur medium. On the one hand, we derive material equations which explicitly includes magnetic field, and reveal why magnetic dipole plays an indispensable role in keeping the symmetry of material equations. On the other hand, we find that the electric quadrupole moment also plays an important role in optical rotation, however, its contribution is indirectly expressed in the material equations, it should be transformed into an equivalent magnetic dipole moment so that the material equations does not lose symmetry.

6. EIT and optical rotation effects predicted by Lorentz’s coupled oscillator model

In the previous section, based on Lorentz’s coupled oscillator model, we have derived the material equations of C₄ symmetrical chiral metamaterials, along with the expressions for dielectric constant ε, magnetic permeability μ and chirality parameter κ in Eqs. (27)–(29). Now in this section, we use them to further verify the EIT effect and optical rotation.

If the metamaterial is regarded as a slab of continuous material, its transmission coefficient t_±± and transmittance T are [15,45]

(48)$${t_{ {\pm}{\pm} }} = \frac{{2{Z_0}Z{e^{ {\pm} i\kappa \sqrt {{\mu _0}{\varepsilon _0}} \omega h}}}}{{2{Z_0}Zcos({kh} )- i({Z_0^2 + {Z^2}} )sin({kh} )}}$$

(49)$$T = {|{{t_{ {\pm}{\pm} }}} |^2}$$

in which k (= [με/(μ₀ε₀)]^1/2k₀) is the wave vector in effective media, h is slab length, Z₀ (= (μ₀/ε₀)^1/2) is the impedance of vacuum, while Z (= [μμ₀/(εε₀)]^1/2) is the impedance of the slab.

Substituting ε, μ, κ into Eq. (48), we calculate the transmittance spectrum and polarization state of the transmitted wave, and show them in Fig. 7. For the purpose of comparison, the simulation results from finite-difference time-domain method are also given in Fig. 7. In the calculation, it is assumed that the metallic bars are in a medium with the dielectric constant ε_FR-4= 4.3 + 0.025i, q²/(mV₀) = 9.3×10¹⁰ A²⋅s²/(kg⋅m³), d = 0.54 mm, ω₀ = 2π×13.7 GHz, γ = 2π×0.1 GHz, ξ = 940-46i (GHz)², θ = 55°, h = 0.8 mm.

Fig. 7. (a) Transmittance spectra and (b) polarization state of the transmitted wave, the incident wave is y-polarized. The blue dashed line is for the calculated result, whereas the red solid line for the simulated one.

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As shown in Fig. 7(a), the calculated transmittance spectrum and the simulated one match relatively well, both display EIT phenomenon at nearly the same frequencies with approximately equal peak transmittances. As for the polarization state of the transmitted wave, optical rotation effect can be seen (noting that the incident wave is y-polarized), but there is discrepancy between the calculation and simulated results, as shown in Fig. 7(b). We estimate this is mainly because the interaction effect between neighboring metallic bars is ignored in the Lorentz’s oscillator model. Besides, treating the metamaterial as a slab also brings some errors. Therefore, the calculated polarization state fail to well match the simulated ones. Despite this discrepancy, we can still think that Lorentz’s oscillator model and our derived equations can well predict the EIT and optical rotation phenomena, this also confirms the effectiveness of the Lorentz’s oscillator model and the correctness of our derivation process.

Metamaterials simultaneously supporting EIT and optical rotation [46,47] may expand the application fields of chiral metamaterials. On the one hand, coupling effect of electric and magnetic responses may lead to EIT phenomenon [35,48]; on the other hand, electric and magnetic responses are indispensable in the optical rotation of chiral metamaterials. Therefore, it is an effective way to use chiral metamaterial to simultaneously achieve EIT and optical rotation effect.

