Characterizing chiroptical properties of 2D/3D structures based on an improved coupled dipole theory

Yuyin Shi; Yuyin Shi; Wei Zhang; Wei Zhang

doi:10.1364/OE.517748

1. Introduction

Chirality refers to the property that a structure can not superpose with its mirror image [1]. Chirality exists extensively in nature, spanning from small molecules such as L-amino acids [2] and D-sugars [3], to macroscale objects such as plant vines and the human body. Chiral structures with mirror symmetry broken have different responses to right circularly polarized (RCP) and left circularly polarized (LCP) light illumination. So chiroptical properties of chiral systems and their enantiomers have attracted tremendous research interest in recent years.

Unlike natural materials, well designed nanostructures provide more opportunities for exploring chiral optics. Artificial chiral structures have been applied widely in molecular chirality enhancement [4,5], material with negative refractive index [6], ultrasensitive detection [7]. They usually exhibit stronger and more tunable optical chirality than the natural counterpart and present new feature in nonlinear effect [8,9], and optical force [10,11] etc.. A commonly used method to introduce chirality is organizing isotropic nanoparticles into three-dimensional (3D) asymmetric structures [12–19], such as nanoparticle dimers [12], tetramers [13], pyramids [14] and helices [15,16].

Due to difficulties in manufacturing process for 3D nanostructures, planar nanostructures have been extensively studied [11,20–34]. Some planar achiral metasurface can performance chiroptical response under oblique incidence, known as extrinsic chirality [21,22,27]. Plum et al. demonstrated that periodic asymmetric split rings under oblique incident wave have strong optical activity and circular dichroism (CD) [27]. Many studies have been performed for two-dimensional (2D) structures with intrinsic chirality at normal incidence, such as chiral oligomers [24–25], gammadion [28], L-shaped metal structures [29,30], asymmetric split rings [31,32], bi-layer configurations [33] and finger-crossed metasurface [34]. The intrinsic chirality with BIC (quasi-BIC) phenomenon has also been investigated [35–37].

Compared with 3D chirality, chiroptical response of 2D systems is more subtle [25,26,38–40]. Fedotov et al. addressed the asymmetric transmission of electromagnetic waves through a planar chiral structure [38]. The importance of absorption/Ohmic dissipation for 2D chiroptical response was emphasized [39,40]. It was pointed out that a planar chiral system may exhibit polarization dependent absorption or scattering losses, even when the extinction is polarization independent [25], which is related to mirror symmetry and reciprocity [41–44]. To have a deep understanding of 2D chiral optics and obtain methods for modulating the chiroptical response, an analytical theory is helpful. Couple dipole theory (CDT) is a very effective method for calculating the 3D optical chirality, which has been demonstrated in our previous work [45–47]. Based on CDT, Najafabadi et al. examined the chiroptical properties of homogeneous and heterogeneous dimers of Ag and Au nanorods (NRs) and discussed the origin of chiroptical activity in optical absorbing/scattering [40,48–50]. Yet, CDT gives vanishing extinction CD and absorption/scattering CD for nanorod dimer of identical NRs even in a 2D mirror symmetry broken configuration.

To reveal the difference/connection between 2D and 3D chiroptical properties, their relation with 2D/3D symmetry and its breaking, and overcome the shortcoming of CDT in characterizing 2D chiroptical responses, we develop an improved couple dipole theory (ICDT) in this paper, inspired by discrete dipole approximation (DDA) [51–53]. Our analytical ICDT provides a clear physical picture and reveals the origin of 2D chiroptical response. Our theory indicates that the inter-particle interaction leads to asymmetric effective polarizability for identical NRs, resulting scattering circular dichroism (SCD) for NRs in configuration with in-plane symmetry broken. Based on ICDT, we demonstrate the relation between geometry/symmetry and chiroptical properties. In particular, a symmetry broken structure may also have vanishing SCD due to the interplay between 2D and 3D symmetry/breaking. The relation with reciprocity has also been addressed.

