Tuning spontaneous radiation of chiral molecules by asymmetric chiral nanoparticles

Dmitry V. Guzatov; Vasily V. Klimov; Hsun-Chi Chan; Guang-Yu Guo

doi:10.1364/OE.25.006036

1. Introduction

Chirality refers to the geometric property of a three-dimensional body not to coincide with its mirror reflection for any shifts and turns. This property, for example, is shown by a human hand, a spring and a DNA molecule. Chirality plays a particularly important role in biology and medicine, since many organic compounds (e.g., amino acids and proteins) have the chiral properties. In particular, a living organism can react quite differently to the different isomers of the same substance. In physics, the primarily interest is in the optical properties of chiral media [1]. For example, the difference in the propagation between the right- and left-handed circularly polarized electromagnetic waves leads to such optical effects as rotation of the polarization plane and circular dichroism. At present, interest in the study of chiral properties continues unabated, because many synthesized metamaterials have the chiral properties [2]. Moreover, new metamaterials allow to substantially enhance optical chirality effects and circular dichroism in particular [3–10]. Furthermore, a new scientific direction of “chiral nanoplasmonics” emerged recently [11], which is dedicated especially to the study of the interaction of chiral nano-objects with electromagnetic fields, atoms and molecules.

A large number of works have been dedicated to the study of the spontaneous emission of atoms and molecules located near a spherical object. The influence of a dielectric microsphere on the spontaneous emission of atoms was considered in [12–14]; the spontaneous radiation of an atom located near a microsphere made of a negative refraction index metamaterial was considered in [15]; the radiation of an atom placed inside a concentric cavity in a spherical particle was considered in [16,17]. In [18–21], the influence of a chiral spherical particle on the radiation of a chiral molecule was investigated.

The scattering of electromagnetic radiation by a particle with chiral concentric spherical shells was previously investigated in details in [22–26]. The properties of a particle with usual (non-chiral) concentric spherical shells in the electromagnetic fields were also investigated [27–34]. However, the problem of scattering of light by an asymmetric chiral particle has not yet received much attention of researchers. Such task is important, because the abilities not only to control the interaction of electric and magnetic fields, but also to form a complex three-dimensional structure of optical fields and probably to create light beams of nontrivial spatio-temporal structure, are exciting (see, for example [35],). In this paper we consider the spontaneous radiation of a chiral molecule located near an asymmetric chiral nanoparticle and as a model of such particle we investigate a spherical dielectric particle with a nonconcentric spherical shell made of a chiral material (i.e., the so-called chiral “nanoegg” particle). Note that the technology for synthesis of asymmetric particles of similar shape (without chirality) has been developed [31]. The problem under consideration in the present paper is a continuation of our previous investigations [18–21], where the spontaneous radiation of chiral molecules near a chiral particle was studied. All analytical results presented in this paper are for arbitrary dimensions, degree of non-concentricity and material composition of the particle with a chiral shell, and also for any relationship between the electric and magnetic dipole moments of a chiral molecule.

The paper is structured as follows. In Section 2 the electromagnetic field of a chiral molecule in the presence of a dielectric spherical particle with a chiral nonconcentric spherical shell is considered. Nevertheless, despite the fact that the chirality can be regarded as a manifestation of spatial dispersion, we use usual (local) boundary conditions at the interface between the two media. A more general approach using nonlocal boundary conditions was developed in [36]. In Section 3 we derive general expressions for the radiative decay rate of the spontaneous emission of a chiral molecule located near a particle with a chiral shell. In Section 4 we present numerical analyses and graphic illustrations of the results to discuss how a nanoparticle influence the spontaneous emission of a chiral molecule. In Section 5 the main findings of this work are summarized.

2. Electromagnetic field of a chiral molecule in the presence of an asymmetric chiral particle

To solve the problem of the radiation of a molecule near a spherical particle with a nonconcentric spherical shell, we use the T-matrix method which is often used to describe a cluster of spherical particles. This method is accurate and based on the Mie theory for each particle and the addition theorem for vector spherical harmonics [37–46]. The method was initially developed by Waterman [47] and further improved in [48–52]. A bibliography on the T-matrix method can be found in [53].

In the present case of a spherical core particle with a spherical nonconcentric shell, the T-matrix method makes use of two coordinate systems centered at the shell center and the core center, respectively, for easy treatment of the two spherical boundary conditions (see Fig. 1). For simplicity, we assume that these coordinate systems have the common z-axis. The coordinates in the coordinate system centered at the shell center are denoted by an additional index s = 1, and the coordinates with respect to the coordinate system centered at the core center are denoted by index s = 2. Further, we consider a chiral particle with the shell of radius a and made of the chiral material defined by the permittivity $ε_{1}$ and permeability $μ_{1}$ as well as by the dimensionless chirality parameter $χ_{1}$ .

Fig. 1 Geometry of a chiral molecule with electric and magnetic dipole moments d₀ and –im₀, located near an asymmetric chiral particle consisting of a spherical core (blue circle) with a spherical nonconcentric shell (green region).

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To describe the electric and magnetic fields in the shell of a chiral medium, we use the constitutive equations for a bi-isotropic material in the Drude-Born-Fedorov form [54–56]:

k_{0} D^{s h e l l} = ε_{1} (k_{0} E^{s h e l l} + χ_{1} rot E^{s h e l l}), k_{0} B^{s h e l l} = μ_{1} (k_{0} H^{s h e l l} + χ_{1} rot H^{s h e l l}),

where

D^{s h e l l}

,

E^{s h e l l}

and

B^{s h e l l}

,

H^{s h e l l}

are the induction and strength of the electric and magnetic fields in the shell, respectively;

k_{0} = ω / c

is the wavenumber in vacuum, where

ω

is the frequency, and c is the speed of light; and the time dependence of the fields is determined by the factor

\exp (- i ω t)

, which is omitted hereafter. It should be noted that Eq. (1) is not the only possible way to describe chiral media. The question of choice of the constitutive equations is considered in details, for example, in the monograph [57]. We note that this particular form of the constitutive equations leads to the boundary conditions which do not prevent the solution of the problem by the method of separation of variables. Therefore, we use these relations [Eq. (1)] for which the boundary conditions are reduced to the continuity of the tangential components of the electric and magnetic fields.

The core of the particle, which is non-chiral and is made of a dielectric with the permittivity $ε_{2}$ and permeability $μ_{2}$ , has the radius $b < a$ and its center is placed at the distance $h < (a - b)$ from the center of the shell (i.e., the core does not intersect the particle surface). A more complicated case where the core intersects the surface of the particle associated with complicated boundary conditions, can be studied numerically. However, we do not consider this case in the present work. To be specific, we consider the core shifted downwards on the z-axis (see Fig. 1). The permittivity and permeability outside the shell take unit values (i.e., the particle is in vacuum). The case of the particle with a chiral core and a chiral shell can be considered in an analogous way.

The system shown in Fig. 1 has rich physics due to the interesting properties of the constituent parts that arise from the asymmetric combination of the dielectric spherical core embedded in the chiral sphere. The dielectric core has well known resonances which are called whispering gallery modes (WGMs) because light can propagate along the surface of the sphere [58]. The chiral spherical shell made of different metamaterials has even more interesting structure of resonances especially in the case of a shell made of DNG metamaterials where unusual narrow DNG plasmon resonances can be excited together with WGMs [15]. Adding chirality results in splitting of wave vectors of these modes. Finally, the asymmetric composition of the dielectric core and the chiral shell mixes all these resonances into an intoxicating potpourri.

To solve the problem of the radiation of a molecule near a particle with a chiral shell we need to write general expressions for the fields inside the core of the particle, inside the chiral shell and outside the particle with allowance for the emitting molecule. Then using the boundary conditions on the surface of the particle we derive the coefficients of the expansions. Let us start with the consideration of the fields outside the particle (in vacuum).

