1 Introduction

It was shown in various works (e.g., [1, 2]) that Raman signals in molecules produced by incident EM fields are strongly amplified when the molecules are inserted in the interstitial gaps between nanoparticles due to the very strong EM fields induced in these gaps ("hot spots"). Various experimental results on surface enhanced Raman scattering (SERS) from molecules on aggregates of nanoparticles were interpreted by hot spot mechanism [3,4,5]. Special studies were made on Raman signals enhancement in dimers (two nanoparticles) [6, 7]. It was found that the Raman signals of spherical dimers are strongly enhanced when the incident polarization is parallel to the inter particle axis of the dimer (parallel polarization) [8]. In this case the opposite charges of polarization produce strong electro-magnetic (EM) fields by facing each other at the small gap. On the other hand, when the incident EM field is polarized in direction perpendicular to the inter particle axis (perpendicular polarization) the induced charges are in directions different from that of the gap. Therefore, in this case, individual local surface plasmons (LSP) in the dimer do not interact strongly with each other. As a result, EM field interaction is approximately compared in this case to that of isolated particles.

It was found that the signal in SERS is proportional to the fourth power of the amplified EM field for parallel polarization [9, 10]. Similar results are obtained by two-photons-induced luminescence (TPI-PL) [8, 11,12,13]. Raman scattering and TPI-PL phenomena are amplified by by many orders of magnitude and there is large amount of literature on these effects. Due to the EM fields amplified at the hot spots many non-linear effects can be observed [1,2,3,4,5,6,7,8,9,10,11], where these effects cannot be observed under ordinary EM fields. It is, therefore, very important to understand the mechanism by which hot spots are produced and their properties.

The enhancement of EM fields by their interaction with local metallic surface plasmons are well known from various studies. For example, for EM field incident on metallic films with extremely small holes it was found that the transmitted light is enhanced by many orders of magnitudes relative to the integrated light intensity over the small holes areas [14, 15]. It has been shown that Laplace equation solutions for single metallic sphere interacting with homogenous EM field lead to electrostatic field of a dipole located at the center of the sphere (e.g., [16]). In the present work we treat the enhancement of the EM field by its interaction with two nearby metallic spheres placed with their centers on the symmetric \(z\)-axis. In this case the induced EM strength depends strongly on the inter particle distance. For cases in which the distance between the two spheres is large relative to their radius the interaction between the two spheres can be treated by conventional theories about dipole–dipole interactions [17]. Experiments were found to be in agreement with such theory (e.g., [18]). For cases in which the distance between the two spheres is much smaller than the radius-\(R\), the analysis becomes quite complicated.

There are many methods for analyzing hot spots phenomena including discrete dipole approximation (DDA) [19,20,21,22], boundary element method (BEM) and finite difference in time domain (FDTD) method [23, 24]. The optical modelling of nanoparticles relies on the solution of Maxwell's equations, for each specific geometry and set of illumination conditions. These methods are based on adaptive grids that assign finer discretization to regions, where the fields are expected to vary more rapidly. Along these methods the formation and control of hot spots with retardation effects can be understood by following the articles of Van Dyne group [25, 26] and more recent Review [27]. However, in the limit of extremely small nano-particles gaps the electromagnetic interaction between the different parts of the nano-particles is instantaneous. Then, Maxwell's equations are leading to the condition that the enhanced electric field can be obtained by Laplace equation solutions. The magnetic response of the particles is then negligible at optical frequencies for very small particles, and the electric field \(\vec{E}\) is given as \(\vec{E} = - \nabla \psi\), where \(\psi\) is solution of Laplace equation considered as potential. In the present electrostatic limit we get superposition of Maxwell's electrostatic solutions having global properties. Accurate calculations for absorbance and scattering of metallic spheres and other small particles can be obtained by Mie theory [28, 29]. This method does not give analytical results as it requires the summation of large number of complicated terms which include retardation effects, where such retardations are neglected here.

