Second harmonic generation in three-dimensional structures based on homogeneous centrosymmetric metallic spheres

Jinying Xu; Xiangdong Zhang

doi:10.1364/OE.20.001668

1. Introduction

During the past two decades,nonlinear optical responses of small particles have been subject of intensive studies due to the development of nanofabrication techniques [1–6]. In particular, second harmonic generation (SHG) has provided a powerful tool for probing physical and chemical properties of the surface and interface of these materials [7–12]. Thus, many experimental and theoretical studies have been done for the SHG of the nanostructures [1–12].

Recently, some novel metamaterials have been fabricated successfully [13–16]. In these materials, the localized electromagnetic excitations induced by the plasmonic oscillations of the conduction electrons inside the metal can play important role in the process of nonlinear optical responses. Therefore, a renaissance of scientific interests appears in the study on the SHG of these materials. The SHGs from different geometric configurations such as sharp metal tips [17, 18], imperfect spheres [19, 20], split-ring resonators [21, 22], metallodielectric multilayer photonic-band-gap structures [23], T-shaped and L-shaped nanoparticles [9, 24–26], and “fishnet” structures [27] have been observed experimentally. Corresponding to the experimental investigations, some theoretical methods with various kinds of approaches for the SHG have been developed [28–37]. Very recently, a numerical method for two-dimensional metamaterials consisting of centrosymmetric cylinders, which can rigorously describe the general case, has been provided by using the multiple scattering matrix algorithm [38, 39].

In this work, we present a theory of the SHG in three-dimensional structures consisting of arbitrary distributions of metallic spheres made of centrosymmetric materials by means of the multiple scattering of electromagnetic multipole fields. The electromagnetic field at both the fundamental frequency (FF) and second harmonic (SH), as well as the scattering cross section, are calculated in a series of particular cases such as a single metallic sphere, two metallic spheres, chains of metallic spheres, and other ordered distributed arrays of spheres.

2. Theory and method

We consider an ensemble of N spheres (the $n t h$ sphere is marked by $n$ , $n = 1, 2...... N$ ), embedded in a background medium with electric permittivity $ε_{b}$ and magnetic permeability $μ_{b}$ . The $n t h$ sphere has radius a, relative permittivity $ε_{s} (ω)$ , relative permeability $μ_{s} (ω)$ and second-order nonlinear susceptibility ${\vec{P}}^{(2 ω)}$ . The geometry of the problem is shown in Fig. 1 . When the ensemble of spheres is irradiated by a plane wave, the scattering field at the FF and the SH can be obtained by means of the multiple scattering method.

Fig. 1 Geometry of the scattering problem for an ensemble of N spheres consisting of centrosymmetric materials. Here $θ_{n}$ , $ϕ_{n}$ and $r_{n}$ are the coordinate of the $n t h$ sphere in the spherical coordinate system, $Ω = (θ, ϕ)$ represents the solid angle of an arbitrary point P. The x, y and z represent three axis directions in corresponding Cartesian coordinate system.

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2.1 Calculation of the fields at the fundamental frequency

The multiple scattering method to calculate the scattering field at the FF has been discussed in many literatures [40–47], which is summarized in the following. For an arbitrary initial incident time-harmonic plane wave ( ${\vec{E}}^{0} e^{- i ω t}$ ) with complex wave vector ${\vec{K}}_{i}$ and polarization $\vec{\in}$ , the electric field component ${\vec{E}}^{0} = \vec{\in} e^{i {\vec{K}}_{i} \cdot \vec{r}}$ can be expanded in the vector spherical waves around the $n t h$ sphere as follows [40]:

{\vec{E}}_{n}^{0} ({\vec{x}}_{n}) = \sum_{l m} {a_{n l m}^{0, E} {\vec{J}}_{E l m} ({\vec{x}}_{n}) + a_{n l m}^{0, H} {\vec{J}}_{H l m} ({\vec{x}}_{n})}

with

{\vec{J}}_{E l m} (\vec{r}) = \frac{i}{k_{b}} \vec{\nabla} \times j_{l} (k_{b} r) {\vec{X}}_{l m} (\hat{r}), {\vec{J}}_{H l m} (\vec{r}) = j_{l} (k_{b} r) {\vec{X}}_{l m} (\hat{r}),

and

[\begin{array}{l} a_{n l m}^{0, H} \\ a_{n l m}^{0, E} \end{array}] = 4 π e^{i {\vec{K}}_{i} \cdot {\vec{r}}_{n}} i^{l} [\begin{matrix} {\vec{X}}_{l m}^{*} (Ω_{i}) \cdot \vec{\in} \\ {\vec{X}}_{l m}^{*} (Ω_{i}) \cdot (\vec{\in} \times {\vec{Κ}}_{ι}) / k_{b} \end{matrix}],

where

Ω_{i}

is the polar angle of

{\vec{K}}_{i}

and

{\vec{x}}_{n} = \vec{r} - {\vec{r}}_{n}

,

{\vec{r}}_{n}

is the center coordinate of the

n t h

sphere with radius a,

k_{b}

is the wave number of the background,

j_{l}

represents the first-kind spherical Bessel function,

{\vec{X}}_{l m}

are vector spherical harmonics [48]. In fact, the total incident field of the

n t h

sphere consists of two distinct parts, firstly the initial incident wave, and secondly the sum of all the scattered waves from other spheres (

n \neq n^{'}

). The total incident field that strikes the surface of the

n t h

sphere

{\vec{E}}_{n}^{i n c} ({\vec{x}}_{n})

, the scattered field

{\vec{E}}_{n}^{s c} ({\vec{x}}_{n})

