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Abstract
We study the chiral optical properties of one-dimensional arrays of plasmonic twisted nanorod dimers. By using finite-difference time-domain (FDTD) simulation and analytical approach based on the coupled dipole model, we have revealed unusual chiral optical responses due to the cooperation of local structure and collective effect. It is found that one-dimensional arrays of achiral unit may show chiral optical responses. Moreover, besides the classical bisignate lineshape of circular dichroism (CD) induced by localized surface plasmon resonance, a new CD peak/dip appears, originating from Wood anomaly. Near the Wood anomaly frequency, the optimal twist angle to achieve the highest CD has been shifted compared with that of single twisted nanorod dimer. The universal geometric configurations of the strongest chiral optical responses have been found.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Chirality plays important roles in biological processes and many biomolecules, such as L-amino acids, D-sugar, protein, DNA, are of chiral structure, i.e., they can not superpose with their mirror images [1]. These chiral objects may show interesting chiral optical spectra, which have important applications. For example, the circular dichroism (CD) could be used to detect the secondary structure of proteins. Usually, the chiral optical response of natural molecules is in the ultraviolet (UV) range and the signal is weak [2]. Composite nanostructures (with subunits)- “artificial molecules” (with “artificial atoms”), which play the role of basic functional and constitutional unit like molecules in natural material, provide more opportunities for exploring the optical processes, including the chiral optics. With the help of the variable shape and size of the subnanostructure (“artificial atom”) as well as the relative spacial arrangement between “atoms”, novel optical properties can be achieved. [3–5].
Recently, there are many studies on the chiral optics based on hybrid nanostructures fabricated by top-down/bottom-up routine [6–9]. These hybrid nanostructures show tunable optical activity in the visible light to the infrared range and the chiral response is quite strong due to the plasmon resonance. For natural molecules, it is almost impossible to modulate the relative position of atoms. However, in hybrid nanostructures (“artificial molecules”), the relative position of the subunit (“artificial atoms”) can be tuned. For example, with the help of molecular linkers/DNA origami, complicated geometrical configuration of plasmonic particles (nanospheres and nanorods) can be realized [7,10–15]. The property of artificial molecules’ variable geometry can be utilized to explore the geometrical chirality and corresponding optical chiral property, originating from electromagnetic interaction of the “atoms” [16].
The chiral optical response, such as circular dichroism (CD), is due to the combined effects from electric dipole and magnetic dipole moments [17,18], which can be modulated by tuning the geometric structure. Aiming at exploring the relationship between chiral optical spectra and geometrical chirality of “artificial molecules”, lots of plasmonic structures have been studied in details [7,10–15,19–23]. Kuzyk et al explored the left- and right-hand nanohelix formed by gold particles, which showed strong bisignate dip-peak circular dichroism (CD) signals in visible light range [7]. Shen et al constructed the tetramer structure with gold nanoparticles, which showed plasmonic CD signals [19]. Ferry et al theoretically showed that with changing the length of one edge of tetrahedral nanoparticle frame, the handness of the structure changed and the CD signals had the opposite sign [20]. Hentschel et al analyzed the plasmonic hybridized modes and corresponding CD signs under geometrical perturbation with two stacked L-shaped resonators [21]. Many studies have been performed on the chiral optical properties of one-/two-/three-dimensional plasmonic structures with distinct shapes [22–26], which may lead to giant plasmonic CD [26,27]. Several research groups have addressed the relationship between the optical and geometrical chirality [28–30].
Though there have been many studies on the chiral optical properties of nanostructures, so far, most studies have focused on the local structure/local interaction. Yet, the collective effect/long range interaction may lead to interesting physical effect, in particular in periodical systems. Some previous studies have shown interesting optical properties due to lattice resonance [31–34], for example, the ultranarrow reflectance/extinction spectra. Relatively little attention has been paid to the collective effect in chiral optical processes. In this paper, we moved a step toward the understanding of the collective effect in chiral plasmonics. By coupled dipole approach and finite-differential time-dominant (FDTD) simulation, we have performed systematic studies of one-dimensional (1D) chains of twisted nanorod dimers, paying attention to the collective effect, in particular, the lattice resonance or Wood anomaly. Our studies reveal that the interplay between local structure/near field interaction and collective effect/far field interaction leads to quite different optical activity than that of the local nanostructures. Interestingly, it is found that the one-dimensional arrays of achiral objects show chiral responses due to the collective effect. Moreover, the universal optimal condition for the strongest CD has been found, which is quite different from that of local nanostructures. Our studies not only deepen the understanding of chiral optics but also provide useful guidance for the design of sensors based on optical activity.
