Giant circular dichroism enhancement and chiroptical illusion in hybrid molecule-plasmonic nanostructures

Yineng Liu; Rongyao Wang; Xiangdong Zhang

doi:10.1364/OE.22.004357

1. Introduction

Chirality is an inherent phenomenon, which plays a pivotal role in biochemistry and the evolution of life itself [1, 2]. Circular dichroism (CD) spectroscopy is one of the central methods to probe chiral nature of molecules through describing the difference in absorption of right- and left-handed circularly-polarized photons [3, 4]. In general, the CD signal of biomolecules is typically weak, thus, chiral analyses by such a spectroscopic technique have usually restricted to the analyst at a relatively high concentration [1–4]. This is frequently an impediment for a practical use because an ultrasensitive probe of tiny amounts of chiral substance is highly demanding for practical biosensing applications in biomedical and pharmaceutical fields.

Recently, the above obstacle is expected to be solved by plasmon-based nanotechnology [5–9]. Some metallic nanoparticles (NPs) present a unique property, known as localized surface plasmon resonances (SPRs), under resonance excitations by external fields [10]. The excitation of SPRs produces intense electric field enhancement in close proximity to the surfaces of metallic NPs. These enhancements have been used to improve nanoparticle-assisted biosensing [11–13], Raman scattering [14–16] and light absorption in solar cells [17, 18]. They have also been applied to the CD probe of molecular chirality [19–37]. Optical transitions of biomolecules are normally in ultraviolet (UV) region (150−300 nm) [1–4]. The previous investigations have shown that a weak CD signal in the UV region from chiral molecules can be both enhanced and transferred to the visible wavelength range by the field-enhanced plamon-molecule electromagnetic interactions [19–37]. Thus, it is very beneficial for the ultrasensitive probe of chiral molecules.

In this work we extend the CD theory, proposed recently in [37] for a chiral molecule inserted into a plasmonic hot spot of a dimer, to the system consisting of arbitrary distributions of metallic nanospheres by means of multiple scattering of electromagnetic multipole fields. Based on such a method, we have calculated CD spectra for a chiral molecule inserted into clusters with various metallic nanospheres. We find that the molecule CD signal in the UV region can be improved 2 orders of magnitude by using plasmonic particle clusters. In contrast to the previous studies [19–28], we show here the enhanced molecular CD is away from the plasmon resonance region, which originates from the strong modification of the optical electric field inside a molecule due to the presence of nanostructures. We also find that the enhancement is sensitive to structure and symmetry of the cluster. Thus, it brings some advantages to perform direct ultrasensitive detection on the chiral molecule in the UV region. Furthermore, quantum size correction on the phenomenon has also been considered.

2. Theory and method

We consider here a hybrid system consisting of a molecule and a cluster with N metallic nanospheres as shown in Fig. 1, which is excited by circularly polarized light ${\vec{E}}_{0} (ω)$ . The total absorption rate of the system can be expressed as [37]

Q = Q_{m o l e c u l e} + Q_{N P},

where

Q_{m o l e c u l e} = ω_{0} ρ_{22} \cdot γ_{22}

represents the absorption rate from the molecule,

ω_{0}

is the frequency of molecular transition,

γ_{i j}

is the relaxation term and

ρ_{i j}

is the matrix element of density matrix, which are obtained by solving the master equation for quantum states of molecules using the rotating wave approximation in the linear regime [37].

Q_{N P} (ω) = 2 ω \sum_{i} I m (ε_{N P}) \int {\vec{E}}_{i, t o t}^{i n} \cdot {\vec{E}}_{i, t o t}^{i n *} d V

is the absorption rate from NPs, in which

{\vec{E}}_{i, t o t}^{i n}

denotes the total field inside the

i

th nanosphere (

i = 1, 2...... N

),

ε_{N P}

is the permittivity of NPs and the integral is taken over nanosphere volumes,

ε_{r, 0}

is the relative dielectric constant of background. A CD signal for the system is defined as the difference between absorption of left- and right-handed polarized light, which can be written as

C D = {〈 Q_{+} - Q_{-} 〉}_{Ω},

where the averaging over the solid angle,

{〈 \cdot \cdot \cdot 〉}_{Ω}

, is needed since hybrid nanosphere complexes have random orientations.

〈 Q_{+} 〉

and

〈 Q_{-} 〉

represent the rate of absorption for left- and right-handed polarized light, respectively.

