Enhancing chiroptical responses in the nanoparticle system by manipulating the far-field and near-field couplings

Qian Zhao; Houjiao Zhang; Zhang-Kai Zhou; Xue-Hua Wang; Xue-Hua Wang

doi:10.1364/OE.484851

1. Introduction

Circular dichroism (CD), as the representative of chiroptical response, can be defined as the differential spectral response in the chiral system excited by asymmetric electromagnetic fields, such as the left-handed and right-handed circularly polarized (LCP and RCP) lights [1–4]. Due to its advantages in exhibiting the structure, dynamics and thermodynamics information of macromolecules, the chiroptical spectroscopy has long been cultivated as a powerful tool in physics, chemistry, and biology [5–7]. However, chiroptical responses from natural molecules are typically very weak and thus limit their further applications in biomolecular detection and sensing [8–13].

With the ability to enhance the interaction between asymmetric light (circularly polarized light) and confined optical modes, nanoparticles and the systems composed of nanoparticles are widely accepted as important platforms to generate strong chiroptical response, including the helical, G-shaped, and dimer nanostructure systems [14–29]. In various chiroptical structures, a twisted nanorod dimer is theoretically considered to be a basic model for the emergence of chiroptical response [30–35] and also experimentally regarded as the building blocks for integrated chiroptical photonic device [36–39]. Therefore, the optical chirality of the twisted nanorods has been extensively studied. However, most of these analyses about this system are based on classical electrodynamics or numerical simulation [40,41], which is difficult to obtain the simplified expression of maximum CD response, leading to problems for guiding fast structure optimizations of chiroptical devices. Therefore, in order to realize the advanced photonic devices with optimized chiroptical response, the approaches for analyzing chiroptical response based on simplified quantitative calculations are highly desired.

Here we take the twisted nanorod dimer system as an example to provide a quantitative analytical approach based on the temporal coupled-mode theory (TCMT) [42,43]. Using this approach, we can intuitively and deeply analyze its optical chirality characteristics from the perspective of mode coupling, and calculate the expression of CD in this system. Thereafter, we clearly obtain the relationship between the optical chirality response and the basic parameters of this system, providing the possibility for fast designing of the chiroptical systems with giant CD response. Our findings not only clarify the origin of strong optical chirality in the twisted nanorod dimer, but also provide the guidance to reach the optimal CD responses, paving the way for theoretical research in optical chirality of nanosystems.

2. Theory and results

Figure 1(a) illustrates the system under consideration, which is the Ag rod system consisting of two twisted Ag rods (defined as Ag rod a and Ag rod b, respectively) periodically arranged with the lengths of Px and Py (Px = Py). Ag rod b rotates around the endpoint of Ag rod a. The two Ag rods are separated by a gap with the distance of G and twisted by an angle of θ. The dimensions of the two Ag rods were set as: length L, width w, and height h. The bottom Ag rod is aligned parallel to the x axis. The Ag rods are embedded in vacuum (silicon dioxide can be considered in the experiment). The LCP and RCP lights were obtained by superimposing x- and y-polarized plane wave sources with a phase difference of 90° and -90°, respectively.

Fig. 1. Enhancement of chiroptical response derived from the mode coupling of two twisted Ag rods. (a) Schematic view of the twisted Ag rods and the definition of the geometrical parameters. (b) TCMT model describing the mode coupling of Ag rods. The Ag rod a (amplitude ${\Psi _a}({\omega ,t} )$ and total loss γ_a) couples to Ag rod b (amplitude ${\Psi _b}({\omega ,t} )$ and total loss γ_b) with near and far field coupling strength g_ab. Ag rods exchange energy with the incoming s(1)+ and outgoing s(1,2)- waves associated with the two ports with coupling constants d(1,2), respectively. (c) The CD and absorption spectra of the twisted Ag rods systems obtained by the FDTD (solid red) and TCMT (blue dotted) methods. The relevant parameters of the two nanorods were set as: length L_a = L_b = 150 nm, width w_a = w_b = 30 nm, height h_a = h_b = 30 nm, distance G = 40 nm, angle θ = 40°, and periodic size Px = Py = 400 nm, absorption losses ${\gamma _{(abs - a)}}$ = ${\gamma _{(abs - b)}}$ = 22.32 eV, radiative losses ${\gamma _{(rad - a)}}$ = ${\gamma _{(rad - b)}}$ = 69.44 meV, and near-field coupling ${g_0}$ = 106.64 meV. (d) The contributions of near and far field coupling to the chiroptical response of this system in Fig. 1(c).