2. Model and theoretical approach
Our system is a 1D array of silver nanorod (modelled as ellipsoid) dimers of “fingers crossed” structure, as shown in Fig. 1. The geometry of the unit of “fingers crossed” structure is characterized by the parameters: the shape and size of the silver ellipsoid, the twist angle $\varphi$ between the long axis of the two ellipsoids, the central distance between the silver ellipsoid $h$. The plasmonic optical activity of the twisted silver ellipsoid dimers can be modulated by tuning the geometric parameters of the structures [35–38]. The global geometry is characterized by the lattice constant (LC) $d$. The correlation between the local structure and the global periodic structure is described by the units’ rotated angle $\theta$ with respect to the x-axis (the chain is along the y-axis). In this paper, we focus on collective chirality due to the interplay between the local intra-dimer interaction within the unit and the inter-dimer interaction among the units.
We consider the silver ellipsoid with long axis diameter 80nm and short axis diameter 30nm. The plasmonic dipole mode along the long axis is described by the Kuwata’s semianalytical formula of ellipsoid’s polarizability $\alpha$ [39]. The sliver’s bulk permittivity is taken from Ref. [40]. The interaction between the silver ellipsoids is described by the dipole interaction for the parameter range we considered [41,42]. We have used the parameter $h$=100nm, the lattice constant $d$=540nm if not specifically noted. The background is the vacuum with the relative refractive index 1.
The particle chain is irradiated by a left/right circularly polarized light with electric field $\textbf {E}=E_0(\hat {e}_x+i\eta \hat {e}_y) e^{i(kz-wt)}$($\eta =+1/-1$ for left/right circular polarization) ($k,w$ the wavevector, frequency of the incident field, $\hat {e}_x$, $\hat {e}_y$ the unit vector along x- and y-axis). In the coupled dipole model (CDM), the excited electric dipoles of the silver ellipsoids at $\textbf {r}_n^A=(0, nd, 0)$ or $\textbf {r}_n^B=(0, nd, h)$ are $\textbf {P}^A_n=\alpha \textbf {E}^A_{loc,n}$ or $\textbf {P}^B_n= \alpha \textbf {E}^B_{loc,n}$, $\textbf {E}^{A/B}_{loc,n}$ is the superposition of the incident field and the radiative electric dipole field from other silver nanoparticles. $\textbf {P}^A_n$ and $\textbf {P}^B_n$ satisfy the coupled equations:
To the first order of inter-unit interaction, the angle dependence of $\bar {D}$ may be neglected, which is also justified by our FDTD simulation. From Eq. (6), it is easy to see that the intensity of CD depends on three terms “A”, “B”, “C” (proportional to A, B, C). The “A” term is due to the interaction within the unit and terms “B”, “C” originate from the inter-unit interaction. Our analytical calculation based on the CDM provides a clear physical picture of the interplay between the local intra-unit interaction and the far field inter-unit interaction, as shown in more detail in the next section. To further support our analytical analysis, we have performed FDTD simulation. In the numerical FDTD calculation, we have used the mesh as 0.9nm. The boundary conditions are periodic conditions in the y-direction and perfectly matched layer (PML) conditions in the other two directions. The light is incident along z-direction as in CDM.
3. Results and discussions
To show the new collective plasmonic chirality, it is helpful to give a complete comparison with the chiral properties of the local unit. We first discuss the chiral property of a single twisted silver nanorod dimer in subsection 3.1. In subsection 3.2, the chirality of 1D twisted dimer chains and new phenomenon due to inter-unit interaction will be discussed in detail. And subsection 3.3 focuses on the chiral responses near lattice resonance. Both the calculation results based on CDM and FDTD are presented.