, and the internal field

{\vec{E}}_{n}^{i n} ({\vec{x}}_{n})

can also be expanded as series of the vector spherical harmonics in the coordinate frame center at the

n t h

sphere:

{\vec{E}}_{n}^{i n c} ({\vec{x}}_{n}) = \sum_{l m} {a_{n l m}^{i n c, E} {\vec{J}}_{E l m} ({\vec{x}}_{n}) + a_{n l m}^{i n c, H} {\vec{J}}_{H l m} ({\vec{x}}_{n})},

{\vec{E}}_{n}^{s c} ({\vec{x}}_{n}) = \sum_{l m} {a_{n l m}^{s c, E} {\vec{H}}_{E l m} ({\vec{x}}_{n}) + a_{n l m}^{s c, H} {\vec{H}}_{H l m} ({\vec{x}}_{n})},

{\vec{E}}_{n}^{i n} ({\vec{x}}_{n}) = \sum_{l m} {a_{n l m}^{i n, E} {\vec{J}}_{_{E l m}}^{s} ({\vec{x}}_{n}) + a_{n l m}^{i n c, H} {\vec{J}}_{_{H l m}}^{s} ({\vec{x}}_{n})}

with

{\vec{J}}_{_{E l m}}^{s} (\vec{r}) = \frac{i}{k_{s}} \vec{\nabla} \times j_{l} (k_{s} r) {\vec{X}}_{l m} (\hat{r}), {\vec{J}}_{_{H l m}}^{s} (\vec{r}) = j_{l} (k_{s} r) {\vec{X}}_{l m} (\hat{r}),

{\vec{H}}_{E l m} (\vec{r}) = \frac{i}{k_{b}} \vec{\nabla} \times h_{l} (k_{b} r) {\vec{X}}_{l m} (\hat{r}), {\vec{H}}_{H l m} (\vec{r}) = h_{l} (k_{b} r) {\vec{X}}_{l m} (\hat{r}),

where

k_{s}

is the wave number inside the sphere,

h_{l}

is the first-kind spherical Hankel function. The expansion coefficients are determined by the following boundary conditions:

{\vec{E}}_{t}^{i n c} + {\vec{E}}_{t}^{s c} = {\vec{E}}_{t}^{i n},

{\vec{D}}_{r}^{i n c} + {\vec{D}}_{r}^{s c} = {\vec{D}}_{r}^{i n},

{\vec{H}}_{t}^{i n c} + {\vec{H}}_{t}^{s c} = {\vec{H}}_{t}^{i n},

{\vec{B}}_{r}^{i n c} + {\vec{B}}_{r}^{s c} = {\vec{B}}_{r}^{i n} .

Here the subscripts $r$ and $t$ refer, respectively, to components perpendicular and parallel to the surface of the sphere. The fields ${\vec{D}}^{i n c}$ , ${\vec{D}}^{s c}$ and ${\vec{D}}^{i n}$ are related to the ${\vec{E}}^{i n c}$ , ${\vec{E}}^{s c}$ and ${\vec{E}}^{i n}$ by the usual linear response. The corresponding magnetic fields ${\vec{H}}^{i n c}$ , ${\vec{H}}^{s c}$ and ${\vec{H}}^{i n}$ are related to the ${\vec{B}}^{i n c}$ , ${\vec{B}}^{s c}$ and ${\vec{B}}^{i n}$ , respectively. Solution of the boundary value problem allows one to express the relations between the expansion coefficients as

T_{l}^{E} = a_{n l m}^{s c, E} / a_{n l m}^{i n c, E}, T_{l}^{H} = a_{n l m}^{s c, H} / a_{n l m}^{i n c, H}, C_{l}^{E} = a_{n l m}^{i n, E} / a_{n l m}^{i n c, E}, C_{l}^{H} = a_{n l m}^{i n, H} / a_{n l m}^{i n c, H},

where

T_{l}^{E}

,

T_{l}^{H}

,

C_{l}^{E}

and

C_{l}^{H}

represent the Mie coefficients for a single sphere, which their explicit forms can be found elsewhere [28, 41]. Thus, with the use of the addition theorem [49], the expansion coefficients

a_{n l m}^{i n c, E}

and

a_{n l m}^{i n c, H}

can be expressed as [50]

a_{n l m}^{i n c, E} = a_{n l m}^{0, E} + \sum_{n^{'} \neq n} \sum_{l^{'} m^{'}} (Ω_{n l m, n^{'} l^{'} m^{'}}^{E E} a_{n^{'} l^{'} m^{'}}^{s c, E} + Ω_{n l m, n^{'} l^{'} m^{'}}^{E H} a_{n^{'} l^{'} m^{'}}^{s c, H}),

a_{n l m}^{i n c, H} = a_{n l m}^{0, H} + \sum_{n^{'} \neq n} \sum_{l^{'} m^{'}} (Ω_{n l m, n^{'} l^{'} m^{'}}^{H E} a_{n^{'} l^{'} m^{'}}^{s c, E} + Ω_{n l m, n^{'} l^{'} m^{'}}^{H H} a_{n^{'} l^{'} m^{'}}^{s c, H}),

where

Ω_{n l m, n^{'} l^{'} m^{'}}^{P P^{'}}

(

P, P^{'} = E, H

) represent the free-space propagator functions, which their explicit forms can be found in [50]. Inserting Eqs. (11) and (12) into Eq. (10), we obtain

