1. Introduction

Chiral structures are ones that cannot be made to coincide with their mirror images [1–3]. These structures show interesting optical chirality, such as circular dichroism (CD) and asymmetric transmission (AT) effects. The CD effect is the transmission intensity difference between right-hand circular polarized (RCP, + ) light and left-hand circular polarized (LCP, −) light [4–6]. The AT effect is defined as the difference in the conversion of linearly and circularly polarized light into opposite polarized light [7–9]. Owing to the strong interaction between light and noble metals, artificial chiral plasmonic nanostructures (ACPNs) are extensively studied in biological monitoring [10–12], analytical chemistry [13–15], circular polarization conversion [16], and negative refractive index media [17–19].

Recently, many researchers have proposed different ACPNs to produce chirality and explore the reason for chirality. For plasmonic nanostructures, 2D achiral plasmonic nanostructures can achieve a chiral response under oblique excitation [20–22]. However, 2D monolayer ACPNs can achieve CD [23, 24] or AT effects [25–27] under normal excitation. Three dimensional (3D) helical ACPNs show strong CD [28–30] or AT [31–33], which is due the introduction of spatial asymmetry in ACPNs. However, these ACPNs, such as layer-by-layer ACPNs prepared by multiple-steps electron beam lithography (EBL) cannot achieve strong CD and circular AT effects simultaneously [30, 34]. The circular AT is achieved under the condition of $\left|t_{+-}\right|\ne \left|t_{-+}\right|\cap t_{++}=t_{--}$, at which $\text{CD}\text{=}{\left|t_{++}\right|}^2-{\left|t_{--}\right|}^2=0$ [35].

Simultaneously achieving CD and AT is important in the field of negative refractive index media [18] and perfect polarization converters [36]. However, at the resonant wavelength of usual 2D and 3D ACPNs, T₊₊ and T₋₋ consistently present valleys limiting the magnitude of the CD effect, meanwhile T₋₊ and T₊₋ consistently present peaks limiting the magnitude of the circular AT effect [33]. The magnitudes of T₋₊ and T₊₋ are also weaker than those of T₊₊ and T₋₋, respectively, which limits circular polarization conversion.

In this paper, we designed a type of chiral metamaterial nanostructure on the basis of theoretical analysis. The fundamental difference between the previous ACPNs and our designed ACPNs is that our designed ACPNs can simultaneously achieve CD and AT effects. The chiral response of twist nanoslit–nanorod arrays (TNNAs) was investigated by finite element method (FEM) to verify the theoretical prediction. Experimentally, the TNNAs can be prepared by EBL method. FEM calculation results show that the TNNAs can achieve large CD and AT effects simultaneously. The effects of the rotation angle of the nanorod relative to the nanoslit, the length of the nanorod, and the separation between the nanoslit and the nanorod on the CD and AT effects were also studied.

2. Theoretical analysis

Given the incident plane wave propagation along the + z direction, the Jones matrix T_circ of circular polarization [33, 34] is described as

We use x, y, + , and − to describe the polarized states of linearly x-polarized, linearly y-polarized, RCP light, and LCP light, respectively. Im(t_xx) and Re(t_xx) represent the real part and imaginary part of t_xx, respectively. The CD of the circularly polarized light is usually defined as

We propose the composed structure of nanoslits and nanorods to satisfy Eq. (4). According to previous literature [38], extraordinary optical transmission of nanoslits only occurs under linear polarization perpendicular to the slits. The Jones matrix of nanoslit arrays on x direction ($T_{\text{l}in-Slit}^{+z}$) can be written as

In accordance with Eq. (4), the condition of $t_{x\text{x}}=t_{xy}=0$ show that TNNAs will produce CD and AT simultaneously. The CD and AT can be written as

When light illuminates the proposed structure backward, the Jones matrix $T_{\text{l}in-ACPN}^{\text{-}z}$ can be written as

Equations (11) and (12) show that when light is incident from the −z direction, the CD is the same as that incident from the + z direction, whereas the AT effect is the opposite of that incident from the + z direction. Therefore, TNNAs produce $\left|C{D}^{+z}\right|=\left|C{D}^{\text{-}z}\right|=\left|{\Delta }_{circ}^{+z}\right|=\left|{\Delta }_{circ}^{\text{-}z}\right|$ simultaneously.