3.1 Chirality of single twisted silver nanorod dimer
We calculate the CD spectra of a single twist silver ellipsoid dimer based on Eqs. (1)–(5) within CDM. The CD spectra are shown in Fig. 2. The CD depends on the shape of silver nanoparticles (related to plasmonic resonance) and geometric configuration ($\propto \sin (2\varphi )$). The dimer configurations with $\varphi =0, \pi /2$ possess mirror symmetry and have vanishing CD, as shown in Figs. 2(a) and (b). The single twisted dimer has a classical plasmonic bisignate dip-peak (related to plasmonic resonance) CD spectrum as shown in Fig. 2(b). The two structures with twist angle $\varphi =\pi /4$ and $\varphi =3\pi /4$ are enantiomer, therefore the corresponding CD signals (blue and red lines) have opposite signs, i.e., $CD(\varphi =\pi /4)=-CD(\varphi =3\pi /4)$.
Fig. 2. CD spectra of a single twisted silver ellipsoid dimer based on CDM. (a)The dependence of CD on the twist angle $\varphi$ and wavelength; (b)The plasmonic bisignate CD spectra as a function of wavelength for $\varphi =0$ (green line), $\varphi =\pi /4$ (blue line), $\varphi =\pi /2$ (purple line) and $\varphi =3\pi /4$ (red line) twist angle; (c) CD as a function (Sine-like function) of the twist angle $\varphi$ for the incident light wavelength 505nm(blue line) and 400nm(red line, scaled 30 times);(d) The extinction cross section of single chiral dimer($\varphi =3\pi /4$) irradiated by a left (blue line)/right (red line) circularly polarized light (LCP/RCP).
From the perspective of pure geometrical chirality, the geometric configurations with twist angle $\varphi =0, \pi /2$ keeps mirror symmetry. The structures with $\varphi = \pi /4, 3\pi /4$ could be configurations with most geometric mirror symmetry breaking. Then one may expect related phenomena in the chiral optics aspect, i.e., the dimer with $\varphi = \pi /4, 3\pi /4$ may have the largest |CD| value. Indeed, as seen in Fig. 2(c), the maximal chiral response is achieved at $\varphi = \pi /4, 3\pi /4$. The much stronger CD of the configuration for $\lambda =505nm$ than that for $\lambda =400nm$ is due to plasmonic resonance. The CD signal has a sine-like shape curve in the range $(0,\pi )$, which is frequency independent. In addition, from Eq. (6), we can obtain the CD for single twisted silver nanorod dimer as $CD \propto \sin (kh)\sin (2\varphi )$ (by setting $\bar {B}=0$, $\bar {C}=0$), which agrees well with the results in Fig. 2(c). And in Fig. 2(d), extiction cross section of single chiral dimer ($\varphi =3\pi /4$) is shown.
3.2 Chirality of 1D chains of twisted silver nanorod dimers
Besides the local/near field interaction within each unit in the chain, there is global/far field interaction between the units, which results in novel chiral optical properties. The optical activity depends on the local structure (such as the twist angle $\varphi$), the global parameter (the lattice constant $d$), the correlation between the local structure and global structure, i.e., the relative orientation of the unit with respect to the chain direction.
The CD spectra of 1D chains are displayed in Fig. 3, which show different features than those of single nanorod dimer. As discussed in the previous section, there are two configurations of a single dimer (as the unit in the 1D chain) with $\varphi =0$ or $\varphi =\pi /2$, which possess mirror symmetry and have vanishing CD. The 1D chains made of these two types of achiral unit have different chiral optical properties. For 1D chain with $\varphi =0$, CD=0. Interestingly, a chain with achiral unit ($\varphi =\pi /2$) shows nonvanishing CD (when $\theta \neq 0$). The optical activity of 1D chain made of achiral unit depends not only on the local structure of the unit (see the difference between the cases of $\varphi =0$ and $\varphi =\pi /2$), but also on the correlation between local structure and global structure, characterized by the parameter $\theta$. As seen from Eq. (6), for 1D chain with achiral unit (i.e., twist angle $\varphi =\pi /2$), $CD \propto \sin (2\theta )$. Here we note that the collective chiral responses in simple 1D straight-chain (with achiral unit $\varphi =\pi /2$) are different from that in Ref. [7], where the global helix lattice structure is the key factor for the optical activity (Note that the 1D straight chain of sphere nanoparticles shows no chiral responses). Also, the collective chirality appears in the presence of normal incidence, which is quite different from the extrinsic chirality as a result of oblique incidence [44].