Q_{\pm} = Q_{m o l e c u l e \pm} + Q_{N P \pm}

, where

Q_{m o l e c u l e \pm}

and

Q_{N P \pm}

are the absorption rate for the molecule and the sphere cluster, respectively. In general, the total

C D

of the system can be divided into two parts,

C D_{t o t a l} = C D_{m o l e c u l e} + C D_{N P},

where

C D_{m o l e c u l e} = {〈 Q_{m o l e c u l e +} - Q_{m o l e c u l e -} 〉}_{Ω}

and

C D_{N P} = {〈 Q_{N P +} - Q_{N P -} 〉}_{Ω}

. Performing the averaging over

Ω

for

C D_{m o l e c u l e}

[37], we obtain

C D_{m o l e c u l e} = \frac{8 \sqrt{ε_{r}}}{3 c} \frac{ω_{0} γ_{21} {| E_{0} |}^{2} Im [{\vec{m}}_{21} . ({\hat{P}}^{+} {\vec{μ}}_{12})]}{{| ℏ (ω - ω_{0}) + i γ_{21} - G |}^{2}}

with

G = \frac{1}{4 π ε_{0}} {\vec{μ}}_{21} \cdot \sum_{i} \nabla ({\vec{Φ}}_{i}^{o u t} . {\vec{μ}}_{12}) | {_{r = r}}_{_{d}},

where the function

G

originates from plasmonic surface charges of

N P s

induced by the molecule dipole, the vector

{\vec{Φ}}_{i}^{o u t} = \sum_{l m} {\vec{D}}_{l m, i} \frac{1}{{| r - r_{i} |}^{l + 1}} Y_{l m} (θ_{r - r_{i}}, ϕ_{r - r_{i}})

defines the electric potential induced by the surface charges of the

i

th sphere,

Y_{l m}

is spherical harmonic,

{\vec{D}}_{l m, i}

is expansion coefficient, which can be determined by the boundary conditions (see Appendix). The

c

represents light velocity in vacuum,

{\vec{μ}}_{i j}

and

{\vec{m}}_{i j}

are matrix elements of the electric and magnetic dipolar moments [37]. Here

\hat{P}

is the field-enhancement matrices inside a molecule, which describes strong changes of the optical electric field inside a molecule due to the presence of NPs and is expressed as

\hat{P} = [\begin{matrix} p_{x x} & p_{x y} & p_{x z} \\ p_{y x} & p_{y y} & p_{y z} \\ p_{z x} & p_{z y} & p_{z z} \end{matrix}] .

The matrix elements (

p_{β α}

) in Eq. (6) are determined from the relation:

p_{β α} =({\vec{E}}_{s c} + {\vec{E}}_{0})_{β} / {({\vec{E}}_{0})}_{α}

,

{\vec{E}}_{s c}

is the scattering field at the position of the molecule from NPs in the cluster, which can be obtained by the multiple scattering method [38]. According to analysis in [37], the

C D_{N P}

mainly comes from two terms:

C D_{N P} = C D_{N P, d i p o l e - f i e l d} + C D_{N P, d i p o l e - d i p o l e}

. From the definition of

Q_{N P}

, we arrive at [37]

C D_{N P, d i p o l e - f i e l d} = \frac{2 ω \sqrt{ε_{r}}}{3 c π ε_{0}} \sum_{i} (Im ε_{N P}) Im \int \frac{{\vec{m}}_{21} . [{\hat{K}}^{+} (r) \nabla (\sum_{i} {\vec{Φ}}^{i, t o t} . {\vec{μ}}_{21})]}{ℏ (ω - ω_{0}) + i γ_{21} - G} d V,

C D_{N P, d i p o l e - d i p l o e} = - \frac{8 ω \sqrt{ε_{r}}}{3 c} \frac{{| E_{0} |}^{2} Im [{\vec{m}}_{21} . ({\hat{P}}^{+} {\vec{μ}}_{12})]}{{| ℏ (ω - ω_{0}) + i γ_{21} - G |}^{2}} Im (G),

where

Φ^{i,}^{t o t} (r) = Φ_{_{0}}^{(i)} + Φ_{_{i}}^{i n} + \sum_{j \neq i} Φ_{_{j}}^{o u t},

Φ_{_{j}}^{o u t} = \sum_{l m} \sum_{l^{'} m^{'}} {\vec{D}}_{l^{'} m^{'}, j} g_{l^{'} m^{'}, l m} (r_{j} - r_{i}) {(r - r_{i})}^{l} Y_{l m} (θ_{r - r_{i}}, ϕ_{r - r_{i}})

with

g_{l^{'} m^{'}, l m} (r_{j} - r_{i}) = \sum_{\begin{array}{l} l_{2} m_{2} \\ l_{2} - l_{1} = l^{'} \end{array}} \sqrt{\frac{4 π}{(2 l^{'})!}} {(- 1)}^{l_{2}} \sqrt{\frac{(2 l_{2})!}{(2 l_{1} + 1)!}} \frac{1}{{| r_{j} - r_{i} |}_{}^{l_{2} + 1}} C_{l m l_{2} m_{2}}^{l^{'} m^{'}} Y_{l_{2} m_{2}} (θ_{r_{j} - r_{i}}, ϕ_{r_{j} - r_{i}}) .