The electric and magnetic fields of the chiral molecule with the electric and magnetic dipole moments $d_{0}$ and $- i m_{0}$ , respectively, are defined in the free space by the electric and magnetic Hertz vectors [59], and in the coordinate system pinned to the center of the shell and denoted by left superscript s = 1, have the form:

{}^{1}E_{0} = {(d_{0} \cdot \nabla) \nabla + d_{0} k_{0}^{2} - i k_{0} [(- i m_{0}) \times \nabla]} \frac{e^{i k_{0} | {}^{1}r - {}^{1}r_{0} |}}{| {}^{1}r - {}^{1}r_{0} |}, {}^{1}H_{0} = \frac{1}{i k_{0}} [\nabla \times {}^{1}E_{0}],

where

{}^{1}E_{0}

and

{}^{1}H_{0}

are the electric and magnetic fields of the chiral molecule, respectively;

\nabla

is the gradient operator;

{}^{1}r

and

{}^{1}r_{0}

are the position vectors of the observation point and the chiral molecule, respectively. Note that the above-used phase relationship between the electric

d_{0}

and magnetic

- i m_{0}

dipole moments is caused by the choice of the chiral molecule in the form of a helix (for details, see, e.g., [60]). In this case the magnetic dipole moment is purely imaginary. Note also that the Cartesian components of the vectors

d_{0}

and

- i m_{0}

, as well as the Cartesian components of the gradient and rotor operators will not change in passing from one local coordinate system to another, because the axes of the local systems are parallel and equally directed. For this reason the index s = 1 in them is also omitted.

The electric field of the chiral molecule [Eq. (2)] for further calculations is conveniently written in spherical coordinates:

{}^{1}E_{0} = {\begin{matrix} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} ({}^{1}A_{m n}^{(0)} {}^{1}N ψ_{m n}^{(0)} + {}^{1}B_{m n}^{(0)} {}^{1}M ψ_{m n}^{(0)}), & | {}^{1}r | < | {}^{1}r_{0} |, \\ \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} ({}^{1}C_{m n}^{(0)} {}^{1}N ζ_{m n}^{(0)} + {}^{1}D_{m n}^{(0)} {}^{1}M ζ_{m n}^{(0)}), & | {}^{1}r | > | {}^{1}r_{0} |, \end{matrix}

where the vector spherical harmonics

{}^{1}N ψ_{m n}^{(0)}

,

{}^{1}M ψ_{m n}^{(0)}

,

{}^{1}N ζ_{m n}^{(0)}

and

{}^{1}M ζ_{m n}^{(0)}

are equal to the harmonics

{}^{1}N ψ_{m n}^{(0)}

,

{}^{1}M ψ_{m n}^{(0)}

,

{}^{1}N ζ_{m n}

and

{}^{1}M ζ_{m n}

, respectively, in [11]; the coefficients

{}^{1}A_{m n}^{(0)}

,

{}^{1}B_{m n}^{(0)}

,

{}^{1}C_{m n}^{(0)}

and

{}^{1}D_{m n}^{(0)}

are given in [11,21]. The magnetic field of a chiral molecule can be found with the help of Eq. (2).

For the electric and magnetic fields outside the particle ( $| {}^{1}r | > a$ ) we can write (s = 1):

{}^{1}E^{o u t} = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} ({}^{1}C_{m n} {}^{1}N ζ_{m n}^{(0)} + {}^{1}D_{m n} {}^{1}M ζ_{m n}^{(0)}), {}^{1}H^{o u t} = \frac{1}{i k_{0}} [\nabla \times {}^{1}E^{o u t}],

where

{}^{1}E^{o u t}

and

{}^{1}H^{o u t}

are the electric and magnetic fields induced near the particle, respectively; the coefficients

{}^{1}C_{m n}

and

{}^{1}D_{m n}

can be found from the boundary conditions.

For the electric and magnetic fields inside the particle core ( $| {}^{2}r | < b$ ) we can write (s = 2):

{}^{2}E^{c o r e} = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} ({}^{2}A_{m n} {}^{2}N ψ_{m n}^{(2)} + {}^{2}B_{m n} {}^{2}M ψ_{m n}^{(2)}), {}^{2}H^{c o r e} = \frac{1}{i k_{2} Z_{2}} [\nabla \times {}^{2}E^{c o r e}],

where

{}^{2}E^{c o r e}

and

{}^{2}H^{c o r e}

are the electric and magnetic fields in the core, respectively; the vector spherical harmonics

{}^{2}N ψ_{m n}^{(2)}

and

{}^{2}M ψ_{m n}^{(2)}

can be found from

{}^{1}N ψ_{m n}^{(0)}

and

{}^{1}M ψ_{m n}^{(0)}

, respectively, with the help of a wavenumber substitution

k_{0} \to k_{2} = k_{0} \sqrt{ε_{2} μ_{2}}

and also by substitution of the local coordinates associated with the external sphere (s = 1) to the coordinates associated with the internal sphere (s = 2);

Z_{2} = \sqrt{μ_{2} / ε_{2}}

is the impedance; the coefficients

{}^{2}A_{m n}

and

{}^{2}B_{m n}

can be found from the boundary conditions.

In accordance to the T-matrix method, the fields inside the chiral shell can be represented as the sum of the partial fields induced by the inner and outer spheres, expressed in the local coordinates of the spheres. Consequently, we can write:

E^{s h e l l} = {}^{1}E + {}^{2}E, H^{s h e l l} = {}^{1}H + {}^{2}H,

where in the case of the bi-isotropic material (see Eq. (1)), and after using the Bohren transformation [61], we have (s = 1, 2):

{}^{s}E = {}^{s}Q_{L} - i Z_{1} {}^{s}Q_{R}, {}^{s}H = - \frac{i}{Z_{1}} {}^{s}Q_{L} + {}^{s}Q_{R},

where

Z_{1} = \sqrt{μ_{1}} / \sqrt{ε_{1}}

is the impedance;

{}^{s}Q_{L}

and

{}^{s}Q_{R}

are the fields of the left-handed (the index “L”) and the right-handed (the index “R”) circularly polarized electromagnetic waves excited in the chiral medium, and they can be written as follows:

\begin{array}{l} {}^{1}Q_{L} = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} {}^{1}A_{m n} ({}^{1}N ψ_{m n}^{(L)} + {}^{1}M ψ_{m n}^{(L)}), {}^{1}Q_{R} = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} {}^{1}B_{m n} ({}^{1}N ψ_{m n}^{(R)} - {}^{1}M ψ_{m n}^{(R)}), \\ {}^{2}Q_{L} = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} {}^{2}C_{m n} ({}^{2}N ζ_{m n}^{(L)} + {}^{2}M ζ_{m n}^{(L)}), {}^{2}Q_{R} = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} {}^{2}D_{m n} ({}^{2}N ζ_{m n}^{(R)} - {}^{2}M ζ_{m n}^{(R)}), \end{array}

where

{}^{1}N ψ_{m n}^{(J)}

,

{}^{1}M ψ_{m n}^{(J)}

,

{}^{2}N ζ_{m n}^{(J)}

and

{}^{2}M ζ_{m n}^{(J)}

(J = L, R) are the vector spherical harmonics in the bi-isotropic medium in coordinates s = 1 and s = 2 (see [11]); the coefficients

{}^{1}A_{m n}

,

{}^{1}B_{m n}

,

{}^{2}C_{m n}

and

{}^{2}D_{m n}

can be found from the boundary conditions.