2. Model and theoretical approach

2.1 Couple dipole theory (CDT)

To explore the fundamental feature of 2D/3D chiropical process, we adopt the basic model of coupled nanorod dimer as shown in Fig. 1. The fingers-crossed configuration and corner-staked configuration have been used to study the 3D optical activity based on numerical simulation and CDT [45,46,54–57]. We first present sample calculation of extinction/ scattering/absorption cross section and CD to illustrate the principle and limitation of CDT. We consider a 3D fingers-crossed dimer composed of two identical Au NRs shown in Fig. 1(a). According to CDT, each NR (viewed as an ellipsoid) with size smaller than the wavelength is represented by a dipole. The dipole moments satisfy the coupled equations [46,47,58]

(1)$$\begin{aligned} & p_A = \alpha_A (E_1 + G_{AB} p_B) \\ & p_B = \alpha_B (E_2 + G_{AB}p_A) \end{aligned}$$

with the incident field in the +z direction on the two dipoles as

(2)$$E_1 = E_0,\quad E_2 = E_0 e^{i \eta\beta} e^{ikd}.$$

Here $\alpha _A$ and $\alpha _B$ are the polarizabilities of the NRs [59], $\eta =\pm 1$ for LCP/RCP light (LCPL/RCPL), and $G_{AB}$ is the interaction constant between dipoles [60]

(3)$$G_{AB} ={-}\dfrac{e^{ikd}}{d^3}(1-ikd-k^2d^2)\cos\beta.$$

The extinction cross section and scattering cross section are

(4)$$C_{ext} = \dfrac{4\pi k}{|E_{0}|^2}[Im (E^*_1 \cdot p_A)+Im (E^*_2 \cdot p_B)],$$

(5)$$C_{sca} = \dfrac{k^4}{|E_{0}|^2} \int d \Omega | [\overrightarrow p_A -\hat n(\hat n \cdot \overrightarrow p_A)] e^{{-}ik \hat n \cdot \overrightarrow r_1} +[\overrightarrow p_B - \hat n (\hat n \cdot \overrightarrow p_B)] e^{{-}ik \hat n \cdot \overrightarrow r_2} |^2.$$

Fig. 1. (a) 3D fingers-crossed Au NR dimer. The line connecting the center of the NRs is perpendicular to the long axis of the NR. The long and short axis of the NRs is a and b, the vertical gap between two NRs is d, and the angle between their long axis is $\beta$. (b) Staked two identical Au NRs with their long axis perpendicular to $\hat {k}$ and vertical gap d. (c) 2D NR dimer structure.

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So the extinction CD is (see support information (SI) for more details of the calculation)

(6)$$CD_{ext} = C_{ext}|_{\eta=1} - C_{ext}|_{\eta={-}1} ={-}8\pi k \sin\beta \sin kd \cdot Im ( \dfrac{\alpha_A \alpha_B G_{AB}}{1- \alpha_A \alpha_B G_{AB}^2} ).$$

Extinction CD means the difference between extinction cross section under RCP incidence and LCP incidence. Note that $(1-\alpha _A \alpha _B G_{AB}^2)\sim 1$ at the first order approximation for weak inter-particle interaction and $G_{AB}\sim \cos (\beta )$. Then we obtain

(7)$$CD_{ext}\propto \sin 2 \beta.$$

So the extinction CD varies sinusoidal with angle $2\beta$ [61,62], $CD_{ext}$ vanishes when $\beta =0, \pi /2$ for mirror symmetric configurations and reaches maximum when $\beta =45^\circ$ and $\beta =135^\circ$ for the configurations with maximum mirror symmetry breaking. Scattering and absorption CD can be calculated in the similar way, and have the similar behavior as that of extinction CD. The CDT works well and CD gives the optical characterization of mirror symmetry (breaking) for 3D systems with particle size smaller than the wavelength (see more results in the next section). Yet, CDT fails to give the characterization of 2D mirror symmetry (breaking).

