Optical response of linear chains of metal nanospheroids and nanospheroids

Abstract

A semi-analytical method for computing the electric field surrounding a finite linear chain of metal nanospheres and nanospheroids is described. In treating chains or clusters of spheres, a common approach is to use the spherical-harmonic addition theorem to relate the multipole expansion coefficients between different spheres. A method is described here that avoids the use of spherical-harmonic addition theorems, which are not applicable to spheroidal chains. Simulations are given that illustrate the large field enhancements that can occur in the gaps between silver nanoparticles arising from plasmon resonances.

1. INTRODUCTION

The problem of predicting the electromagnetic field in the gaps between metal nanoparticles under optical illumination has attracted interest in recent years because of the very large field enhancements induced in the particle gaps arising from surface plasmon resonances. Such enormous field enhancements underlie, for example, the phenomenon of surface-enhanced Raman scattering (SERS). Since the first application of the SERS technique was developed for chemical analysis [1], our laboratory has extensively investigated the SERS technology for biochemical sensing and medical diagnostics [2,3]. It is believed that the anomalously strong Raman signal originates from “hot spots,” i.e., regions where clusters of several closely spaced nanoparticles are concentrated in a small volume [4–7]. To further investigate this effect, we consider in this report the general problem of calculating the electric field surrounding a finite chain of metal nanospheres or nanospheroids when illuminated with coherent light.

The chain structure is assumed to consist of nanoparticles aligned closely with small gaps between them (Fig. 1). A. multipole analysis of this problem has been used by several authors for treating clusters or chains of spherical particles [8–13]. In this approach, the scattered field from a particular sphere is represented in a multipole expansion in a coordinate system centered on that sphere, and the multipole coefficients in these systems are then related using translational formulas based on the spherical-harmonic addition theorem [14]. Unfortunately, this method cannot be applied to spheroidal chains [15]. Here we describe a method applicable to both spheres and spheroids which avoids the use of translational formulas at the expense of the numerical, but straightforward, evaluation of certain simple integrals.

Fig. 1.

(a) Linear chain of spheres; (b) the ith and jth pair in the chain. The electric field is directed along the z axis.

In this paper, we will assume the quasi-static approximation, but the basic approach can be extended to the full-wave problem, in which retardation affects are accounted for. In the quasi-static approximation, the particles are assumed much smaller than an optical wavelength so that the incident electric field may be regarded as uniform over the dimensions of the particle. We shall also assume that the incident electric field is polarized in the direction, parallel to the linear chain axis, which is the most important case if our aim is to create large fields in the gaps between the particles. This polarization direction also renders the problem axially symmetric, which further simplifies the analysis. Under optical illumination, this requires linearly polarized light with a direction of propagation normal to the axis of the chain.

We illustrate our approach by computing the electric field in the gaps between two spheres and between two spheroids over a range of frequencies so that the induced plasmon resonances are evident. At frequencies matching the plasmon resonances, very large field enhancements can occur. We also show how the field enhancement varies with the aspect ratio of the prolate spheroids.

In the simulations, we use the following dispersion model for the dielectric constant for silver [17]:

$$\epsilon (\omega )=1+\sum _{k=1}^6\frac{\mathrm{\Delta }{\epsilon }_k}{-a_k{\omega }^2-i{b}_k\omega +c_k},$$ (1)

where Δε_k, a_k, b_k, and c_k are constants that provide the best fit for various metals [18].

2. CHAIN OF NANOSPHERES

Consider M colinear spheres as illustrated in Fig. 1(a) and assume that the incident electric field is directed along the line joining the centers of the spheres (the z axis). Suppose the ith sphere has radius r_i with dielectric constant ε_i(ω) and denote by L_i the z coordinate of the center of this sphere. We let L_ij ≡ L_j−L_i represent the (signed) distance between the centers of the ith and jth spheres (i.e., L_ji = −L_ij). We define spherical coordinate systems centered on each sphere; thus, for example, a point r may be represented in the coordinate system whose origin is the center of the ith sphere by writing r = (r_i,θ_i), or r = (r_i,u_i), where for brevity u_i ≡ cos θ_i. From Fig. 1(b), we have

$$r_i=\sqrt{L_{ij}^2+r_j^2+2L_{ij}{r}_j{u}_j},$$ (2)

$$u_i=\frac{r_j{u}_j+L_{ij}}{\sqrt{L_{ij}^2+r_j^2+2L_{ij}{r}_j{u}_j}},$$ (3)

and conversely,

$$r_j=\sqrt{L_{ij}^2+r_i^2-2L_{ij}{r}_i{u}_i},$$ (4)

$$u_j=\frac{r_i{u}_i-L_{ij}}{\sqrt{L_{ij}^2+r_i^2-2L_{ij}{r}_i{u}_i}},$$ (5)

where the latter two equations can be obtained from Eqs. (2) and (3) by interchanging i and j and using L_ji = −L_ij.

In the quasi-static approximation the electric field is expressed as the gradient of a potential. Let ${\psi }_p^{(i)}(r_i,u_i),{\psi }_s^{(i)}(r_i,u_i)$, and ${\psi }_{in}^{(i)}(r_i,u_i)$ denote, respectively, the potential functions associated with the primary (incident) field, the scattered field, and the field internal to the ith sphere measured with respect to the spherical coordinate system (r_i,u_i) centered at this sphere. Solutions to Laplace’s equation can then be represented as expansions in Legendre polynomials as follows. For the ith sphere, these fields are

$${\psi }_p^{(i)}=-E_0{r}_i{P}_1(u_i),$$ (6)

$${\psi }_s^{(i)}=\sum _{n=1}^{\infty }{a}_n^{(i)}\frac{1}{r_i^{n+1}}{P}_n(u_i),$$ (7)

$${\psi }_{in}^{(i)}=\sum _{n=1}^{\infty }{b}_n^{(i)}{r}_i^n{P}_n(u_i),$$ (8)

where P_n is the Legendre polynomial of degree n. It can be shown that the sums begin with n = 1 if there is no net charge on the spheres. For a chain of M spheres, the total field at a point r external to the spheres is given by the sum of the incident field and the M scattered fields.