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The chiroptical response of this system can be understood on the basis of TCMT. In this framework, the incident light can be considered as the input channels of the system, and it is coupled with the optical modes excited within these Ag rods. Figure 1(b) depicts the schematic diagram of TCMT model about the Ag rods systems. Let ${\Psi _a}({\omega ,t} )$ and ${\Psi _b}({\omega ,t} )$ represent the amplitude of the resonant modes of Ag rod a and Ag rod b, and the dynamic equations of these Ag rods excited by LCP and RCP lights can be written as [44,45]

(1)$$\begin{array}{c}\frac{{d{\varPsi _a}({\omega ,t} )}}{{dt}} = ({ - i{\omega_a} - {\gamma_a}} ){\varPsi _a}({\omega ,t} )- i{g_{ab}}{\varPsi _b}({\omega ,t} )+ \sqrt {\frac{{{A_c}}}{{{A_1}}}} {d_a}{s_{1a + }}\\= ({ - i{\omega_a} - {\gamma_a}} ){\varPsi _a}({\omega ,t} )- i{g_{ab}}{\varPsi _b}({\omega ,t} )- i\sqrt {\frac{{{A_c}}}{{{A_1}}}} \sqrt {{\gamma _{rad - a}}} {s_{1 + }}\end{array}$$

(2)$$\begin{array}{c}\frac{{d{\varPsi _b}({\omega ,t} )}}{{dt}} = ({ - i{\omega_b} - {\gamma_b}} ){\varPsi _b}({\omega ,t} )- i{g_{ab}}{\varPsi _a}({\omega ,t} )+ \sqrt {\frac{{{A_c}}}{{{A_1}}}} {d_b}{s_{1b + }}\\= ({ - i{\omega_b} - {\gamma_b}} ){\varPsi _b}({\omega ,t} )- i{g_{ab}}{\varPsi _a}({\omega ,t} )- i\sqrt {\frac{{{A_c}}}{{{A_1}}}} \sqrt {{\gamma _{rad - b}}} {e^{i[{ - k({G + h} )\pm \theta } ]}}{s_{1 + }}\\ {\kern 6cm}- \theta \to \textrm{LCP},\; + \theta \to \textrm{RCP}\end{array}$$

where i is the imaginary unit, the resonant frequencies of the Ag rod a and Ag rod b are denoted by ${\omega _a}$ and ${\omega _b}$, respectively. The total losses of the Ag rod a and Ag rod b are denoted by ${\gamma _a}$ and ${\gamma _b}$, respectively. The radiative and absorption losses of the Ag rod a / Ag rod b are denoted by ${\gamma _{rad - a}}$/${\gamma _{rad - b}}$ and ${\gamma _{abs - a}}$ /${\gamma _{abs - b}}$, respectively, while it satisfies as ${\gamma _a} = {\gamma _{abs - a}} + {\gamma _{rad - a}}$ and ${\gamma _b} = {\gamma _{abs - b}} + {\gamma _{rad - b}}$. $- i\sqrt {{\gamma _{rad - a}}} $ ($- i\sqrt {{\gamma _{rad - b}}} $) is the coupling constants ${d_a}$ (${d_b}$) between the Ag rod a (Ag rod b) and the input channel ${s_{1a + }}$ (${s_{1b + }}$), ${A_c}$ is the effective aperture of the Ag rod and ${A_1}$ is the spot size of the excitation light [44], ${g_{ab}}$ is the coupling coefficient between the modes of Ag rod a and Ag rod b.