2.2 Improved couple dipole theory (ICDT)

We develop an improved couple dipole theory(ICDT), which can provide a clear physics picture and uncover the underlying mechanism of chiroptical response associated with 2D mirror symmetry breaking. In general our system is a dimer of two identical Au NRs with their long axis perpendicular to $\hat {k}$ as shown in Fig. 1(b) (The 2D structure is shown in Fig. 1(c)). The distances between middle points of NRs and z axis are $g_1$ and $g_2$. The vertical distance between two NRs is d and the angle between long axis of NRs is $\beta$. To manifest the role of 2D in-plane symmetry breaking for chiral response, we have not included the effect from substrate which leads to mirror symmetry breaking in the direction perpendicular to the 2D plane and associated chiral responses. In our theory, each NR is considered as two dipoles as shown in Fig. 1(b)/(c). The dipole moments are determined by the following coupling equations [46,47,58]

(8)$$\begin{aligned} & p_1 = \alpha(E_1 + G_{12}p_2 + G_{13}p_3 + G_{14}p_4) \\ & p_2 = \alpha(E_1+ G_{12}p_1 + G_{23}p_3 + G_{24}p_4 )\\ & p_3 = \alpha(E_2 + G_{13} p_1+ G_{23} p_2 + G_{12}p_4) \\ & p_4 = \alpha(E_2 + G_{14}p_1 + G_{24}p_2 + G_{12}p_3), \end{aligned}$$

where $\alpha$ is the polarizability of half a NR, and $G_{ij}$ ($G_{13},G_{14},G_{23},G_{24}$) is the interaction constant between dipoles of different NRs

(9)$$\begin{aligned} G_{ij} = & -\dfrac{e^{ikr_{ij}}}{r_{ij}^3} \{ k^2[ (d_i-d_j \cos\beta)(d_i\cos \beta-d_j)-r_{ij}^2 \cos\beta ] \\ & + \dfrac{1-ikr_{ij}}{r_{ij}^2}[r_{ij}^2 \cos\beta -3(d_i-d_j \cos\beta)(d_i \cos\beta -d_j)] \} \end{aligned}$$

and the interaction between two dipoles within the same NR is

(10)$$G_{12} = \dfrac{1}{\alpha}-\dfrac{2}{\alpha_L},$$

where $\alpha _L$ is the polarizability of the entire NR.

By Solving Eq. (8), we have

(11)$$\begin{aligned} p_A = p_1+p_2 = \widetilde \alpha_A (E_1+ \widetilde G p_B) \\ p_B = p_3+p_4 = \widetilde \alpha_B (E_2+ \widetilde G p_A). \end{aligned}$$

Here $p_A$ and $p_B$ are the whole dipole moments of each NR, this equation has the same form as Eq. (1). The equivalent polarizability of each Au NR and the equivalent interaction constant between the whole electric dipoles are

(12)$$\widetilde \alpha_A = \dfrac{2\alpha}{1-\alpha G_{12}- \dfrac{\alpha}{2} G^I B^I}, \quad \widetilde \alpha_B = \dfrac{2\alpha}{1-\alpha G_{12}- \dfrac{\alpha}{2} G^{II} A^{II}},$$

(13)$$\widetilde G = \dfrac{\overline G[4(1+\alpha G_{12})^2 - \alpha^2(G^{III})^2 ] +\alpha^2 G^I G^{II} G^{III}}{4[4(1+\alpha G_{12})^2 - \alpha^2 (G^{III})^2 ]},$$

and

(14)$$\begin{aligned} CD_{sca} = & -\dfrac{k^4}{|1-\widetilde \alpha_A \widetilde \alpha_B \widetilde G^2|} \{ \dfrac{8 i \pi \sin\beta}{3} [|\widetilde \alpha_A|^2 (\widetilde \alpha_B^* \widetilde G^* e^{{-}ikd}-\widetilde \alpha_B \widetilde G e^{ikd}) + |\widetilde \alpha_B|^2 (\widetilde \alpha_A \widetilde G e^{{-}ikd}-\widetilde \alpha_A^* \widetilde G^* e^{ikd})] \\ & +i\sin\beta(\cos\beta Q_1-Q_2)(\widetilde \alpha_A \widetilde \alpha_B^* e^{{-}ikd}-\widetilde \alpha_B \widetilde \alpha_A^* e^{ikd}) \}. \end{aligned}$$

