Plane wave scattering from a plasmonic nanowire-film system with the inclusion of non-local effects

Rahul Trivedi; Yashna Sharma; Anuj Dhawan

doi:10.1364/OE.23.026064

1. Introduction

Surface Plasmons are collective coherent oscillations of the conduction band electrons at metal-dielectric interfaces of metallic thin films, gratings, or nanostructures [1–7]. When surface plasmons are excited at certain frequencies at metal-dielectric interfaces, this excitation is accompanied by an increase in the optical near-fields in the vicinity of the metallic structures as well as a decrease in the fraction of power reflected from the metallic structures such as metallic thin films [1–5].

In recent years, there has been an increased interest in studying the optical properties of structures consisting of plasmonic nanostructures such as nanoparticles or nanowires spaced at a certain distance, typically 1 nm to 10 nm, from the plasmonic thin films [8–10] by a dielectric layer. These plasmonic structures have a far–field spectra which exhibits resonances that depend strongly on the thickness of the spacer between the nanostructures and the plasmonic film. This property of such a plasmonic structure makes it favorable for many applications such as position and thickness sensing [8]. It may also be used for vibration or strain sensing or as a highly sensitive gyroscope.

Moreover, these plasmonic structures can be employed for biological and chemical sensing applications by measuring the change of thickness – of the spacer layer between the plasmonic nanostructures and the underlying plasmonic film – upon attachment of an analyte molecule to a binding site on the spacer layer. These analyte molecules could be chemical molecules -such as alkanethiols containing different carbon atoms, and therefore different chain lengths as was experimentally demonstrated by Ciraci et. al. [10] - or biological molecules such as nucleic acids (DNA and RNA molecules) or proteins. At the plasmon resonance, the fields in the spacer material typically become very large, which could enable the use of such nanostructures for developing sensors based on Surface-Enhanced Raman scattering [11–17] and plasmonics-enhanced fluorescence [10]. In order to employ such a plasmonic structure for different sensing applications, it is important to obtain an in-depth understanding of the behavior of the optical properties of the plasmonic structure, which could be achieved by an accurate theoretical analysis of the electromagnetic response of this structure.

The analytical solution presented in this paper is a full-wave solution which can be used to compute the fields scattered by the nanostructure on illumination by a plane electromagnetic wave. Several authors have presented analytical solutions of the electromagnetic fields around individual metallic nanoparticles or nanowires and their assemblies as well as from assemblies of dielectric cylinders [18–26], but very few authors have presented solutions which take into account the presence of the plane substrate underneath the nanostructures [27, 28]. However, there is no previous report of an analytical solution of this problem accounting for both the dispersive and non-local nature of the plasmonic medium. The analytical solution provided in our paper accounts for the non-local nature of the plasmonic media. In order to demonstrate the accuracy of our analytical method, we present an exhaustive comparison of our results for the case of local plasmonic media with a FDTD analysis of the same plasmonic structure.

Numerical analysis of a plasmonic nanostructure-spacer-film system has been done before using methods such as FDTD, Green’s function method and Finite Element Method [11]. Recently, there have been some reports of inclusion of the non local effects into the conventional numerical methods such as FEM [29,30]. Although numerical methods enable us to model and analyze complicated geometries, the analytical solution has its advantages in accuracy, better convergence, lower computational time and simple inclusion of non-local effects.

The nanowire-film plasmonic system studied in this paper allows the development of very small gaps between the two plasmonic elements - the plasmonic nanowires array and the underlying plasmonic film - in the vertical direction, these gaps being even less than 1 nm [8, 11]. Plasmonic nanowires and their arrays, or other 1-D or 2-D plasmonic nanostructures could be developed on top of a thin dielectric spacer layer - which is deposited on top of a plasmonic thin film - to achieve the plasmonic nanostructures studied in this paper using focused ion beam milling [31], electron beam lithography [32], deep UV lithography or nanoimprint lithography. While the development of sub-10 nm lateral spacings between the plasmonic nanostructures has been recently demonstrated [33–35], the development of sub-1 nm spacings in the lateral direction has not been achieved thus far. On the other hand, sub-5 nm spacers can be fabricated in the vertical direction by developing self-assembled monolayers (SAMs) of amineterminated alkanethiols [10], layer-by-layer (LBL) deposition of polyelectrolytes [8, 9], atomic layer deposition [11], or chemical vapor deposition.

2. Analytical method

2.1. Electromagnetic fields inside a non-Local medium

The non-locality of the medium is captured using the hydrodynamic model [38]. A description of the hydrodynamic model is given as follows: By the Helmholtz theorem [39–41], the electric field (E) inside a source free non-local medium can be expressed as a sum of a divergence-less field (referred to as the transverse field E_T) and a curl-less field (referred to as the longitudinal field E_L) (All fields are assumed to have a time dependance of exp(−iωt)):

E (r, ω) = E_{T} (r, ω) + E_{L} (r, ω) where \nabla \cdot E_{T} (r, ω) = 0 and \nabla \times E_{L} (r, ω) = 0

The non-local medium responds to these two components differently. The transverse fields see a k independent permittivity function while the longitudinal fields see a k dependant permittivity function. The electric displacement vector D can be calculated as follows (here $\tilde{v} (k)$ denotes the three dimensional Fourier transform of v(r)):

\tilde{D} (k, ω) = {\tilde{D}}_{T} (k, ω) + {\tilde{D}}_{L} (k, ω)

{\tilde{D}}_{T} (k, ω) = ε_{0} ε_{T} (ω) {\tilde{E}}_{T} (k, ω)