Here

Φ^{i,}^{t o t} (r)

is the total electric potential function inside the

i

th sphere,

Φ_{_{0}}^{(i)}

is the potential induced by the isolated molecule,

Φ_{_{i}}^{i n}

is the potential inside the

i t h

sphere induced by the induced charge,

C_{l m l_{2} m_{2}}^{l^{'} m^{'}}

is the expanded coefficient. Equations (9)-(11) are obtained by the addition theorem (see Appendix). The position-dependent matrix

\hat{K} (r)

determines the field inside

N P s

, which can be expressed as

\hat{K} = [\begin{matrix} k_{r r} & k_{r θ} & k_{r ϕ} \\ k_{θ r} & k_{θ θ} & k_{θ ϕ} \\ k_{ϕ r} & k_{ϕ θ} & k_{ϕ ϕ} \end{matrix}] .

The matrix elements (

k_{β α}

) in Eq. (12) are determined from the relation:

k_{β α} =({\vec{E}}_{s c})_{β} / {({\vec{E}}_{0})}_{α}

. Based on the above equations, the CD of hybrid systems consisting of a molecule and nanosphere clusters can be obtained through numerical calculations.

Fig. 1 Geometry of a hybrid single molecule-plasmonic nanostructure. The system is scattered by an incident polarized light. Here $θ_{i}$ , $ϕ_{i}$ and $r_{i}$ are the coordinates of the $i t h$ sphere in the spherical coordinate system, and the coordinates of the chiral molecule are marked by $θ_{d}$ , $ϕ_{d}$ and $r_{d}$ .

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3. Giant CD in the UV region

We first consider the case of two metallic spheres and a molecule. The CD for such a system has been discussed very well in [37], which allows us to validate our calculated method. Figure 2(a) shows our calculated CD signals as a function of wavelengths for a hybrid system consisting of a molecule and a gold dimer. Here the radii of two gold spheres are taken as 17.5nm and the separation between them is $d_{1} = 4 n m$ . The molecular linker located at the nanogap is idealized to be a single molecular dipole that is oriented at $θ = π / 3$ as shown in inset of Fig. 2(a), $θ$ is the angle between the orientation of molecular dipole and y axis. The parameters of the molecular dipole are taken according to [1–37]: $μ_{12} = | e | r_{12}$ and ${\vec{μ}}_{12} \cdot {\vec{m}}_{21} / μ_{12} = i | e | r_{0} ω_{0} r_{21} / 2$ . In our calculations, we use $r_{12} = 2 \overset{0}{A}$ , $r_{0} = 0.05 \overset{0}{A}$ and $γ_{12} = 0.3 e V$ . For the dielectric functions of Au, the Palik’s data were adopted [39], the permittivity of water is taken as $ε_{r, 0} = 1.8$ . The black line and red line correspond to $C D_{m o l e c u l e}$ and $C D_{N P}$ , respectively, the total CD is described by the green line. The resonance of molecules appears at $λ = 300 n m$ , and the plasmon resonance for Au sphere is larger than 520nm, which can be seen clearly from extinction of the Au dimer (black line in Fig. 2(d)). Although a molecular exciton is off the plasmon resonance of a Au dimer, a small enhancement of the CD still appears. This is because the interference of incident and induced fields (Fano effect) [37]. Our calculated results are identical with those in [37].

Fig. 2 Calculated CD signals as a function of wavelengths for molecule-NP complexes with two spheres (a), two large spheres and one small sphere with $d_{2} = 2 n m$ (b), two large spheres and one small sphere with $d_{2} = 1.5 n m$ (c). Here $d_{2}$ represent the distance between the molecule and small sphere, and $d_{1} = 4 n m$ is the distance between two large spheres. The radii of two large Au spheres are taken as 17.5nm, the radius for small Au sphere is taken as 4nm. In inset, $θ = π / 3$ represents the angle between the orientation of molecular dipole and y axis. (d) Calculated extinctions of Au NPs.

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However, the situation becomes different, if we put another Au sphere with radius 4nm near the dimer as shown in inset of Fig. 2(b). Figure 2(b) and 2(c) display calculated results of CD signals for the three-sphere system with $d_{2} = 2 n m$ and $d_{2} = 1.5 n m$ , respectively. Here $d_{2}$ represents the distance between the molecule and the third sphere. It is seen clearly that the $C D_{m o l e c u l e}$ (black line) is improved largely although the $C D_{N P}$ (red line) is almost unchanged with the introduction of the third sphere, which results in the strong enhancement of total CD signals (green line) around $λ = 300 n m$ . For example, the CD is improved 10 times at $d_{2} = 2 n m$ for the present case, it reaches 40 times at $d_{2} = 1.5 n m$ . Such an improvement of CD signal depends on the orientation of molecular dipole, which can be seen clearly from Fig. 3.

Fig. 3 (a) Calculated CD signals as a function of wavelengths for molecule-NP complexes with two Au large spheres and one Au small sphere at various orientations of a molecular dipole. Here $d_{2} = 2 n m$ . (b) The corresponding CD signals for (a) as a function of orientations of a molecular dipole. The other parameters are identical with those in Fig. 2. The corresponding CD signals as a function of wavelengths and orientations of a molecular dipole for Ag sphere systems are given in (c) and (d), respectively. The sizes of Ag spheres and cluster structure are identical with those of three-sphere Au system.