The mode coupling between Ag rod a and Ag rod b includes the near-field and far-field coupling and the coupling coefficient can be written as

(3)$${g_{ab}} = {g_0} - i\sqrt {\frac{{{A_c}}}{{{A_1}}}} \sqrt {{\gamma _{rad - a}}} \sqrt {{\gamma _{rad - b}}} cos\theta {e^{i[{ - k({G + h} )} ]}}$$

where the first term ${g_0}$ is the near-field coupling between two modes originating from the overlapping of strongly localized evanescent waves of Ag rods [45]. For the origin of the far-field coupling of the system, we can understand it from the perspective of TCMT. The periodic Ag rods will excite the radiation channels at the upper and lower interfaces [45,46]. The Ag rod a will be coupled by the radiation channel excited at the lower interface of the Ag rod b ($- i\sqrt {\frac{{{A_c}}}{{{A_1}}}} \sqrt {{\gamma _{rad - a}}} \sqrt {{\gamma _{rad - b}}} cos\theta {e^{i[{ - k({G + h} )} ]}}{\varPsi _b}({\omega ,t} )$), and the Ag rod b will be coupled by the radiation channel excited at the upper interface of the Ag rod a ($- i\sqrt {\frac{{{A_c}}}{{{A_1}}}} \sqrt {{\gamma _{rad - a}}} \sqrt {{\gamma _{rad - b}}} cos\theta {e^{i[{ - k({G + h} )} ]}}{\varPsi _a}({\omega ,t} )$). So the far-field coupling between two modes from the radiation channels of the Ag rods is $- i\sqrt {\frac{{{A_c}}}{{{A_1}}}} \sqrt {{\gamma _{rad - a}}} \sqrt {{\gamma _{rad - b}}} cos\theta {e^{i[{ - k({G + h} )} ]}}$, corresponding to the second term of Eq. (3).

By taking $\frac{{d{\varPsi _a}({\omega ,t} )}}{{dt}} ={-} i\omega {\varPsi _a}({\omega ,t} )$ and $\frac{{d{\varPsi _b}({\omega ,t} )}}{{dt}} ={-} i\omega {\varPsi _b}({\omega ,t} )$ (steady state), the resonance-mode amplitude ${\varPsi _a}(\omega )$ and ${\varPsi _b}(\omega )$ under steady state are calculated to be

(4a)$${\varPsi _a}(\omega )={-} {s_{1 + }}\sqrt {\frac{{{A_c}}}{{{A_1}}}} \frac{{{g_{ab}}\sqrt {{\gamma _{rad - b}}} {e^{i[{ - k({G + h} )\pm \theta } ]}} + i\sqrt {{\gamma _{rad - a}}} [{ - i({\omega - {\omega_b}} )+ {\gamma_b}} ]}}{{[{ - i({\omega - {\omega_a}} )+ {\gamma_a}} ][{ - i({\omega - {\omega_b}} )+ {\gamma_b}} ]+ {{({{g_{ab}}} )}^2}}}$$

(4b)$${\varPsi _b}(\omega )={-} {s_{1 + }}\sqrt {\frac{{{A_c}}}{{{A_1}}}} \frac{{{g_{ab}}\sqrt {{\gamma _{rad - a}}} + i\sqrt {{\gamma _{rad - b}}} {e^{i[{ - k({G + h} )\pm \theta } ]}}[{ - i({\omega - {\omega_a}} )+ {\gamma_a}} ]}}{{[{ - i({\omega - {\omega_a}} )+ {\gamma_a}} ][{ - i({\omega - {\omega_b}} )+ {\gamma_b}} ]+ {{({{g_{ab}}} )}^2}}}$$