{\tilde{D}}_{L} (k, ω) = ε_{0} ε_{L} (k, ω) {\tilde{E}}_{L} (k, ω)

where

ε_{T} (ω) = ε_{b} (ω) - \frac{ω_{P}^{2}}{ω^{2} + i γ ω}

ε_{L} (k, ω) = ε_{b} (ω) - \frac{ω_{P}^{2}}{ω^{2} + i γ ω - β^{2} | k |^{2}}

Here ε_b(ω) is the permittivity due to the bound-electrons in the metal and its frequency dependence can be captured using the following model [42–44]:

ε_{b} (ω) = ε_{\infty} - \sum_{n = 1}^{\infty} \frac{Δ ε_{n}}{ω^{2} + i γ_{n} ω - ω_{n}^{2}}

γ_n and ω_n being the damping constant and the natural frequency of the n^th oscillator mode used to describe the bound electrons. γ in Eq. (4) is the damping constant as seen by the conduction electrons and ω_P is the plasma frequency of the metal. Inclusion of frequency dependance of the response of bound electrons through the term ε_b(ω) allows us to approximately capture certain phenomena such as d electron shielding in nobel metals like Au and Ag, which is otherwise ignored in the conventional Drude Model [36, 37]. β is a constant used to model the non-local effects and is of the order of 0.77v_F, v_F being the fermi velocity of the electrons [38]. Using the Maxwell’s equations ∇ × E(r,ω) = iμ₀ωH(r,ω) and ∇ × H(r,ω) = −iωD(r,ω) to eliminate H(r,ω) and obtain:

\nabla \times (\nabla \times E (r, ω)) = μ_{0} ω^{2} D (r, ω)

In the k domain, Eq. (6) translates to:

k \times (k \times \tilde{E} (k, ω)) = - μ_{0} ω^{2} \tilde{D} (k, ω)

or

k \times (k \times ({\tilde{E}}_{T} (k, ω) + {\tilde{E}}_{L} (k, ω))) = - μ_{0} ω^{2} ({\tilde{D}}_{T} (k, ω) + {\tilde{D}}_{L} (k, ω))

Noting that from Eq. (1) k · Ẽ_T = 0 and k × Ẽ_L = 0 and using Eq. (2)

- | k |^{2} {\tilde{E}}_{T} (k, ω) = - μ_{0} ε_{0} ω^{2} (ε_{T} (ω) {\tilde{E}}_{T} (k, ω) + ε_{L} (k, ω) {\tilde{E}}_{L} (k, ω))

which is equivalent to

- | k |^{2} {\tilde{E}}_{T} (k, ω) = - μ_{0} ω^{2} ε_{0} ε_{T} (ω) {\tilde{E}}_{T} (k, ω) or | k |^{2} = μ_{0} ε_{0} ε_{T} (ω) ω^{2}

0 = - μ_{0} ε_{0} ε_{L} (k, ω) ω^{2} {\tilde{E}}_{L} (k, ω) or ε_{L} (k, ω) = 0

Physically Eq. (10) is a statement of the fact that the longitudinal component of the electric field is composed of plane waves with wave number k_L and the transverse component of the electric field is composed of the plane waves with wave number k_T where:

k_{T} = \frac{ω}{c} \sqrt{ε_{T} (ω)}

k_{L} = \frac{ω_{P}}{β} \sqrt{1 + \frac{i ω γ}{ω_{P}^{2}} - \frac{1}{ε_{b} (ω)}}

Further, since E_L(r,ω) is a curl less field, it can be expressed as the curl of a scalar function ψ_L(r,ω) [39–41]:

E_{L} (r, ω) = \nabla ψ_{L} (r, ω)

Moreover, since E_L(r,ω) is a superposition of plane electromagnetic waves with wavenumber k_L, it satisfies the helmholtz’s equation $\nabla^{2} E_{L} (r, ω) + k_{L}^{2} E_{L} (r, ω) = 0$ . In terms of ψ_L(r,ω), this condition translates to:

\nabla^{2} ψ_{L} (r, ω) + k_{L}^{2} ψ_{L} (r, ω) = 0

Also note that since ε_L(k,ω) = 0 for the plane wave components of the longitudinal electric field, ${\tilde{D}}_{L} (k, ω) = 0$ and hence D(r,ω) = D_T (r,ω) = ε_T (ω)E_T (r,ω).

Consider now a 2D electric field polarised in the x − y plane with its magnetic field given by

H (r, ω) = H_{z} (r, ω) \hat{z}

The transverse component of the electric field can be expressed in terms of the magnetic field using the Ampere’s Law:

E_{T} (r, ω) = \frac{i}{ε_{0} ε_{T} (ω) ω} \nabla \times (H_{z} (r, ω) \hat{z})

Since ∇ × E(r,ω) = iμ₀ωH(r,ω) and ∇ × E_L(r,ω) = 0,

\nabla \times E_{T} (r, ω) = i μ_{0} ω H_{z} (r, ω) \hat{z}

Together with Eq. (15), Eq. (16) gives:

\nabla^{2} H_{z} (r, ω) + k_{T}^{2} H_{z} (r, ω) = 0

Also, using Eqs. (15) and (12), the net electric field can be expressed as:

E (r, ω) = \frac{i}{ε_{0} ε_{T} (ω) ω} \nabla \times (H_{z} (r, ω) \hat{z}) + \nabla ψ_{L} (r, ω)