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Figure 3(a) shows calculated CD signals as a function of wavelengths for the three-sphere system at various orientations of the molecular dipole, the corresponding results as a function of orientation of the molecular dipole are described in Fig. 3(b). The smaller the value for $θ$ , the greater increasing for CD. More than 2 orders of magnitude CD enhancement in the UV region are observed at small $θ$ . Such a phenomenon does not own by Au particle clusters alone. Figure 3(c) and 3(d) show the corresponding results for silver sphere clusters. Here the radii of two large and one small Ag spheres are taken identical with those of three-sphere Au system, the separations among spheres are also the same. The Palik’s data are still adopted for the dielectric functions of Ag. Similar phenomenon is observed again.

Such a phenomenon is not caused by plasmon resonances. This can be seen clearly from Fig. 2(d). The red line and green line in Fig. 2(d) represent extinctions for the case at $d_{2} = 2 n m$ and $d_{2} = 1.5 n m$ , respectively. Comparing them with the two-sphere case, we cannot find new plasmon resonance around $λ = 300 n m$ with introducing the third sphere. In fact, the phenomena are related to symmetry of the structure. If we put the fourth Au sphere with the same parameters to the third sphere at the symmetric position on the left side of the structure as shown in inset of Fig. 4, the phenomenon for enhanced CD disappears. Figure 4displays such results. The black line and red line in Fig. 4(a) represent the calculated results for the four-sphere system at $d_{2} = 2 n m$ and $d_{2} = 1.5 n m$ , respectively. The corresponding extinctions are also described by the black line and red line in Fig. 4(b). Comparing these results with those in Fig. 2 for the two-sphere system, we find that the change of CD is minimal with symmetrically introducing two small spheres. This is in contrast to the case of three-sphere system.

Fig. 4 (b) Calculated CD signals as a function of wavelengths for molecule-NP complexes with four spheres as shown in Inset. The radii of two large Au spheres are taken as 17.5nm. The red line and black line represent the results with $d_{2} = 1.5 n m$ and $d_{2} = 2 n m$ , respectively. The other parameters are identical with those in Fig. 2. (a) The corresponding extinctions of 4-sphere system.

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In order to disclose the physical origin of the phenomenon, we calculate electric field distributions around molecular positions in three-sphere and four-sphere systems. Figure 5(a) and 5(b) present the comparison for the spatial profile of electric field amplitudes in two systems at wavelength $λ = 300 n m$ . Comparing them, we find that the electric field intensity at molecular position in the four-sphere system is not smaller than that in the three-sphere system. However, the components of $| {\hat{P}}^{+} {\vec{μ}}_{12} |$ along y and z directions for two systems exhibit very large differences. Because the orientation of our studied molecules is in the yz plane, other components of $| {\hat{P}}^{+} {\vec{μ}}_{12} |$ are zero in addition to $| {\hat{P}}^{+} {\vec{μ}}_{12} |_{z}$ and $| {\hat{P}}^{+} {\vec{μ}}_{12} |_{y}$ . The black line, red line and green line in Fig. 5(c) describe the real part of $| {\hat{P}}^{+} {\vec{μ}}_{12} |_{z}$ for two-sphere, three-sphere and four-sphere system, respectively, the corresponding results for $| {\hat{P}}^{+} {\vec{μ}}_{12} |_{y}$ are given in Fig. 5(d). As we can see that the real part of $| {\hat{P}}^{+} {\vec{μ}}_{12} |_{y}$ in the three-sphere system is enhanced largely around $λ = 300 n m$ in comparing with those in two-sphere and four-sphere systems. This is the reason why the above phenomena appear. This means that large CD effect of chiral molecules in the UV region can be created by introducing nanostructures to modify matrix elements of field-enhancement matrices. Its physical origin is different from the previous reported plasmon-resonance-enhanced CD phenomenon [19–37].

Fig. 5 The spatial profile of the electric field amplitude in yz plane for the three-sphere system (a) and four-sphere system (b) at wavelength $λ = 300 n m$ and $d_{2} = 2 n m$ . The real parts of matrix elements $| {\hat{P}}^{+} {\vec{μ}}_{12} |_{z}$ (c) and $| {\hat{P}}^{+} {\vec{μ}}_{12} |_{y}$ (d) as a function of wavelengths for molecule-NP complexes with two spheres (black line), three sphere (red line) and four spheres (green line). The other parameters are identical with those in Fig. 2.

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At the same time, the nanostructure can also cause asymmetry of CD spectra. Figure 6(a) describes the case of three-sphere system at $θ = π / 3$ when the small sphere locates at the right side of the dimer (left-opened structure), whereas the corresponding results are given in Fig. 6(b) when the small sphere locates at the left side of the dimer (right-opened structure). Comparing them, we find that the CD spectra exhibit different features although extinctions for two kinds of structure without chiral molecules are the same. This also leads to another kind of interesting phenomenon, that is, the nanostructure causes mirror symmetry breaking of CD spectra.