Fig. 3. (a) CD of 1D chain versus twist angle $\varphi$ and wavelength for $\theta =\pi /4$ and lattice constant $d$=540nm; (b) The CD spectra for $\theta =\pi /4$, lattice constant $d$=540nm, twist angle $\varphi =0$ (green line), $\varphi =\pi /4$ (blue line), $\varphi =\pi /2$ (purple line), and $\varphi =3\pi /4$ (red line); (c) Comparison between CD of single dimer (for $\lambda$=500nm (red line) and 542nm (purple line and scaled 50 times)) and that of a dimer chain (for $\lambda$=500nm (blue line) and 542nm (green line));(d) The extinction cross section (per unit) of the dimer chain with achiral unit cell ($\varphi =\pi /2$) irradiated by a left(blue line)/right (red line) circularly polarized light (LCP/RCP).
We’ve seen that $\theta$ (the angle describing the relative orientation of the local unit) plays an important role in the modulation of the collective CD spectrum. In the special condition of $\theta =0$, $CD\propto \sin (2\varphi )$, showing the same dependence on the local twist angle $\varphi$ as for the single dimer. For other value of $\theta$, 1D chain shows different $\varphi$ dependence from that of a single dimer. Besides the appreciable CD for $\varphi =\pi /2$ as discussed above and shown in Fig. 3(b), $CD(\varphi =\pi /4)\neq -CD(\varphi =3\pi /4)$ (see Fig. 3(b) and compare with Fig. 2(b)). Moreover, unlike the case of single dimer, the configuration of maximum |CD| is different from those with $\varphi =\pi /4, 3\pi /4$, as seen from Fig. 3(c). The extinction cross section of dimer chain with achiral unit cell are presented in Fig. 3(d). It is clear that the resonant sharp peak induced by lattice resonance/Wood anomaly has different response for left/right circularly polarized light.
3.3 Chiral responses near the lattice resonance
In a periodic system, there is an important physical effect from lattice resonance (LR). When the light’s wavelength near the lattice constant, strong collective inter-unit interaction occurs and a new third sharp peak/dip is induced by LR in the CD spectrum as seen in Fig. 3(b). It is related to super-narrow lineshape originating from Wood anomaly effects induced by LR [31].
The variation of CD signals is important and useful in applications. We introduce a new quantity $\Delta CD \equiv CD_{max}(\varphi )-CD_{min}(\varphi )$ in each frequency. $\Delta CD$ versus wavelength for systems with different lattice constants is shown in the Fig. 4. We’ve seen sharp peaks of $\Delta CD$ appearing at the wavelength near the lattice constant due to LR, which may have applications in sensing based on chiral optics.
Fig. 4. $\Delta CD$ versus wavelength for 1D twisted dimer chain with different lattice constants 522nm, 530nm and 540nm and orientation angle $\theta =\pi /4$.