As compared to the local case, a non-local problem has an additional function ψ_L(r,ω) to be determined. This calls for the introduction of an additional boundary condition. The usually employed boundary condition is derived from the fact that the normal component of the conduction current in the non-local medium is zero at the medium boundary (since the free electrons cannot escape the medium) [38, 45]. To derive a condition on the electric field, consider the decomposition of the polarisation vector P(r,ω) as shown:

P (r, ω) = P_{f} (r, ω) + P_{b} (r, ω)

P_f (r,ω) is the polarisation due to the conduction electron and is related to the conduction electron density J_f (r,ω) through:

J_{f} (r, ω) = - i ω P_{f} (r, ω)

P_b(r,ω) is the polarisation due to the bound electrons, which in the hydrodynamic model are assumed to behave locally. Therefore:

P_{b} (r, ω) = ε_{0} (ε_{b} (ω) - 1) E (r, ω)

Since D(r,ω) = ε₀E(r,ω) + P(r,ω), using eqs. (19), (20) and (21):

J_{f} (r, ω) = - i ω (D (r, ω) - ε_{0} ε_{b} (ω) E (r, ω))

Consider now an interface between a metal and a vacuum. At the surface of the two medium inside the metal, $J_{f} (r, ω) \cdot \hat{n} |_{m e t a l} = 0$ . Thus

D (r, ω) \cdot \hat{n} |_{m e t a l} - ε_{0} ε_{b} (ω) E (r, ω) \cdot \hat{n} |_{m e t a l} = 0

But since $\nabla \cdot D (r, ω) = 0, D (r, ω) \cdot \hat{n}$ is continuous across the boundary. Therefore

D (r, ω) \cdot \hat{n} |_{v a c u u m} = ε_{0} E (r, ω) \cdot \hat{n} |_{v a c u u m} = D (r, ω) \cdot \hat{n} |_{m e t a l}

and thus

E (r, ω) \cdot \hat{n} |_{v a c u u m} = ε_{b} (ω) E (r, ω) \cdot \hat{n} |_{m e t a l}

Equation (25) is the required boundary condition.

2.2. Analysis of the coupled nanowire-film system

A schematic of the coupled nanowire and film system is shown in Fig. 1(b). The gold film-vacuum interface at x = d is modelled by its plane wave reflection coefficient Γ(k_y) (derived in section 2.3). Consider a plane wave excitation of this structure, where the incident plane wave is described by:

H_{z}^{(i)} = H_{0} \exp (i k_{0} (x \cos ϕ_{0} + y \sin ϕ_{0}))

Fig. 1 (a) A metallic nanowire coupled to a metallic thin film, thereby confining the electromagnetic fields in between the metallic structures (b) Schematic of the coupled nanowire and film system

Download Full Size | PDF

This plane wave will be directly reflected off the x = d plane surface to produce a reflected field. The net excitation field $H_{z}^{(e)}$ , defined as the field that would have been produced in the absence of the nanowire, is the sum if the incident and the directly reflected field and is given by:

\begin{array}{l} H_{z}^{(e)} = H_{0} \exp (i k_{0} (x \cos ϕ_{0} + y \sin ϕ_{0})) + \\ H_{0} Γ (k_{0} \sin ϕ_{0}) \exp (2 i k_{0} d \cos ϕ_{0}) \exp (i k_{0} (- x \cos ϕ_{0} + y \sin ϕ_{0})) \end{array}

Owing to the cylindrical symmetry of the situation, it is convenient to expand this field in terms of the cylindrical wave functions using the Jacobi Anger expansion [39–41]:

H_{z}^{(e)} = H_{0} \sum_{n = - \infty}^{\infty} I_{n} (ϕ_{0}) J_{n} (k_{0} r) \exp (i n ϕ)

where

I_{n} (ϕ_{0}) = i^{n} (\exp (- i n ϕ_{0}) + {(- 1)}^{n} Γ (k_{0} \sin ϕ_{0}) \exp (2 i k_{0} d \cos ϕ_{0}) \exp (i n ϕ_{0}))

The scattered fields can be split into two components - one that is directly radiated by the dipole oscillations in the nanowire $H_{z}^{(r)}$ and is given by

H_{z}^{(r)} = H_{0} \sum_{n = - \infty}^{\infty} S_{n} H_{n}^{(1)} (k_{0} r) \exp (i n ϕ)

and one that is due to the reflection of

H_{z}^{(r)}

off the plane interface

H_{z}^{(r e)}

. The method used for calculating

H_{z}^{(r e)}

in terms of

H_{z}^{(r)}

is as follows [27, 28]: We express

H_{z}^{(r)}

as a superposition of a continuum of plane waves. The reflected wave due to each of the plane waves can be calculated using the reflection coefficient Γ(k_y) and these reflected waves can be superimposed to obtain

H_{z}^{(r e)}

. To do so, note that [46]:

H_{n}^{(1)} (k_{0} r) \exp (i n ϕ) = \int_{- \infty}^{\infty} g_{n} (k_{y}) \exp (i (\sqrt{k_{0}^{2} - k_{y}^{2}} x + k_{y} y)) d k_{y} for x > 0

where

\begin{array}{l} g_{n} (k_{y}) = \frac{1}{i π} \frac{{(\sqrt{k_{y}^{2} - k_{0}^{2}} + k_{y})}^{n}}{k_{0}^{n} \sqrt{k_{y}^{2} - k_{0}^{2}}} for | k_{y} | > k_{0} \\ = \frac{1}{π} \frac{\exp (- i n \cos^{- 1} (k_{y} / k_{0}))}{\sqrt{k_{0}^{2} - k_{y}^{2}}} for | k_{y} | < k_{0} \end{array}