To find the coefficients of expansions (4), (5) and (8) we use the boundary conditions of continuity of the tangential components of the electric and magnetic fields on the surfaces of the inner and outer spheres of the particle [60]. In this case, it is necessary to use the addition theorem for vector spherical harmonics, which allow the harmonics written in some local coordinates to be represented in the form of expansions in harmonics written in other local coordinates, as pointed out in the Appendix. In particular, for the use of the boundary conditions on the surface of the core it is necessary to express the fields $E^{s h e l l}$ and $H^{s h e l l}$ in coordinates s = 2. To do this, we should initially write ${}^{1}E$ and ${}^{1}H$ (s = 1) in the forms of expansions in vector spherical harmonics in coordinates s = 2 with the help of the addition theorem, and we should then add the resultant expressions to the fields ${}^{2}E$ and ${}^{2}H$ (s = 2). Similarly, for the use of the boundary conditions on the external surface of the particle, we should represent the fields $E^{s h e l l}$ and $H^{s h e l l}$ in coordinates s = 1 with the help of the addition theorem.

As a result, for the coefficients ${}^{1}A_{m n}$ , ${}^{1}B_{m n}$ , ${}^{2}C_{m n}$ and ${}^{2}D_{m n}$ we can write the system of equations:

\begin{array}{l} {}^{L}A_{n}^{(2)} (b) \sum_{q = | m |}^{\infty} (V_{m n q}^{(L)} + W_{m n q}^{(L)}) {}^{1}A_{m q} - i Z_{1} {}^{R}A_{n}^{(2)} (b) \sum_{q = | m |}^{\infty} (V_{m n q}^{(R)} - W_{m n q}^{(R)}) {}^{1}B_{m q} \\ + {}^{L}C_{n}^{(2)} (b) {}^{2}C_{m n} - i Z_{1} {}^{R}C_{n}^{(2)} (b) {}^{2}D_{m n} = 0, \\ {}^{L}B_{n}^{(2)} (b) \sum_{q = | m |}^{\infty} (V_{m n q}^{(L)} + W_{m n q}^{(L)}) {}^{1}A_{m q} + i Z_{1} {}^{R}B_{n}^{(2)} (b) \sum_{q = | m |}^{\infty} (V_{m n q}^{(R)} - W_{m n q}^{(R)}) {}^{1}B_{m q} \\ + {}^{L}D_{n}^{(2)} (b) {}^{2}C_{m n} + i Z_{1} {}^{R}D_{n}^{(2)} (b) {}^{2}D_{m n} = 0, \end{array}

and

\begin{array}{l} {}^{L}V_{n}^{(0)} (a) {}^{1}A_{m n} - i Z_{1} {}^{R}V_{n}^{(0)} (a) {}^{1}B_{m n} + {}^{L}X_{n}^{(0)} (a) \sum_{q = | m |}^{\infty} {(- 1)}^{n + q} ({\tilde{V}}_{m n q}^{(L)} - {\tilde{W}}_{m n q}^{(L)}) {}^{2}C_{m q} \\ - i Z_{1} {}^{R}X_{n}^{(0)} (a) \sum_{q = | m |}^{\infty} {(- 1)}^{n + q} ({\tilde{V}}_{m n q}^{(R)} + {\tilde{W}}_{m n q}^{(R)}) {}^{2}D_{m q} = \frac{i Z_{1}}{{(k_{0} a)}^{2}} {}^{1}A_{m n}^{(0)}, \\ {}^{L}W_{n}^{(0)} (a) {}^{1}A_{m n} + i Z_{1} {}^{R}W_{n}^{(0)} (a) {}^{1}B_{m n} + {}^{L}Y_{n}^{(0)} (a) \sum_{q = | m |}^{\infty} {(- 1)}^{n + q} ({\tilde{V}}_{m n q}^{(L)} - {\tilde{W}}_{m n q}^{(L)}) {}^{2}C_{m q} \\ + i Z_{1} {}^{R}Y_{n}^{(0)} (a) \sum_{q = | m |}^{\infty} {(- 1)}^{n + q} ({\tilde{V}}_{m n q}^{(R)} + {\tilde{W}}_{m n q}^{(R)}) {}^{2}D_{m q} = \frac{i Z_{1}}{{(k_{0} a)}^{2}} {}^{1}B_{m n}^{(0)}, \end{array}

where the lower limit of the summation should be assumed equal to 1 for

m = 0

and equal to

| m |

for

m \neq 0

; the functions

V_{m n q}^{(J)}

,

W_{m n q}^{(J)}

,

{\tilde{V}}_{m n q}^{(J)}

,

{\tilde{W}}_{m n q}^{(J)}

,

{}^{J}A_{n}^{(p)} (ξ)

,

{}^{J}B_{n}^{(p)} (ξ)

,

{}^{J}C_{n}^{(p)} (ξ)

,

{}^{J}D_{n}^{(p)} (ξ)

,

{}^{J}V_{n}^{(p)} (ξ)

,

{}^{J}W_{n}^{(p)} (ξ)

,

{}^{J}X_{n}^{(p)} (ξ)

and

{}^{J}Y_{n}^{(p)} (ξ)

(where p = 0, 2,

ξ = a, b

and J = L, R) are given in the Appendix. The coefficients

{}^{2}A_{m n}

,

{}^{2}B_{m n}

,

{}^{1}C_{m n}

and

{}^{1}D_{m n}

can be expressed in the coefficients

{}^{1}A_{m n}

,

{}^{1}B_{m n}

,

{}^{2}C_{m n}

and

{}^{2}D_{m n}

. Here we do not give these expressions because they are cumbersome.

Note that in the particular case of a concentric shell, i.e., in the case $h = 0$ , Eqs. (9) and (10) can be significantly simplified, because in this case we can put: $W_{m n q}^{(J)} = {\tilde{W}}_{m n q}^{(J)} = 0$ and $V_{m n q}^{(J)} = {\tilde{V}}_{m n q}^{(J)} = δ_{q n}$ , where $δ_{q n}$ is the Kronecker delta. We also note that as seen from Eqs. (9) and (10), the structure of thus-obtained equations is that the subscripts n and q change, while the subscript m is fixed. In the numerical solution of the system [Eqs. (9) and (10)] use is made of the truncated equations with $n, q \leq N_{\max} > > 1$ . In this case, the greater the $N_{\max}$ , the more accurate the results for the coefficients ${}^{1}A_{m n}$ , ${}^{1}B_{m n}$ , ${}^{2}C_{m n}$ and ${}^{2}D_{m n}$ , because they tend to zero as $n \to \infty$ [45].

3. Radiative decay rate of the spontaneous emission of a chiral molecule near an asymmetric chiral particle

The relative radiative decay rate of the spontaneous emission of a chiral molecule located near a dielectric spherical particle with a nonconcentric spherical shell made of a chiral material can be calculated as the ratio of the radiation power $P_{r a d}$ of the molecule plus particle system to the radiation power $P_{r a d, 0}$ of the molecule in the absence of the particle [62]. The power $P_{r a d}$ is given by the formula [60]:

P_{r a d} = \frac{c}{8 π} \int_{S} d S Re ([({}^{1}E^{o u t} + {}^{1}E_{0}) \times ({}^{1}H^{o u t *} + {}^{1}H_{0}^{*})] \cdot n),

where the integration is performed over a closed surface S, covering the molecule and the particle; n is the outward normal to this surface; the asterisk means the complex conjugation. The power

P_{r a d, 0}

can be found from Eq. (11) if we put the induced fields equal to zero:

P_{r a d, 0} = \frac{c}{8 π} \int_{S} d S Re ([{}^{1}E_{0} \times {}^{1}H_{0}^{*}] \cdot n) = \frac{c k_{0}^{4}}{3} ({| d_{0} |}^{2} + {| m_{0} |}^{2}) .

The obtained expression Eq. (12) coincides with the total radiation power of the electric and magnetic dipoles, because they exhibit no interference with each other.