Scattering CD means the difference between scattering cross section under RCP incidence and LCP incidence. The detailed information of $G^{I},G^{II},G^{III},\bar {G},A^{II},B^{I}$ and the integral values $Q_1$, $Q_2$ are included in the SI. The above formalism of ICDT reveals one important effect: the interaction between NRs may lead to asymmetric effective dielectric polarizabilities (i.e. $\widetilde \alpha _A \neq \widetilde \alpha _B$) for identical NRs. It has important physical consequences and brings about applications in characterization of planar chiral/mirror symmetry (breaking) as discussed in more detail later. From Eq. (6), it is seen that $CD_{ext}=0$ for a planar NR dimer even with 2D chiral symmetry broken. It is known that the scattering/adsorption spectrum may provide additional information [41,42]. For the planar structures with $d=0$ (as shown in Fig. 1(c)), the SCD is

(15)$$\begin{aligned} CD_{sca} = & -\dfrac{k^4}{|1-\widetilde \alpha_A \widetilde \alpha_B \widetilde G^2|^2} \{ \dfrac{8 i \pi \sin\beta}{3} [|\widetilde \alpha_A|^2 (\widetilde \alpha_B^* \widetilde G^*-\widetilde \alpha_B \widetilde G) + |\widetilde \alpha_B|^2 (\widetilde \alpha_A \widetilde G-\widetilde \alpha_A^* \widetilde G^*)] \\ & +i\sin\beta(\cos\beta Q_1-Q_2)(\widetilde \alpha_A \widetilde \alpha_B^*-\widetilde \alpha_B \widetilde \alpha_A^*) \}. \end{aligned}$$

In the framework of CDT, $CD_{sca}=0$ for two identical NRs ($\widetilde \alpha _A=\widetilde \alpha _B=\alpha _L$) even for 2D chiral asymmetric configuration ( $g_1\neq g_2$). Importantly, ICDT may give nonzero $CD_{sca}$ due to interaction-induced asymmetry $\widetilde \alpha _A \neq \widetilde \alpha _B$. Thus, unlike CDT, ICDT is able to provide clear physical picture of chiroptical characterization of 2D planar chiral symmetry breaking. The ICDT also applies to dielectric systems with small size nanoparticles, where the dipole modes dominate. In addition, ICDT can give insight on the (non)reciprocity.

2.3 Scattering CD and reciprocity

Our analytical ICDT can not only give a clear picture of 2D in-plane mirror symmetry breaking induced chiroptical response, but also provide insight on the reciprocity. Here we consider the generalized "reciprocity"/"nonreciprocity": identical/asymmetric optical response for incident lights in opposite direction [41,42,63]. For example, reciprocity means extinction (or scattering/absorption) cross section are the same for incident light with wave vector $\pm \overrightarrow {k}$, either for LCPL or RCPL, resulting same extinction(or scattering/absorption) CD for incident light propagation in opposite direction. We note that physical quantities may show different reciprocal properties [41,42].

Based on Eq. (4) one can show that extinction cross section and CD maintain reciprocity. While for scattering, it should be discussed carefully. Straightforward calculation shows that

(16)$$\begin{aligned} C_{sca}^{{+}L}-C_{sca}^{{-}L} = & \dfrac{k^4 \sin(\beta+kd)}{|1-\widetilde \alpha_A \widetilde \alpha_B \widetilde G^2|^2} \{ \dfrac{16\pi}{3} ( |\widetilde \alpha_A|^2 \cdot Im [\widetilde\alpha_B \widetilde G] -|\widetilde \alpha_B|^2 \cdot Im [\widetilde\alpha_A \widetilde G]) \\ & +i (\cos\beta Q_1-Q_2) \cdot (\widetilde\alpha_A \widetilde\alpha_B^*-\widetilde\alpha_A^*\widetilde\alpha_B) \}, \end{aligned}$$

where $C_{sca}^{\pm L}$ means scattering cross section at LCPL incident from $\overrightarrow {\pm k}$ direction. Similar expression can be obtain for RCPL with $\beta$ replaced by $-\beta$.