Therefore, if a cylindrical wave $H_{z}^{(I)} = H_{n}^{(1)} (k_{0} r) \exp (i n ϕ)$ is incident on the surface x = d, the reflected wave $H_{z}^{(R)}$ is given by

H_{z}^{(R)} = \int_{- \infty}^{\infty} Γ (k_{y}) g_{n} (k_{y}) \exp (i (\sqrt{k_{0}^{2} - k_{y}^{2}} (2 d - x) + k_{y} y)) d k_{y}

The Jacobi-Anger expansion can be used to expand the term $\exp (i (- x \sqrt{k_{0}^{2} - k_{y}^{2}} + k_{y} y))$ in terms of the cylindrical wave functions as shown:

\exp (i (- x \sqrt{k_{0}^{2} - k_{y}^{2}} + k_{y} y)) = \sum_{m = - \infty}^{\infty} \exp (- i m \cos^{-}^{1} (k_{y} / k_{0})) J_{m} (k_{0} r) \exp (i n ϕ)

Substituting Eq. (34) into Eq. (33), we obtain

H_{z}^{(R)} = \sum_{m = - \infty}^{\infty} γ_{n + m} J_{m} (k_{0} r) \exp (i m ϕ)

where

γ_{n} = \int_{- \infty}^{\infty} Γ (k_{y}) g_{n} (k_{y}) \exp (2 i \sqrt{k_{0}^{2} - k_{y}^{2}} d) d k_{y}

Using Eq. 35, the reflected field $H_{z}^{(r e)}$ can be written as

H_{z}^{(r e)} = H_{0} \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} S_{n} γ_{m + n} J_{m} (k_{0} r) \exp (i m ϕ) = H_{0} \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} S_{m} γ_{m}_{+ n} J_{n} (k_{0} r) \exp (i n ϕ)

As is described in section I, the fields need to be described by two functions $H_{z}^{(c)}$ and $ψ_{L}^{(c)}$ which satisfy Eq. (17) and Eq. (13) respectively (The superscript refers to the fact that these functions describe the fields inside the cylindrical nanowire). Solutions of these equations which are finite at the origin are given by

H_{z}^{(c)} = H_{0} \sum_{n = - \infty}^{\infty} A_{n} J_{n} (k_{T} r) \exp (i n ϕ)

ψ_{L}^{(c)} = H_{0} \sum_{n = - \infty}^{\infty} B_{n} J_{n} (k_{L} r) \exp (i n ϕ)

To determine the unknowns S_n, A_n and B_n, we impose the boundary conditions of the continuity of H_z, E_ϕ and ε_b(ω)E_r at r = a. Doing so we obtain:

I_{n} (ϕ_{0}) J_{n} (k_{0} a) + \sum_{m = - \infty}^{\infty} γ_{m + n} S_{m} J_{n} (k_{0} a) + S_{n} H_{n}^{(1)} (k_{0} a) = A_{n} J_{n} (k_{T} a)

\begin{array}{r} \frac{i k_{0}}{ω ε_{0}} [I_{n} (ϕ_{0}) {J^{'}}_{n} (k_{0} a) + \sum_{m = - \infty}^{\infty} γ_{m + n} S_{m} {J^{'}}_{n} (k_{0} a) + S_{n} (H_{n}^{{(1)}^{'}}) (k_{0} a)] = \\ \frac{i k_{T}}{ω ε_{0} ε_{T} (ω)} A_{n} {J^{'}}_{n} (k_{T} a) - \frac{i n}{a} B_{n} J_{n} (k_{L} a) \end{array}

\begin{array}{r} \frac{n}{ω ε_{0} a} [I_{n} (ϕ_{0}) {J^{'}}_{n} (k_{0} a) + \sum_{m = - \infty}^{\infty} γ_{m + n} S_{m} {J^{'}}_{n} (k_{0} a) + S_{n} H_{n}^{{(1)}^{'}} (k_{0} a)] = \\ \frac{n ε_{b} (ω)}{ω ε_{0} ε_{T} (ω) a} A_{n} J_{n} (k_{T} a) - ε_{b} (ω) k_{L} B_{n} {J^{'}}_{n} (k_{L} a) \end{array}

Eliminating A_n and B_n from Eq. (40) we obtain the following equation:

\sum_{m = - \infty}^{\infty} γ_{m + n} S_{m} + Q_{n} S_{n} = - I_{n} (ϕ_{0})

where

Q_{n} = \frac{H_{n}^{(1)} (k_{0} a) [{J^{'}}_{n} (k_{T} a) + δ_{n} J_{n} (k_{T} a)] - \sqrt{ε_{T} (ω)} J_{n} (k_{T} a) H_{n}^{{(1)}^{'}} (k_{0} a)}{J_{n} (k_{0} a) [{J^{'}}_{n} (k_{T} a) + δ_{n} J_{n} (k_{T} a)] - \sqrt{ε_{T} (ω)} J_{n} (k_{T} a) {J^{'}}_{n} (k_{0} a)}

where

δ_{n} = \frac{n^{2} J_{n} (k_{L} a)}{k_{T} k_{L} a^{2} {J^{'}}_{n} (k_{L} a)} (\frac{ε_{T} (ω)}{ε_{b} (ω)} - 1)

2.3. Reflection of a plane wave from the surface of a non-local medium

The analysis presented in section 2.2 requires the evaluation of the reflection coefficient Γ [47] which we calculate in this section. Note that the y component of the wave vector k_y remains conserved during the reflection and transmission of the wave and therefore Γ is expressed as a function of k_y and the other material parameters of the structure (such as permittivity of metal and so on). The metal is assumed to occupy the volume described by x > d. The incident wave in vacuum is given by