As for the surface S, over which the integration in Eq. (11) is performed, it is convenient to take a sphere of infinite radius centered at the center of the outer sphere of the particle. After performing the intergration in Eq. (11), we find for the radiative rate $γ_{r a d}$ the next expression:

\frac{γ_{r a d}}{γ_{0}} = \frac{P_{r a d}}{P_{r a d, 0}} = \frac{3}{k_{0}^{6} ({| d_{0} |}^{2} + {| m_{0} |}^{2})} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} \frac{n (n + 1)}{4 n + 2} \frac{(n + m)!}{(n - m)!} ({| {}^{1}C_{m n}^{(0)} + {}^{1}C_{m n} |}^{2} + {| {}^{1}D_{m n}^{(0)} + {}^{1}D_{m n} |}^{2}),

where

γ_{0} = 4 P_{r a d, 0} / (ℏ ω)

is the spontaneous emission decay rate of the chiral molecule in free space (in a vacuum). It should be noted that Eq. (13) corresponds to the transition of the molecule from an excited state to the ground state, i.e., to the case of a two-level molecule. In this case,

d_{0}

and

- i m_{0}

should be considered as the electric and magnetic dipole moments of the transition at frequency

ω

. To take into account the possibility of transition into several states, it is only necessary to sum the corresponding partial line widths in Eq. (13).

Note that in the case of a non-absorbing particle (i.e., without the Joule losses) Eq. (13) coincides with an analogous expression for the total (the radiative plus nonradiative) decay rate of the spontaneous emission. In this paper, we consider only the radiative decay rate (the rate of the spontaneous radiation), because this quantity can be measured by a detector. The nonradiative channel of the spontaneous emission decay, i.e., the channel associated with the Joule losses in the spherical system, will be discussed in a separate publication.

4. Numerical results and discussion

The process of the spontaneous emission of a chiral molecule located near an asymmetric chiral particle is very complex. For clarity here we present graphic illustrations of some possible regimes of interaction of the chiral molecule with the particle of various asymmetricy and material compositions. We restrict ourselves to the most interesting case where the chiral molecule is on the z-axis passing through the centers of the internal and external spheres of the particle (see Fig. 1). In this geometry, only the coefficients ${}^{1}A_{m n}^{(0)}$ , ${}^{1}B_{m n}^{(0)}$ , ${}^{1}C_{m n}^{(0)}$ and ${}^{1}D_{m n}^{(0)}$ with $m = 0, \pm 1$ (see Eq. (3)), are nonzero. That allows one to reduce the number of equations to be solved in the system [Eqs. (9) and (10)], i.e., to simplify significantly the mathematical task under consideration. Below, we consider a chiral molecule with equal Cartesian projections of the electric dipole moment (i.e., the case of $d_{0 x} = d_{0 y} = d_{0 z}$ ) and with equal Cartesian projections of the magnetic dipole moment (i.e., the case of $m_{0 x} = m_{0 y} = m_{0 z}$ ). This choice of the molecule allows one to study all the possible features of the spontaneous emission at the same time. In this case, we call the molecule with parallel $d_{0}$ and $m_{0}$ the “right” molecules, and the molecule with antiparallel $d_{0}$ and $m_{0}$ the “left” molecules.

Since the problem of the spontaneous radiation of a chiral molecule near a dielectric spherical particle with a chiral concentric spherical shell is insufficiently studied, we first present the results for this case (i.e., in the case of a symmetric chiral particle). Figure 2 shows the radiative decay rate of the spontaneous emission of a chiral molecule located near a dielectric spherical particle with a concentric spherical shell made of a chiral dielectric ( $ε_{1} > 0$ and $μ_{1} > 0$ ) [Fig. 2(a)] and a double negative (DNG) metamaterial with negative refraction index ( $ε_{1} < 0$ and $μ_{1} < 0$ ) [Fig. 2(b)] as a function of $k_{0} a$ . Since chiral DNG metamaterials are currently not widely used in experiments (see, however [63–65], for recent experimental works), we have chosen these values for these parameters mainly because our previous works indicated the presence of resonances in these parameter value regions.

Fig. 2 Radiative decay rate of the spontaneous emission of a chiral molecule located close to the surface ( ${}^{1}r \to a$ ) of a spherical particle with a dielectric core ( $ε_{2} = 2$ , $μ_{2} = 1$ and $b / a = 0.75$ ) and a concentric spherical shell as the function of $k_{0} a$ . (a) The shell is made of a chiral dielectric ( $ε_{1} = 3$ , $μ_{1} = 1$ and $χ_{1} = 0.1$ ). (b) The shell is made of a chiral DNG-metamaterial ( $ε_{1} = - 3$ , $μ_{1} = - 1$ and $χ_{1} = 0.1$ ). The solid lines correspond to the “right” molecule ( $m_{0} = 0.1 d_{0}$ ), and the dashed lines correspond to the “left” molecule ( $m_{0} = - 0.1 d_{0}$ ). The cases of the pure dielectric ( $b = a$ ) and pure chiral ( $b = 0$ ) spheres are also presented.

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As seen in Fig. 2(a), in the case of the shell with a chiral dielectric, the radiative decay rate is an oscillating function of $k_{0} a$ . Qualitatively, these oscillations are similar in nature to the WGMs in the dielectric spheres [58]. In this case, an increase in $k_{0} a$ causes an increase both in the number of oscillations and in their Q-factors. A solid chiral particle exhibits a similar dependence (i.e., the case of $b = 0$ in Fig. 2(a), see also [20]). However, the positions of the maxima in these two cases occur in different places (see Fig. 2(a)). Figure 2(a) also shows that the radiative decay rates of the spontaneous emission of the “right” and “left” molecule enantiomers can differ significantly from each other.

In the case of a dielectric spherical particle with a shell made of a chiral DNG-metamaterial (see Fig. 2(b)) the dependence of the radiative decay rate on $k_{0} a$ is more complex than in the case of a chiral dielectric shell. Clearly, the radiative decay rate in this case is not an oscillating function of $k_{0} a$ . Instead, the decay rate spectra consist of broad features decorated with a number of distinct high-Q peaks. The positions, shapes and sizes of these peaks are different for different shell thicknesses [cf. dependencies for $b = 0$ and $b = 0.75 a$ in Fig. 2(b)]. These aperiodic peaks are due to the excitation of well-known DNG plasmons [15,20]. For the case of $b = a$ , the system simply reduces to a solid dielectric spherical particle of small size in comparison with the wavelength. In such particles the resonances are of bad quality and can hardly be observed.

For further detailed analyses, the influence of the thickness of the chiral concentric shell on the radiative decay rate of the spontaneous emission of a chiral molecule is shown in Fig. 3, where the decay rates are plotted as a function of ratio $b / a$ . Below we restrict ourselves to the case of the particles of relatively small sizes ( $k_{0} a \leq 1$ ), because they are of great interest for practical applications.

Fig. 3 Radiative decay rate of the spontaneous emission of a chiral molecule located close to the surface ( ${}^{1}r \to a$ ) of a spherical particle with a dielectric core ( $ε_{2} = 2$ and $μ_{2} = 1$ ) and a concentric spherical shell as the function of $b / a$ for the selected values of $k_{0} a$ . (a) The shell is made of a chiral dielectric ( $ε_{1} = 3$ , $μ_{1} = 1$ and $χ_{1} = 0.1$ ). (b) The shell is made of a chiral DNG-metamaterial ( $ε_{1} = - 3$ , $μ_{1} = - 1$ and $χ_{1} = 0.1$ ). The solid lines correspond to the “right” molecule ( $m_{0} = 0.1 d_{0}$ ), and the dashed lines correspond to the “left” molecule ( $m_{0} = - 0.1 d_{0}$ ).

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One can see from Fig. 3(a) that for the case of a dielectric particle with a chiral dielectric shell, an increase in the core radius at the fixed outer radius of the shell causes a decrease in the radiative decay rate, because the permittivity of the core is less than the permittivity of the shell. Moreover, a reduction of the chiral shell thickness leads to a decrease in the difference between the decay rates of the “right” and “left” chiral molecule enantiomers.