\sum_{n^{'} l^{'} m^{'}} [δ_{n n^{'}} δ_{l l^{'}} δ_{m m^{'}} - T_{l}^{E} (Ω_{n l m, n^{'} l^{'} m^{'}}^{E E} a_{n^{'} l^{'} m^{'}}^{s c, E} + Ω_{n l m, n^{'} l^{'} m^{'}}^{E H} a_{n^{'} l^{'} m^{'}}^{s c, H})] = T_{l}^{E} a_{n l m}^{0, E},

\sum_{n^{'} l^{'} m^{'}} [δ_{n n^{'}} δ_{l l^{'}} δ_{m m^{'}} - T_{l}^{H} (Ω_{n l m, n^{'} l^{'} m^{'}}^{H E} a_{n^{'} l^{'} m^{'}}^{s c, E} + Ω_{n l m, n^{'} l^{'} m^{'}}^{H H} a_{n^{'} l^{'} m^{'}}^{s c, H})] = T_{l}^{H} a_{n l m}^{0, H} .

The above introduction about the method for the FF only focuses on the case that all spheres have the same radius. In fact, it is convenient to discuss the scattering problem with different radii of spheres by using such a method. The detailed descriptions about solving the linear system of the above equations have been given in [40–47]. After solving Eqs. (13) and (14), the expansion coefficients $a_{n l m}^{s c, E}$ and $a_{n l m}^{s c, H}$ are obtained. Then, the fields inside the spheres can be determined by using Eqs. (10) and (6).

2.2 Calculation of the fields at the second harmonic

After the FF field in the cluster with N spheres is obtained, we calculate the electromagnetic field at the SH. We first determine the source of the field at the SH, namely, the nonlinear polarization induced by the field at the FF. The nonlinear polarization vector ( ${\vec{P}}^{(2 ω)}$ ) includes surface ( ${\vec{P}}_{s u r f a c e}^{(2 ω)} (\vec{r})$ ) and bulk ( ${\vec{P}}_{b}^{(2 ω)} (\vec{r})$ ) nonlinear polarization components. For some metal systems, the SHG is dominated by the surface component and the bulk nonlinear source polarization can be neglected [51]. Here we treat the nonlinearity of the interface as a sheet of current or, equivalently, as a sheet of nonlinear source polarization according to Refs [2,3,52],

{\vec{P}}_{s u r f a c e}^{(2 ω)} (\vec{r}) = {\vec{P}}_{s}^{(2 ω)} (θ, φ) δ (r - a) = {\overset{\leftrightarrow}{χ}}_{s}^{(2)} : {\vec{E}}^{(ω)} (\vec{r}) {\vec{E}}^{(ω)} (\vec{r}) δ (r - a),

where

{\overset{\leftrightarrow}{χ}}_{s}^{(2)}

is the surface second-order susceptibility tensors. The nonlinear polarization vector (

{\vec{P}}_{s}^{(2 ω)} (θ, φ)

) can be expanded in terms of the spherical harmonics

Y_{l m} (θ, φ)

and vector spherical harmonics,

{\vec{P}}_{s}^{(2 ω)} (θ, φ) = \sum_{l m} G_{r, n l m} Y_{l m} (θ, φ) \hat{r} + G_{M, n l m} {\vec{X}}_{l m} (θ, φ) + G_{E, n l m} \hat{r} \times {\vec{X}}_{l m} (θ, φ),

where

G_{r, n l m}

,

G_{M, n l m}

and

G_{E, n l m}

are the expansion coefficients. Similar to the case of the FF, the SH source field inside the

n t h

sphere can be expanded as

{\vec{E}}_{n}^{int, (2 ω)} ({\vec{x}}_{n}) = \sum_{l m} {A_{n l m}^{int, E} {\vec{J}}_{E l m}^{s, (2 ω)} ({\vec{x}}_{n}) + A_{n l m}^{int, H} {\vec{J}}_{H l m}^{s, (2 ω)} ({\vec{x}}_{n})},

where

{\vec{J}}_{E l m}^{s, (2 ω)} ({\vec{x}}_{n})

and

{\vec{J}}_{H l m}^{s, (2 ω)} ({\vec{x}}_{n})

are given by Eq. (7) by replacing

k_{s}

with

K_{s} = \frac{2 ω}{c} \sqrt{ε_{s} (2 ω) μ_{s} (2 ω)}

. The SH source field outside the

n t h

sphere are expressed as

{\vec{E}}_{n}^{o u t, (2 ω)} ({\vec{x}}_{n}) = \sum_{l m} {A_{n l m}^{o u t, E} {\vec{H}}_{E l m}^{(2 ω)} ({\vec{x}}_{n}) + A_{n l m}^{o u t, H} {\vec{H}}_{H l m}^{(2 ω)} ({\vec{x}}_{n})},

where

{\vec{H}}_{E l m}^{(2 ω)} ({\vec{x}}_{n})

and

{\vec{H}}_{H l m}^{(2 ω)} ({\vec{x}}_{n})

are given by Eq. (8) by replacing

k_{b}

with

K_{b} = \frac{2 ω}{c} \sqrt{ε_{b} (2 ω) μ_{b} (2 ω)}

. The expansion coefficients in Eqs. (17) and (18)

A_{n l m}^{int, E}

,

A_{n l m}^{int, H}

,

A_{n l m}^{o u t, E}

and

A_{n l m}^{o u t, H}

are determined by the following four boundary conditions [2]

{\vec{E}}_{t}^{o u t, (2 ω)} - {\vec{E}}_{t}^{i n, (2 ω)} = - \frac{4 π}{ε_{1} (2 ω)} {\vec{\nabla}}_{t} P_{s, r}^{(2 ω)},

D_{r}^{o u t, (2 ω)} - D_{r}^{i n, (2 ω)} = - 4 π {\vec{\nabla}}_{t} \cdot {\vec{P}}_{s}^{(2 ω)},

{\vec{H}}_{t}^{o u t, (2 ω)} - {\vec{H}}_{t}^{i n, (2 ω)} = 4 π i \frac{2 ω}{c} \hat{r} \times {\vec{P}}_{s}^{(2 ω)},

B_{r}^{o u t, (2 ω)} - B_{r}^{i n, (2 ω)} = 0.