Fig. 6 Calculated CD signals as a function of wavelengths for molecule-NP complexes with two large spheres and one small sphere at $d_{2} = 2 n m$ . (a) left-opened structure with molecular orientation at $θ = π / 3$ , (b) right-opened structure with molecular orientation at $θ = π / 3$ , (c) right-opened structure with molecular orientation at $θ = 4 π / 3$ . The other parameters are identical with those in Fig. 2.

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In general, an enantiomeric pair of chiral molecule possesses CD spectra with a mirror symmetry, as shown in inset of Fig. 6(c). Here the enantiomeric pair are denoted by the molecular dipoles with opposite sign of ${\vec{μ}}_{12}$ . Figure 6(c) shows calculated CD signals for the right-opened structure at $θ = 4 π / 3$ . Comparing Fig. 6(c) with Fig. 6(a), we find that the mirror symmetry of CD spectra does not exist. Originally a chiral molecule such as left-handed molecule should possess a negative CD signal around $λ = 300 n m$ , when the nanostructure introduces, it exhibits a positive CD signal. Such a reversion of CD band means that nanostructures can cause chiroptical illusion of molecular CD signature. Also noteworthy that native CD response of a chiral molecule is normally of a broad band-width spanning the range of molecular resonant absorptions. In contrast, the present enhanced CD signal shows a very sharp peak around the wavelength of molecule resonances. This advantage may offer a very big superiority for a high spectral resolution of chiral probe of biomolecules, since biomolecules like proteins usually have very complicated structures of CD spectrum.

4. Effect of quantum size on CD

Recent investigations show that as the nanoparticle radii below 10 nm or the gap between two metal spheres is smaller than 0.8nm, the effect of quantum size on the plasmon resonance becomes important [40–44]. For above some cases, we consider 4nm radius sphere, quantum corrections are required due to the size of the particles. In the following, we consider the effect of quantum size on the CD signals. Considering quantum size effects of small particles, we model the conduction electrons as a free electron gas constrained by infinite potential barriers at the physical edges of the particle according to [42–44]. The standard Drude model is recast with Lorentzian terms that can be defined quantum mechanically, the particle permittivity can be expressed as [45]:

ε (ω)= ε_{I B} + ω_{p}^{2} \sum_{i} \sum_{f} \frac{S_{i f}}{ω_{i f}^{2} - ω^{2} - i γ ω},

where

ε_{I B}

is a frequency-dependent correction term to account for the contribution of the d-band valence electrons to interband transitions at higher energies,

ω_{p}

is the plasma frequency, and

γ

is the scattering frequency, dependent on the nanosphere dimension (particle radius) and the empirical constant.

ω_{i f}

and

S_{i f}

represent electron transition frequencies and oscillator strengths, respectively.

Based on Eq. (13), we recalculate CD signals for the three-sphere system with $d_{2} = 1.5 n m$ as given in Fig. 2. The calculated results are plotted in Fig. 7. The red dashed line in Fig. 7 represents the result with the quantum size correction, comparing it with classical result (black line) we find that blue shift of enhanced peak occurs, the amplitude of the peak also decreases. However, we change slightly the position of the third sphere, for example, as $d_{2} = 1.4 n m$ , the same amplitude enhancement of CD (green dotted line) is observed again. This means that the phenomena disclosed in the above calculations always appear even considering the quantum effect.

Fig. 7 Comparison between classical and QM-corrected results for calculated CD signals as a function of wavelengths for molecule-NP complexes with two large spheres and one small sphere at $d_{2} = 1.5 n m$ . The black line corresponds to classical results, red line is QM-corrected results, green line is the results with QM correction at $d_{2} = 1.4 n m$ .

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5. Summary

We have presented a multiple scattering method to calculate CD signals of the hybrid system consisting of a molecule and nanosphere cluster. The CD spectra for a chiral molecule inserted into clusters with various metallic nanospheres have been calculated. Our calculated results show that giant enhanced CD of the molecule in the UV region, more than 2 orders of magnitude enhancement in some cases, can be realized by using metallic nanosphere clusters. The enhanced CD resonances are very sensitive to structures and symmetries of the cluster, and they are also very sharp around the wavelength of molecule resonances. Thus, it brings some advantages to perform direct ultrasensitive detection on the chiral molecule in the UV region, which need not transfer to the visible wavelength. Furthermore, we have also found that the mirror symmetry of CD signals can be broken by introducing nanostructures. Therefore, the chiroptical illusion phenomenon of CD signals caused by nanostructures has been disclosed. In addition, quantum size correction on the phenomenon has also been considered. The physical origin for the phenomena has been analyzed. The proposed effects offer an alternative avenue for the ultrasensitive probing of chiral molecules in the UV region.

Appendix

In this appendix we present formalism and solution for ${\vec{D}}_{l m, i}$ , $Φ^{i,}^{t o t} (r)$ and $G$ . First let us consider electric potential induced only by molecular dipole. Supposing the molecule is placed in the position ${\vec{r}}_{d} = (r_{d}, θ_{d}, ϕ_{d})$ , the molecular dipole moment is $\vec{d}$ , then the electric potential is expressed as

φ_{0} = \frac{1}{4 π ε_{0}} \frac{\vec{d} \cdot \vec{r}}{r^{3}} .