7. Conclusion

In summary, we have designed and fabricated a C₄-symmetry chiral metamaterial, which simultaneously exhibits EIT phenomenon and optical rotation. Compared with other metamaterial-based structure for optical rotation, the device we designed has high transmission because of the EIT effect. Using analytical equivalent circuit model, we reveal how EIT phenomenon is generated through analyzing electric and magnetic resonance modes and their coupling effects, and provide a way about how to produce EIT effect in chiral metamaterials. What is more, we employ Lorentz's coupled oscillator model to explain optical rotation effect and derive the material equations which are similar to those of Pasteur medium. Importantly, we find that, magnetic dipole and electric quadrupole play important roles in optical rotation and keeping the symmetry of material equations. Besides, we use the relevant parameters in the derived material equations to further predict and verify the EIT and optical rotation effects.

We believe our work offers a better understanding of the optical rotation and EIT in chiral metamaterials. What is more, combing EIT effect and optical rotation into a single metamaterial expands the application fields of chiral metamaterials, and enriches the domain of multifunctional metamaterials.

Appendix 1

This appendix gives the detailed calculation and derivation processes for getting Eqs. (16)–(18), i.e., the expressions for far-zone vector potential A, polarization P₁, and magnetization M₁.

For mode 1, we use the formula about the far-zone vector potential A, i.e., Eq. (15)

(50)$${\textbf A}({\textbf r} )\approx \frac{{{\mu _0}}}{{4\pi }}\smallint \frac{{{\textbf J}({{\textbf r^{\prime}}} ){e^{ik({R - {{\boldsymbol e}_{\boldsymbol R}} \cdot {\textbf r^{\prime}}} )}}}}{{R - {{\hat{{\boldsymbol e}}}_{\boldsymbol R}} \cdot {\textbf r^{\prime}}}}{d^3}r^{\prime}$$

and substitute Eq. (14) for current density J₁ into Eq. (50). It should be mentioned that J₁ is only present in the metallic bars and not in the dielectric region, therefore the volume integral in Eq. (50) is only over the metallic region. By approximation, we have

(51)$$\begin{array}{l} \; \; {\textbf A}({\textbf r} )\; \approx \frac{{{N_0}{V_m}{\mu _0}}}{{4\pi R}}\left[ {{J_1}{e^{ik\left( {R - {z_0} - \frac{d}{2}} \right)}}{{\hat{{\textbf r}}}_{\textbf 1}} + {J_1}{e^{ik\left( {R - {z_0} + \frac{d}{2}} \right)}}{{\hat{{\textbf r}}}_{\textbf 2}}} \right]\\ \; \; \; \; \; \; \; \; \; \; \; = \frac{{{N_0}{V_m}{\mu _0}{e^{ik({R - {z_0}} )}}}}{{4\pi R}}\left( {{J_1}{e^{ - ik\frac{d}{2}}}{{\hat{{\textbf r}}}_{\textbf 1}} + {J_1}{e^{ik\frac{d}{2}}}{{\hat{{\textbf r}}}_{\textbf 2}}} \right) \end{array}$$

where r = r(0,0,R), ${\hat{{\boldsymbol e}}_{\boldsymbol R}} = ({\textbf r} - {{\textbf r}_{\textbf 0}})/|{{\textbf r} - {{\textbf r}_{\textbf 0}}} |= \hat{{\boldsymbol z}}$, $\hat{{\boldsymbol x}}$ and ŷ are the unit vectors along the x-axis and y-axis, respectively; ${\hat{{\textbf r}}}_{\textbf 1}$ and ${\hat{{\textbf r}}}_{\textbf 2}$ are the unit vectors along the current direction in the upper and lower metallic bars, respectively; V_m is the volume of a single metallic bar. The metallic bar-pairs are arranged periodically, N₀ is the number of periods, N₀V_m represents the volume of the upper or lower metallic bars, and the first and second terms in the brackets on the right side of Eq. (51) are for the upper and lower metallic bars, respectively.