Inserting Eq. (16) into Eqs. (19)a)-(19d), we obtain,

A_{n l m}^{int, E} = \frac{\frac{ε_{b} (2 ω)}{K_{b}} h_{l}^{(1)} (K_{b} a) A_{n l m}^{o u t, E} + 4 π i G_{E, n l m}}{ε_{b} (2 ω) j_{l} (K_{1} a) / K_{1}},

A_{n l m}^{int, H} = \frac{h_{l}^{(1)} (K_{b} a)}{j_{l} (K_{1} a)} A_{n l m}^{o u t, H},

where

A_{n l m}^{out, E}

and

A_{n l m}^{out, H}

are given in [52]. If we make no assumptions about the nature of the interface other than that it exhibits isotropic symmetry with a mirror plane perpendicular to the interface, the surface nonlinear susceptibility tensor

{\overset{\leftrightarrow}{χ}}_{s}^{(2)}

then has three nonvanishing and independent elements:

χ_{⊥ ⊥ ⊥}

,

χ_{⊥ ∥ ∥}

,

χ_{∥ ⊥ ∥} = χ_{∥ ∥ ⊥}

,where

⊥

and

∥

refer to the local spatial components perpendicular and parallel to the surface. The susceptibility

{\overset{\leftrightarrow}{χ}}_{s}^{(2)}

can be written, in terms of the unit vectors for the spherical coordinate system, as the triadic [5]

{\overset{\leftrightarrow}{χ}}_{s}^{(2)} = χ_{⊥ ⊥ ⊥} \hat{r} \hat{r} \hat{r} + χ_{⊥ ∥ ∥} \hat{r} (\hat{θ} \hat{θ} + \hat{φ} \hat{φ}) + χ_{∥ ⊥ ∥} (\hat{θ} \hat{r} \hat{θ} + \hat{φ} \hat{r} \hat{φ} + \hat{θ} \hat{θ} \hat{r} + \hat{φ} \hat{φ} \hat{r}) .

Hence, the nonlinear source polarization for the surface is

{\vec{P}}_{s}^{(2 ω)} = \hat{r} (χ_{⊥ ⊥ ⊥} E_{r}^{(ω)} E_{r}^{(ω)} + χ_{⊥ ∥ ∥} {\vec{E}}_{t}^{(ω)} \cdot {\vec{E}}_{t}^{(ω)}) + 2 χ_{∥ ⊥ ∥} E_{r}^{(ω)} {\vec{E}}_{t}^{(ω)} .

Inserting Eq. (23) into Eq. (16), the expansion coefficients $G_{r, n l m}$ , $G_{M, n l m}$ and $G_{E, n l m}$ can be obtained, which are given in [52].

Similar to the linear scattering problem, the total incident SH field of the $n t h$ sphere can be viewed as consisting of two distinct components: a source field from other spheres without being scattered by any of the spheres, and the sum of all the SH scattered waves from other spheres. As for the SH scattered field ( ${\vec{E}}_{n}^{s c a, (2 ω)} ({\vec{x}}_{n})$ ) of the $n t h$ sphere, it can be expanded as

{\vec{E}}_{n}^{s c a, (2 ω)} ({\vec{x}}_{n}) = \sum_{l m} {A_{n l m}^{s c a, E} {\vec{H}}_{E l m}^{(2 ω)} ({\vec{x}}_{n}) + A_{n l m}^{s c a, H} {\vec{H}}_{H l m}^{(2 ω)} ({\vec{x}}_{n})},

where

A_{n l m}^{s c a, E}

and

A_{n l m}^{s c a, H}

are the expansion coefficients. The total incident SH field(

{\vec{E}}_{n}^{l o c, (2 ω)} ({\vec{x}}_{n})

) of the

n t h

sphere is written as

{\vec{E}}_{n}^{l o c, (2 ω)} ({\vec{x}}_{n}) = \sum_{n^{'} \neq n} [{\vec{E}}_{n}^{o u t, (2 ω)} ({\vec{x}}_{n^{'}}) + {\vec{E}}_{n}^{s c a, (2 ω)} ({\vec{x}}_{n^{'}})],

with

{\vec{E}}_{n}^{s c a, (2 ω)} ({\vec{x}}_{n^{'}}) = \sum_{l m} {A_{n^{'} l m}^{s c a, E} {\vec{H}}_{E l m}^{(2 ω)} ({\vec{x}}_{n^{'}}) + A_{n^{'} l m}^{s c a, H} {\vec{H}}_{H l m}^{(2 ω)} ({\vec{x}}_{n^{'}})},

Here ${\vec{E}}_{n}^{o u t, (2 ω)} ({\vec{x}}_{n^{'}})$ and ${\vec{E}}_{n}^{s c a, (2 ω)} ({\vec{x}}_{n^{'}})$ correspond to the source field and the SH scattered field from the $n^{'} t h$ sphere, respectively. Inserting Eqs. (18) and (26) into Eq. (25), the total incident SH field of the $n t h$ sphere is rewritten as

{\vec{E}}_{n}^{l o c, (2 ω)} ({\vec{x}}_{n}) = \sum_{n^{'} \neq n} \sum_{l m} {(A_{n l m}^{o u t, E} + A_{n^{'} l m}^{s c a, E}) {\vec{H}}_{E l m}^{(2 ω)} ({\vec{x}}_{n^{'}}) + (A_{n l m}^{o u t, H} + A_{n^{'} l m}^{s c a, H}) {\vec{H}}_{H l m}^{(2 ω)} ({\vec{x}}_{n^{'}})} .