H_{z}^{(i)} = H_{0} \exp (i (k_{y} y + k_{0 x} x))

where

k_{0 x} = \sqrt{k_{0}^{2} - k_{y}^{2}}

. The reflected wave is given by

H_{z}^{(r)} = H_{R} \exp (i (k_{y} y - k_{0 x} x))

The fields in the metal would again be described by two functions $H_{z}^{(c)}$ and $ψ_{L}^{(c)}$ . Since the waves in the metal would be propagating in the +x direction, the following functional forms for $H_{z}^{(c)}$ and $ψ_{L}^{(c)}$ are assumed:

H_{z}^{(c)} = H_{T} \exp (i (k_{y} y + k_{T x} x))

ψ_{L}^{(c)} = ψ_{0} \exp (i (k_{y} y + k_{L x} x))

where

k_{T x} = \sqrt{k_{T}^{2} - k_{y}^{2}}

and

k_{L x} = \sqrt{k_{L}^{2} - k_{y}^{2}}

. To evaluated the unknowns Γ, H_T and ψ₀, we impose the boundary conditions of continuity of H_z, E_y and ε_b(ω)E_x at x = d.

H_{0} \exp (i k_{0 x} d) + H_{R} \exp (- i k_{0 x} d) = H_{T} \exp (i k_{T x} d)

\frac{k_{0}_{x}}{ε_{0} ω} (H_{0} \exp (i k_{0 x} d) - H_{R} \exp (- i k_{0 x} d)) = \frac{k_{T x} H_{T}}{ε_{0} ε_{T} (ω) ω} \exp (i k_{T x} d) + i k_{y} ψ_{0} \exp (i k_{L x} d)

\frac{k_{y}}{ε_{0} ω} (H_{0} \exp (i k_{0 x} d) + H_{R} \exp (- i k_{0 x} d)) = [\frac{k_{y} H_{T}}{ε_{0} ε_{T} (ω) ω} \exp (i k_{T x} d) - i k_{L x} ψ_{0} \exp (i k_{L x} d)] ε_{b} (ω)

Using Eqs. (48) to eliminate H_T and ψ₀ we obtain:

\frac{H_{R}}{H_{0}} = [\frac{ε_{T} (ω) k_{0 x} - k_{T x} + \frac{k_{y}^{2}}{k_{L x}} (\frac{ε_{T} (ω)}{ε_{b} (ω)} - 1)}{ε_{T} (ω) k_{0 x} + k_{T x} - \frac{k_{y}^{2}}{k_{L x}} (\frac{ε_{T} (ω)}{ε_{b} (ω)} - 1)}] \exp (2 i k_{0 x} d)

The reflection coefficient Γ can thus be calculated as:

\begin{array}{l} Γ (k_{y}) = \frac{H_{z}^{(r)} (x = d, y)}{H_{z}^{(i)} (x = d, y)} = [\frac{ε_{T} (ω) k_{0 x} - k_{T x} + \frac{k_{y}^{2}}{k_{L x}} (\frac{ε_{T} (ω)}{ε_{b} (ω)} - 1)}{ε_{T} (ω) k_{0 x} + k_{T x} - \frac{k_{y}^{2}}{k_{L x}} (\frac{ε_{T} (ω)}{ε_{b} (ω)} - 1)}] \\ = \frac{ε_{T} (ω) \sqrt{k_{0}^{2} - k_{y}^{2}} - \sqrt{k_{T}^{2} - k_{y}^{2}} + \frac{k_{y}^{2}}{\sqrt{k_{L}^{2} - k_{y}^{2}}} (\frac{ε_{T} (ω)}{ε_{b} (ω)} - 1)}{ε_{T} (ω) \sqrt{k_{0}^{2} - k_{y}^{2}} + \sqrt{k_{T}^{2} - k_{y}^{2}} - \frac{k_{y}^{2}}{\sqrt{k_{L}^{2} - k_{y}^{2}}} (\frac{ε_{T} (ω)}{ε_{b} (ω)} - 1)} \end{array}

3. Results and discussion

3.1. Validation of the analytical method

While the analytical method described in section 2 is exact, in order to obtain numerical estimates of the near and far fields scattered by the plasmonic structure, it is essential to truncate Eq. (41) beyond an upper limit N_m on m as shown:

\sum_{m = - N_{m}}^{N_{m}} γ_{m + n} S_{m} + Q_{n} S_{n} = - I_{n} (ϕ_{0})

Equation (51) is a set of 2N_m +1 linear equations that can be solved numerically to yield S_n, and by extension the electromagnetic fields in space. The integral in Eq. (36) cannot, in general, be evaluated analytically but has to be estimated numerically. This can be done using the gaussian quadrature rules for numerical evaluation of integrals. Additionally, N_m should be chosen large enough so as to ensure the convergence of the electromagnetic fields in space. Figure 2(a) and 2(b) show that this is indeed possible, i.e. the electromagnetic fields in space do converge for sufficiently large values of N_m. Figure 2(a) shows the plot of |S_n| vs n for different N_m and it can be seen that |S_n| does not change significantly on increasing N_m from 10 to 15. Figure 2(b) shows the variation of the field enhancement (η) between cylinder and the gold film with N_m. Here η is defined as (Z₀ ≈ 377Ω is the impedance of free space)

η = \frac{E_{P}^{2}}{E_{0}^{2}} where E_{P} = | E (x = d / 2, y = 0) | and E_{0} = Z_{0} H_{0}

Fig. 2 (a) Plot of |S_n| vs n for N_m = 5,10 and 15. The dimensions of the structure used are - t = 2 nm, a = 40 nm, ϕ₀ = π/6 and λ = 650 nm. (b) Plot of η vs N_m for the same dimensions.