The absorption of the two Ag rods system can be given by

(5)$$A(\omega )= {\gamma _{abs - a}}{\left|{\frac{{{\varPsi _a}(\omega )}}{{{s_{1 + }}}}} \right|^2} + {\gamma _{abs - b}}{\left|{\frac{{{\varPsi _b}(\omega )}}{{{s_{1 + }}}}} \right|^2}$$

The optical chirality is defined as $\textrm{CD} = {\textrm{A}_{\textrm{LCP}}} - {\textrm{A}_{\textrm{RCP}}}$, where ${\textrm{A}_{\textrm{LCP}}}$ and ${\textrm{A}_{\textrm{RCP}}}$ are the absorption under LCP and RCP excitation, respectively.

The optical chirality CD can be written as

(6)$$\textrm{CD}(\omega )= {\textrm{A}_{\textrm{LCP}}}(\omega )- {\textrm{A}_{\textrm{RCP}}}(\omega )= {\textrm{A}_{({ - \theta } )}}(\omega )- {\textrm{A}_{({ + \theta } )}}(\omega )$$

Here we consider that the two Ag rods have the identical sizes so as to satisfy the mode matching and facilitate theoretical analysis. The CD and absorption spectra of the twisted Ag rods systems are obtained by the FDTD and TCMT methods (see Supplement 1), which are in great agreement with each other (Fig. 1(c)). For the twisted Ag rods systems, two extreme values at λ₁ = 740 nm (mode 1) and λ₂ = 825 nm (mode 2) are observed in the CD spectra, which can correspond to the resonances of antibonding modes and bonding modes of absorption spectrum under linearly polarized light, respectively (see Fig. 1(c)). The antibonding modes and bonding modes of Ag rods arise from the near and far coupling between the modes of Ag rods and correspond to two extreme values of the optical chirality in this system. It can be concluded that the gained optical chirality of the system is caused by the mode coupling of Ag rod.

For λ₂ = 825 nm (mode 2), the different electric field and charge distributions for LCP and RCP lights give rise to the distinct interaction between Ag rods, thus forming the differential absorption of Ag rods and producing a significant CD response in the system (see Supplement 1). On the contrary, for λ₁ = 740 nm (mode 1), the near-field characteristics under the LCP and RCP lights are similar, leading to a weaker CD response than that of mode 2 (see Supplement 1).

Figure 1(d) shows the contribution of near and far field coupling between the Ag rods to the chiroptical response obtained by the TCMT calculations. It can be seen from Fig. 1(d) that the near field coupling contributes to the chiroptical responses and determines the resonant frequencies of mode 1 and 2, and the main contribution of the chiroptical responses originates from the far field coupling.

As it is well-known that, calculating expressions about the maximum CD are very important for enhancing the chiroptical response, which is also a key task for designing and fabricating photonic devices with large CD responses. From the above, we can know that the strongest chiroptical response of this system occurs at the resonant frequency of either mode 1 or 2 (${({\textrm{CD}(\omega )} )_{max}} = |{{{\{{\textrm{CD}({{\omega_1}} ),\textrm{CD}({{\omega_2}} )} \}}_{max}}} |$). Here we can calculate the resonance frequency of mode 1 and 2 to obtain the expression of the maximum CD signal.

The two resonance frequencies of mode 1 and 2 can be obtained (see Supplement 1)

(7a)$${\omega _1} = \frac{{{\omega _a} + {\omega _b}}}{2} + {g_0} - \sqrt {\frac{{{A_c}}}{{{A_1}}}} \sqrt {{\gamma _{rad - a}}} \sqrt {{\gamma _{rad - b}}} cos\theta sin[{k({G + h} )} ]$$

(7b)$${\omega _2} = \frac{{{\omega _a} + {\omega _b}}}{2} - {g_0} + \sqrt {\frac{{{A_c}}}{{{A_1}}}} \sqrt {{\gamma _{rad - a}}} \sqrt {{\gamma _{rad - b}}} cos\theta sin[{k({G + h} )} ]$$