By inserting in Eq. (26) the Graf formula [49], ${\vec{E}}_{l o c}^{(2 ω)} ({\vec{x}}_{n})$ can be expressed as

{\vec{E}}_{n}^{l o c, (2 ω)} ({\vec{x}}_{n}) = \sum_{n l m} {A_{n l m}^{l o c, E} {\vec{J}}_{E l m}^{(2 ω)} ({\vec{x}}_{n}) + A_{n l m}^{l o c, H} {\vec{J}}_{H l m}^{(2 ω)} ({\vec{x}}_{n})},

where

A_{n l m}^{l o c, E} = \sum_{n^{'} \neq n} \sum_{l^{'} m^{'}} {Ω_{n l m, n^{'} l^{'} m^{'}}^{E E} (A_{n^{'} l^{'} m^{'}}^{o u t, E} + A_{n^{'} l^{'} m^{'}}^{s c a, E}) + Ω_{n l m, n^{'} l^{'} m^{'}}^{E H} (A_{n^{'} l^{'} m^{'}}^{o u t, H} + A_{n^{'} l^{'} m^{'}}^{s c a, H})},

A_{n l m}^{l o c, H} = \sum_{n^{'} \neq n} \sum_{l^{'} m^{'}} {Ω_{n l m, n^{'} l^{'} m^{'}}^{H E} (A_{n^{'} l^{'} m^{'}}^{o u t, E} + A_{n^{'} l^{'} m^{'}}^{s c a, E}) + Ω_{n l m, n^{'} l^{'} m^{'}}^{H H} (A_{n^{'} l^{'} m^{'}}^{o u t, H} + A_{n^{'} l^{'} m^{'}}^{s c a, H})} .

At the same time, the total incident and scattered SH fields of the $n t h$ sphere still satisfy the following relations:

A_{n l m}^{s c a, E} = T_{l}^{E} A_{n l m}^{l o c, E},

A_{n l m}^{s c a, H} = T_{l}^{H} A_{n l m}^{l o c, H} .

From Eqs. (29)-(32), we obtain [52]

\sum_{n^{'} l^{'} m^{'}} [δ_{n n^{'}} δ_{l l^{'}} δ_{m m^{'}} - T_{l}^{E} (Ω_{n l m, n^{'} l^{'} m^{'}}^{E E} A_{n^{'} l^{'} m^{'}}^{s c a, E} + Ω_{n l m, n^{'} l^{'} m^{'}}^{E H} A_{n^{'} l^{'} m^{'}}^{s c a, H})] = \sum_{n^{'} l^{'} m^{'}} T_{l}^{E} [Ω_{n l m, n^{'} l^{'} m^{'}}^{E E} A_{n^{'} l^{'} m^{'}}^{o u t, E} + Ω_{n l m, n^{'} l^{'} m^{'}}^{E H} A_{n^{'} l^{'} m^{'}}^{o u t, H}],

\sum_{n^{'} l^{'} m^{'}} [δ_{n n^{'}} δ_{l l^{'}} δ_{m m^{'}} - T_{l}^{H} (Ω_{n l m, n^{'} l^{'} m^{'}}^{H E} A_{n^{'} l^{'} m^{'}}^{s c a, E} + Ω_{n l m, n^{'} l^{'} m^{'}}^{H H} A_{n^{'} l^{'} m^{'}}^{s c a, H})] = \sum_{n^{'} l^{'} m^{'}} T_{l}^{H} [Ω_{n l m, n^{'} l^{'} m^{'}}^{E E} A_{n^{'} l^{'} m^{'}}^{o u t, E} + Ω_{n l m, n^{'} l^{'} m^{'}}^{E H} A_{n^{'} l^{'} m^{'}}^{o u t, H}] .

Solving the self-consistent equations, Eqs. (33) and (34), we can obtain the SH scattered coefficients $A_{n l m}^{s c a, E}$ and $A_{n l m}^{s c a, H}$ . Then the SH field can be calculated from the following relations:

{\vec{E}}^{(2 ω)} (\vec{r}) = \sum_{n l m} (A_{n l m}^{o u t, E} + A_{n l m}^{s c a, E}) {\vec{H}}_{E l m}^{(2 ω)} ({\vec{r}}_{n}) + (A_{n l m}^{o u t, H} + A_{n l m}^{s c a, H}) {\vec{H}}_{H l m}^{(2 ω)} ({\vec{r}}_{n}) .

The above description about the method focuses on plane-wave sources. If we use focused beams or near-field dipole sources, the theory is also applicable. Similar to the case of the FF, the multiple scattering method for the SH is accurate and can reach rapid convergence in the undepleted-pump approximation. Based on such a method, it is convenient to study the scattering problem for the systems consisting of non overlapping spheres. The method allows one to determine not only the spatial distribution of the electromagnetic field but also a series of important physical quantities, such as the scattering cross section.