We consider three cases here. First, for the 3D "fingers-crossed" configuration with $g_1=g_2=0$ and $d\neq 0$ as shown in Fig. 1(a), we have $r_{13}=r_{24}$, $r_{14}=r_{23}$. This hidden geometric symmetry leads to $G_{13}=G_{24}$, $G_{14}=G_{23}$ and $\tilde{\alpha}_A=\tilde{\alpha}_B=\alpha_L$. In this case, the difference between the results from ICDT and CDT is very small. This structure maintains reciprocity for scattering process ($C_{sca}^{+L}=C_{sca}^{-L}$ and $C_{sca}^{+R}=C_{sca}^{-R}$).

For the second case of general 2D planar structure with $d=0$ and $g_1\neq g_2$ as shown in Fig. 1(c) (with 2D mirror symmetry broken), we have $C_{sca}^{+L}-C_{sca}^{-L} = -C_{sca}^{+R}+C_{sca}^{-R}\neq 0$, and $CD_{sca}^{+k} = -CD_{sca}^{-k}$. Thus we get:

(17)$$\begin{aligned} & C_{sca}^{{-}LCP}=C_{sca}^{{+}RCP} \\ & C_{sca}^{{+}LCP}=C_{sca}^{{-}RCP}. \end{aligned}$$

For 2D configurations with mirror symmetry broken, the scattering process breaks reciprocity, though the extinction maintains reciprocity.

Finally, for 2D symmetric structures with $g_1=g_2$, we have $r_{14}=r_{23}$ and $G_{14}=G_{23}$, then $\widetilde \alpha _A = \widetilde \alpha _B$. The scattering processes of these structures maintain reciprocity.

3. More results and discussions

In this section we present more results and discussions on 2D/3D chiroptical properties and the relation with geometry/symmetry, paying attention to extinction/scattering/absorption CD and reciprocity. Our studies based on the analytical approach of ICDT provide clear physical picture, which is further supported by FDTD simulation. In the calculation of optical properties of difference arrangement of Au NRs, the long axis of the NR is fixed at a=60nm, and the short axis of the NR is b=12nm. The dielectric constants of Au are taken from Ref. [64]. In the FDTD simulation, the mesh is chosen as 0.5nm.

3.1 Chiroptical response of 3D finger-crossed NR dimer

Using ICDT and FDTD simulation, we perform calculation of the chiroptical properties of the 3D fingers-crossed NR dimer to validate our ICDT on one hand, and to compare with the properties of planar structures on the other hand. As pointed out in the section 2.3, the difference between the results from ICDT and CDT is negligible due to the "hidden symmetry" of "fingers-crossed" configuration. We calculate the extinction/scattering/absorption CD spectra by CDT/ICDT for 3D structures with parameters $d=200nm$, $g_1=g_2=0$, angle $\beta$ varying from $0^\circ$ to $180^\circ$. It is seen from Fig. 2(a) that CDs vanish for twist angles $\beta =0^\circ, 90^\circ$ associated with mirror symmetry. The maximum CD appears at $\beta = 45^\circ$ due to the maximum degree of symmetry breaking [45,46]. The FDTD results agree with those from CDT/ICDT. The twist angle dependent chiroptical response has also been verified by experiment [65]. The extinction, scattering and absorption CDs for general 3D structure with $g_1=20$, $g_2=44nm$, $d=200nm$ have also been calculated by CDT, ICDT and FDTD as shown in Fig. 2(b). There is slight difference between the results by CDT and ICDT for typical 3D structures. The results from FDTD simulation agree with those from CDT and ICDT.

Fig. 2. (a) Extinction, scattering, and absorption CDs calculated by CDT/ICDT and FDTD for "finger-crossed" configurations. (b) Extinction, scattering, and absorption CDs calculated by CDT, ICDT and FDTD for general 3D structure.