If we define the center of

i t h

NP as

{\vec{r}}_{1} = (r_{i}, θ_{i}, ϕ_{i})

, the electric potential is written as

Φ_{0}^{(i)} = \sum_{l m} {\vec{B}}_{l m, i} (\frac{4 π}{2 l + 1}) \frac{{| r - r_{i} |}^{l}}{r_{d, i}^{l + 2}} Y_{l m} (θ_{r - r_{i}}, ϕ_{r - r_{i}}) (r_{d, i} = | r_{d} - r_{i} |) .

By the formula

\frac{\vec{r}}{r^{3}} =- \nabla (\frac{1}{r})

and

\frac{1}{| \vec{r} - {\vec{r}}^{'} |} = \sum_{l = 0}^{\infty} \sum_{m =- l}^{l} (\frac{4 π}{2 l + 1}) \frac{r_{<}^{l}}{r_{>}^{l + 1}} Y_{l}^{m *} (θ^{'}, ϕ^{'}) Y_{l}^{m} (θ, ϕ),

we can obtain the coefficients:

\begin{array}{l} B^{θ}_{l m, i} = \frac{\partial}{\partial θ^{'}} {[Y_{l}^{m *} (θ^{'}, ϕ^{'})]}_{θ^{'} = θ_{r_{d} - r_{i}}, ϕ^{'} = ϕ_{r_{d} - r_{i}}} \\ B^{ϕ}_{l m, i} = \frac{1}{\sin θ^{'}} \frac{\partial}{\partial ϕ^{'}} {[Y_{l}^{m *} (θ^{'}, ϕ^{'})]}_{θ^{'} = θ_{r_{d} - r_{i}}, ϕ^{'} = ϕ_{r_{d} - r_{i}}} \\ B^{r}_{l m, i} =- (l +1) {[Y_{l}^{m *} (θ^{'}, ϕ^{'})]}_{θ^{'} = θ_{r_{d} - r_{i}}, ϕ^{'} = ϕ_{r_{d} - r_{i}}} \end{array}

and

[\begin{matrix} B_{l m, i}^{x} \\ B_{l m, i}^{y} \\ B_{l m, i}^{z} \end{matrix}] = {[\begin{matrix} \sin θ \cos ϕ & \cos θ \cos ϕ & - \sin ϕ \\ \sin θ \sin ϕ & \cos θ \sin ϕ & \cos ϕ \\ \cos θ & - \sin θ & 0 \end{matrix}]}_{θ^{'} = θ_{r_{d} - r_{i}}, ϕ^{'} = ϕ_{r_{d} - r_{i}}} [\begin{matrix} B_{l m, i}^{r} \\ B_{l m, i}^{θ} \\ B_{l m, i}^{ϕ} \end{matrix}] .

Now we investigate the electric potential induced by dipole moment in the presence of NP cluster. The electric potential functions for $i$ th and $j$ th spheres induced by dipole moment, $Φ_{i}$ and $Φ_{j}$ , can be expressed as

\begin{array}{l} Φ_{i} {={}_{\sum_{l m} {\vec{D}}_{l m, i} \frac{1}{{| r - r_{i} |}^{l + 1}} Y_{l m} (θ_{r - r_{i}}, ϕ_{r - r_{i}}) (| r - r_{i} | 〉 R_{i}), \to Φ_{i}^{o u t}}^{\sum_{l m} {\vec{D}}_{l m, i} \frac{{| r - r_{i} |}^{l}}{R_{i}^{2 l + 1}} Y_{l m} (θ_{r - r_{i}}, ϕ_{r - r_{i}}) (| r - r_{i} | 〈 R_{i}), \to Φ_{i}^{i n}} \\ Φ_{j} {={}_{\sum_{l m} {\vec{D}}_{l m, j} \frac{1}{{| r - r_{j} |}^{l + 1}} Y_{l m} (θ_{_{r - r_{j}}}, ϕ_{r - r_{j}}) (| r - r_{j} | 〉 R_{j}), \to Φ_{j}^{o u t}}^{\sum_{l m} {\vec{D}}_{l m, j} \frac{{| r - r_{j} |}^{l}}{R_{j}^{2 l + 1}} Y_{l m} (θ_{r - r_{j}}, ϕ_{_{r - r_{j}}}) (| r - r_{j} | 〈 R_{j}), \to Φ_{j}^{i n}} . \end{array}

By using the following addition formalism for potentials:

{Y_{l_{1}} (θ_{1}, ϕ_{1}) \otimes Y_{l_{2}} (θ_{2}, ϕ_{2})}_{l m} = \sum_{m_{1}, m_{2}} C_{l_{1} m_{1} l_{2} m_{2}}^{l m} Y_{l_{1} m_{1}} (θ_{1}, ϕ_{1}) Y_{l_{2} m_{2}} (θ_{2}, ϕ_{2}),