2.3 Calculation of scattering cross sections

The field distribution provides the essential information regarding the properties of the optical near field, the scattering cross sections characterize the process of energy transfer from the incident wave into the far-field. The total scattering cross section, $C_{s} (ϖ)$ , is defined as

C_{s} (ϖ) = \int q_{s} (ϖ; Ω) d Ω,

where

Ω

is the solid angle,

q_{s} (ϖ; Ω)

is the differential cross section, which is expressed as

q_{s} (ϖ; Ω) = \lim_{r \to \infty} r^{2} Re {[{\vec{E}}^{s} (ϖ) \times {\vec{H}}^{s} (- ϖ)] \cdot \hat{r}},

where

{\vec{E}}^{s}

and

{\vec{H}}^{s}

represent the FF (

ϖ = ω

) or the SH (

ϖ = 2 ω

) scattered field. As

r \to \infty

, they can be written as

\lim_{r \to \infty} {\vec{E}}^{s} (\vec{r}) = \vec{f} (Ω) \frac{e^{i k r}}{r},

\lim_{r \to \infty} {\vec{H}}^{s} (\vec{r}) = \hat{r} \times \vec{f} (Ω) \frac{e^{i k r}}{r} .

Here

\vec{f} (Ω) = \sum_{n} e^{- i k_{b} \hat{r} \cdot {\vec{r}}_{n}} i^{- l - 1} \sum_{l m} [{\vec{X}}_{l m} (Ω) a_{n l m}^{H} / k - \hat{r} \times {\vec{X}}_{l m} (Ω) a_{n l m}^{E} / k],

where

k = k_{b}

,

a_{n l m}^{E} = a_{n l m}^{s c, E}

and

a_{n l m}^{H} = a_{n l m}^{s c, H}

are for the FF;

k = K_{b}

,

a_{n l m}^{E} = A_{n l m}^{s c a, E}

and

a_{n l m}^{H} = A_{n l m}^{s c a, H}

for the SH. Inserting Eqs. (38) and (39) into Eq. (37), the differential cross section is expressed as

q_{s} (ϖ; Ω) = {| \vec{f} (Ω) |}^{2} .

Based on Eqs. (36), (40), (41) and the multiple scattering formula, the total scattering cross section can be obtained by performing numerical calculations.

3. Numerical results and discussion

In this section, we present numerical results for the linear and nonlinear scatterings from a set of homogeneous centrosymmetric metallic spheres.

3.1 A single metallic sphere

We first consider the case of a single metallic sphere. This is an important case because it has an analytical solution, which allows us to validate our numerical method. Figure 2 (I) and (II) show the calculated results of the scattering cross sections for the FF and the SH scattering from the single metallic sphere as a function of the FF photon energy. In the calculation, a circularly polarized incident beam ( $\vec{\in} =(1, i)/ \sqrt{2}$ ) is taken and the wave vector of the incident wave is assumed along x axis ( $Ω_{i} = (\frac{π}{2}, 0)$ ). The permittivity of the metallic sphere is described by the following Drude model

ε_{s} (ω) = 1 - \frac{ω_{p}^{2}}{ω (ω + i ν)},

where

ω_{p}

and

ν

are the plasma and damping frequency, respectively. As specific values for these parameters we choose

ω_{p} = 1.35 \times 10^{16} r a d / s

and

ν = 0.03 ω_{p}

, which correspond to the metal Ag [53]. Because the SHG is dominated by the surface components for the present systems and the contributions of the bulk nonlinear polarization can be neglected, in the calculations we take the surface second-order susceptibility

{\hat{χ}}_{⊥ ⊥ ⊥} = 2.79 \times 10^{- 18} m / V

,

{\hat{χ}}_{∥ ∥ ⊥} = {\hat{χ}}_{∥ ⊥ ∥} = 3.98 \times 10^{- 20} m / V

and

{\hat{χ}}_{⊥ ∥ ∥} = 0

[54]. Different sizes of spheres are considered. The solid line, dashed line and dotted line correspond to a = 10nm, 30nm and 50nm, respectively. It can be seen in the figures that new plasmon resonances appear in the SH spectra comparing with those in the FF spectra. The bigger the radius of the sphere is, the more resonance peaks appear. The above results are only for the case with the circularly polarized incident beam. If the linearly polarized incident beam is used, similar phenomena can be observed. The comparative study for two kinds of incident beam had been provided in the previous investigations [3, 5, 6]. Here our calculated results are in agreement with them. These results are for the case of the permittivity of the material being taken as Drude model. We have also performed calculations with different form of the permittivity such as measured values, the similar phenomena can also be observed.

Fig. 2 The spectra of the scattering cross sections for metal spheres of radii a = 10nm (dashed line), 30 (solid line), and 50 nm (dotted line). (I): the FF; (II): the SH.

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The properties of resonance peaks can be revealed, in part, by calculating the distribution of the scattering field. The panels A and B in Fig. 3 display the spatial distribution of the amplitude of the electric field (a = 10 nm) for the FF and the SH at $ℏ ω = 5.0$ eV, respectively, which corresponds to one of the maxima in Fig. 2(I) and (II). The amplitudes of the electric field around the sphere exhibit local maxima for both the FF and the SH, which is a signature of the excitation of localized surface plasmon-polariton (SPP) modes. This is a type of SPP resonances excited at the SH and the FF simultaneously. On the other hand, the SPP resonances at low frequency at $ℏ ω = 2.7$ eV in the panel D of Fig. 3, are due to the excitation of SPPs at the SH, with no such localized modes being excited at the FF (see panel C in Fig. 3). Here the panels A and B correspond to the dipolar excitation, and D to the quadrupolar excitation as has been described in [3, 5]. The distinction between the two types of SPP resonances also appears in more complex scattering geometries as well.