When it satisfies $\frac{{{\mathrm{\Delta }^2} + \sqrt {\frac{{{A_c}}}{{{A_1}}}} {\gamma _0}{\gamma _{rad}}|{cos\theta cos[{k({G + h} )} ]} |\; }}{{{\mathrm{\Delta }^2} + \frac{1}{4}{{\left( {{\gamma_0} + \sqrt {\frac{{{A_c}}}{{{A_1}}}} {\gamma_{rad}}|{cos\theta cos[{k({G + h} )} ]} |\; } \right)}^2}}}\; \approx 1$, the maximum CD can be approximately simplified as (see Supplement 1)

(8)$${|{\textrm{CD}} |_{max}}\; \approx \frac{{2\frac{{{A_c}}}{{{A_1}}}{\gamma _{abs}}{\gamma _{rad}}|{sin\theta sin[{k({G + h} )} ]} |}}{{{{\left( {{\gamma_{abs}} + {\gamma_{rad}} - \sqrt {\frac{{{A_c}}}{{{A_1}}}} {\gamma_{rad}}|{cos\theta cos[{k({G + h} )} ]} |} \right)}^2}\; }}$$

where ${\gamma _{abs}}$ and ${\gamma _{rad}}$ are the absorption and radiative losses of the Ag rod a (Ag rod b) with identical sizes, and $\mathrm{\Delta } = {g_0} - \sqrt {\frac{{{A_c}}}{{{A_1}}}} {\gamma _{rad}}cos\theta sin[{k({G + h} )} ]$.

From Eq. (8), we can find that the maximum CD signal is mainly related to the intrinsic parameters of Ag rod nanosystem: twist angle $\theta $, distance G, and the radiative loss of the Ag rod ${\gamma _{rad}}$. Therefore, we can manipulate these parameters to explore the dependence relationships between those intrinsic parameters and the chiroptical response of Ag rod nanosystem, so as to obtain the guidance for optimizing the CD signal of this system.

Next, we turned to study the relationship between the structure parameters (such as the $\theta $, G, and ${\gamma _{rad}}$) and the maximum CD response in the Ag rod nanosystem. We began our investigations with the parameters of $\theta $ and G, and the simulation and calculation results are given in Fig. 2.

Fig. 2. Chiroptical response tuned by the twist angle θ and distance G. (a), (b) CD spectra depending on the twist angle θ. (c) The maximum of |CD| as a function of the twist angle θ. (d), (e) CD spectra depending on the distance G. (f) The maximum of |CD| as a function of the distance G.

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The strong dependence of the chiroptical response in the system with the twist angle $\theta $ are shown in Fig. 2(a)–2(b), where Fig. 2(a) is the CD spectra for different angles ($\theta = $ 30°, 60°, 90°, 120° and 150°) and Fig. 2(b) is a two-dimensional map of the CD signal as a function of the twist angle and the wavelength of the incident light. It is obvious in Fig. 2(a) that when $\theta $ is less (greater) than 90°, the extreme value of CD is greater (less) than 0, corresponding to ${\textrm{A}_{\textrm{LCP}}} > {\textrm{A}_{\textrm{RCP}}}$ (${\textrm{A}_{\textrm{LCP}}} < {\textrm{A}_{\textrm{RCP}}}$). The white dotted lines in Fig. 2(b) represent the hybrid mode 1 and mode 2 with the extreme value of CD spectra in the system. One can clearly see that the CD response respectively reaches its extreme value at the resonant frequency of mode 2 or mode 1 (see the red and blue region in Fig. 2(b)), when $\theta $ is from 0° to 90° or is from 90° to 180°. These facts verify the relationship between maximum CD response and twisted angle $\theta $, and the reasons are also discussed in details (see Supplement 1).