\frac{1}{r^{l + 1}} Y_{l m} (θ, ϕ) = \sqrt{\frac{4 π}{(2 l)!}} \sum_{\begin{array}{l} l_{1}, l_{2} = 0 \\ l_{2} - l_{1} = l \end{array}}^{l} {(- 1)}^{l_{2}} \sqrt{\frac{(2 l_{2})!}{(2 l_{1} + 1)!}} \frac{r_{1}^{l_{1}}}{r_{2}^{l_{2} + 1}} {Y_{l_{1}} (θ_{1}, ϕ_{1}) \otimes Y_{l_{2}} (θ_{2}, ϕ_{2})}_{l m},

performing the following coordinate transformation:

\begin{array}{l} \frac{1}{{| r - r_{j} |}^{l + 1}} Y_{l m} (θ_{_{r - r_{j}}}, ϕ_{r - r_{j}}) \\ = \frac{1}{{| (r - r_{i})-(r_{j} - r_{i}) |}^{l + 1}} Y_{l m} (θ_{_{(r - r_{i})-(r_{j} - r_{i})}}, ϕ_{(r - r_{i})-(r_{j} - r_{i})}) \\ = \sqrt{\frac{4 π}{(2 l)!}} \sum_{\begin{array}{l} l_{1}, l_{2} = 0 \\ l_{2} - l_{1} = l \end{array}}^{l} {(- 1)}^{l_{2}} \sqrt{\frac{(2 l_{2})!}{(2 l_{1} + 1)!}} \frac{{| r - r_{i} |}_{}^{l_{1}}}{{| r_{j} - r_{i} |}_{}^{l_{2} + 1}} \sum_{m_{1}, m_{2}} C_{l_{1} m_{1} l_{2} m_{2}}^{l m} Y_{l_{1} m_{1}} (θ_{r - r_{i}}, ϕ_{r - r_{i}}) Y_{l_{2} m_{2}} (θ_{r_{j} - r_{i}}, ϕ_{r_{j} - r_{i}}), \\ = \sum_{l_{1} m_{1}} g_{l m, l_{1} m_{1}} (r_{j} - r_{i}) {(r - r_{i})}^{l_{1}} Y_{l_{1} m_{1}} (θ_{r - r_{i}}, ϕ_{r - r_{i}}) \end{array}

with

g_{l m, l_{1} m_{1}} (r_{j} - r_{i}) = \sum_{\begin{array}{l} l_{2} m_{2} \\ l_{2} - l_{1} = l \end{array}} \sqrt{\frac{4 π}{(2 l)!}} {(- 1)}^{l_{2}} \sqrt{\frac{(2 l_{2})!}{(2 l_{1} + 1)!}} \frac{1}{{| r_{j} - r_{i} |}_{}^{l_{2} + 1}} C_{l_{1} m_{1} l_{2} m_{2}}^{l m} Y_{l_{2} m_{2}} (θ_{r_{j} - r_{i}}, ϕ_{r_{j} - r_{i}}),

we obtain:

\begin{array}{l} \sum_{l^{'} m^{'}} {\vec{D}}_{l^{'} m^{'}, j} \frac{1}{{| r - r_{j} |}^{l^{'} + 1}} Y_{l^{'} m^{'}} (θ_{r - r_{j}}, ϕ_{r - r_{j}}) \\ = \sum_{l^{'} m^{'}} {\vec{D}}_{l^{'} m^{'}, j} \frac{1}{{| (r - r_{i})-(r_{j} - r_{i}) |}^{l^{'} + 1}} Y_{l^{'} m^{'}} (θ_{(r - r_{i})-(r_{j} - r_{i})}, ϕ_{(r - r_{i})-(r_{j} - r_{i})}) \\ = \sum_{l m} \sum_{l^{'} m^{'}} {\vec{D}}_{l^{'} m^{'}, j} g_{l^{'} m^{'}, l m} (r_{j} - r_{i}) {(r - r_{i})}^{l} Y_{l m} (θ_{r - r_{i}}, ϕ_{r - r_{i}}) \end{array}

with

g_{l^{'} m^{'}, l m} (r_{j} - r_{i}) = \sum_{\begin{array}{l} l_{2} m_{2} \\ l_{2} - l_{1} = l^{'} \end{array}} \sqrt{\frac{4 π}{(2 l^{'})!}} {(- 1)}^{l_{2}} \sqrt{\frac{(2 l_{2})!}{(2 l_{1} + 1)!}} \frac{1}{{| r_{j} - r_{i} |}_{}^{l_{2} + 1}} C_{l m l_{2} m_{2}}^{l^{'} m^{'}} Y_{l_{2} m_{2}} (θ_{r_{j} - r_{i}}, ϕ_{r_{j} - r_{i}}) .