$$\psi (\mathbf{r})={\psi }_p(\mathbf{r})+\sum _{j=1}^M{\psi }_s^{(j)}({\mathbf{r}}_j).$$ (9)

The boundary conditions on the surface of the ith sphere are then

$${\left[{\psi }_p^{(i)}+{\psi }_s^{(i)}+\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^M{\psi }_s^{(j)}\right]}_{r_i=R_i}={{\psi }_{in}^{(i)}\mid }_{r_i=R_i},$$ (10)

$${\epsilon }_0\frac{\partial }{\partial r_i}{\left[{\psi }_p^{(i)}+{\psi }_s^{(i)}+\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^M{\psi }_s^{(j)}\right]}_{r_i=R_i}={{\epsilon }_i\frac{\partial }{\partial r_i}{\psi }_{in}^{(i)}\mid }_{r_i=R_i}.$$ (11)

Substituting the potentials (6)–(8) into relations (10) and (11), multiplying both equations by P_m(u_i), integrating with respect to u_i between −1 and 1, and using the Legendre orthogonality relation

$${\int }_{-1}^1{P}_m(u)P_n(u)\text{d}u=\frac{2}{2m+1}{\delta }_{mn}$$ (12)

results in the following system of equations

$$-E_0{R}_i{\delta }_{1m}+\frac{1}{R_i^{m+1}}{a}_m^{(i)}+\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^M\sum _{n=1}^{\infty }{Q}_{mn}^{(ij)}{a}_n^{(j)}=R_i^m{b}_m^{(i)},$$ (13)

$$-E_0{R}_i{\delta }_{1m}-\frac{(m+1)}{R_i^{m+1}}{a}_m^{(i)}+\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^M\sum _{n=1}^{\infty }{R}_{mn}^{(ij)}{a}_n^{(j)}=m{\gamma }_i{R}_i^m{b}_m^{(i)},$$ (14)

where γ_i ≡ ε_i/ε₀,

$$Q_{mn}^{(ij)}\equiv c_m{\int }_{-1}^1{\left[\frac{1}{r_j^{n+1}}{P}_n(u_j)\right]}_{r_i=R_i}{P}_m(u_i)\text{d}{u}_i,$$ (15)

$$R_{mn}^{(ij)}\equiv c_m{R}_i{\int }_{-1}^1\frac{\partial }{\partial r_i}{\left[\frac{1}{r_j^{n+1}}{P}_n(u_j)\right]}_{r_i=R_i}{P}_m(u_i)\text{d}{u}_i,$$ (16)

and c_m ≡ (2m + 1)/2. In Eqs. (13) and (14), the index i ranges from 1 to M. In Appendix A, we show that the following relation holds between the matrix elements:

$$R_{mn}^{(ij)}=m{Q}_{mn}^{(ij)}.$$ (17)

The symmetry $Q_{mn}^{(ij)}={(-1)}^{m+n}{Q}_{mn}^{(ji)}$ can also be demonstrated from the definition of $Q_{mn}^{(ij)}$ given by Eq. (15). To show this, we interchange i and j in Eq. (15) (and recall that L_ji = −L_ij), change the sign of the integration variable, and employ the result P_n(−u) = (−1)ⁿP_n(u).

In a numerical solution, we will truncate the sum on n to N terms. That is, we will consider multipoles up to order N. Substituting relation (17) into relation (14) and eliminating the terms involving $b_m^{(i)}$ in relations (13) and (14), we obtain the following set of linear equations in the unknown coefficients $a_n^{(i)}$:

$${\lambda }_m^{(i)}{a}_m^{(i)}+\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^M\sum _{n=1}^N{X}_{mn}^{(ij)}{a}_n^{(j)}=p_m^{(i)},$$ (18)

where i = 1,2,…,M, m = 1,2,…,N, and

$$p_m^{(i)}\equiv E_0{R}_i({\gamma }_i-1){\delta }_{1m},$$ (19)

$${\lambda }_m^{(i)}\equiv \frac{m(1+{\gamma }_i)+1}{R_i^2},$$ (20)

$$X_{mn}^{(ij)}\equiv m({\gamma }_i-1)R_i^{m-1}{Q}_{mn}^{(ij)}.$$ (21)

This is a system of MN equations that can be solved for the MN coefficients $a_m^{(i)}$.

After calculating the coefficients $a_n^{(j)}$, the total scattered field is then computed from

$${\psi }_s(\mathbf{r})=\sum _{j=1}^M\sum _{n=1}^N{a}_n^{(j)}\frac{1}{r_j^{n+1}}{P}_n(u_j).$$ (22)

The scattered electric field is then given by

$${\mathbf{E}}_s(\mathbf{r})=\sum _{j=1}^M{\mathbf{E}}_s^{(j)}(\mathbf{r})=-\sum _{j=1}^M{\nabla }_j{\psi }_s^{(j)}(\mathbf{r}),$$ (23)

where ${\mathbf{E}}_s^{(j)}(\mathbf{r})$ denotes the field scattered from the jth sphere, and from Eq. (22) we have defined

$${\psi }_s^{(j)}(\mathbf{r})\equiv \sum _{n=1}^N{a}_n^{(j)}\frac{1}{r_j^{n+1}}{P}_n(u_j).$$ (24)

The subscript j on ∇_j in Eq. (23) signifies that the gradient is defined with respect to the jth coordinate system. The field scattered from the jth sphere is then.