In the present electrostatics approach, where retardation effects are negligible we use the analytical method of bi-spherical coordinates [30,31,32,33,34,35] for analyzing hot spots effects in the two metallic spheres system. Solution \(\psi\) of Laplace equation in the bi-spherical coordinates considered as the potential has the form given as [32]

$$ \psi = G(\eta ,\alpha )V(\eta )\theta (\alpha )\Lambda (\phi ) $$
(1)

where

$$ G\left( {\eta ,\alpha } \right) = \left( {\cosh \eta - \cos \alpha } \right)^{1/2} . $$
(2)

The general solution of Laplace equation is given by superposition of solutions of the form of Eq. (1) described later in the article as: \(\psi = \sum\limits_{n} {\psi_{n} } (n = 1,2,3, \ldots )\). Here \(\psi\) is function of the radial bi-spherical coordinates \(\eta\), the polar angle \(\alpha\) and azimuthal angle \(\phi\). Solutions for \(\psi\) were developed in the literature for various cases [31,32,33,34,35]. In the present article we treat the solutions of \(\psi\) in the electrostatic approach for the system of two metallic spheres with equal radius (\(R_{1} = R_{2}\)) and incident EM field polarized in the symmetric \(z\)-axis (parallel polarization), as described in Fig. 1. We apply the method of bi-spherical coordinates and analyze hot spots effects in a different way from that presented in the literature.

Fig. 1
figure 1

Two metallic spheres with dielectric constant \(\varepsilon (\omega )\) with radius R and the surrounding medium with dielectric constant \(\varepsilon_{1}\), under external EM field \(E_{z}\). Various parameters are described in the present \(x,z\) coordinates system

Full size image

Bi-spherical coordinates are useful for exploiting the spherical symmetries of metallic nano-particles. The advantages obtained by the use of bi-spherical coordinates specialized to the present dimers systems are explained as follows: (a) Laplace equation solution-\(\psi\), considered as the potential, is almost separable function of these coordinates represented by superposition of solutions of the form of Eq. (1). However, we have a coupling between the coordinates \(\eta\) and \(\alpha\) given by Eq. (2). (b) For particles possessing axial symmetry we can expand \(\psi\) with azimuthal components \(\exp (im\varphi )\). The analysis becomes separable for different values of \(m\). For the dimers system with parallel polarization the electrostatic potential has cylindrical symmetry about the \(z\)-axis, so that it is independent of the azimuthal angle \(\phi\) and only the term \(m = 0\) has to be considered. Under this symmetry the function \(\Lambda (\phi )\) is omitted from Eq. (1). In addition, in the solution \(\psi_{n}\) of order \(n\), \(\theta (\alpha )\) is reduced to Legendre polynomial: \(P_{n} (\cos \alpha )\). (c) The bi-spherical coordinates under the present cylindrical symmetry are obtained in the \(x,z\) plane, where \(\eta\) and \(\alpha\) are the bi-spherical radial and polar coordinates, respectively. The bi-spherical coordinates result from rotating this two-dimensional bi-spherical coordinates around the \(z\)-axis, as described in Fig. 2. This figure describes the symmetric case, in which the radiuses of the two metallic spheres are equal i. e. \(R_{1} = R_{2}\). (d) Bi-spherical coordinates are very useful in deriving the boundary conditions between the metallic spheres and their surroundings. Using first boundary condition the potential \(\psi\) is not changed in transmission through the metallic spherical surfaces given as

$$ \left[ {\psi^{(1)} } \right]_{{\eta = \eta_{0} }} = \left[ {\psi^{(2)} } \right]_{{\eta = \eta_{0} }} . $$
(3)
Fig. 2
figure 2

Bi-spherical coordinates in the \(x,z\) plane showing circles of constant bi-spherical radial coordinate \(\eta_{0}\), and curves of constant polar angular \(\alpha\)

Full size image

Here, the locus of all points on the metallic spheres surfaces, are given by \(\eta = \pm \eta_{0}\). In addition to the first boundary condition we need to use the second boundary condition for the derivative of \(\psi\) perpendicular to the metallic surfaces. This boundary condition becomes very complicated in orthogonal \(x,y,z\) coordinates, since the angle between the perpendicular to the sphere surface and the z-axis representing the electric field direction is changed in quite complicated way, i.e., as function of the location of the boundary point on the sphere surface. This second boundary condition is simplified in bi-spherical coordinates, since the locus of all the points on the metallic sphere surfaces is obtained by certain bi-spherical coordinate \(\pm \eta_{0}\) and this boundary condition has the special form:

$$ \varepsilon_{1} \left[ {\frac{\partial }{\partial \eta }\psi^{(1)} } \right]_{{\eta = \eta_{0} }} = \varepsilon \left( \omega \right)\left[ {\frac{\partial }{\partial \eta }\psi^{(2)} } \right]_{{\eta = \eta_{0} }} . $$
(4)

Here, \(\varepsilon \left( \omega \right)\) and \(\varepsilon_{1}\) are the dielectric constants of the metallic spheres and theirs surrounding, \(\psi^{(2)}\) and \(\psi^{(1)}\) are the potentials in the metallic spheres and its surrounding, respectively. The derivative \(\frac{\partial }{\partial \eta }\) with its value at \(\eta_{0}\) is obtained for any point on the sphere surface performing the derivative in direction perpendicular to the surface. We take into account that \(\frac{\partial }{\partial \eta }\psi\) is proportional to the gradient in the bi-spherical \(\eta\) direction.