In order to further explore the manipulation of the maximum CD by tuning the angle $\theta $, the max |CD| values versus different angles are respectively obtained by FDTD and Eq. (8) (Fig. 2(c)), and the results are in good accordance with each other. When the angle is close to 0° or 180°, consequently there is a little difference in the absorption of the system between the excitation with the LCP and RCP light (see Eq. (2)), which results in CD ≈ 0. When the angle is close to 90°, the absence of far-field coupling in the system leads to a small CD signal. It should be mentioned that Eq. (8) can give the function relationship between the max |CD| value and $\theta $. So, by solving the extreme value of this function, the optimum angle $\theta $ for the extreme chiroptical responses can be obtained (i.e., $\theta = $ 40° and 140°, see Fig. 2(c)) within a few minutes. Therefore, it is found that Eq. (8) can be a fast guide for optimizing the CD response in the twisted Ag rods system.

The CD spectral evolutions with varied distance G display the similar trend as those with the angle $\theta $ (Fig. 2(d)–2(e)). Figure 2(f) shows the max |CD| values versus different distances obtained by FDTD and Eq. (8). It can be seen that when the distance is between 100 and 200 nm, these results are not very consistent with each other. The reason is that the far-field and near- field couplings of the system with these distances are very weak, so it does not meet the conditions that Eq. (8) can replace the max |CD| value (more detailed discussions can be found in Supplement 1). However, it is noticed that the optimum distance from FDTD and Eq. (8) can match well, so Eq. (8) can also be a guide to optimize the CD response in the system by tuning the distance G.

Since the radiation loss of Ag rod is related to the far field coupling of the system, the optical chirality can be enhanced by manipulating the radiation loss. So, we sequentially go to study the manipulation of CD signals induced by tuning the radiation loss of Ag rod, which is affected by the periodic size P of this system (Fig. 3).

Fig. 3. Chiroptical response tuned by the period P. (a) CD spectra depending on the period P. (b) The maximum of |CD| and radiation loss of Ag rod as a function of the period P.

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Figure 3(a) shows the CD spectra of this system as the periodic size P changes from 300 to 500 nm. It can be seen that the periodic size P has rarely influenced on the lineshape of CD spectra and maximum CD changes slowly with P. To clearly illustrate the manipulation of CD signals by tuning the radiation loss of Ag rod, the maximum of |CD| values of the system and the radiation loss of the Ag rod versus different periodic size P are obtained by FDTD in Fig. 3(b). We find that the maximum of |CD| value decreases slowly (blue line) and the radiation loss of the Ag rod decreases rapidly as the increasing of P (red line). This phenomenon can be understood from the far field coupling of the system. Although the radiation loss of Ag rod will increase the far field coupling of the system, it will also increase the total loss of the system, so that CD signal changes slowly with the radiation loss. Based on the results in Fig. 3, it can be concluded the chiroptical response of this system is less sensitive to the radiation loss of Ag rod than the angle $\theta $ and distance $G$. Moreover, the system with large radiation loss can obtain the large CD more easily.

It can be seen from the Eq. (6) that manipulating the resonant frequency of Ag rod can also modify the CD response of our proposed system. The resonance frequency of Ag rod can be changed by varying the length of Ag rod, thus we next turn to investigate the CD signals of the Ag rod nanosystem with different rod lengths (Fig. 4).

Fig. 4. Chiroptical response tuned by the length La of Ag rod a. (a), (b) CD spectra depending on the length La. (c) The maximum of |CD| value as a function of the length La.

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Figure 4(a) shows the CD spectra of the system for various lengths of Ag rod a (L_a = 110, 130, 150, 170, and 190 nm). When the length of the Ag rod a is far from 150 nm (length of Ag rod b), the mode coupling of Ag rods induced by the large resonance frequency detuning gives rise to the weak CD signals. For the sake of more complete and clear characterizations, the CD spectra versus wavelength and the length of Ag rod a changing from 110 to 190 nm is depicted in Fig. 4(b). The maximum CD responses appear at the resonant frequencies of mode 2 because the angle $\theta $ and distance G are the same as those in Fig. 1(c).