$${\mathbf{E}}_s^{(j)}(\mathbf{r})=-{\nabla }_j{\psi }_s^{(j)}(\mathbf{r})=-\left[{\widehat{r}}_j\frac{\partial {\psi }_s^{(j)}}{\partial r_j}+{\widehat{\theta }}_j\frac{1}{r_j}\frac{\partial {\psi }_s^{(j)}}{\partial {\theta }_j}\right],$$ (25)

where r̂_j and θ̂_j are unit vectors in the jth coordinate system. Substituting Eq. (24) into Eq. (25) gives

$${\mathbf{E}}_s^{(j)}(\mathbf{r})=\sum _{n=1}^N{a}_n^{(j)}\left[{\widehat{r}}_j\frac{(n+1)}{r_j^{n+2}}{P}_n(u_j)+{\widehat{\theta }}_j\frac{\sqrt{1-u_j^2}}{r_j^{n+2}}{P}_n^{\prime }(u_j)\right],$$ (26)

where the prime on P_n denotes derivative with respect to its argument.

Now define for brevity the components

$$E_r^{(j)}(\mathbf{r})\equiv \sum _{n=1}^N{a}_n^{(j)}\frac{(n+1)}{r_j^{n+2}}{P}_n(u_j),$$ (27)

$$E_{\theta }^{(j)}(\mathbf{r})\equiv \sum _{n=1}^N{a}_n^{(j)}\frac{\sqrt{1-u_j^2}}{r_j^{n+2}}{P}_n^{\prime }(u_j),$$ (28)

so that Eq. (26) may be written

$${\mathbf{E}}_s^{(j)}(\mathbf{r})={\widehat{r}}_j{E}_r^{(j)}(\mathbf{r})+{\widehat{\theta }}_j{E}_{\theta }^{(j)}(\mathbf{r}).$$ (29)

As an example, the total scattered field represented in Cartesian coordinates is

$${\mathbf{E}}_s(\mathbf{r})=\widehat{x}{E}_x(\mathbf{r})+\widehat{z}{E}_z(\mathbf{r})=\sum _{j=1}^M[{\widehat{r}}_j{E}_r^{(j)}(\mathbf{r})+{\widehat{\theta }}_j{E}_{\theta }^{(j)}(\mathbf{r})].$$ (30)

From this, we see that

$$E_x(\mathbf{r})=\sum _{j=1}^M[(\widehat{x}·{\widehat{r}}_j)E_r^{(j)}(\mathbf{r})+(\widehat{x}·{\widehat{\theta }}_j)E_{\theta }^{(j)}(\mathbf{r})],$$ (31)

$$E_z(\mathbf{r})=\sum _{j=1}^M[(\widehat{z}·{\widehat{r}}_j)E_r^{(j)}(\mathbf{r})+(\widehat{z}·{\widehat{\theta }}_j)E_{\theta }^{(j)}(\mathbf{r})],$$ (32)

where x̂·r̂_j = sin θ_j, x̂·θ̂_j = cos θ_j, ẑ·r̂_j = cos θ_j and ẑ·θ̂_j = −sin θ_j.

We conclude this section by noting that the correctness of the integral (15) and the relation (17) have been checked using the spherical-harmonic addition theorem derived in [19].

3. DIPOLE LIMIT

It is instructive to consider the case where the first term in the sum on m in Eq. (18) dominates. This is the dipole term, which will dominate when the sphere separations are large compared to the sphere diameters. Keeping only the first term in the sum (18) (that is, setting N = 1 and m = 1 and n = 1) gives the equation

$$-E_0{R}_i({\gamma }_i-1)+\frac{(2+{\gamma }_i)}{R_i^2}{a}_1^{(i)}+({\gamma }_i-1)\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^N{Q}_{11}^{(ij)}{a}_1^{(j)}=0,$$ (33)

where i = 1,2,…,M. In this case $Q_{11}^{(ij)}$ can be evaluated analytically. From Eq. (15) and noting that P₁(u) = u, we have

$$Q_{11}^{(ij)}=\frac{3}{2}{\int }_{-1}^1{\left[\frac{u_j}{r_j^2}\right]}_{r_i=R_i}{u}_i\text{d}{u}_i=\frac{3}{2}{\int }_{-1}^1\frac{(R_i{u}_i-L_{ij})u_i\text{d}{u}_i}{{(L_{ij}^2+R_i^2-2L_{ij}{R}_i{u}_i)}^{3/2}},$$ (34)

where the latter integral follows from Eq. (15). This can integrated to give

$$Q_{11}^{(ij)}=-\frac{2R_i}{\mid L_{ij}{\mid }^3}.$$ (35)

Using this result, Eq. (33) becomes

$$-E_0{R}_i({\gamma }_i-1)+\frac{(2+{\gamma }_i)}{R_i^2}{a}_1^{(i)}-2R_i({\gamma }_i-1)\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^M\frac{1}{\mid L_{ij}{\mid }^3}{a}_1^{(j)}=0.$$ (36)

For example, for two spheres with a center-to-center separation L, Eq. (36) reduces to the two equations

$$\begin{array}{l}-E_0{R}_1({\gamma }_1-1)+\frac{(2+{\gamma }_1)}{R_1^2}{a}_1^{(1)}-\frac{2R_1({\gamma }_1-1)}{L^3}{a}_1^{(2)}=0,\\ -E_0{R}_2({\gamma }_2-1)+\frac{(2+{\gamma }_2)}{R_2^2}{a}_1^{(2)}-\frac{2R_2({\gamma }_2-1)}{L^3}{a}_1^{(1)}=0.\end{array}$$