Download Full Size | PDF

It is found that the value of N_m required to achieve convergence is a strong function of the various geometrical parameters of the structure, especially the separation between the cylinder and the substrate t = d − a. It is found that N_m required to achieve convergence increases on decreasing t. For a radius a = 40 nm, the value of N_m to achieve convergence in η within an error of 1% increases from 7 to 18 on decreasing t from 8 nm to 2 nm. N_m also depends, though to a smaller extent, on a - the radius of the nanowire. On increasing a from 40 nm to 60 nm for t = 2 nm, the value of N_m required to achieve convergence in η within an error of 1% increased from 18 to 24.

To validate the analytical solution presented in section II, it is compared with a commercially available FDTD software, LUMERICAL FDTD Solution 8.7.1 for the case of local medium (β = 0 m/s) as shown in Fig. 3.

Fig. 3 (a) Plot of η vs λ for t = 2,4 and 8 nm. a = 40 nm and ϕ₀ = π/6 for all calculations. (b) Plot of η vs λ for a = 40,50 and 60 nm. t = 4 nm and ϕ₀ = π/6 for all calculations..

Download Full Size | PDF

Fig. 4 Field profile (|E(x,y)|/E₀) generated for a = 40 nm, t = 4 nm, ϕ₀ = π/6 and λ = 650 nm using (a) FDTD and (b) Analytical method

Download Full Size | PDF

A grid size of 0.5 nm was used in all the FDTD simulations and perfectly matched layers were used to extend the boundary of the simulation region to infinity. As can be seen from Figs. 3(a) and (b), there is a good agreement between the predictions of the analytical method and the FDTD results for the field enhancement (η) in between the cylinder and the gold film. Figure 4 shows the normalised field profile plot (i.e. plot of |E(x,y)|/E₀) calculated using the analytical solution as well as the FDTD method.

Since the field enhancement η was evaluated to convergence and there is no a priori approximation in the analytical method, the slight difference between the analytical and the FDTD results seems to stem from the use of Perfectly Matched Layers (PML) to extend the boundaries of the simulation region, since some of the plane wave components that should have been reflected are absorbed by the PML at its intersection with the gold film.

3.2. Impact of non-locality of the medium

Figure 5(a) shows the variation of the field enhancement η with the free space wavelength λ for different values of the non-local parameter β. It is observed that the field enhancement in the region between the nanowire and the film is higher in the case of a non-local medium as compared to a local medium. A similar increase in the electric field due to non-locality of in the far-field spectra has been previously numerically analysed and reported by Ruppin [45].

Moreover on increasing β, the resonant wavelength shifts towards the blue region, and the shift is of the order of 30 nm on changing β from 3×10⁶ m/s to 3×10⁷ m/s. This in agreement with the physical picture of surface plasmon resonances, since if the plasmonic medium is considered to be non-local, then the bound charge densities would no longer be confined to the surface, rather they would extend to a small volume beyond the surface. This would result in a larger restoring force on the conduction electrons in the metal, thereby giving a higher resonant frequency or a smaller resonant wavelength.

Figure 5(b) shows the variation of the shift in the resonant wavelength (∆λ_res) on changing β from 0 m/s to 3×10⁷ m/s with the spacer thickness t. Note that the blue shift in resonant wavelength is significant only when the dimensions of the nanowire-film gap are very small (less than ~5 nm). The effect of non-locality is of significance in the design of sub-5 nm plasmonics based sensors which utilise the shift of the resonant wavelength on the change of a geometrical or electromagnetic parameters of the plasmonic structure. For instance, due to non-local effects, the shift in the resonant wavelength of the system on changing the spacer thickness (t) will not be as pronounced as in the case of a local medium, since the blue shift due to non-locality compensates for the red shift due to change of spacer thickness.

Note that the enhancement plot for β = 0 m/s presented in Fig. 5(a), in addition to accounting for the local electromagnetic response of the free electrons, also includes the effect of d electron screening in Au via the inclusion of a frequency dependant bound electron permittivity ε_b(ω) (refer to Eq.(5)). It is well known that d electron screening results in a blue shift of the plasmon resonance as compared to the usual Drude description of metals (ε_b(ω) = 1). Our analysis includes the non-locality in conduction electrons over and above this screening effect. We therefore predict a further blue shift due to the non-locality of the conduction electrons (i.e. for β > 0 m/s, see Fig. 5(b)). We do assume the d electrons to behave locally (i.e. ignore a k dependance in ε_b(ω)), although this assumption is not very inaccurate for the dimensions of interest.

Fig. 5 (a) Plot of η vs λ for t = 2 nm, a = 40 nm, ϕ₀ = π/6 and β = 3×10⁶ m/s, 1×10⁷ m/s and 3 × 10⁸ m/s. (b) Variation of ∆λ_res on increasing spacer thickness t for a = 40 nm and ϕ₀ = π/6.