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3.2 Relationship between CD and symmetry-breaking

For 2D systems, extinction CD vanishes for all configurations due to 3D mirror symmetry and it can not tell the 2D in-plane mirror symmetry breaking. Here we discuss the relationship between 2D symmetry break and the scattering CD based on ICDT, which cannot be accomplished by CDT. We first consider two types of 2D mirror symmetric configurations, the V-shaped structure shown in Fig. 3(a) and the T-shaped structure shown in Fig. 3(b). For the V-shaped structure of 2D mirror symmetry, using the geometry dependent coupling constants Gs (see SI), we have

(18)$$G_{14}=G_{23}, \quad \widetilde G = 0,$$

resulting the same equivalent polarizability $\widetilde \alpha _A=\widetilde \alpha _B$ and the scattering CD=0. For the T-shaped structure of 2D mirror symmetry, we have

(19)$$G_{13}={-}G_{23}, \quad G_{14}={-}G_{24}, \quad \widetilde G = 0,$$

resulting scattering CD=0. As the general configuration with 2D mirror symmetry broken shown in Fig. 3(c), the interaction leads to asymmetric equivalent polarizability $\widetilde \alpha _A \neq \widetilde \alpha _B$, resulting nonvanishing scattering CD (see also section 2.2). Thus our ICDT points out that the scattering CD provides a measure of the 2D mirror symmetry breaking, like the extinction CD measuring the 3D mirror symmetry breaking.

Fig. 3. Schematic diagrams of a V-shaped structure (a), a T-shaped structure (b) and a general structure with 2D mirror symmetry broken (c).

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3.3 Distance-dependent scattering CD: the interplay between symmetry breaking and interaction

For 2D structures without propagation phase change along the direction of light propagation, the extinction cross sections are the same under LCPL and RCPL incidence. Thus there is no CD in the extinction cross section in all cases no matter whether there is 2D in-plane mirror symmetry breaking. Based on ICDT, we have revealed that scattering CD can give the signature of 2D chiral symmetry breaking. We address the factors affecting scattering CD in subsections 3.3, 3.4 and 3.5.

We consider 2D structures of NR dimers as shown in Fig. 1(c). It is clear that the system with $g_1=g_2$ is of in-plane mirror symmetry and $\Delta g=|g_2-g_1|$ could be used as a measure of the degree of symmetry breaking. We fix $g_1$=20nm, $\beta =50^\circ$. As $g_2$ ($g_2>g_1$) increases, $\Delta g$ (the degree of geometric symmetry breaking) increases, which may lead to an enhancement of scattering CD. On the other hand, the coupling strength between NRs are weakened due to longer interaction distance, leading to the decrease of scattering CD. Our calculations based on ICDT and FDTD reveal a structure of maximum scattering CD due to the interplay between these two effects shown in Fig. 4(a). The difference between the results from ICDT and FDTD may be due to the higher order effects. To gain deeper understanding of the mechanism, we perform more analysis of the scattering CD (see SI for details)

(20)$$CD \sim \dfrac{Im(\widetilde \alpha_A^*+\widetilde \alpha_B^*)}{|1-\widetilde \alpha_A \widetilde \alpha_B \widetilde G^2|^2 \cdot r^6} \cdot Re \{(G_{24}-G_{13})(G_{14}-G_{23}) \cdot r^6) \} \equiv II\cdot AD$$

and check the roles of symmetry (breaking) and interaction. Since $G_{ij} \sim 1/r^3$ (r is the distance between the centers of two NRs) indicates the inter-particle distance dependence of interaction and $(G_{24}-G_{13})(G_{14}-G_{23})$ contains the symmetry (breaking) information (see also last section), we show the asymmetry degree $AD\equiv (G_{24}-G_{13})(G_{14}-G_{23}) \cdot r^6$ (black curve) and interaction intensity ($II$, blue curve) in Fig. 4(b). As $g_2$ increases from 40nm to 50nm, the interaction intensity decreases while the degree of asymmetry increases monotonically. The competition between these factors leads to an optimal configuration of maximum scattering CD.

Fig. 4. (a) Scattering CD (based on ICDT and FDTD) and equivalent SCD$=II\cdot AD$ (Eq. (20)) versus $g_2$. (b)Asymmetry degree and interaction intensity versus $g_2$.