We restrict the present theoretical analysis to symmetric dimers but it is straightforward to generalize the system to two metallic spheres which have different radius or core shell structures [36,37,38] and have spherical shapes. Electrostatic solutions can be developed for very small particles which have different shapes (e.g., nano-stars) in which retardation effects are neglected. Symmetries other than that of spherical for such particles can be of help but the use of bi-spherical coordinates is not suitable for the boundary conditions for other symmetries.

The paper is arranged as follows: In Sect. 2 we discuss the relations between the geometry of the present system and bi-spherical coordinates. We add in this section Figs. 1 and 2 which help to visualize these relations. The general forms of Laplace equation solutions for two metallic spheres with parallel polarization are given shortly in Sect. 3. These equations were obtained in the literature [31,32,33,34,35] but the methods used for their derivations will help to follow the new material given in later sections. Boundary conditions for the present system are analyzed in Sect. 4 giving different results from other authors due to the use of certain approximations which are valid for hot spots. The electric field at the hot spots is obtained in Sect. 5, where the dependence on bi-spherical coordinates is transformed to that of orthogonal \(x,y,z\) coordinates. In Sect. 6, we demonstrate the results by some numerical calculations. In Sect. 7, we compare our results with general computational methods. In Sect. 8, we summarize our results and conclusions.

2 The geometry of the present system related to bi-spherical coordinates

We describe the two metallic spheres of equal radius \(R\) with bi-spherical parameters in Fig. 1. We choose the vertical z-axis along the line passing through the centers of the spheres. The perpendicular \(x,y\) plane contains the midpoint between the two spheres. We assume that the distance from the center of one sphere with radius, R (the upper one) to the center of the coordinate system along the \(z\) coordinate is \(+ D\) and that for the other sphere with the same radius R (the lower one) is \(- D\). For the use of bi-spherical coordinates [30,31,32,33,34,35], we define

$$ D = R + \delta ;a = \left[ {D^{2} - R^{2} } \right]^{1/2} . $$
(5)

Here, the shortest distance between the two spheres surfaces is given by \(2\delta\). We treat the case, where the incoming EM field is homogenous and the electric field \(E_{z}\) is along the \(z\)-axis.

This system has cylindrical symmetry under rotation around the \(z\)-axis. We assume dielectric values [39, 40]: \(\varepsilon (\omega )\) for the two metallic spheres which are function of the frequency \(\omega\) and for the surrounding medium with constant value \(\varepsilon_{1}\). We treat the limiting case for which \(\delta\) is much smaller than \(R\), where under this condition hot spots are produced. For simplicity of analysis we assumed that the two spheres have the same radius.

The bi-spherical coordinates are a special three dimensional orthogonal coordinates system defined by coordinates \(\eta ,\alpha ,\phi\) [30,31,32,33,34,35]:

$$ \begin{gathered} x = a\sin \alpha \cos \phi /\left( {\cosh \eta - \cos \alpha } \right), \hfill \\ y = a\sin \alpha \sin \phi /\left( {\cosh \eta - \cos \alpha } \right), \hfill \\ z = a\sinh \eta /\left( {\cosh \eta - \cos \alpha } \right). \hfill \\ \end{gathered} $$
(6)

The inverse transformations are given by

$$ \begin{gathered} \tanh \eta = \frac{2az}{{x^{2} + y^{2} + z^{2} + a^{2} }};\sinh \eta = \frac{2az}{{\sqrt {\left( {x^{2} + y^{2} + z^{2} + a^{2} } \right)^{2} - \left( {2az} \right)^{2} } }}, \hfill \\ \cos \alpha = \frac{{x^{2} + y^{2} + z^{2} - a^{2} }}{{\sqrt {\left( {x^{2} + y^{2} + z^{2} } \right)^{2} - \left( {2az} \right)^{2} } }};\sin \alpha \frac{{2a\left( {x^{2} + y^{2} } \right)^{1/2} }}{{\sqrt {\left( {x^{2} + y^{2} + z^{2} } \right)^{2} - \left( {2az} \right)^{2} } }}. \hfill \\ \end{gathered} $$
(7)