In the case of a shell with chiral DNG-metamaterial properties (see Fig. 3(b)), the dependence of the radiative decay rates on ratio $b / a$ behaves in a more complex way. Initially, the radiative decay rates hardly change as the core radius increases, until $b / a$ reaches a value where the first high-Q peak in the decay rate occurs. Many more high-Q peaks appear as the core radius further increases. The occurrence of these peaks indicates the excitation of high-Q surface modes in the dielectric particle with the chiral DNG-metamaterial shell. Similar phenomena take place in the case of a spherical particle made of a DNG-metamaterial [15,21]. Figure 3(b) shows that the efficiency of the surface mode excitation depends strongly on the shell thickness.

Below we consider the influence of non-concentricity of a chiral shell on the spontaneous radiation of a chiral molecule. Figure 4 shows the radiative decay rate of the spontaneous emission of the chiral molecule located near the dielectric spherical particle with the chiral spherical shell of small non-concentricity ( $h / b = 0.1$ ) as a function of $b / a$ . One can see from Fig. 4(a) that for the shell made of a chiral dielectric, the radiative decay rate as a function of $b / a$ changes very slowly. To some extent, the case of a molecule located near a concentric shell falls in between the cases of a molecules located near the thin and thick parts of a nonconcentric shell. It should be noted also that the difference in the spontaneous decay rates of the “right” and “left” chiral molecule enantiomers decreases as the shell thickness decreases (i.e., as $b / a$ increases).

Fig. 4 Radiative decay rate of the spontaneous emission of a chiral molecule located close to the surface ( ${}^{1}r \to a$ ) of a spherical particle with a dielectric core ( $ε_{2} = 2$ , $μ_{2} = 1$ and $k_{0} b = 0.5$ ) and a nonconcentric spherical shell ( $h / b = 0.1$ ) as the function of $b / a$ . (a) The shell is made of a chiral dielectric ( $ε_{1} = 3$ , $μ_{1} = 1$ and $χ_{1} = 0.1$ ). (b) The shell is made of a chiral DNG-metamaterial ( $ε_{1} = - 3$ , $μ_{1} = - 1$ and $χ_{1} = 0.1$ ). The solid lines correspond to the “right” molecule ( $m_{0} = 0.1 d_{0}$ ), and the dashed lines correspond to the “left” molecule ( $m_{0} = - 0.1 d_{0}$ ). The sketches represent the cases of the molecule position near the thin and thick parts of a shell. The case of a concentric shell ( $h = 0$ ) is also presented.

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The dependence of the radiative decay rate on $b / a$ becomes more complex in the case of a chiral DNG-metamaterial shell [see Fig. 4(b)]. In this case high-Q maxima occur and also the number of these maxima increases when the shell becomes thinner (i.e., as $b / a$ increases). Note that in the case of the nonconcentric shell, more maxima than in the case of a concentric shell occur. This is due to a larger number of surface modes excited in the case of a nonconcentric shell. Note also that the conditions for the occurrence of the maxima (i.e., the values of $b / a$ correspond to the maxima in Fig. 4(b)) for the concentric and nonconcentric shells are different.

The case of a shell with a greater degree of non-concentricity is presented in Fig. 5, where the radiative decay rate of a chiral molecule located near a dielectric spherical particle with a chiral nonconcentric shell is shown as a function of ratio $h / a$ .

Fig. 5 Radiative decay rate of the spontaneous emission of a chiral molecule located close to the surface ( ${}^{1}r \to a$ ) of a spherical particle with a dielectric core ( $ε_{2} = 2$ , $μ_{2} = 1$ and $b / a = 0.5$ ) and a nonconcentric spherical shell as the function of $h / a$ for the selected values of $k_{0} a$ . (a) The shell is made of a chiral dielectric ( $ε_{1} = 3$ , $μ_{1} = 1$ and $χ_{1} = 0.1$ ). (b) The shell is made of a chiral DNG-metamaterial ( $ε_{1} = - 3$ , $μ_{1} = - 1$ and $χ_{1} = 0.1$ ). The solid lines correspond to the “right” molecule ( $m_{0} = 0.1 d_{0}$ ), and the dashed lines correspond to the “left” molecule ( $m_{0} = - 0.1 d_{0}$ ). The sketches represent the cases of the molecule position near the thin and thick parts of a shell.

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Figure 5(a) shows that in the case of a particle with a chiral dielectric shell, a smooth varying dependence of the radiative decay rate on $h / a$ appears. In the case of a chiral DNG-metamaterial shell (see Fig. 5(b)), however, this dependence is more complex. We can see the pronounced maxima corresponding to the excitation of surface modes in the particle. In this case, as can be clearly seen in Fig. 5(b) (as in Fig. 4(b)), there is a significant difference between the radiative decay rates of molecules located near the thin and thick parts of a nonconcentric shell and also near a concentric shell (the point $h = 0$ in Fig. 5(b)) made of a chiral DNG-metamaterial.

Chiral molecules play an important role in biology and in medicine. It is therefore important to be able to carry out the selection of the “right” and “left” enantiomers of a chiral molecule in racemic mixtures. In [18,66] it was shown that the significant difference in the radiative decay rates for the “right” and “left” enantiomers located near a chiral nanoparticle made of metamaterials can serve as a basis for the development of the methods for such selection and also for the development of the nano-bio-sensors for effective detection of a small number of left enantiomers against a high background of right enantiomers. Figure 6(a) shows that a dielectric spherical nanoparticle with a shell made of a chiral DNG-metamaterial with $ε_{1} < 0$ and $μ_{1} < 0$ can accelerate the decay of the “left” molecule and slow it down for the “right” molecule. In contrast, а nanoparticle with a shell made of a DNG-metamaterial with $ε_{1} > 0$ and $μ_{1} < 0$ (see Fig. 6(b)) will accelerate the decay for the “right” molecule and slow it down for the “left” molecule. Figure 6 indicates that this difference in the radiative decay rates of the molecule enantiomers can become a factor of 3 or even 12 in the present cases. Moreover, the location of the molecule near the thin part of the nonconcentric shell can further increase the difference in the radiative decay rates of the “right” and “left” molecules, which would enhance the selection of the enantiomers.

Fig. 6 Radiative decay rate of the spontaneous emission of a chiral molecule located close to the surface ( ${}^{1}r \to a$ ) of a spherical nanoparticle ( $k_{0} a = 0.1$ ) with a dielectric core ( $ε_{2} = 2$ , $μ_{2} = 1$ and $b / a = 0.5$ ) and a chiral ( $χ_{1} = 0.2$ ) nonconcentric spherical shell ( $h / b = 0.1$ ) as the function of the real part of the permittivity $ε_{1} = {ε^{'}}_{1} + i 0.1$ for the selected permeability: (a) $μ_{1} = - 2.3 + i 0.1$ and (b) $μ_{1} = - 3 + i 0.1$ . The solid lines correspond to the “right” molecule ( $m_{0} = 0.1 d_{0}$ ), and the dashed lines correspond to the “left” molecule ( $m_{0} = - 0.1 d_{0}$ ). The colors represent the cases of the molecule position near the thin (blue), thick (red) parts of a shell, and the case of concentric shell (green). The insets show near-field patterns Re(E_z) at (a) ${ε^{'}}_{1} = - 1.33$ and (b) ${ε^{'}}_{1} = 0.833$ on the z = 0 plane. The scale colorbars are in units of V/m.

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To understand the physics of these effects, we also performed independent numerical simulations by using the finite element method as implemented in COMSOL Multiphysics® software [67]. In order to simulate the chiral media, the weak form in the COMSOL RF module is modified to include chirality [20]. The results of our simulations are shown in the insets to Fig. 6 as well as in Figs. 7 and 8. Importantly, Figs. 6(a) and 7 show that the fields are strongly localized on the surfaces of both core and shell, even though the nanoparticle is small in comparison with the wavelength ( $k_{0} a = 0.1$ ). This is possibly because both dielectric core and DNG chiral shell allow the propagation of electromagnetic waves. As a result, the resonances in Fig. 6(a) near ${ε^{'}}_{1} = - 1.33$ are similar to the whispering gallery resonances.