It is noticed that the curve representing the maximum of |CD| values versus different length of the Ag rod a shows an interesting phenomenon (Fig. 4(c)). The optimized length for reaching greatest CD response is L_a = 158 nm rather than the length of Ag rod b (i.e., L_b = 150 nm). This phenomenon can be understood from the Eq. (S8). When the length (frequency) of Ag rod a is equal to that of Ag rod b, the chiroptical response of the system originates from mode coupling with frequency matched. When the length of Ag rod a is slightly higher than that of Ag rod b, the small frequency detuning of Ag rod a will also contribute to the chiroptical response of the system, leading to the increase of CD. When the length of Ag rod a is further increased, the large frequency detuning will weaken the mode coupling of Ag rods and make the CD response decrease. Therefore, we can draw an important conclusion from Fig. 4 that a small detuning between the resonant frequencies of the two twisted Ag rods can help to enhance CD response (see Supplement 1).

The value of theoretical methods lies in analyzing the physical laws of the system and guiding the device design to bring about better applications. Here we will show that the parameters of this chiral system can be optimized to improve the CD response based on our theoretical calculations. In Fig. 5(a), we perform the map of max |CD| in function of the distance G and twist angle θ obtained by the FDTD solutions. Ag rod a and Ag rod b are the same as those in Fig. 1, the periodic size is set as 300 nm, and the results of other periodic sizes are also provided (see Supplement 1). It can be seen that the CD signal in the system can reach ∼ 0.55 by sweeping these parameters (G, θ), but this method is very time-consuming. In our case, the optimum parameters (G, θ) of Fig. 5(a) can only be obtained by sweeping several hundred cases in FDTD, taking more than several days with an advanced computer.

Fig. 5. Enhancing CD response of Ag rods system under the guidance of theory. (a) The map of CD versus orientation distance G and twist angle θ with periodic size of 300 nm. (b) CD spectra depending on the length La. (c) Schematic view of the twisted Ag rods with a metal layer. (d) The maximum of |CD| of the twisted Ag rods with a metal layer.

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On the other side, if we combine our theory calculations with the amendment obtained by FDTD simulations, the time for getting optimized CD response of this Ag nanorod system will be greatly shortened. Two steps are included based on our considerations. Firstly, since Eq. (8) can give the function relationship between the max |CD| value and parameters (G, θ), so the optimum parameters (G, θ) and the extreme chiroptical responses can be obtained by solving this function (the length of Ag rods and periodic size P are the same with those in Fig. 5(a)). Secondly, as we have demonstrated in Fig. 4, a small resonant mismatching of the two twisted Ag rods will help to enhance the CD response, we can further apply FDTD simulations to obtain the CD response of this system under varied lengths of Ag rod a (other structure parameters are unchanged). So, we only need to sweep 6 cases to get the optimized CD response of 0.56 (Fig. 5(b)). Since our theory calculations save lots of time, our optimization processes merely takes a few minutes, greatly increasing the efficiency for designing the chiroptical devices. It should be mentioned that there is a small difference between our considerations and the approach only based on FDTD simulations. In the FDTD methods, since the simulations are based on solving Maxwell equations, the small detuning of resonant frequency caused by twisted angle can be considered. So in the FDTD simulations, one only need to consider two Ag rods with the same size. However, in our theoretical calculations, it is difficult to include the small frequency detuning into Eq. (8). So, we need two steps to obtain the optimized geometry parameters, which are the calculations for optimized result of (G, θ), and FDTD simulations for the final length of Ag rod a (i.e., L_a), which may have a small difference with the length of Ag rod b.

Based on previous studies, for the system of two twisted Ag rods, a CD response of ∼0.56 is close to the limitation of this system [39]. However, with the guidance of our theory, we find that the twisted Ag rods dimer has the possibility of further improving the chiroptical response because the transmitted light from the lower interface of this Ag rods system loses a part of the light energy. To compensate for the loss energy of transmitted light, here we add a metal layer under the twisted Ag rods (Fig. 5(c)), the transmitted light will become the input light again after being reflected by the metal layer, so as to enhance the optical response of the twisted Ag rods.