The two poles with \(\eta = \pm \infty\) are located on the \(z\)-axis at \(z = \pm a\) and denoted in Fig. 1 by \(F_{1}\) and \(F_{2}\) . Surfaces of constant \(\eta\) are given by the spheres:

$$ x^{2} + y^{2} + (z - a\coth \eta )^{2} = \frac{{a^{2} }}{{\sinh^{2} \eta }}. $$
(8)

For constant value of \(\eta\) Eq. (8) represents spheres. The special value \(\eta = + \eta_{0}\) for the metallic spheres with radius \(R\) is defined by the following equivalent equations:

$$ \sinh \eta_{0} = a/R;\cosh \eta_{0} = \frac{1}{R}\sqrt {R^{2} + a^{2} } ;D = a\coth \eta_{0} $$
(9)

By substituting in Eq. (8), the special value \(\eta = \eta_{0}\), we get from Eq. (9):

$$ x^{2} + y^{2} + (z - D)^{2} = R^{2} . $$
(10)

This equation for \(\eta = \eta_{0}\) represents the upper sphere surface with radius \(R\), where its center is moved from the center of the coordinate system by a distance D in the positive \(z\) direction. For the special case of \(\eta = - \eta_{0}\), we use Eqs. (8–9), with \(\sinh \eta_{0} \to \sinh \left( { - \eta_{0} } \right) = - \sinh \eta_{0}\). Then, instead of Eq. (10), we get \(x^{2} + y^{2} + (z + D)^{2} = R^{2}\), where this equation represents the lower sphere surface for \(\eta = - \eta_{0}\) with radius \(R\), where its center is moved from the center of the coordinate system by a distance D in the negative \(z\) direction. Therefore, the locus of all points on the upper sphere surface is given by the parameter \(+ \eta_{0}\) and those on the lower sphere surface by \(- \eta_{0}\).

In the present case of parallel polarization, there is not any dependence on the azimuthal angle \(\phi\) and the bi-spherical coordinates are \(\eta\), and \(\alpha\). The relations between these bi-spherical coordinates and the orthogonal \(x,z\) coordinates are described in Fig. 2. Due to the system cylindrical symmetry the relations between \(\eta ,\alpha\) and the three orthogonal coordinates, \(x,y,z\) is obtained by rotating this two dimensional bi-spherical coordinates around the \(z\)-axis. It is interesting to note that such two dimensional picture of bi-spherical coordinates was applied in recent article about "Transport phenomena in bi-spherical coordinates" [41] which is different field from the present one.

The bi-spherical coordinates are described in Fig. 2 by circles in the \(x,z\) plane for radial constant bi-spherical coordinate \(\eta_{0}\). Special circles are obtained on the right side of the plane for \(\eta_{0} = 0.2,0.4,0.8,1.3\) and the corresponding ones on the left side for \(\eta_{0} = - 0.2,\; - 0.4,\; - 0.8,\; - 1.3\) using the symmetry condition \(R_{1} = R_{2}\). For very large circles there is a cutoff in the figure. The radius of the circles is changed from \(R = \infty\) for \(\eta_{0} = 0\) to \(R \to 0\) for \(\eta_{0} \to \infty\) in the corresponding singular focal points \(F_{1}\) or \(F_{2}\). The curves for constant value \(\alpha\) are described in Fig. 2 for \(\alpha = \pi ,\;\pi /2,\;\pi /3,\;\pi /6\) and they are orthogonal to the circles with constant \(\eta_{0}\) values. These curves end in the singular points for which \(\eta_{0} = \pm \infty\). The external EM field in this figure is propagating in the \(x\) direction and is polarized in the \(z\) direction. The polar angles \(\alpha\) are changing from \(\alpha = 0\) to \(\alpha = \pi\), where the curve for \(\alpha = \pi\) is along the \(z\)-axis ending in the singular focal points.