Fig. 7 Spatial distribution of Re(E_z) for parameters corresponding to Fig. 6a. Upper three panels are for the “right” molecule ( $m_{0} = 0.1 d_{0}$ ), and lower ones correspond to the “left” molecule ( $m_{0} = - 0.1 d_{0}$ ). The scale colorbars are in units of V/m.

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Fig. 8 Spatial distribution of Re(E_z) for parameters corresponding to Fig. 6b. Upper three pictures are for the “right” molecule ( $m_{0} = 0.1 d_{0}$ ), and lower ones correspond to the “left” molecule ( $m_{0} = - 0.1 d_{0}$ ). The scale colorbars are in units of V/m.

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In strong contrast, the insets to Fig. 6(b) and also Fig. 8 indicate that the fields are concentrated mainly in the vicinity of the excited molecule. Such type of localization is due to the fact that for the parameters used for Fig. 6(b) the propagation of electromagnetic waves inside the shell is impossible. Therefore, the resonance in Fig. 6(b) near ${ε^{'}}_{1} = 0.833$ is similar to the magnetic plasmon resonance.

5. Conclusions

In conclusion, we have derived analytical expressions for the radiative decay rate of the spontaneous emission of a chiral (i.e., an optically active) molecule located near a dielectric spherical particle with a chiral nonconcentric spherical shell made of a bi-isotropic material in a variety of cases such as different material compositions, various degrees of non-concentricity of the shell, and also different sizes of the particle. Our numerical analyses of these expressions show that for a chiral molecule near a dielectric spherical particle with a shell made of a chiral metamaterial with simultaneously negative permittivity and permeability, as well as a chiral metamaterial with positive permittivity and negative permeability, the “right” and “left” enantiomers of the molecule can exhibit a pronounced difference between the radiative decay rates. Furthermore, the position of the molecule near the nonsymmetric part of the shell can be used to manipulate the above-mentioned difference. Clearly, the results obtained here can be used to calculate the radiative decay rate of the spontaneous emission of chiral molecules located near a dielectric (a metal) spherical particle with a nonconcentric spherical shell made of a chiral material, for planning experiments on the interaction between chiral molecules and chiral particles, for interpretation of experimental data, and also for designing nano-bio-sensors.

Appendix Mathematical additions

To use the boundary conditions for the fields in Eq. (6) the following theorems for the vector spherical harmonics in Eq. (8) should be applied [43–46] (J = L, R):

\begin{array}{l} {\begin{matrix} {}^{1}N ψ_{m q}^{(J)} \\ {}^{1}M ψ_{m q}^{(J)} \end{matrix}} = \sum_{n = | m |}^{\infty} [V_{m n q}^{(J)} {\begin{matrix} {}^{2}N ψ_{m n}^{(J)} \\ {}^{2}M ψ_{m n}^{(J)} \end{matrix}} + W_{m n q}^{(J)} {\begin{matrix} {}^{2}M ψ_{m n}^{(J)} \\ {}^{2}N ψ_{m n}^{(J)} \end{matrix}}], \\ {\begin{matrix} {}^{2}N ζ_{m q}^{(J)} \\ {}^{2}M ζ_{m q}^{(J)} \end{matrix}} = \sum_{n = | m |}^{\infty} {(- 1)}^{n + q} [{\tilde{V}}_{m n q}^{(J)} {\begin{matrix} {}^{1}N ζ_{m n}^{(J)} \\ {}^{1}M ζ_{m n}^{(J)} \end{matrix}} - {\tilde{W}}_{m n q}^{(J)} {\begin{matrix} {}^{1}M ζ_{m n}^{(J)} \\ {}^{1}N ζ_{m n}^{(J)} \end{matrix}}], | {}^{1}r | > h, \end{array}

where the lower limit of the summation should be assumed equal to 1 for

m = 0

and equal to

| m |

for

m \neq 0

; h is the distance between the origins of the local systems of coordinates (see Fig. 1). The functions in Eq. (14) can be written as follows:

\begin{array}{l} {\begin{matrix} V_{m n q}^{(J)} \\ {\tilde{V}}_{m n q} \end{matrix}} = {\begin{matrix} U_{m n q}^{(J)} \\ {\tilde{U}}_{m n q}^{(J)} \end{matrix}} - k_{J} h (\frac{n - m}{n (2 n - 1)} {\begin{matrix} U_{m, n - 1, q}^{(J)} \\ {\tilde{U}}_{m, n - 1, q}^{(J)} \end{matrix}} + \frac{n + m + 1}{(n + 1) (2 n + 3)} {\begin{matrix} U_{m, n + 1, q}^{(J)} \\ {\tilde{U}}_{m, n + 1, q}^{(J)} \end{matrix}}), \\ {\begin{matrix} W_{m n q}^{(J)} \\ {\tilde{W}}_{m n q}^{(J)} \end{matrix}} = - \frac{i m k_{J} h}{n (n + 1)} {\begin{matrix} U_{m n q}^{(J)} \\ {\tilde{U}}_{m n q}^{(J)} \end{matrix}}, {\begin{matrix} U_{m n q}^{(J)} \\ {\tilde{U}}_{m n q}^{(J)} \end{matrix}} = {[\frac{(n - m)! (q + m)!}{(n + m)! (q - m)!}]}^{1 / 2} \times \\ \times \sum_{σ = | n - q |}^{n + q} i^{n - q - σ} \frac{ψ_{σ} (k_{J} h)}{k_{J} h} {\begin{matrix} {(- 1)}^{m} (2 n + 1) C_{q m n, - m}^{σ 0} C_{q 0 n 0}^{σ 0} \\ (2 σ + 1) C_{q m σ 0}^{n m} C_{q 0 σ 0}^{n 0} \end{matrix}}, \end{array}

where

C_{a α b β}^{c γ}

is the Clebsch-Gordan coefficient [68]; the definition of

k_{J}

and

ψ_{σ} (k_{J} h)

is presented below. In the case of a concentric shell, i.e., when

h = 0

, from Eq. (14) one can obtain:

{}^{1}N ψ_{m n}^{(J)} = {}^{2}N ψ_{m n}^{(J)}

,

{}^{1}M ψ_{m n}^{(J)} = {}^{2}M ψ_{m n}^{(J)}

,

{}^{1}N ζ_{m n}^{(J)} = {}^{2}N ζ_{m n}^{(J)}

and

{}^{1}M ζ_{m n}^{(J)} = {}^{2}M ζ_{m n}^{(J)}

.