By performing Taylor expansion to Eq. (51), we get

(52)$$\begin{array}{l} {\textbf A}({\textbf r} )\; \approx \frac{{{N_0}{V_m}{\mu _0}{e^{ik({R - {z_0}} )}}}}{{4\pi R}}\left[ {{J_1}({{{\hat{{\textbf r}}}_1} + {{\hat{{\textbf r}}}_2}} )+ {J_1}ik\frac{d}{2}({{\hat{{\textbf r}}}_2} - {{\hat{{\textbf r}}}_1})} \right]\\ {\; \; \; \; \; \; \; \; \; } = \frac{{{N_0}{V_m}{\mu _0}{e^{ik({R - {z_0}} )}}}}{{4\pi R}}\left( {2{J_1}cos\frac{\theta }{2}\hat{{\boldsymbol y}} - {J_1}ikd\textrm{sin}\frac{\theta }{2}\hat{{\boldsymbol x}}} \right) \end{array}$$

When Taylor expansion is carried out in Eq. (51), the center point of expansion r₀(0,0,z₀) should be consistent with the center point of expansion in Eqs. (11) and (12).

We can obtain the dipole moment by comparing the actual radiated field with that of an equivalent dipole [24], by doing so, we can find the first term in the brackets on the right side of Eq. (52) corresponds to ∂P₁/∂t, while the second term corresponds to M₁, that is,

(53)$${\textbf A}({\textbf r} )= \frac{{{N_0}{V_0}{\mu _0}{e^{ik({R - {z_0}} )}}}}{{4\pi R}}\left( {\frac{{\partial {{\textbf P}_{\textbf 1}}}}{{\partial t}} + ik{{\hat{{\boldsymbol e}}}_{\boldsymbol R}} \times {{\textbf M}_{\textbf 1}}} \right)$$

where the polarization P₁ and magnetization M₁ for mode 1 are deduced as

(54)$${{\textbf P}_{\textbf 1}} = 2i\frac{1}{\omega }\cos \frac{\theta }{2}{J_1}\frac{{{V_m}}}{{{V_0}}}\hat{{\boldsymbol y}}$$

(55)$${{\textbf M}_{\textbf 1}} = d\sin \frac{\theta }{2}{J_1}\frac{{{V_m}}}{{{V_0}}}\hat{{\boldsymbol y}}$$

Noting that polarization P₁ (or magnetization M₁) is the average electric (or magnetic) dipole moment per unit volume, it is easy to understand the term V_m/V₀ appears in Eqs. (54)–(55) and N₀V₀ appears in Eq. (53). Here V₀ is the volume of a period of the metamaterial.

By now, we obtain the expressions for far-zone vector potential A, polarization P₁, and magnetization M₁, namely, Eqs. (53)–(55), which corresponds to Eqs. (16)–(18), respectively.

Appendix 2

This appendix presents the detailed derivation processes for getting Eqs. (21)–(22), i.e., the expressions for polarization P₁^′, P₂^′, and the total polarization P of the C₄ symmetric metamaterial.

In a Cartesian coordinate system, when an object is rotated counterclockwise by an angle β, the transformation matrix is $\left[ {\begin{array}{cc} {\cos \beta }&{\sin \beta }\\ { - \sin \beta }&{\cos \beta } \end{array}} \right]$. In our case, the metallic bars are rotated by β=90°, then the transformation matrix is $\left[ {\begin{array}{cc} 0&1\\ { - 1}&0 \end{array}} \right]$.

The following shows how we use the transformation matrix to get Eqs. (21)–(22) from Eqs. (17) and (19).

After rewriting P₁ and P₂ (i.e., Eq. (17) and Eq. (19)) in matrix form, we can obtain

(56)$${{\textbf P}_{\textbf 1}} + {{\textbf P}_{\textbf 2}} = \left[ {\begin{array}{c} {{P_x}}\\ {{P_y}} \end{array}} \right] = \left[ {\begin{array}{cc} {2\frac{q}{{{V_0}}}({{F_ + } - {F_ - }} ){{\sin }^2}\frac{\theta }{2}}&{ - ikd\frac{q}{{{V_0}}}({{F_ + } - {F_ - }} )\sin \frac{\theta }{2}\cos \frac{\theta }{2}}\\ { - ikd\frac{q}{{{V_0}}}({{F_ + } + {F_ - }} )\sin \frac{\theta }{2}\cos \frac{\theta }{2}}&{2\frac{q}{{{V_0}}}({{F_ + } + {F_ - }} ){{\cos }^2}\frac{\theta }{2}} \end{array}} \right]\left[ {\begin{array}{c} {{E_x}}\\ {{E_y}} \end{array}} \right]$$