Solving for $a_1^{(1)}$ and $a_1^{(2)}$ gives

$$a_1^{(1)}=\frac{E_0{R}_1^3[({\gamma }_2+2)({\gamma }_1-1)+2({\gamma }_1-1)({\gamma }_2-1)R_2^3/L^3]}{({\gamma }_2+2)({\gamma }_1+2)-4({\gamma }_1-1)({\gamma }_2-1)R_1^3{R}_2^3/L^6},$$ (37)

$$a_1^{(2)}=\frac{E_0{R}_2^3[({\gamma }_1+2)({\gamma }_2-1)+2({\gamma }_2-1)({\gamma }_1-1)R_1^3/L^3]}{({\gamma }_2+2)({\gamma }_1+2)-4({\gamma }_1-1)({\gamma }_2-1)R_1^3{R}_2^3/L^6}.$$ (38)

We can expand these equations to first order in. (R₁/L)³ and (R₂/L)³, giving

$$a_1^{(1)}=\frac{E_0{R}_1^3({\gamma }_1-1)}{{\gamma }_1+2}\left[1+2{(R_2/L)}^3\frac{({\gamma }_2-1)}{({\gamma }_2+2)}\right],$$ (39)

$$a_1^{(2)}=\frac{E_0{R}_2^3({\gamma }_2-1)}{{\gamma }_2+2}\left[1+2{(R_1/L)}^3\frac{({\gamma }_1-1)}{({\gamma }_1+2)}\right].$$ (40)

Note that when R₂ → 0 or when γ₂ → 1, Eq. (39) reduces, as expected, to the correct coefficient for an isolated sphere of radius R₁ given by

$$a_1^{(1)}=\frac{E_0{R}_1^3({\gamma }_1-1)}{{\gamma }_1+2}=\frac{E_0{R}_1^3({\epsilon }_1-{\epsilon }_0)}{{\epsilon }_1+2{\epsilon }_0}.$$ (41)

The scattered electric field from an isolated sphere of radius R₁ assuming a uniform incident electric field directed along the z axis is given by

$${\mathbf{E}}_s^{(1)}=-\nabla {\psi }_s^{(1)}(r_1,u_1)=\frac{a_1^{(1)}}{r_1^3}(2{\widehat{r}}_{1}cos{\theta }_1+{\widehat{\theta }}_{1}sin{\theta }_1),$$ (42)

where

$$a_1^{(1)}=\frac{E_0{R}_1^3({\gamma }_1-1)}{{\gamma }_1+2}.$$ (43)

Setting θ₁ = 0 in. Eq. (42), we have for the field on the z axis

$$E_z^{(1)}=\frac{2a_1^{(1)}}{r_1^3}.$$ (44)

Now note that if a second sphere of radius R₂ lies at a distance L along the z axis from the first sphere, it scatters a field in the direction of the first sphere, which, to first order in (R₂/L)³, will add to the field incident on the first sphere, resulting in a total incident field of magnitude Ẽ₀ given by

$${\stackrel{\sim }{E}}_0=E_0+\frac{2a_1^{(2)}}{L^3},$$ (45)

with

$$a_1^{(2)}=\frac{E_0{R}_2^3({\gamma }_2-1)}{{\gamma }_2+2}.$$ (46)

If we now replace E₀ in Eq. (43) by Ẽ₀, we obtain Eq. (39), valid to first order in (R₂/L)³.

4. CHAIN OF SPHEROIDS

In this section, we extend our analysis to spheroids. Figure 2 shows two spheroids in a chain of M particles. Prolate spheroidal coordinates centered on the ith spheroid are [20]

Fig. 2.

The ith and jth pair of spheroids in a chain.

$${\xi }_i=(r_1^{(i)}+r_2^{(i)})/2d_i,{\eta }_i=(r_1^{(i)}-r_2^{(i)})/2d_i,$$ (47)

where

$$r_1^{(i)}=\sqrt{{(z-L_i+d_i)}^2+x^2},$$ (48)

$$r_2^{(i)}=\sqrt{{(z-L_i-d_i)}^2+x^2},$$ (49)

and 2d_i is the separation between the foci. In the above, L_i is the z coordinate of the center of the ith spheroid. Inverting these equations, one finds

$$x=d_i\sqrt{({\xi }_i^2-1)(1-{\eta }_i^2)},$$ (50)

$$z=d_i{\xi }_i{\eta }_i+L_i.$$ (51)

Setting the “radial” spheroidal coordinate ξ_i to a constant ξ̄_i, together with the values of L_i, and d_i, defines the surface of the ith spheroid. If a_i and b_i are the lengths of the major and minor semi-axes of the ith spheroid, we have the relations a_i=d_iξ̄_i and $b_i=d_i\sqrt{{\overline{\xi }}_i^2-1}$. Or, given a_i and b_i these can be inverted to yield $d_i=\sqrt{a_i^2-b_i^2}$ and ξ̄_i=a_i/d_i. In the limit as d_i → 0 and ξ_i → ∞, the spheroidal coordinates reduce to spherical coordinates, and $d_i{\xi }_i\to \sqrt{{(z-L_i)}^2+x^2}=r_i$ and ${\eta }_i\to (z-L_i)/\sqrt{{(z-L_i)}^2+x^2}=cos{\theta }_i=u_i$ in our earlier notation. In spheroidal coordinates, the gradient is

$$\nabla \psi =\widehat{\xi }\frac{1}{h_{\xi }}\frac{\partial \psi }{\partial \xi }+\widehat{\eta }\frac{1}{h_{\eta }}\frac{\partial \psi }{\partial \eta }+\widehat{\phi }\frac{1}{h_{\phi }}\frac{\partial \psi }{\partial \phi },$$ (52)

where ξ̂,η̂, and φ̂ are spheroidal unit vectors and the spheroidal scale factors are [19]

$$h_{\xi }=d\sqrt{({\xi }^2-{\eta }^2)/({\xi }^2-1)},$$ (53)

$$h_{\eta }=d\sqrt{({\xi }^2-{\eta }^2)/(1-{\eta }^2)},$$ (54)

$$h_{\phi }=d\sqrt{({\xi }^2-1)/(1-{\eta }^2)}.$$ (55)