The functions in Eq. (10) have the form (p = 0, 2, $ξ = a, b$ and J = L, R):

\begin{array}{l} {\begin{matrix} {}^{J}A_{n}^{(p)} (ξ) \\ {}^{J}B_{n}^{(p)} (ξ) \end{matrix}} = {\begin{matrix} Z_{p} \\ Z_{1} \end{matrix}} \frac{ψ_{n} (k_{J} ξ)}{k_{J} ξ} \frac{{ψ^{'}}_{n} (k_{p} ξ)}{k_{p} ξ} - {\begin{matrix} Z_{1} \\ Z_{p} \end{matrix}} \frac{{ψ^{'}}_{n} (k_{J} ξ)}{k_{J} ξ} \frac{ψ_{n} (k_{p} ξ)}{k_{p} ξ}, \\ {\begin{matrix} {}^{J}C_{n}^{(p)} (ξ) \\ {}^{J}D_{n}^{(p)} (ξ) \end{matrix}} = {\begin{matrix} Z_{p} \\ Z_{1} \end{matrix}} \frac{ζ_{n} (k_{J} ξ)}{k_{J} ξ} \frac{{ψ^{'}}_{n} (k_{p} ξ)}{k_{p} ξ} - {\begin{matrix} Z_{1} \\ Z_{p} \end{matrix}} \frac{{ζ^{'}}_{n} (k_{J} ξ)}{k_{J} ξ} \frac{ψ_{n} (k_{p} ξ)}{k_{p} ξ}, \end{array}

and

\begin{array}{l} {\begin{matrix} {}^{J}V_{n}^{(p)} (ξ) \\ {}^{J}W_{n}^{(p)} (ξ) \end{matrix}} = {\begin{matrix} Z_{p} \\ Z_{1} \end{matrix}} \frac{ψ_{n} (k_{J} ξ)}{k_{J} ξ} \frac{{ζ^{'}}_{n} (k_{p} ξ)}{k_{p} ξ} - {\begin{matrix} Z_{1} \\ Z_{p} \end{matrix}} \frac{{ψ^{'}}_{n} (k_{J} ξ)}{k_{J} ξ} \frac{ζ_{n} (k_{p} ξ)}{k_{p} ξ}, \\ {\begin{matrix} {}^{J}X_{n}^{(p)} (ξ) \\ {}^{J}Y_{n}^{(p)} (ξ) \end{matrix}} = {\begin{matrix} Z_{p} \\ Z_{1} \end{matrix}} \frac{ζ_{n} (k_{J} ξ)}{k_{J} ξ} \frac{{ζ^{'}}_{n} (k_{p} ξ)}{k_{p} ξ} - {\begin{matrix} Z_{1} \\ Z_{p} \end{matrix}} \frac{{ζ^{'}}_{n} (k_{J} ξ)}{k_{J} ξ} \frac{ζ_{n} (k_{p} ξ)}{k_{p} ξ}, \end{array}

where

ψ_{n} (k_{J} ξ) = {(π k_{J} ξ / 2)}^{1 / 2} J_{n + 1 / 2} (k_{J} ξ)

and

ζ_{n} (k_{J} ξ) = {(π k_{J} ξ / 2)}^{1 / 2} H_{n + 1 / 2}^{(1)} (k_{J} ξ)

are the Riccati-Bessel functions. Here

J_{n + 1 / 2} (k_{J} ξ)

and

H_{n + 1 / 2}^{(1)} (k_{J} ξ)

are the Bessel function and the Hankel function of the 1-st kind, respectively [69]; the stroke means the derivative;

Z_{0} = 1

,

Z_{1} = \sqrt{μ_{1}} / \sqrt{ε_{1}}

and

Z_{2} = \sqrt{μ_{2} / ε_{2}}

;

k_{2} = k_{0} \sqrt{ε_{2} μ_{2}}

;

k_{L} = \frac{k_{0} \sqrt{ε_{1}} \sqrt{μ_{1}}}{1 - χ_{1} \sqrt{ε_{1}} \sqrt{μ_{1}}}, k_{R} = \frac{k_{0} \sqrt{ε_{1}} \sqrt{μ_{1}}}{1 + χ_{1} \sqrt{ε_{1}} \sqrt{μ_{1}}}

are the wavenumbers of the left-handed (the index “L”) and the right-handed (the index “R”) circularly polarized electromagnetic waves. Note that in the present paper we put

\sqrt{ε_{1} μ_{1}} = \sqrt{ε_{1}} \sqrt{μ_{1}}

, because such definition is important to provide the correct refractive index, as well as the positive coefficient of extinction in the case of the DNG-metamaterial shell.

Funding

The authors are grateful to the Advanced Research Foundation (Contract 7/004/2013-2018 by 23/12/2013) and the Russian Foundation for Basic Research (Grants 14-02-00290 and 15-52-52006) as well as the Ministry of Science and Technology (Grant MOST 104-2923-M-002-004-MY3) and the National Center for Theoretical Sciences of Taiwan for financial support of this work. VVK also thanks the Competitiveness Program of NRNU MEPhI for financial support.

References and links

1. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House Publishers, 1994).

2. Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011). [CrossRef] [PubMed]

3. Y. Zhao, A. N. Askarpour, L. Sun, J. Shi, X. Li, and A. Alù, “Chirality detection of enantiomers using twisted optical metamaterials,” Nat. Commun. 8, 14180 (2017). [CrossRef] [PubMed]

4. C. S. Ho, A. Garcia-Etxarri, Y. Zhao, and J. Dionne, “Enhancing enantioselective absorption using dielectric nanospheres,” ACS Photonics, in press (2016).

5. C. Kramer, M. Schäferling, T. Weiss, H. Giessen, and T. Brixner, “Analytic optimization of near-field optical chirality enhancement,” ACS Photonics, in press (2017).

6. A. García-Etxarri and J. A. Dionne, “Surface-enhanced circular dichroism spectroscopy mediated by nonchiral nanoantennas,” Phys. Rev. B 87(23), 235409 (2013). [CrossRef]

7. A. O. Govorov, Z. Fan, P. Hernandez, J. M. Slocik, and R. R. Naik, “Theory of circular dichroism of nanomaterials comprising chiral molecules and nanocrystals: plasmon enhancement, dipole interactions, and dielectric effects,” Nano Lett. 10(4), 1374–1382 (2010). [CrossRef] [PubMed]

8. M. L. Nesterov, X. Yin, M. Schäferling, H. Giessen, and T. Weiss, “The role of plasmon-generated near fields for enhanced circular dichroism spectroscopy,” ACS Photonics 3(4), 578–583 (2016). [CrossRef]

9. Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104(16), 163901 (2010). [CrossRef] [PubMed]

10. M. Schäferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Phys. Rev. X 2(3), 031010 (2012). [CrossRef]

11. S. V. Boriskina and N. I. Zheludev, Singular and Chiral Nanoplasmonics (Pan Stanford Publishing, 2015).

12. H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87(2), 1355–1360 (1987). [CrossRef]

13. H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A Gen. Phys. 38(7), 3410–3416 (1988). [CrossRef] [PubMed]

14. V. V. Klimov and V. S. Letokhov, “Electric and magnetic dipole transitions of an atom in the presence of spherical dielectric interface,” Laser Phys. 15(1), 61–73 (2005).

15. V. V. Klimov, “Spontaneous emission of an excited atom placed near a “left-handed” sphere,” Opt. Commun. 211(1–6), 183–196 (2002). [CrossRef]

16. V. V. Klimov and V. S. Letokhov, “Enhancement and inhibition of spontaneous emission rates in nanobubbles,” Chem. Phys. Lett. 301(5–6), 441–448 (1999). [CrossRef]

17. J. Enderlein, “Spectral properties of a fluorescing molecule within a spherical metallic nanocavity,” Phys. Chem. Chem. Phys. 4(12), 2780–2786 (2002). [CrossRef]

18. V. V. Klimov, D. V. Guzatov, and M. Ducloy, “Engineering of radiation of optically active molecules with chiral nano-meta-particles,” Europhys. Lett. 97(4), 47004 (2012). [CrossRef]

19. D. V. Guzatov and V. V. Klimov, “The influence of chiral spherical particles on the radiation of optically active molecules,” New J. Phys. 14(12), 123009 (2012). [CrossRef]

20. V. V. Klimov, I. V. Zabkov, A. A. Pavlov, and D. V. Guzatov, “Eigen oscillations of a chiral sphere and their influence on radiation of chiral molecules,” Opt. Express 22(15), 18564–18578 (2014). [CrossRef] [PubMed]

21. D. V. Guzatov and V. V. Klimov, “Spontaneous emission of a chiral molecule near a cluster of two chiral spherical particles,” Quantum Electron. 45(3), 250–257 (2015). [CrossRef]

22. C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62(4), 1566–1571 (1975). [CrossRef]

23. P. L. E. Uslenghi, “Scattering by impedance sphere coated with a chiral layer,” Electromagn. 10(1–2), 201–211 (1990). [CrossRef]

24. E. K. N. Yung and B. J. Hu, “Scattering by conducting sphere coated with chiral media,” Microw. Opt. Technol. Lett. 36(4), 288–293 (2002). [CrossRef]

25. S. Arslanagic, R. W. Ziolkowski, and O. Breinbjerg, “Analytical and numerical investigation of the radiation from concentric metamaterial spheres excited by an electric Hertzian dipole,” Radio Sci. 42(6), RS6S16 (2007).