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3.4 Angel-dependent scattering CD

We discuss the dependence of scattering CD on the angle $\beta$. First consider the case of $g_1=0nm$, $g_2=60nm$. The scattering CD is shown in Fig. 5(a). When $\beta =90^\circ$, corresponding to "T-shaped" structure with in-plane mirror symmetry, the scattering CD is zero. The scattering CDs for $\beta =90^\circ \pm \delta$ are of the same amplitude and opposite sign, relating to the two structures of mirror image to each other. Though extinction CD vanishes, the scattering CD provide chiroptical characterization of in-plane symmetry (breaking). Fig. 5(b) shows the angle dependence of scattering CD for more general structure with $g_1=20nm$, $g_2=44nm$. Very small CD for large angle $\beta$ is due to weak interaction for large inter-particle distance.

Fig. 5. Scattering CD versus $\beta$. (a) $g_1=0nm$, $g_2=44nm$. (b) $g_1=20nm$, $g_2=44nm$.

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3.5 Scattering CD due to 2D/3D symmetry breaking

We discuss the relationship between the scattering CD and the 2D/3D geometric symmetry breaking. For a symmetric structure with $g_1=g_2=44nm$, $\beta =50^\circ$, the dependence of scattering CD on $d$ is shown Fig. 6(a). The SCD=0 for a 2D symmetric structure ($d=0nm$, with in-plane symmetry). The SCD for the structure with d has the same amplitude and opposite sign of that for the structure with -d (the 3D image structure). For the other asymmetric system with $g_1(=20nm) \neq g_2(=44nm)$, different behavior is found. When $d=0$, the 2D (in-plane) symmetry is broken and $SCD \neq 0$ (see Fig. 6(b)). Also $SCD(d)\neq -SCD (-d)$. The surface plasmon resonances (SPRs) of nanorods leads to the resonant peak in the scattering spectra and strong chiroptical responses. The change of the aspect ratio of nanorods leads to resonance wavelength shifted. We note that the nanosphere dimer doesn’t show chiroptical response due the mirror symmetry. Importantly, at a particular value of d(=-9nm), SCD=0 for incident light in the +k direction (see Fig. 6(c)), even for the system with 3D symmetry broken. It is due to the cancellation between the contribution to SCD from 2D (in-plane) symmetry breaking ($g_1 \neq g_2$) and that from 3D symmetry breaking. Also the system and its image may not have the same amplitude of SCD, see Fig. 6(c) (d=-9nm) and 6(d) (d=+9nm) for an example. For large $|d|$, the SCD due to 3D symmetry breaking dominates, which results in $SCD (d) \simeq -SCD (-d)$ (see 6(a)).

Fig. 6. (a) The d dependence of SCD for symmetric systems (black solid line) and asymmetric systems (red dash line). (b) Scattering cross sections of a 2D (d = 0nm) asymmetric structure for LCPL and RCPL at +k incident direction. (c) Scattering cross sections of a 3D (d =-9nm) asymmetric structure for LCPL and RCPL at +k incident direction. (d) Scattering cross sections of a 3D (d =+9nm) asymmetric structure for LCPL and RCPL at +k incident direction. In (b)-(d), blue solid lines refer to scattering cross sections for RCPL and red dash lines refer to those for LCPL.

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Here we address the connection/difference between 2D and 3D chiroptical characteristics. A structure with 3D mirror symmetry has vanishing extinction CD. A 2D structure is a particular case with 3D mirror symmetry and the extinction CD is zero, no matter the 2D in-plane symmetry is broken or not. The SCD is able to manifest the 2D in-plane symmetry (breaking). In general, the 2D chiroptical response is weaker than that of 3D system. It is because the 3D CD is determined by the inter-particle distance and 2D CD depends on the difference of inter-particle distance, see Eqs (6) and (20) for example. Our ICDT captures missed multipole contribution in CDT. In some 3D configuration such as fingers-crossed NR dimer, hidden symmetry leads to negligible difference between the results from CDT and those from ICDT. So CDT worked well in many previous studies. For 2D structure or 3D structure with small length scale in the light propagation direction (as discussed in this section), one should use more accurate ICDT. We would like to point out that the scattering reciprocity is preserved for symmetric systems with $g_1=g_2$, and is broken for asymmetric systems with $g_1 \neq g_2$. There is no direct correlation between SCD and reciprocity for general 3D structures. While for 2D systems with d=0, there is a relation between SCD and reciprocity. For systems with in-plane symmetry, SCD=0 and scattering reciprocity maintains. While for 2D asymmetric systems, $SCD \neq 0$ and scattering reciprocity is broken. The results based on ICDT have also been verified by FDTD simulations.