Fig. 3 (Color online) The spatial profile of the amplitude of the electric field, calculated at $ℏ ω = 5.0$ eV (panels A and B) and $ℏ ω = 2.7$ eV (panels C and D). The radius of the sphere is a = 10 nm. A and C correspond to the FF; B and D to the SH.

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3.2 A spherical metallic dimer

Figure 4 shows the scattering cross sections of two-sphere systems for the different separation distances at the FF ((I) and (III)) and the SH ((II) and (IV)), respectively. Here the radii of two spheres are taken as a = 10nm. Figure 4(I) and (II) correspond to the separation distance d = 22nm, while (III) and (IV) to d = 35nm. The solid lines and dashed lines represent the cases at the incident direction of the circularly polarized wave perpendicular or parallel to the longitudinal axis of the system, respectively. Similar to the case of the single sphere, the SPP resonances at the SH can be also divided into two types, those induced by the resonant excitation of SPP modes at the FF and those that are associated with the excitation of SPP modes solely at the SH.

Fig. 4 The spectra of the scattering cross sections for the metallic two-sphere system. The radii of two spheres are taken as a = 10 nm. (I) and (II) correspond to the separation distance d = 22nm, (III) and (IV) to d = 35nm. Solid line and dashed line correspond to the case at the incident direction of the wave perpendicular or parallel to the longitudinal axis of the system, respectively. (I) and (III): the FF; (II)and (IV): the SH.

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In contrast to the case of the single sphere, the spectra of the two-sphere system exhibit richer phenomena. On the one hand, the excitation spectra for the FF and the SH are related to the angle of incidence. This can be found from the comparison between the solid lines and dashed lines in the figures. For example, there is no localized SPP excitation for the parallel incidence at $ℏ ω = 4.1$ eV and d = 22nm, whereas strong excitations for both the FF and the SH can be observed for the normal incidence (A and B points in Fig. 4(I) and (II)).

On the other hand, the excitation spectra also depend on the separation distance between two spheres. When the distance between two spheres is big, the coupling between them is weak. In such a case the excitation feature for the two-sphere system is similar to the case of the single sphere. This can be seen more clearly from the panels C and D in Fig. 5 . The panels C and D in Fig. 5 show the electric field distributions of the two-sphere system at the normal incidence. The panel A for the FF and B for the SH correspond to the case at $ℏ ω = 4.1$ eV and d = 22nm, while C and D to that at $ℏ ω = 4.1$ eV and d = 35nm. Comparing them, we find that different excitation features appear when the distance between two spheres decreases and the coupling becomes stronger. The corresponding cases for the parallel incidence are shown in the panels E, F, G and H of Fig. 5. The changes of the excitation modes with the separation distance are observed again, which means that the coupling between two spheres play an important role in the excitation spectra. Due to the parallel incidence, some asymmetric properties of the field distribution around two spheres are also observed (H in Fig. 5 shows more clearly). In fact, the excitation spectra of the systems are also related to the polarization of the incident beam. The above discussions only focus on the circularly polarized incident beam. If we use other polarized incident beams such as linearly polarized beams (perpendicular or parallel to the longitudinal axis of the systems), the change in the excitation spectra will take place. However, the coupling phenomena exhibited between two spheres are similar.

Fig. 5 (Color online) The spatial profile of the amplitude of the electric field for the two-sphere system, calculated at $ℏ ω = 4.1$ eV for the normal incidence with d = 22nm (panels A and B) and d = 35nm (panels C and D); calculated at $ℏ ω = 6.54$ eV for the parallel incidence with d = 22nm (panels E and F) and d = 35nm (panels G and H). The radii of the spheres are a = 10 nm. A, C, E and G correspond to the FF; B, D, F and H to the SH. Here A, B, C, D, E, F, G and H correspond to the points marked in Fig. 3.

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3.3 A chain of metallic spheres

Now we apply our numerical method to study the SHG in chains of coupled metallic spheres. Such nanostructures can find important technological applications to subwavelength active optical waveguides and optical nanoantennae [55, 56]. This is because the coupling effect among the spheres leads to the long-range interactions along the chain. Such an interaction plays an important role in determining the global optical response of the structure.

The panels A and B in Fig. 6 depict the electric field patterns along a sphere chain with N = 12 for the FF and the SH, respectively, corresponding to the peaks at $ℏ ω = 5.3$ eV of the scattering cross sections as shown in Fig. 6(I) and (II). Here the circularly polarized incident beam is taken. From the distributions of the field, we can observe clearly the coupling characteristics of the field among the spheres. In some cases, sphere plasmon resonance modes can couple each other to form the coupled modes (propagating modes) along the chain. This can be demonstrated by comparing the field distributions inside the chain with those outside the sphere chain [38]. The panels A and B in Fig. 6 show such cases for both the FF and SH. Such a phenomenon depends on the excitation frequency. In some frequency regions, the coupling effects do not occur. For example, the panel C in Fig. 6 shows the electric field distribution of the FF along the sphere chain at $ℏ ω = 2.82$ eV. There is no coupling effect among the spheres and the field is excited solely around the single sphere. In contrast, the coupling effect for the SH at such a case is observed as shown in the panel D of Fig. 6. This is a kind of case that the chain of metallic spheres supports propagating modes only at the SH. In addition, the new excitations around 4.8eV for both the FF and the SH have also been observed owing to the coupling effect of spheres inside the chain.