We add a metal layer with the system corresponding to the black circle in Fig. 5(b), and the max |CD| of the hybrid system composed of the twisted Ag rods with a metal layer can be approximately written as

(9)$${({{{|{\textrm{CD}} |}_{max}}} )_{hybrid}} \propto {({{{|{\textrm{CD}} |}_{max}}} )_a}\; {|{1 + r{e^{ikH}}} |^2} + {({{{|{\textrm{CD}} |}_{max}}} )_b}\; {|{1 + r{e^{ik({H + G + h} )}}} |^2}$$

where r represents the reflection coefficient at air-metal layer interfaces. ${({{{|{\textrm{CD}} |}_{max}}} )_a}$ and ${({{{|{\textrm{CD}} |}_{max}}} )_b}$ are CD signal of Ag rod a and Ag rod b in the system corresponding to the black circle in Fig. 5(b). The height between the Ag rod a and the metal layer is H.

The max |CD| responses of hybrid system with different height H obtained by FDTD and Eq. (9) are showed in Fig. 5(d) and the results are in great agreement. It can be seen from Fig. 5(d) that adding the metal reflective layer, the CD signal of the Ag rods can be improved to 0.78, which is undoubtedly a giant chiroptical response for the simple nanoparticle system. Also, it is found the curves in Fig. 5(d) exhibit an oscillation shape. The reason for this fact is the interference effect between incident light and transmission light reflected by the metal layer. When the height is close to 200 nm or 600 nm, the incident light are in phase with transmitted light reflected by the metal layer (i.e., the phase difference of the two lights is 0), leading to the enhancement of incident light amplitude. Therefore, the CD signal in the hybrid system can reach 0.78. On the other side, when the height is close to 400 nm or 800 nm, these two lights are out of phase (i.e., the phase difference is π) and the CD signal is almost 0. From the results of Fig. 5(d), the CD signal of the hybrid system can be modulated from 0 to ∼0.78 by manipulating the height H. The above discussions are based on the systems in vacuum. However, our approach can also be applicable to systems with silicon dioxide substrate and the CD value of the twisted Ag rods with metal layer can reach 0.76 under the guidance of this approach. Also, it is found that the incident angle in the x-z plane is insensitive to the CD response of the twisted Ag rods (see Supplement 1).

3. Conclusion

In summary, we provide an analytical approach from the perspective of mode coupling to study the twisted nanorod dimer system, which is considered to be a basic model for the emergence of chiroptical response. First, we can quantitatively calculate the CD response based on this approach and find that the strong chiroptical response of this system originates from the hybrid mode obtained by far-field coupling and near-field coupling of nanoparticles. Then, we studied the factors that can influence the chiroptical responses of the two twisted nanorods, which are the twist angle, distance between nanorods, period constant, and resonance frequency of nanorod. Furthermore, with the help of theoretical guidance, we find that the CD response of twisted Ag rods added with a metal layer can reach ∼ 0.78. Our approach is based on the TCMT, so it is applicable to the systems that can excite the optical modes with clear Lorentz lineshapes, including various plasmonic and dielectric structures. We expect that our approach can provide the chirality analysis of various optical systems and guide the design of large CD systems.

Funding

National Key Research and Development Program of China (2021YFA1400804); National Natural Science Foundation of Chinas of China (11974437, 12222415); Guangdong Natural Science Funds (2020A0505140004); Guangdong Special Support Program (2019JC05X397).

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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Enhancing chiroptical responses in the nanoparticle system by manipulating the far-field and near-field couplings

Abstract

1. Introduction

2. Theory and results

3. Conclusion

Funding

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

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Cited By

Figures (5)

Equations (11)

Optics Express