Let ${\psi }_p^{(i)}({\xi }_i,{\eta }_i),{\psi }_s^{(i)}({\xi }_i,{\eta }_i)$, and ${\psi }_{in}^{(i)}({\xi }_i,{\eta }_i)$ denote, respectively, the incident, scattered, and internal potentials associated with the ith spheroid in the spheroidal coordinates (ξ_iη_i) centered at this spheroid. We then have

$${\psi }_p^{(i)}=-E_0{d}_i{P}_1({\xi }_i)P_1({\eta }_i),$$ (56)

$${\psi }_s^{(i)}=\sum _{n=1}^{\infty }{a}_n^{(i)}{Q}_n({\xi }_i)P_n({\eta }_i),$$ (57)

$${\psi }_{in}^{(i)}=\sum _{n=1}^{\infty }{b}_n^{(i)}{P}_n({\xi }_i)P_n({\eta }_i).$$ (58)

In a chain of M spheroids, the boundary conditions on the ith spheroid are

$${\left[{\psi }_p^{(i)}+{\psi }_s^{(i)}+\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^M{\psi }_s^{(j)}\right]}_{{\xi }_i={\overline{\xi }}_i}={{\psi }_{in}^{(i)}\mid }_{{\xi }_i={\overline{\xi }}_i},$$ (59)

$${\epsilon }_0\frac{\partial }{\partial {\xi }_i}{\left[{\psi }_p^{(i)}+{\psi }_s^{(i)}+\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^M{\psi }_s^{(j)}\right]}_{{\xi }_i={\overline{\xi }}_i}={{\epsilon }_i\frac{\partial {\psi }_{in}^{(i)}}{\partial {\xi }_i}\mid }_{{\xi }_i={\overline{\xi }}_i}.$$ (60)

Substituting Eqs. (56)–(58) into the boundary conditions, multiplying by P_m(η_i), and integrating with respect to η_i from −1 to 1, we obtain

$$-E_0{d}_i{\overline{\xi }}_i{\delta }_{1m}+Q_m({\overline{\xi }}_i)a_m^{(i)}+\sum _{n=1}^{\infty }{Q}_{mn}^{(ij)}{a}_n^{(j)}=P_m({\overline{\xi }}_i)b_m^{(i)},$$ (61)

$$-E_0{d}_1{\overline{\xi }}_i{\delta }_{1m}+{\overline{\xi }}_i{Q}_m^{\prime }({\overline{\xi }}_i)a_m^{(i)}+\sum _{n=1}^{\infty }{R}_{mn}^{(ij)}{a}_n^{(j)}={\gamma }_i{\overline{\xi }}_i{P}_m^{\prime }({\overline{\xi }}_i)b_m^{(i)},$$ (62)

where γ_i ≡ ε_i/ε₀

$$Q_{mn}^{(ij)}\equiv c_m{\int }_{-1}^1{[Q_n({\xi }_j)P_n({\eta }_j)]}_{{\xi }_i={\overline{\xi }}_i}{P}_m({\eta }_i)\text{d}{\eta }_i,$$ (63)

$$R_{mn}^{(ij)}\equiv c_m{\overline{\xi }}_i{\int }_{-1}^1\frac{\partial }{\partial {\xi }_i}{[Q_n({\xi }_j)P_n(u_j)]}_{{\xi }_i={\overline{\xi }}_i}{P}_m({\eta }_i)\text{d}{\eta }_i,$$ (64)

and c_m ≡(2m + 1)/2. We show in Appendix A that the following relationship holds between the matrix elements:

$$R_{mn}^{(ij)}=\frac{{\overline{\xi }}_i{P}_m^{\prime }({\overline{\xi }}_i)}{P_m({\overline{\xi }}_i)}{Q}_{mn}^{(ij)}.$$ (65)

Substituting Eq. (65) into Eq. (62), eliminating the terms involving $b_m^{(i)}$ in Eqs. (61) and (62), and truncating the sum to N terms leaves the following system of equations for $a_m^{(i)}$:

$${\lambda }_m^{(i)}{a}_m^{(i)}+\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^M\sum _{n=1}^N{X}_{mn}^{(ij)}{a}_n^{(j)}=p_m^{(i)},$$ (66)

where m = 1, …,N, i = 1, …,M, and

$$X_{mn}^{(ij)}\equiv ({\gamma }_i-1)P_m^{\prime }({\overline{\xi }}_i)Q_{mn}^{(ij)},$$ (67)

$${\lambda }_m^{(i)}\equiv {\gamma }_i{Q}_m({\overline{\xi }}_i)P_m^{\prime }({\overline{\xi }}_i)-Q_m^{\prime }({\overline{\xi }}_i)P_m({\overline{\xi }}_i),$$ (68)

$$p_m^{(i)}\equiv E_0{d}_i{\overline{\xi }}_i({\gamma }_i-1){\delta }_{1m}.$$ (69)

We can use the identity $P_m(\xi )Q_m^{\prime }(\xi )-P_m^{\prime }(\xi )Q_m(\xi )=-1/({\xi }^2-1)$, to write relation (68) as

$${\lambda }_m^{(i)}=({\gamma }_i-1)Q_m({\overline{\xi }}_i)P_m^{\prime }({\overline{\xi }}_i)+1/({\overline{\xi }}_i^2-1).$$ (70)

In computing the matrix element $Q_{mn}^{(ij)}$ using relation (63), the variable of integration is η_i and one needs to express the variables ξ_j and η_j in terms of η_i. This is done using the relations (47) through (51).