26. Y. L. Geng, C. W. Qiu, and N. Yuan, “Scattering by an impedance sphere coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag. 57(2), 572–576 (2009). [CrossRef]

27. V. I. Rozenberg, ““Diffraction and scattering of electromagnetic waves by an inhomogeneous sphere,” Izv. VUZ,” Radiofiz. 13(3), 337–348 (1970).

28. F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. A 9(8), 1327–1335 (1992). [CrossRef]

29. F. Borghese, P. Denti, R. Saija, M. A. Iati, and O. I. Sindoni, “Optical resonances of spheres containing an eccentric spherical inclusion,” J. Opt. 29(1), 28–34 (1998). [CrossRef]

30. V. Griaznov, I. Veselovskii, P. Di Girolamo, B. Demoz, and D. N. Whiteman, “Numerical simulation of light backscattering by spheres with off-center inclusion. Application to the lidar case,” Appl. Opt. 43(29), 5512–5522 (2004). [CrossRef] [PubMed]

31. H. Wang, Y. Wu, B. Lassiter, C. L. Nehl, J. H. Hafner, P. Nordlander, and N. J. Halas, “Symmetry breaking in individual plasmonic nanoparticles,” Proc. Natl. Acad. Sci. U.S.A. 103(29), 10856–10860 (2006). [CrossRef] [PubMed]

32. M. W. Knight and N. J. Halas, “Nanoshells to nanoeggs to nanocups: symmetry breaking beyond the quasistatic limit in complex plasmonic nanoparticles,” New J. Phys. 10(10), 105006 (2008). [CrossRef]

33. J. Zhang and A. Zayats, “Multiple Fano resonances in single-layer nonconcentric core-shell nanostructures,” Opt. Express 21(7), 8426–8436 (2013). [CrossRef] [PubMed]

34. F. Mangini, N. Tedeschi, F. Frezza, and A. Sihvola, “Electromagnetic interaction with two eccentric spheres,” J. Opt. Soc. Am. A 31(4), 783–789 (2014). [CrossRef] [PubMed]

35. G. Pariente and F. Quéré, “Spatio-temporal light springs: extended encoding of orbital angular momentum in ultrashort pulses,” Opt. Lett. 40(9), 2037–2040 (2015). [CrossRef] [PubMed]

36. A. A. Golubkov and V. A. Makarov, “Boundary conditions for electromagnetic field on the surface of media with weak spatial dispersion,” Phys. Uspekhi 38(3), 325–332 (1995). [CrossRef]

37. M. Quinten and U. Kreibig, “Absorption and elastic scattering of light by particle aggregates,” Appl. Opt. 32(30), 6173–6182 (1993). [CrossRef] [PubMed]

38. F. Borghese, P. Denti, G. Toscano, and O. I. Sindoni, “Electromagnetic scattering by a cluster of spheres,” Appl. Opt. 18(1), 116–120 (1979). [CrossRef] [PubMed]

39. J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres part I. Multipole expansion and ray-optical solutions,” IEEE Trans. Antenn. Propag. 19(3), 378–390 (1971). [CrossRef]

40. K. A. Fuller and G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. I: Linear chains,” Opt. Lett. 13(2), 90–92 (1988). [CrossRef] [PubMed]

41. A. K. Hamid, “Modeling the scattering from a dielectric spheroid by a system of dielectric spheres,” J. Electromagn. Waves Appl. 10(5), 723–729 (1996). [CrossRef]

42. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11(11), 2851–2861 (1994). [CrossRef]

43. K. A. Fuller, “Scattering and absorption cross sections of compounded spheres. I. Theory for external aggregation,” J. Opt. Soc. Am. A 11(12), 3251–3260 (1994). [CrossRef]

44. T. J. Dufva, J. Sarvas, and J. C.-E. Sten, “Unified derivation of the translational addition theorems for the spherical scalar and vector wave functions,” Progr. Electromag. Res. B 4, 79–99 (2008). [CrossRef]

45. Y. A. Ivanov, Diffraction of Electromagnetic Waves on Two Bodies (Nauka i tekhnika, 1968).

46. J. M. Gérardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25(6), 4204–4229 (1982). [CrossRef]

47. P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D Part. Fields 3(4), 825–839 (1971). [CrossRef]

48. M. I. Mishchenko and L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109(1–2), 16–21 (1994). [CrossRef]

49. M. I. Mishchenko, D. W. Mackowski, and L. D. Travis, “Scattering of light by bispheres with touching and separated components,” Appl. Opt. 34(21), 4589–4599 (1995). [CrossRef] [PubMed]

50. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transf. 55(5), 535–575 (1996). [CrossRef]

51. D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13(11), 2266–2278 (1996). [CrossRef]

52. D. W. Mackowski, “An effective medium method for calculation of the T matrix of aggregated spheres,” J. Quant. Spectrosc. Radiat. Transf. 70(4–6), 441–464 (2001). [CrossRef]

53. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Goddard Institute for Space Studies, 2004).

54. P. Drude, Lehrbuch der Optik (Verlag von S. Hirzel, 1906).

55. M. Born, Optik (Springer-Verlag, 1972).

56. B. V. Bokut’, A. N. Serdyukov, and F. I. Fedorov, “Phenomenological theory of optically active crystals,” Sov. Phys. Crystallogr. 15, 871–874 (1971).

57. F. I. Fedorov, Theory of Gyrotropy (Nauka i tekhnika, 1976).

58. L. Rayleigh, Theory of Sound, vol. II (Macmillan, 1878); Optical Effects Associated with Small Particles, R. K. Chang and P. W. Barber, eds. (World Scientific Publishing, 1988).

59. L. A. Vainshtein, Electromagnetic Waves (Radio i sviaz, 1990).

60. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

61. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29(3), 458–462 (1974). [CrossRef]

62. V. V. Klimov and M. Ducloy, “Spontaneous emission rate of an excited atom placed near a nanofiber,” Phys. Rev. A 69(1), 013812 (2004). [CrossRef]

63. B. Wang, J. Zhou, T. Koschny, and C. M. Soukoulis, “Nonplanar chiral metamaterials with negative index,” Appl. Phys. Lett. 94(15), 151112 (2009). [CrossRef]

64. E. Plum, J. Zhou, J. Dong, V. A. Fedotov, T. Koschny, C. M. Soukoulis, and N. I. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79(3), 035407 (2009). [CrossRef]

65. S. Zhang, Y.-S. Park, J. Li, X. Lu, W. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009). [CrossRef] [PubMed]

66. V. V. Klimov and D. V. Guzatov, “Using chiral nano-meta-particles to control chiral molecule radiation,” Phys. Uspekhi 182(10), 1130–1135 (2012).

67. https://www.comsol.com/

68. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonsky, Quantum Theory of Angular Momentum (Nauka, 1975, in Russian).

69. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Tuning spontaneous radiation of chiral molecules by asymmetric chiral nanoparticles

Abstract

1. Introduction

2. Electromagnetic field of a chiral molecule in the presence of an asymmetric chiral particle

3. Radiative decay rate of the spontaneous emission of a chiral molecule near an asymmetric chiral particle

4. Numerical results and discussion

5. Conclusions

Appendix Mathematical additions

Funding

References and links

Cited By

Figures (8)

Equations (18)

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