Then

\begin{array}{l} φ_{j}^{o u t} = \sum_{_{l m}} {\vec{D}}_{l m, j} \frac{1}{{| r - r_{j} |}^{l + 1}} Y_{l m} (θ_{_{r - r_{j}}}, ϕ_{r - r_{j}}) \\ = \sum_{l m} \sum_{l^{'} m^{'}} {\vec{D}}_{l^{'} m^{'}, j} g_{l^{'} m^{'}, l m} (r_{j} - r_{i}) {(r - r_{i})}^{l} Y_{l m} (θ_{r - r_{i}}, ϕ_{r - r_{i}}), \end{array}

where

g_{l^{'} m^{'}, l m} (r_{j} - r_{i}) = \sum_{\begin{array}{l} l_{2} m_{2} \\ l_{2} - l_{1} = l^{'} \end{array}} \sqrt{\frac{4 π}{(2 l^{'})!}} {(- 1)}^{l_{2}} \sqrt{\frac{(2 l_{2})!}{(2 l_{1} + 1)!}} \frac{1}{{| r_{j} - r_{i} |}_{}^{l_{2} + 1}} C_{l m l_{2} m_{2}}^{l^{'} m^{'}} Y_{l_{2} m_{2}} (θ_{r_{j} - r_{i}}, ϕ_{r_{j} - r_{i}}) .

By using boundary conditions:

ε_{N P} \frac{\partial φ_{i}}{\partial r} |_{r = R - 0 +} = ε_{0} \frac{\partial φ_{i}}{\partial r} |_{r = R + 0 +}

, we obtain

\begin{array}{l} ε_{N P} {{\vec{B}}_{l m, i} (\frac{4 π}{2 l + 1}) \frac{l {| r - r_{i} |}^{l - 1}}{r_{d, i}^{l +2}} + {\vec{D}}_{l m, i} \frac{l {| r - r_{i} |}^{l - 1}}{R_{i}^{2 l + 1}}} Y_{l m} (θ_{r - r_{i}}, ϕ_{r - r_{i}}) + ε_{N P} [\sum_{l^{'} m^{'}} \sum_{j \neq i} {\vec{D}}_{l^{'} m^{'}, j} g_{l^{'} m^{'}, l m} (r_{j} - r_{i})] l {(r - r_{i})}^{l - 1} Y_{l m} (θ_{r - r_{i}}, ϕ_{r - r_{i}}) \\ = ε_{0} {{\vec{B}}_{l m, i} (\frac{4 π}{2 l + 1}) \frac{l {| r - r_{i} |}^{l - 1}}{r_{d, i}^{l +2}} + {\vec{D}}_{l m, i} \frac{- (l + 1)}{{| r - r_{i} |}^{l + 2}}} Y_{l m} (θ_{r - r_{i}}, ϕ_{r - r_{i}}) + ε_{0} [\sum_{l^{'} m^{'}} \sum_{j \neq i} {\vec{D}}_{l^{'} m^{'}, j} g_{l^{'} m^{'}, l m} (r_{j} - r_{i})] l {(r - r_{i})}^{l - 1} Y_{l m} (θ_{r - r_{i}}, ϕ_{r - r_{i}}) |_{r - r_{i} = R_{i}} \\ \Rightarrow \\ ε_{N P} {\vec{B}}_{l m, i} (\frac{4 π}{2 l + 1}) \frac{l R_{i}^{l - 1}}{r_{d, i}^{l +2}} + {\vec{D}}_{l m, i} \frac{ε_{N P} \cdot l}{R_{i}^{l + 2}} + ε_{N P} [\sum_{l^{'} m^{'}} \sum_{j \neq i} {\vec{D}}_{l^{'} m^{'}, j} g_{l^{'} m^{'}, l m} (r_{j} - r_{i})] l R_{i}^{l - 1} \\ = ε_{0} {\vec{B}}_{l m, i} (\frac{4 π}{2 l + 1}) \frac{l R_{i}^{l - 1}}{r_{d, i}^{l +2}} + {\vec{D}}_{l m, i} \frac{- (l + 1) \cdot ε_{0}}{R_{i}^{l + 2}} + ε_{0} [\sum_{l^{'} m^{'}} \sum_{j \neq i} {\vec{D}}_{l^{'} m^{'}, j} g_{l^{'} m^{'}, l m} (r_{j} - r_{i})] l R_{i}^{l - 1} . \end{array}

To solve Eq. (29), we obtain

{\vec{D}}_{l m, i}

, then

Φ^{i,}^{t o t} (r)

and

G

can be obtained.

Acknowledgments

We wish to thank Hui Zhang for useful discussions. This work was supported by the National Natural Science Foundation of China (Grant No. 11274042) and the National Key Basic Research Special Foundation of China under Grant 2013CB632704.

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Giant circular dichroism enhancement and chiroptical illusion in hybrid molecule-plasmonic nanostructures

Abstract

1. Introduction

2. Theory and method

3. Giant CD in the UV region

4. Effect of quantum size on CD

5. Summary

Appendix

Acknowledgments

References and links

Cited By

Figures (7)

Equations (29)

Optics Express