Download Full Size | PDF

The applicability of the hydrodynamic model in describing the plasmon resonances of nanoparticles and nanowires has been a subject of great debate in the recent years [48, 49]. In this context, we would like to remark that our analytical treatment ignores quantum mechanical spill out of conduction electrons beyond the metal surface. The predictions of our analytical solution are expected to deviate from the more accurate quantum mechanical calculations (e.g. calculations based on time–dependent density functional theory [49]) if (a) the gap between the nanowire and the gold film is reduced below 1 nm, in which case quantum mechanical tunnelling between the nanowire and the film would become a dominant factor in governing the resonances of the system or (b) the nanowire itself is made very thin, thereby making the quantum mechanical spill out of electrons outside the nanowire surface significant. These effects are not included in the hydrodynamic model, which assumes the confinement of electrons within the metal. The accuracy of our analysis is contingent on the dimensions of the system being sufficiently large (> 1 nm), so as to allow us to neglect such quantum mechanical effects.

4. Conclusion

In this paper, we have presented a full electromagnetic solution to determine the optical fields on the excitation of a plasmonic nanowire coupled to a plasmonic film. The problem of computing the spatial fields inside and outside the plasmonic material was formulated as a boundary value problem, treating the plasmonic material as non-local. The analytical solution was compared to an independent numerical method (FDTD) for the case when the plasmonic medium was treated as local in nature, and it was observed that the analytical solution agreed with the simulation results. The analytical solution was also used to analyze the impact of non-localities of the plasmonic medium, which could not be done using the FDTD simulations. An accurate analytical solution helps gain insight and an in-depth understanding of the behaviour of the optical properties of the plasmonic nanowire-film system and hence would be an invaluable resource for designing devices suited to various position and material sensing applications.

Acknowledgments

The authors would like to thank the sponsors of this work: Department of Electronics and Information Technology (DEITY), Ministry of Communications and Information Technology (MCIT) of the Government of India under grant number RP02395 as well as the Department of Biotechnology of the Government of India under grant number RP02829 for their support.

References and links

1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

3. H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, 1957).

4. T. Okamoto, Near-Field Optics and Surface Plasmon Polaritons, ed. S. Kawata, ed. (Springer-Verlag, 2001).

5. S. A. Maier, Plasmonics: Fundamentals and Applications ( Springer, New York2007).

6. M. Born and E. Wolf, Principles of Optics, 7. (Cambridge University, 1999). [CrossRef]

7. J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3(4), 485–491 (2003). [CrossRef]

8. R. T. Hill, J. J. Mock, A. Hucknall, S. D. Wolter, N. M. Jokerst, D. R. Smith, and A. Chilkoti, “Plasmon Ruler with Angstrom Length Resolution,” ACS Nano 6(10), 9237–9246 (2012). [CrossRef] [PubMed]

9. J. J. Mock, R. T. Hill, A. Degiron, S. Zauscher, A. Chilkoti, and D. R. Smith, “Distance-dependent plasmon resonant coupling between a gold nanoparticle and gold film,” Nano Lett. 8 (8), 2245–2252 (2008). [CrossRef] [PubMed]

10. C. Circai, R. T. Hill, J. J. Mock, Y. Urzhumov, A. I. Fernndez-Domnguez, S. A. Maier, J. B. Pendry, A. Chilkoti, and D. R. Smith, “Probing the Ultimate Limits of Plasmonic Enhancement,” Science 337(6098), 1072–1074 (2012). [CrossRef]

11. S. Mubeen, S. Zhang, N. Kim, S. Lee, S. Kramer, H. Xu, and M. Moskovits, “Plasmonic Properties of Gold Nanoparticles Separated from a Gold Mirror by an Ultra-thin Oxide,” Nano Lett. 12(4), 2088–2094 (2012). [CrossRef] [PubMed]

12. R. K. Chang and T. E. Furtak, Surface-Enhanced Raman Scattering (Plenum, New York, 1982). [CrossRef]

13. A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, “Surface-enhanced Raman scattering”, J. Phys: Condensed Matter 4(5), 1143 (1992).

14. H. Xu, E. J. Bjereld, M. Kall, and L. Borjesson, “Spectroscopy of single Hemoglobin molecules by surface enhanced Raman scattering”, Phys. Rev. Lett. 83, 4357 (1999). [CrossRef]

15. K. Kneipp, M. Moskovits, and H. Kneipp, Surface-Enhanced Raman Scattering: Physics and Applications, (Springer, Berlin, 2006). [CrossRef]

16. S. Nie and S. R. Emory, “Probing Single Molecules and Single Nanoparticles by Surface-Enhanced Raman Scattering”, Science 275 (5303), 1102–1106 (1997). [CrossRef] [PubMed]

17. P. K. Aravind and H. Metiu, “The enhancement of Raman and fluorescent intensity by small surface roughness. Changes in dipole emission”, Chem. Phys. Lett. 74 (2), 301–305 (1980). [CrossRef]

18. K. Ohtaka, T. Ueta, and K. Amemiya, “Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods,” Phys. Rev. B 57, 2550 (1998). [CrossRef]

19. R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 6 (1999). [CrossRef]

20. J. Pritz and L. M. Woods, “Surface plasmon polaritons in concentric cylindrical structures,” Solid State Commun . 146 (7–8), 345–350 (2008). [CrossRef]

21. R. Gomez-Medina and J. J. Saenz, “Unusually strong optical interactions between particles in quasi-one-dimensional geometries,” Phys. Rev. Lett. 93, 243602 (2004). [CrossRef]

22. M. Laroche, S. Albaladejo, R. Gmez-Medina, and J. J. Saenz, “Tuning the optical response of nanocylinder arrays: An analytical study,” Phys. Rev. B 74, 245422 (2006). [CrossRef]

23. R. Gomez-Medina, M. Laroche, and J. J. Saenz, “Extraordinary optical reflection from sub-wavelength cylinder arrays,” Opt. Express 14(9), 3730–3737 (2006). [CrossRef] [PubMed]