The scattered electric field is obtained by computing the gradient of the potential in spheroidal coordinates. This field is

$${\mathbf{E}}_s(\mathbf{r})=-\sum _{j=1}^M{\nabla }_j{\psi }_s^j(\mathbf{r}),$$ (71)

where

$${\psi }_s^{(j)}(\mathbf{r})=\sum _{n=1}^N{a}_n^{(j)}{Q}_n({\xi }_j)P_n({\eta }_j).$$ (72)

The scattered field from the jth spheroid is

$${\mathbf{E}}_s^{(j)}=-{\nabla }_j{\psi }_s^{(j)}=-\left[{\widehat{\xi }}_j\frac{1}{h_{{\xi }_j}}\frac{\partial {\psi }_s^{(j)}}{\partial {\xi }_j}+{\widehat{\eta }}_j\frac{1}{h_{{\eta }_j}}\frac{\partial {\psi }_s^{(j)}}{\partial {\eta }_j}\right],$$ (73)

where ξ̂_j and η̂_i are unit vectors in the jth system. From Eqs. (72) and (71), we have

$${\mathbf{E}}_s^{(j)}=\sum _{n=1}^N{a}_n^{(j)}\left[{\widehat{\xi }}_j\frac{1}{h_{{\xi }_j}}{Q}_n^{\prime }({\xi }_j)P_n({\eta }_j)+{\widehat{\eta }}_j\frac{1}{h_{{\eta }_j}}{Q}_n({\xi }_j)P_n^{\prime }({\eta }_j)\right].$$ (74)

This can be written

$${\mathbf{E}}_s(\mathbf{r})={\widehat{\xi }}_j{E}_{\xi }^{(j)}+{\widehat{\eta }}_j{E}_{\eta }^{(j)},$$ (75)

where

$$E_{\xi }^{(j)}=\sum _{n=1}^N{a}_n^{(j)}\frac{1}{h_{{\xi }_j}}{Q}_n^{\prime }({\xi }_j)P_n({\eta }_j),$$ (76)

$$E_{\eta }^{(j)}=\sum _{n=1}^N{a}_n^{(j)}\frac{1}{h_{{\eta }_j}}{Q}_n({\xi }_j)P_n^{\prime }({\eta }_j).$$ (77)

In Cartesian coordinates, we write

$${\mathbf{E}}_s(\mathbf{r})=\widehat{x}{E}_x(\mathbf{r})+\widehat{z}{E}_z(\mathbf{r})=\sum _{j=1}^N[{\widehat{\xi }}_j{E}_{\xi }^{(j)}+{\widehat{\eta }}_j{E}_{\eta }^{(j)}],$$ (78)

from which we see that

$$E_x(\mathbf{r})=\sum _{n=1}^N[(\widehat{x}·{\widehat{\xi }}_j)E_{\xi }^{(j)}+(\widehat{x}·{\widehat{\eta }}_j)E_{\eta }^{(j)}],$$ (79)

$$E_z(\mathbf{r})=\sum _{n=1}^N[(\widehat{z}·{\widehat{\xi }}_j)E_{\xi }^{(j)}+(\widehat{z}·{\widehat{\eta }}_j)E_{\eta }^{(j)}].$$ (80)

In Appendix B we show that

$$\widehat{x}·{\widehat{\xi }}_j=d_j{\xi }_j/h_{{\eta }_j},$$ (81)

$$\widehat{x}·{\widehat{\eta }}_j=-d_j{\eta }_j/h_{{\xi }_j},$$ (82)

$$\widehat{z}·{\widehat{\xi }}_j=d_j{\eta }_j/h_{{\xi }_j},$$ (83)

$$\widehat{z}·{\widehat{\eta }}_j=d_j{\xi }_j/h_{{\eta }_j},$$ (84)

where the h are the spheroidal scaling factors (53)–(55).

5. SIMULATIONS

Consider the two spheres and two spheroids shown in Fig. 3. As a sample calculation, we show in Figs. 4–6 the calculated values of the magnitude of the electric field between the two spheres and between the two spheroids with two different aspect ratios. In these figures, the plots show the calculated value of the field magnitude over a range of wavelengths at a point on axis in the gap midway between the two particles. The magnitude of the incident electric field is unity; thus, the plots show the field enhancement relative to the incident field. The peaks correspond to the frequencies of the plasmon resonances. Because of the assumption of a uniform, incident electric field (the quasi-static approximation), the enhancement is scale invariant; that is, the enhancement depends only on the ratio of the gap width to the particle size (e.g., the radius of a sphere or, for a spheroid, the lengths of the semi-major and semi-minor axes).

Fig. 3.

Cross-sectional view of a sphere (dashed) and two spheroids of volume equal to that of the sphere. The spheroids have aspect ratios of (a) 2 and (b) 4.

Fig. 4.

Field in the gap between two spheres (dashed curve) and two spheroids of aspect ratios 2 and 4. In all three cases, the gap is 10% of the diameter of the sphere (of unit radius). The spheroids are equal in volume to that of the sphere.

Fig. 6.

Field in the gap between two spheres (dashed curve) and two spheroids of aspect ratios 2 and 4. The gap is 2% of the diameter of the sphere. The spheroids are equal in volume to that of the sphere.

In the calculations, we compare three pairs of particles with different gaps between them: a pair of identical spheres of unit radius, and a pair of prolate spheroids with two different aspect ratios but equal in volume to the sphere, as illustrated in Fig. 3. Figure 3(a) shows a spheroid with an aspect ratio of 2 (i.e., a/b = 2) and with a volume equal to that of a sphere of unit radius (a = 1.588, b = 0.794). Figure 3(b) is a spheroid with an aspect ratio of 4 of the same volume (a = 2.520. b = 0.630) as its corresponding sphere.