Fig. 6 (Color online) The top two panels show the spectra of the scattering cross section corresponding to a chain of N = 12 metallic spheres under the parallel incidence. The radius of the sphere is a = 10 nm and the separation distance is d = 22nm. (I): the FF; (II): the SH. The spatial profile of the amplitude of the electric field, calculated at $ℏ ω = 5.3$ eV ((A) and (B)) and $ℏ ω = 2.82$ eV ((C) and (D)), is presented in the bottom panels. The panels A and C correspond to the FF, whereas the panels B and D correspond to the SH. Here A and B correspond to the dipolar excitation, and D to the quadrupolar excitation [3, 5].

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The results shown in Fig. 6 are only for the case at the incident direction of the wave parallel to the axis of the chain of spheres. For the case at the normal incidence, the phenomena exhibited are more remarkable. Figure 7 shows the corresponding calculated results for such a case. In contrast to no coupling effect (panel C in Fig. 7), the standing waves in the chain of spheres are excited for the FF (panel A in Fig. 7) and the SH (panels B and D in Fig. 7) due to the long-range interactions among the metallic spheres, which are also in agreement with the spectra of the scattering cross sections as shown in Fig. 7(I) and (II).

Fig. 7 (Color online) The same as in Fig. 6, but for the normal incidence. The spatial profile of the amplitude of the electric field, calculated at $ℏ ω = 5.4$ eV ((A) and (B)) and $ℏ ω = 2.59$ eV ((C) and (D)). Here A and B correspond to the dipolar excitation, and D to the quadrupolar excitation [3, 5].

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3.4 A cluster of metallic spheres

In the following we consider a cluster consisting of multi-layer spheres as shown in Fig. 8(I) . This is a typical 3D structure consisting of three layers, every layer includes $4 \times 4$ metallic spheres. The radii and the separation distances of spheres are taken as a = 10 nm and d = 22nm, respectively. The calculated results of the scattering cross sections for such a structure with various layers under the incidence of the circularly polarized plane wave from the left side of the sample are plotted in Fig. 8(II) and (III) for the FF and the SH, respectively. The dot-dashed line, dashed line and solid line correspond to the case with one-layer, two-layer and three-layer, respectively. Comparing them, we find that more excited modes appear with the increase of the layer number. This means that the interlayer coupling leads to new excitations. Such excitations also include two types: the resonant excitation of SPP modes at both the SH and the FF, and the excitation of SPP modes solely at the SH. The features of the excitations in such a structure exhibit 3D coupling effects because of three-direction interactions, which can be seen more clearly from the field distributions inside the sample.

Fig. 8 (I) The 3D sample and cross sections. (II) and (III) The spectra of the scattering cross sections corresponding to the cluster as shown in (I) under the incidence of the plane wave from the left side of the sample with a = 10 nm and d = 22nm. (II): the FF; (III): the SH.

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Without loss of generality, we take two cross-sections inside the sample as shown in Fig. 8(I) to analyze the distributions of the FF and SH fields. The first cross-section goes through the middle “vacant space” between two layers as illustrated by the shaded plane and the second goes across the centers of the spheres inside one layer as shown by dot-dashed lines (white color plane). The FF and SH fields at the first cross-section with $ℏ ω = 2.21$ eV (corresponding to the points A and B in Fig. 8(II) and (III)) are plotted in Fig. 9(a) and (b) , respectively, while the corresponding fields at the second cross-section are described in Fig. 9(c) and (d). The calculated results for the other systems have shown that the excitations of SPPs at the SH can be observed when the localized modes at the FF are not excited. The motivation to choose the energy $ℏ ω = 2.21$ eV is to test whether or not the phenomenon exists in such a complex structure. Due to the transverse and the longitudinal coupling interactions, the twist patterns for the SH (see Fig. 9(b) and (d)) are observed clearly. In contrast, there is no such a phenomenon for the FF (see Fig. 9(a) and (c)). This means that the collective SH excitations along different directions in 3D cluster can be realized, even the FF excitation does not exist.

Fig. 9 The spatial profiles of the amplitude of the electric field, calculated at $ω = 2.21$ eV (corresponding to the points A and B in Fig. 7 (II) and (III)). (a) and (b) correspond to the FF and the SH fields at the cross-section as shown by shadow in Fig. 7 (I), respectively, whereas (c) and (d) correspond to the FF and the SH fields at the cross-section as shown by dot-dashed lines in Fig. 7 (I). The other parameters are identical with those in Fig. 7.

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4. Summary

To summarize, we have developed in this paper a theory of SHG, based on the multiple scattering method, for studying the linear and nonlinear scattering effects in the 3D metamaterials made of centrosymmetric spheres. Based on such a method, the electromagnetic field at both the FF and the SH, as well as the scattering cross section, are calculated in a series of particular cases, a single metallic sphere, metallic two-sphere system, a chain of metallic spheres, and a finite 3D array of spheres. Our results show that the linear and nonlinear optical responses of all ensembles of metallic spheres were strongly influenced by the excitation of SPP resonances. The physical origin for such a phenomenon has also been elucidated and discussed. Finally, we would like to point out that our method can apply to any 3D structures consisting of the spheres, even for random systems, although our calculations focus on several special 3D structures.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No.10825416) and the National Key Basic Research Special Foundation of China under Grant 2007CB613205.

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Second harmonic generation in three-dimensional structures based on homogeneous centrosymmetric metallic spheres

Abstract

1. Introduction

2. Theory and method

2.1 Calculation of the fields at the fundamental frequency

2.2 Calculation of the fields at the second harmonic

2.3 Calculation of scattering cross sections

3. Numerical results and discussion

3.1 A single metallic sphere

3.2 A spherical metallic dimer

3.3 A chain of metallic spheres

3.4 A cluster of metallic spheres

4. Summary

Acknowledgments

References and links

Cited By

Figures (9)

Equations (48)

Optics Express