By rotating Fig. 2 around the \(z\)-axis, the circles are transformed to spheres with the same radius. Under the transformation \(z \to - z\), we get in this figure the transformation \(\eta \to - \eta\), where the electric field is in the \(z\) direction. Metallic spheres with radius \(R\) are described by the circles corresponding to: \(\eta = \pm \eta_{0}\), where the relation between \(R\) and \(\eta_{0}\) is given by Eq. (9). We used the bi-spherical coordinates \(\eta ,\alpha\) in the \(x,z\) plane which is a special bi-spherical coordinates system [31,32,33,34,35].

It is straightforward to generalize the present analysis to the case in which the two metallic spheres have different radius but the analysis will involve complicated calculations. For example the simple definition given by Eq. (7) for parameter \(a\) should be changed to more complicated equation (see, e.g., [32] Eq. (2)). In addition, for two metallic spheres which have different radius we must use the more complicated relations between \(R_{1}\) and \(\eta_{01}\), and between \(R_{2}\) and \(\eta_{02}\) representing the surfaces of two metallic spheres with different radius [31,32,33,34,35].

3 Laplace equations solutions for two metallic spheres with incident EM field parallel to the symmetric z coordinate

The general electrostatic solutions for the present dimer case are given by superposition of bi-spherical solutions [31,32,33,34,35]: (a) In the region surrounding the spheres with external EM field potential \(- Ez\):

$$ \psi^{(1)} = \sum\limits_{n = 0}^{\infty } {\psi_{n}^{(1)} } = \left( {\cosh \eta - \cos \alpha } \right)^{1/2} \sum\limits_{n = 0}^{\infty } {A_{n} V_{n}^{(1)} (\eta )P_{n} \left( {\cos \alpha } \right)} - Ez, $$
(11)

(b) In the metallic spheres by

$$ \psi^{(2)} = \sum\limits_{n = 0}^{\infty } {\psi_{n}^{\left( 2 \right)} } = \left( {\cosh \eta - \cos \alpha } \right)^{1/2} \sum\limits_{n = 0}^{\infty } {B_{n} V_{n}^{\left( 2 \right)} (\eta )P_{n} \left( {\cos \alpha } \right)} . $$
(12)

Laplace equation solutions are given here by superposition of solutions of order \(n\), where in practical calculations \(n\) is between \(n = 0\) and maximal value \(n = n_{\max .}\) and where the number of solutions to be considered increases for shorter distances between the two spheres. \(P_{n} \left( {\cos \alpha } \right)\) is Legendre polynomial of order \(n\) and where \(A_{n}\) and \(B_{n}\) are certain coefficients corresponding to the solutions outside the metallic spheres and inside them, respectively. These coefficients are obtained in the next section using the boundary conditions in a way which is different from that obtained by other authors. \(V_{n}^{(1)}\) , and \(V_{n}^{(2)}\) are parts of the potentials \(\psi^{(1)}\) and \(\psi^{(2)}\), respectively. These functions include superposition of amplified and attenuated exponentials with exponents \(\pm \left( {n + 1/2} \right)\eta\). For the potential \(\psi^{(2)}\) in the metallic spheres which include the singular points \(\eta = \pm \infty\) the amplified component diverges at the singular points so that this component vanishes and we have the relation [32, 35]:

$$ V_{n}^{(2)} = \exp \left[ { - (n + 1/2)\left| \eta \right|} \right]. $$
(13)

The potential due to the external field is given by \(- Ez\), where the external field \(E_{ext} = E_{z}\) is written in short notation as \(E\). This potential is antisymmetric with reflectance through the \(x,y\) plane, i.e., for \(z \to - z\) or \(\eta \to - \eta\), so that it imposes the relation \(V_{n}^{(1)} ( - \eta ) = - V_{n}^{(1)} (\eta )\). Then, we find [32, 35] that \(V_{n}^{(1)} (\eta )\) is reduced to antisymmetric \(\sinh\) function:

$$ V_{n}^{(1)} (\eta ) = \sinh \left[ {(n + 1/2)\eta } \right]. $$
(14)

Equations (11) and (12) are then transformed using Eqs. (13–14), respectively, as

$$ \begin{gathered} \psi^{(1)} = \left( {\cosh \eta - \cos \alpha } \right)^{1/2} \hfill \\ \sum\limits_{n = 0}^{\infty } {P_{n} \left( {\cos \alpha } \right)A_{n} \sinh \left\{ {\left( {n + 1/2} \right)\eta } \right\} - Ez} , \hfill \\ \end{gathered} $$
(15)
$$ \psi^{(2)} = \left( {\cosh \eta - \cos \alpha } \right)^{1/2} \sum\limits_{n = 0}^{\infty } {B_{n} \exp \left[ { - \left| \eta \right|(n + 1/2)} \right]P_{n} \left( {\cos \alpha } \right)} . $$
(16)

4 Boundary conditions applied to the present system in comparison with different conditions applied by other authors

We derive the coefficients \(A{}_{n}\) and \(B_{n}\) in Laplace equation solutions using the boundary conditions in a way which is different from that used by other authors.