In Fig. 4, the dashed curve (labeled 1) is the field in the gap between two identical spheres of unit radius, where the gap is 10% of the sphere diameter. The curve labeled 2 is the field between two identical spheroids each with an aspect ratio of 2, and the curve labeled 4 is the result for spheroids with an aspect ratio of 4. Here the gap between the spheroids is the same as that of the sphere in both cases. Note that the plasmon resonance is red-shifted with increasing aspect ratio. Figure 5 is similar to Fig. 4 except now the gap between all particles was reduced to 5% of the sphere diameter, again with the spheroid aspect ratios labeled 2 and 4. In Fig. 6 the gap has been further reduced to 2% of the sphere diameter.

Fig. 5.

Field in the gap between two spheres (dashed curve) and two spheroids of aspect ratios 2 and 4. The gap is 5% of the diameter of the sphere. The spheroids are equal in volume to that of the sphere.

Note that for a given gap width, the two spheroids produce a noticeably larger enhancement than the two spheres. This is expected, since the smaller curvature at the spheroid ends creates a larger surface charge density and a larger field. The increased field at the ends can be attributed to the “lightning rod effect” [21]. The prolate spheroids with the 2% gap and an aspect ratio of 4 show more than a three-order-of-magnitude field enhancement.

6. CONCLUSION

Two simple algorithms have been developed for computing the electric field surrounding linear chains of nanospheres and nanospheroids. Here the quasi-static approximation is used, which holds under the assumption that the incident field is uniform over the size of the particle. When the electric field is polarized along the axis of the chain, the direction of the incident light will be normal to this axis, implying that the quasi-static approximation should hold when the wavelength is large compared with the diameter of one particle. As noted, this polarization gives rise to the largest field enhancement in the gaps between the particles. One feature of the quasi-static approximation is that the multipole coefficients $a_n^{(i)}$ are independent of frequency. The only frequency dependence in the problem enters through that of the dielectric function ε(ω). As a result, the computational times involved are many orders of magnitude shorter than those of full-wave algorithms, such as finite-element-based approaches or the finite-difference time-domain method. In summary, the approach here is simple, fast and accurate, but limited to axially symmetric systems of spheres or spheroids.

The example given above for a pair of prolate spheroids with an aspect ratio of 4 and a 2% gap shows an electric field enhancement in the gap of over 2000 at the peak of the plasmon resonance. In SERS, the total signal is approximately proportional to the fourth power of the electric field magnitude [4–7], giving in the latter case a total SERS enhancement of over 10¹³. This SERS enhancement is close to what is required for single molecule detection at the hot spot in the gap [22]. A spatially averaged enhancement will, of course, be much less than this peak value.

APPENDIX A: OF RELATIONS (17) AND (65)

To demonstrate the relation (17), let ψ_i and ψ_j denote two solutions to Laplace’s equation, i.e., ∇²ψ_i = 0 and ∇²ψ_i=0, which are regular (source-free) in and on a sphere of radius R_i. Now multiply the former equation by ψ_j and the latter equation by ψ_i and subtract, which gives ψ_j∇²ψ_i− ψ_i∇²ψ_j=∇ · [ψ_j∇ψ_i−ψ_i∇ψ_j] = 0. Integrating this over the volume of a sphere of radius R_i; and using the divergence theorem results in

$$\int \int {\left[{\psi }_j\frac{\partial {\psi }_i}{\partial r}-{\psi }_i\frac{\partial {\psi }_j}{\partial r}\right]}_{r=R_i}{\text{d}}^2\mathbf{r}=0,$$ (A1)

which is an integral over the surface of a sphere of radius R_i. Now substitute into Eq. (A1) the following solutions to Laplace’s equation: ${\psi }_i=r_i^m{P}_m(u_i)$ and ${\psi }_j=P_n(u_j)/r_j^{n+1}$, where r_i, r_j, u_i, and u_j, are defined as in Fig. 1(b). Substituting into Eq. (A1), letting r=r_i, and noting that ${\text{d}}^2\mathbf{r}=2\pi R_i^{2}sin{\theta }_i\text{d}{\theta }_i=-2\pi R_i^2\text{d}{u}_i$, (A1) reduces to $m{Q}_{mn}^{(ij)}-R_{mn}^{(ij)}=0$, where $Q_{mn}^{(ij)}$ and $R_{mn}^{(ij)}$ are defined by relations (15) and (16), which confirms relation (17).

To demonstrate the relation (65) for the spheroidal case, the argument parallels that given above, but, we employ spheroidal coordinates. Let ψ_i and ψ_j denote two solutions to Laplace’s equation that are regular on and inside a spheroid defined by d_i and ξ̄_i. In this case, Eq. (A1) is replaced, by

$$\int \int [{\psi }_j\nabla {\psi }_i-{\psi }_i\nabla {\psi }_j]·{\widehat{\xi }}_i{\mid }_{{\xi }_i={\overline{\xi }}_i}{\text{d}}^2\mathbf{r}=0,$$ (A2)

where the integral is over the surface of the spheroid. The surface element in spheroidal coordinates is d²r =h_ηh_φd_ηdφ, and we also have

$$\nabla \psi ·\widehat{\xi }=\frac{1}{h_{\xi }}\frac{\partial \psi }{\partial \xi }.$$ (A3)