Figure 1 shows the proposed TNNAs. The arrays and cell of the TNNAs are shown in Figs. 1(a) and 1(b), respectively. The nanoslits and nanorods are gold, and their frequency dependent permittivities are obtained from Ref. 40. The nanoslits (yellow) and nanorods (yellow) are separated by a SiO₂ layer (blue) with a refractive index of 1.45 and thickness g. The periods of the TNNAs are 1000 nm and 700 nm in the x and y directions, respectively. The height and width of the nanoslits are 100 nm each. The nanorods are located below the nanoslits. The length, width, and height of the nanorods are L, 60 nm, and 60 nm, respectively. The twist angle between the nanoslits and nanorods is α, as shown in the inset in Fig. 1. Incident RCP (aqua circle-arrow) and LCP (pink circle-arrow) light illuminates normally to the structure and can be partially converted into opposite handedness. The solid black arrows labeled by k indicate the directions of wave vectors, which are oriented along the + z or −z direction. FEM software COMSOL Multiphysics is used to calculate the transmittance of the TNNAs.

 

Fig. 1 (a) Schematic of TNNAs with g = 60 nm, L = 360 nm, and α = 45°. (b) Its unit cell with the associated geometric features.

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3. Results and discussion

Figure 2(a) shows the $T_{circ}^{\text{+}z}$, ${\text{CD}}_{}^{\text{+}z}$, and ${\Delta }_{circ}^{\text{+}z}$ of TNNAs with g = 60 nm, L = 360 nm, and α = 45°. For the transmitted spectra of T₊₊ (${\left|t_{++}^{}\right|}^2$), T₊₋ (${\left|t_{+-}^{}\right|}^2$), T₋₊ (${\left|t_{-+}^{}\right|}^2$), and T₋₋ (${\left|t_{--}^{}\right|}^2$), obvious peaks or valleys are observed at resonance wavelengths λ_I = 1.64 µm, λ_II = 1.23 µm, and λ_III = 1.05 µm. For simplicity, we divide the resonance modes into modes I, II, and III. At around resonance wavelengths, the spectra of T₊₊ and T₋₋ correspondingly present peaks and valleys leading to a large CD effect, whereas the spectra of T₊₋ and T₋₊ correspondingly present valleys and peaks leading to a large AT effect. The spectrum of CD is coincident with that of AT. Figure 2(b) shows the $T_{circ}^{-z}$,${\text{CD}}_{}^{-z}$, and ${\Delta }_{circ}^{-z}$ of the TNNAs with the same structural parameters as those in Fig. 2(a). T₊₊ and T₋₋ are the same as those in Fig. 2(a), which is consistent with Eq. (11). T₊₋ and T₋₊ are exchanged with respect to Fig. 2(a), which is consistent with Eq. (12). Under the excitation of LCP light, the magnitude of T₊₋ (transmittance of RCP light) is larger than that of T₋₋ (transmittance of LCP light), which achieves a large converter.

 

Fig. 2 Simulated (T)_circ matrix, CD spectrum, and AT spectrum of TNNAs for + z-direction light excitation (a) and −z-direction light excitation (b). The three resonances are labeled mode I, II, and III.

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The normalized charge distributions of the nanoslit (lower surface) and the nanorod (upper surface) at λ_I = 1.64 µm, λ_II = 1.23 µm, and λ_III = 1.05 µm are calculated to investigate the origin of the modes in Fig. 2, as shown in Fig. 3. The red color indicates positive charge, and the blue color indicates negative charge. Figures 3(a), 3(d), and 3(g) show the normalized charge distributions of the nanoslit. The magenta solid arrows represent the equivalent electric dipole moments of nanoslits. Figures 3(b), 3(e), and 3(h) show the normalized charge distributions of the nanorod. The charges mainly distribute on the two ends of the nanorod. The green arrows represent the equivalent electric dipole moments of nanorods.

 

Fig. 3 The charge distributions of (a, d, and g) nanoslits and (b, e, and h) nanorods. (c, f, and i) Born-Kuhn modes of TNNAs for RCP at three modes under −z-direction light excitation.