Though the single dimer configuration with most mirror symmetry breaking (having largest |CD|) is $\varphi =\pi /4$ or $3\pi /4$, the 1D chains show quite different behavior than the single dimer. Here we explore the configuration of the strongest chiral response. In the parameter range near LR, the "B" term in Eq. (6) dominates. By solving the equation $\partial CD/\partial \theta =0$, we obtain the optimal condition of $\theta$ for fixed $\varphi$
where $\theta \in (0,\pi )$ and n is an integer. Interestingly this condition is independent on the material parameter and geometric parameter of the nanorod. Equation (7) can be rewritten as $\theta +\varphi /2=n\pi /2$, which means the angular bisector of twisted dimer should be parallel with the chain’s periodic direction.To check the result above, we consider the case with a local twist angle $\varphi =\pi /3$ as an example and study the dependence of CD on the angle $\theta$. We choose the wavelength close to the lattice constant for a stronger CD signal (see Fig. 4). As shown in Fig. 5(a), the extreme values appear at the positions $\theta =\pi /3$ and $5\pi /6$ as predicted by Eq. (7), which are marked with dash lines.
Fig. 5. CD based on CDM(lines) and FDTD(stars). (a) CD versus the orientation angle $\theta \in (0,\pi )$ with local twist angle $\varphi = \pi /3$, $d$= 540nm and the wavelength of the incident light is 542nm. The dash lines are determined by Eq. (7). (b) CD versus local twist angle $\varphi \in (0,\pi )$ with orientation angle $\theta = \pi /6$, $d$= 540nm and the wavelength of the incident light is 542nm. The dash lines are determined by Eq. (8).
Next, we consider the $\varphi$ dependence. From $\partial CD/\partial \varphi =0$, we have
To verify above equation, we set the orientation angle $\theta = \pi /6$ and change the local twist angle $\varphi$, as an example. As shown in Fig. 5(b), the extreme values of CD appear approximately at $\varphi =\pi /6$ and $2\pi /3$. The minor derivation is partially due to the contribution from other terms.To further support our calculation and analysis based on CDM, we have also performed full wave electromagnetic field simulation by FDTD method. The results are shown with stars in Fig. 5, which indicates that our results based on CDM are solid.
In Fig. 6, we show the CD map of $\theta$ and $\varphi$ in the wavelength (542nm) near the lattice resonance. There is a symmetry ( $\varphi \rightarrow \pi -\varphi$, $\theta \rightarrow \pi -\theta$ and $CD \rightarrow -CD$) in the map. This symmetry of CD map originates from that the structure with $\pi -\varphi$ and $\pi -\theta$ forms an enantiomer of that with $\varphi$ and $\theta$. Based on the Eqs. (7) and (8), we can solve the position ($\varphi ,\theta$) on the map to achieve extreme of the CD signal. Clearly $(\pi /3,5\pi /6)$ and $(2\pi /3,\pi /6)$ are the positions ($\varphi ,\theta$) of CD’s extremes, which agrees well with the results in Fig. 6. We have found the universal geometric configurations (independent of the material parameter and geometric parameter of the nanorod) with the strongest chiral optical responses due to collective effect.
Fig. 6. The map of CD versus orientation angle $\theta$ and local twist angle $\varphi$ with lattice constant of 540nm and incident light wavelength of 542nm.
Finally, we would like to discuss possible experimental proposals. By using the nanofabrication technique based on DNA origami, it is now possible to arrange the nanorod at a precise position. The nanorods could be placed on the two-side of DNA origami sheet to form dimers. The twist angle and the inter-particle distance can be tuned by careful design [11,45,46]. We would like to point out that the two-dimensional arrays of nanorod dimers have periods (lattice constants) in two directions and the resulting LRs may bring about new features in plasmonic chirality, which will be presented in our future paper.
4. Conclusion
Based on CDM and FDTD simulation, the optical properties of 1D chains of twisted Ag nanorod dimers have been studied systematically. It is found that the combination of the local geometric configuration and the collective effect leads to novel chiral optical properties. Interestingly, the 1D chain made of achiral unit may lead to a significant chiral signature. The lattice resonance results in a sharp chiral CD peak. Moreover, the universal optimal configurations of the strongest CD have been found. Our results not only deepen our understanding of the optical chirality at the nanometer scale, but also provide useful guidance for the design of nano-sensors with high sensitivity.
Funding
National Key Research and Development Program of China (2017YFA0303400); National Natural Science Foundation of China (11774036); National Natural Science Foundation of China-Research Grants Council (11861161002).
Disclosures
The authors declare no conflicts of interest.
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