3.6 Chiroptical response of 2D square array of NR dimer

We have explored the scattering properties of NR dimer. For a metasurface of arrays of NR dimer, the chiroptical response shows similar feature as that of a NR dimer. We calculate the reflection coefficients of a metasurface with unit cell of NR dimer, which was studied in Section 3.3–3.5. The metasurface is made of a 2D array of NR dimer with lattice period $p_x=p_y=300nm$. The reflection coefficient can be calculated using the formula

(21)$$R = |p_A|^2+|p_B|^2+p_A p_B^*\cos{\beta}+p_B p_A^*\cos{\beta}.$$

We analyze two cases: the 2D symmetric and asymmetric unit cell. The calculation results based on ICDT and FDTD are presented in Fig. 7. It is seen that the results based on ICDT and FDTD agree well and they show the same physical picture analyzed above: SCD vanishes for the metasurface with symmetric unit cell and nonzero SCD can be observed for the metasurface with asymmetric unit cell. The difference between FDTD simulation and ICDT is due to higher order/multipole contribution, which is small for nanoparticle’s size smaller than the wavelength.

Fig. 7. (a) Reflection coefficient of 2D array composed by 2D achiral unit cells ($g_1=g_2=50nm,\beta =50^\circ$). The result are calculated by ICDT. (b) Reflection coefficient of the same structure as that in (a) calculated by FDTD. (c) Reflection coefficient of 2D array composed by 2D chiral unit cells ($g_1=20nm, g_2=50nm, \beta =50^\circ$). The result are calculated by ICDT. (d) Reflection coefficient of the same structure as that in (c) calculated by FDTD.

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4. Conclusion

We have developed an analytical ICDT to explore the connection between 2D/3D geometry/symmetry and chiroptical response. Our ICDT can capture the missing information in CDT and provide quantitative results, supported by FDTD simulation. We point out that the scattering CD can characterize the 2D in-plane mirror symmetry breaking, though the extinction CD vanishes. Based on the uncovered mechanism of 2D chiroptical response, we find an optimal structure with maximum SCD due to the competition between symmetry breaking and interaction. Different nature of reciprocity of 2D/3D structure is analysed based on ICDT. Interestingly, a symmetry broken structure with vanishing SCD is found due to the interplay between 2D/3D symmetry breaking. Our results not only provide a general theoretical framework for chiral optics, but also deepen our understanding of the different chiroptical response related to 3D/2D symmetry/breaking.

Funding

National Natural Science Foundation of China (12174032); National Natural Science Foundation of China-Research Grants Council (11861161002); National Key Research and Development Program of China (2017YFA0303400).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Characterizing chiroptical properties of 2D/3D structures based on an improved coupled dipole theory

Abstract

1. Introduction

2. Model and theoretical approach

2.1 Couple dipole theory (CDT)

2.2 Improved couple dipole theory (ICDT)

2.3 Scattering CD and reciprocity

3. More results and discussions

3.1 Chiroptical response of 3D finger-crossed NR dimer

3.2 Relationship between CD and symmetry-breaking

3.3 Distance-dependent scattering CD: the interplay between symmetry breaking and interaction

3.4 Angel-dependent scattering CD

3.5 Scattering CD due to 2D/3D symmetry breaking

3.6 Chiroptical response of 2D square array of NR dimer

4. Conclusion

Funding

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (7)

Equations (21)

Optics Express