Based on Eq. (56) and the transformation matrix, we can get P₁^′+P₂^′ by using coordinate transformation because polarization P₁^′ and P₂^′ are for the case when the metallic bars are rotated by 90°. Then, we have

(57)$$\begin{aligned} &\mathbf{P}_{1}^{\prime}+\mathbf{P}_{2}^{\prime}=\left[\begin{array}{c} P_{x}^{\prime} \\ P_{y}^{\prime} \end{array}\right]\\ &=\left[\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right]^{-1}\left[\begin{array}{cc} 2 \frac{q}{V_{0}}\left(F_{+}-F_{-}\right) \sin ^{2} \frac{\theta}{2} & -i k d \frac{q}{V_{0}}\left(F_{+}-F_{-}\right) \sin \frac{\theta}{2} \cos \frac{\theta}{2} \\ -i k d \frac{q}{V_{0}}\left(F_{+}+F_{-}\right) \sin \frac{\theta}{2} \cos \frac{\theta}{2} & 2 \frac{q}{V_{0}}\left(F_{+}+F_{-}\right) \cos ^{2} \frac{\theta}{2} \end{array}\right]\left[\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right]\left[\begin{array}{c} E_{x} \\ E_{y} \end{array}\right]\\ &=\left[\begin{array}{ccc} 2 \frac{q}{V_{0}}\left(F_{+}+F_{-}\right) \cos ^{2} \frac{\theta}{2} & i k d \frac{q}{V_{0}}\left(F_{+}+F_{-}\right) \sin \frac{\theta}{2} \cos \frac{\theta}{2} \\ i k d \frac{q}{V_{0}}\left(F_{+}-F_{-}\right) \sin \frac{\theta}{2} \cos \frac{\theta}{2} & 2 \frac{q}{V_{0}}\left(F_{+}-F_{-}\right) \sin ^{2} \frac{\theta}{2} \end{array}\right]\left[\begin{array}{l} E_{x} \\ E_{y} \end{array}\right] \end{aligned}$$

By adding Eq. (56) and Eq. (57), we obtain the total polarization P of the C₄ symmetric metamaterial, that is

(58)$${\textbf P} = \left[ {\begin{array}{c} {{P_x}}\\ {{P_y}} \end{array}} \right] + \left[ {\begin{array}{c} {{P_x}^\prime }\\ {{P_y}^\prime } \end{array}} \right] = \left[ {\begin{array}{cc} {2\frac{q}{{{V_0}}}({{F_ + } + {F_ - }\cos \theta } )}&{ikd\frac{q}{{{V_0}}}{F_ - }\sin \theta }\\ { - ikd\frac{q}{{{V_0}}}{F_ - }\sin \theta }&{2\frac{q}{{{V_0}}}({{F_ + } + {F_ - }\cos \theta } )} \end{array}} \right]\left[ {\begin{array}{c} {{E_x}}\\ {{E_y}} \end{array}} \right]$$

One can readily find that Eq. (58) is the matrix form of Eq. (21).

In a similar way, we can obtain Eq. (22) for the total magnetization M of the C₄ symmetric metamaterial.

Funding

National Natural Science Foundation of China (61675074, 61771215).

Disclosures

The authors declare no conflicts of interest.

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Optical rotation and electromagnetically induced transparency in a chiral metamaterial with C₄ symmetry

Abstract

1. Introduction

2. Device structure

3. EIT phenomenon and optical rotation observed in simulation and experiment

4. Investigation of EIT effect by analytical equivalent circuit model

5. Interpretation of optical rotation by Lorentz's coupled oscillator model

5.1 Derivation of current density J

5.2 Calculation of polarization P and magnetization M

5.3 Material equations and the role of magnetic dipole moment in material equations

5.4 Role of electric quadrupole in optical rotation

6. EIT and optical rotation effects predicted by Lorentz’s coupled oscillator model

7. Conclusion

Appendix 1

Appendix 2

Funding

Disclosures

References

Cited By

Figures (7)

Equations (58)

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