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At λ_I = 1.64 µm, tilted effective dipole P_I-S appears around the slit. P_I-S can be decomposed into P_I-S⊥ and P_I-S∥. The decomposition can be performed for each mode. P_I-S⊥ is perpendicular to the slit, which contributes to the transmission of energy through the slit [38]. As shown in Fig. 3(c), P_I-S⊥ and P_I-R form a bonding mode in the Born–Kuhn oscillator model [41, 42]. At λ_II = 1.23 µm, effective dipole electron oscillation P_II-S is also tilted around the slit. P_II-S⊥ is perpendicular to the slit. As shown in Fig. 3(f), P_II-S⊥ and P_II-R form an antibonding mode in the Born–Kuhn oscillator model [42]. At λ_III = 1.05 µm [Fig. 3(g)], except the region around the slit, a large electric field distributes along the line perpendicular to the slit, which shows that λ_III = 1.05 µm is attributed to the electrons oscillating in the direction perpendicular to the slit. In the region around the slit, effective dipole P_III-S exhibits a smaller tilted angle related to the slit than those shown in Figs. 3(a) and 3(d). As shown in Fig. 3(i), P_III-S⊥ and P_III-R form an antibonding mode.

We vary the thickness of SiO₂ to investigate the influences of the coupling between nanoslit and nanorod on the CD and AT effects of TNNAs. Figure 4(a) shows the simulated ${\text{CD}}_{}^{-z}$ and ${\Delta }_{circ}^{-z}$ spectra of different thickness g with fixed L = 360 nm and α = 45°. Modes I, II, and III are observed in ${\text{CD}}_{}^{-z}$ and ${\Delta }_{circ}^{-z}$ spectra with different thickness g. Mode I blue shifts with an increase in g. Modes II and III red shift as g increases. Given that mode I is ascribed to the bonding mode between P_I-S⊥ and P_I-R, attraction is presented between P_I-S⊥ and P_I-R. An increase in g will decrease the attraction between them, leading to a decrease in the equivalent electric dipole moments and the blue shift of the resonant wavelength [43]. On the contrary, modes II and III are attributed to the antibonding mode between the electron oscillations around the slit and on the nanorod; this finding indicates that repulsion exists between them. An increase in g will decrease the repulsion between them, leading to an increase in the equivalent electric dipole moments and the red shift of the resonant wavelength [43].

 

Fig. 4 ${\text{CD}}_{}^{-z}$ and ${\Delta }_{circ}^{-z}$ spectra of TNNAs with different (a) thickness g, (b) angle α, and (c) length L.

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We show the simulated ${\text{CD}}_{}^{-z}$ and ${\Delta }_{circ}^{-z}$ spectra of different angle α with fixed L = 360 nm and g = 60 nm in Fig. 4(b) to investigate the effects of angle α on CD and AT. Given α = 0° and 90°, TNNA is achiral, and the three peaks disappear. When α is increased from 15° to 75°, modes I and III red shift, whereas mode II blue shifts. With an increase in α, the component of P_I-S⊥ increases, thereby resulting in the red shift of mode I. On the contrary, the component of P_II-S⊥ decreases as α increases, which results in the blue shift of mode II. The component of P_III-S⊥ increases as α increases, which results in the red shift of mode III.

We also present the simulated ${\text{CD}}_{}^{-z}$ and ${\Delta }_{circ}^{-z}$ spectra of different length L with fixed g = 60 nm and α = 45° in Fig. 4(c) to investigate the effects of length L on CD and AT. The three modes are present in ${\text{CD}}_{}^{-z}$ and ${\Delta }_{circ}^{-z}$ spectra, and all red shift with an increase in L. With an increase in L, the equivalent electric dipole moments on the rod and around the slit increase, resulting in the red shift of the three modes.

4. Conclusions

In this study, we present TNNAs comprising nanoslits and twist nanorods to achieve CD and AT effects simultaneously by analyzing the transmission matrix. The transmission properties of the TNNAs are calculated by FEM. The results show that at resonant wavelengths, the spectra of T₊₊ and T₋₋ correspondingly present peaks and valleys leading to a large CD effect, meanwhile one of the spectra for T₋₊ and T₊₋ presents valleys and another presents peaks leading to a large AT effect. The magnitude of CD is equivalent to that of AT. When light illuminates the proposed structure backward, the magnitude of T₊₋ is stronger than that of T₋₋, which achieves a large converter. The CD and AT effects strongly depend on the geometric parameters of the TNNAs. Overall, these results are useful for designing negative refractive index media and perfect polarization converters. Furthermore, the method of designing nanostructures based on an analysis of the transmission matrix can be employed in negative refractive index media and perfect polarization converters.