Thus, Eq. (A2) may be written

$$\begin{array}{l}2\pi {\int }_{-1}^1{\left[{\psi }_j\frac{\partial {\psi }_i}{\partial \xi }-{\psi }_i\frac{\partial {\psi }_j}{\partial \xi }\right]}_{\xi =\overline{\xi }}\frac{h_{\eta }{h}_{\phi }}{h_{\xi }}\text{d}\eta =\pi d({\overline{\xi }}^2-1)\\ \times {\int }_{-1}^1{\left[{\psi }_j\frac{\partial {\psi }_i}{\partial \xi }-{\psi }_i\frac{\partial {\psi }_j}{\partial \xi }\right]}_{\xi =\overline{\xi }}\text{d}\eta =0,\end{array}$$ (A4)

where the φ integration just gives 2π due to the axial symmetry. Now use the following solutions to Laplace’s equation—ψ_i=P_m(ξ_i)P_m(η_i) and ψ_j=Q_n(ξ_j)P_n(η_j)—and set ξ=ξ_i, η= η_i and ξ̄=ξ̄_i in Eq. (A4). This gives

$$\begin{array}{l}{P}_m^{\prime }({\overline{\xi }}_i){\int }_{-1}^1{[Q_n({\xi }_j)P_n({\eta }_j)]}_{{\xi }_i={\overline{\xi }}_i}{P}_m({\eta }_i)\text{d}{\eta }_i\\ =P_m({\overline{\xi }}_i){\int }_{-1}^1\frac{\partial }{\partial {\xi }_i}{[Q_n({\xi }_j)P_n({\eta }_j)]}_{{\xi }_i={\overline{\xi }}_i}{P}_m({\eta }_i)\text{d}{\eta }_i,\end{array}$$ (A5)

from which relation (65) follows.

APPENDIX B: DERIVATION OF COEFFICIENTS (81)–(84)

One way to compute the coefficients (81)—(84) is to equate the gradient in the two coordinate systems:

$$\nabla \psi =\widehat{x}\frac{\partial \psi }{\partial x}+\widehat{z}\frac{\partial \psi }{\partial z}=\overline{\xi }\frac{1}{h_{\xi }}\frac{\partial \psi }{\partial \xi }+\widehat{\eta }\frac{1}{h_{\eta }}\frac{\partial \psi }{\partial \eta }.$$ (B1)

First let ψ=x and then ψ=z, where x and z in spheroidal coordinates are given by relations (50) and (51). Dotting both sides of Eq. (B1) first with x̂ and then with ẑ and computing the derivatives with respect to ξ and η leads to the coefficients (81)–(84).

APPENDIX C: SOME NUMERICAL CONSIDERATIONS

In this appendix, we make two suggestions that can improve the performance of a numerical algorithm using the above formulas. First, in calculating the integrals (15) and (63) numerically, one can show that the integrand varies most rapidly near u_i =1 and η_i =1, respectively. By changing the variable of integration from u_i = cos θ_i to θ_i in integral (15) and from η_i ≡ cos ϑ_i to ϑ_i in integral (63), the integrands will vary more slowly in this region and a quadrature algorithm will require fewer function evaluations.

Second, the system of equations defined by relation (66) for the spheroidal problem is poorly conditioned as written, whereas the system (18) for the spherical case is well conditioned, as can be checked by computing the singular-value decomposition of the matrix (21). The conditioning of relation (66) can be noticeably improved by scaling system (66) so that it reduces to the spherical system (18) in the limit as d_i → 0. As d_i → 0, we have ξ_i → ∞ and ξ̄_i → ∞, such that d_i ξ̄_i → R_i, d_i ξ_i → r_i, and η_i → u_i = cos θ_i. The scaling is performed by exploiting the following asymptotic formulas for large ξ [23]:

$$Q_n(\xi )\to \frac{{\alpha }_n}{{\xi }^{n+1}}=\frac{{\alpha }_n{d}^{n+1}}{{(\xi d)}^{n+1}}\to \frac{{\alpha }_n{d}^{n+1}}{r^{n+1}}$$ (C1)

$$P_n(\xi )\to {\beta }_n{\xi }^n=\frac{{\beta }_n}{d_n}{(\xi d)}^n\to \frac{{\beta }_n}{d^n}{r}^n,$$ (C2)

where α_n = n!/(2n + 1)!! and β_n + (2n − 1)!!/n! We define

$${\stackrel{\sim }{a}}_m^{(i)}={\alpha }_m{d}_i^{m+1}{a}_m^{(i)},$$ (C3)

$${\stackrel{\sim }{X}}_{mn}^{(ij)}=\frac{d_i^{m-1}}{{\alpha }_n{\beta }_m{d}_j^{m+1}}{X}_{mn}^{(ij)},$$ (C4)

$${\stackrel{\sim }{\lambda }}_m^{(i)}=\frac{1}{d_i^2{\alpha }_m{\beta }_m}{\lambda }_m^{(i)}=\frac{2m+1}{d_i^2}{\lambda }_m^{(i)},$$ (C5)

where $a_m^{(i)},X_{mn}^{(ij)}$, and ${\lambda }_m^{(i)}$ are given in relations (18)–(21). $p_m^{(i)}$ will remain the same. Then relation (66) becomes

$${\stackrel{\sim }{\lambda }}_m^{(i)}{\stackrel{\sim }{a}}_m^{(i)}+\sum _{\begin{array}{l}j=1\\ j\ne i\end{array}}^M\sum _{n=1}^N{\stackrel{\sim }{X}}_{mn}^{(ij)}{\stackrel{\sim }{a}}_n^{(j)}=p_m^{(i)}.$$ (C6)

The scattered potential (57) for the ith spheroid now becomes, using Eq. (C3).

$${\psi }_s^{(i)}({\xi }_i,{\eta }_i)=\sum _{n=1}^N{a}_n^{(i)}{Q}_n({\xi }_i)P_n({\eta }_i)=\sum _{n=1}^N{\stackrel{\sim }{a}}_n^{(i)}\frac{Q_n({\xi }_i)}{{\alpha }_n{d}_i^{n+1}}{P}_n({\eta }_i).$$ (C7)

With this scaling, the above equations reduce to those for the sphere, given by relations (18)–(21), as d_i → 0.