24. K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52(10), 2603–2611 (2004). [CrossRef]

25. K. Ohtaka and H. Numata, “Multiple scattering effects in photon diffraction for an array of cylindrical dielectrics,” Phys. Lett. A , 73 (5–6), 411–413 (1979). [CrossRef]

26. T. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” Prog. Electromag. Res. 29, 69–85 (2000). [CrossRef]

27. R. Borghi, F. Gori, and M. Santarsiero, “Plane-wave scattering by a perfectly conducting circular cylinder near a plane surface: cylindrical-wave approach,” J. Opt. Soc. Am. A , Vol. 13(3), 2441–2452 (1996). [CrossRef]

28. R. Borghi, F. Gori, and M. Santarsiero, “Plane-wave scattering by a set of perfectly conducting circular cylinders in the presence of a plane surface,” J. Opt. Soc. Am. A 13(12), 2441–2452 (1996). [CrossRef]

29. S. Raza, G. Toscano, A. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412 (2011). [CrossRef]

30. G. Toscano, S. Raza, A. P. Jauho, N. A. Mortensen, and M. Wubs, “Modified field enhancement and extinction in plasmonic nanowire dimers due to nonlocal response,” Opt. Express 20 (4), 4176–4188 (2012). [CrossRef]

31. S. Li, M. L. Pedano, S. Chang, C. A. Mirkin, and G. C. Schatz, “Gap structure effects on surface-enhanced Raman scattering intensities for gold gapped rods,” Nano Lett. 10(5), 1722–1727 (2010). [CrossRef] [PubMed]

32. S. S. Acimovic, M. P. Kreuzer, M. U. Gonzlez, and R. Quidant, “Plasmon Near-Field Coupling in Metal Dimers as a Step toward Single-Molecule Sensing,” ACS Nano 3(5), 1231–1237 (2009). [CrossRef] [PubMed]

33. M. D. Fischbein and M. Drndic, “Sub-10 nm Device Fabrication in a Transmission Electron Microscope,” Nano Lett. 7(5), 1329–1337 (2007). [CrossRef] [PubMed]

34. V. Auzelyte, C. Dais, P. Farquet, D. Grtzmacher, L. J. Heyderman, and F. Luo, “Extreme ultraviolet interference lithography at the Paul Scherrer Institut,” J. Micro/Nanolith. MEMS MOEMS 8(2), 021204 (2009). [CrossRef]

35. B. D. Gates, Q. Xu, M. Stewart, D. Ryan, C. G. Willson, and G. M. Whitesides, “New Approaches to Nanofabrication: Molding, Printing, and Other Techniques,” Chem. Rev. 105(4), 1171–1196 (2005). [CrossRef] [PubMed]

36. A. Liebsch, “Surface plasmon dispersion of Ag,” Phys. Rev. Lett. 71, 145 (1993). [CrossRef] [PubMed]

37. A. Liebsch, “Surface-plasmon dispersion and size dependence of Mie resonance: Silver versus simple metals,” Phys. Rev. B 48, 11317 (1993). [CrossRef]

38. C. Cirací, J. B. Pendry, and D. R. Smith, “Hydrodynamic model for plasmonics: a macroscopic approach to a microscopic problem,” ChemPhysChem 14(6), 1109–1116 (2013). [CrossRef] [PubMed]

39. H. J. Weber and G. B. Arfken, Mathematical Methods for Physicists, (Harcourt, Academic Press, 2003).

40. H. W. Wyld, Mathematical Methods for Physics, (Advanced Books Classics, 1973).

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

42. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

43. A. Dhawan, S. J. Norton, Michael D. Gerhold, and Tuan Vo-Dinh, “Comparison of FDTD numerical computations and analytical multipole expansion method for plasmonics-active nanosphere dimers,” Opt. Express 17(12), 9688–9703 (2009). [CrossRef] [PubMed]

44. R. Trivedi and A. Dhawan, “Full-wave electromagentic analysis of a plasmonic nanoparticle separated from a plasmonic film by a thin spacer layer,” Opt. Express 22 (17), 19970–19989 (2014). [CrossRef] [PubMed]

45. A. Moreau, C. Cirací, and D. R. Smith, “Impact of nonlocal response on metallodielectric multilayers and optical patch antennas,” Phys. Rev. B 87, 045401(2013). [CrossRef]

46. G. Cincotti, F. Gori, and M. Santarsiero, “Plane wave expansion of cylidrical functions,” Opt. Commun. 95(4–6), 192–198 (1993). [CrossRef]

47. R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Commun. 190(1–6), 205–209 (2001). [CrossRef]

48. R. C. Monreal, T. J. Antosiewicz, and S. P. Apell, “Competition between surface screening and size quantization for surface plasmons in nanoparticles,” New J. Phys. 15, 083044 (2013). [CrossRef]

49. T. V. Teperik, P. Nordlander, J. Aizpurua, and A. G. Borisov, “Quantum effects and nonlocality in strongly coupled plasmonic nanowire dimers,” Opt. Express 21(22), 27306–27325 (2013). [CrossRef] [PubMed]

Plane wave scattering from a plasmonic nanowire-film system with the inclusion of non-local effects

Abstract

1. Introduction

2. Analytical method

2.1. Electromagnetic fields inside a non-Local medium

2.2. Analysis of the coupled nanowire-film system

2.3. Reflection of a plane wave from the surface of a non-local medium

3. Results and discussion

3.1. Validation of the analytical method

3.2. Impact of non-locality of the medium

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (60)

Optics Express