For the purpose of matching the boundary conditions we use the relation [32]:

$$ z = a\left( {\cosh \eta - cos\alpha } \right)^{1/2} \sum\limits_{n = 0}^{\infty } {\sqrt 2 \left( {2n + 1} \right)} P_{n} \left( {\cos \alpha } \right)\exp \left[ { - \left( {n + 1/2} \right)\eta } \right]. $$
(17)

Then, Eq. (15) is transformed to

$$ \psi^{(1)} \left( {\eta ,\alpha } \right) = \left( {\cosh \eta - \cos \alpha } \right)^{1/2} \sum\limits_{n = 0}^{\infty } {P_{n} \left( {\cos \alpha } \right)\left[ {A_{n} \sinh \left( {n + \frac{1}{2}} \right)\eta - Ea2^{1/2} \left( {2n + 1} \right)e^{{ - \eta \left( {n + 1/2} \right)}} } \right]} . $$
(18)

For \(\eta = \eta_{0}\), Eq. (18) is transformed to

$$ \psi^{(1)} \left( {\eta_{0} ,\alpha } \right) = \left( {\cosh \eta_{0} - \cos \alpha } \right)^{1/2} \sum\limits_{n = 0}^{\infty } {P_{n} \left( {\cos \alpha } \right)\left[ {A_{n} \sinh \left( {n + \frac{1}{2}} \right)\eta_{0} - Ea2^{1/2} \left( {2n + 1} \right)e^{{ - \left( {n + 1/2} \right)\eta_{0} }} } \right]} . $$
(19)

Using the equality of Eq. (3), and comparing the corresponding expressions in Eqs. (19) and (15) for \(\eta = \eta_{0}\) and for each \(n\) value, we obtain [32, 35]:

$$ B_{n} \exp^{{ - \left( {n + \frac{1}{2}} \right)\eta_{0} }} = A_{n} \sinh \left( {n + \frac{1}{2}} \right)\eta_{0} - 2^{1/2} Ea\left( {2n + 1} \right)e^{{ - (n + 1/2)n_{0} }} . $$
(20)

Using Eq. (4) and Eqs. (16, 18), we get

$$ \begin{gathered} \varepsilon (\omega )\left[ {\frac{\partial }{\partial \eta }\left\{ {\left( {\cosh \eta - \cos \alpha } \right)^{1/2} \sum\limits_{n = 0}^{\infty } {B_{n} \exp \left[ { - \left| \eta \right|(n + 1/2)} \right]P_{n} \left( {\cos \alpha } \right)} } \right\}} \right]_{{\eta = \eta_{0} }} \hfill \\ = \varepsilon_{1} \frac{\partial }{\partial \eta }\left\{ {\left( {\cosh \eta - \cos \alpha } \right)^{1/2} \sum\limits_{n = 0}^{\infty } {\left[ {A_{n} \sinh \left( {n + \frac{1}{2}} \right)\eta - Ea2^{1/2} \left( {2n + 1} \right)e^{{ - \left( {n + 1/2} \right)\eta }} } \right]} P_{n} \left( {\cos \alpha } \right)} \right\}_{{\eta = \eta_{0} }} . \hfill \\ \end{gathered} $$
(21)

Derivatives in Eq. (21) include derivatives according to \(\eta\) of \(\left( {\cosh \eta - \cos \alpha } \right)^{1/2}\) in addition to the derivatives of the terms in the summations of this equation. The derivatives of \(\left( {\cosh \eta - \cos \alpha } \right)^{1/2}\) include after rearranging the equation, terms that are proportional to: \(\cos \alpha P_{n} (\cos \alpha )\). For these terms one can use the relation:

$$ \cos \left( \alpha \right)P_{n} (\cos \alpha ) = \frac{n}{2n + 1}P_{n - 1} (\cos \alpha ) + \frac{n + 1}{{2n + 1}}P_{n + 1} (\cos \alpha ). $$
(22)

Then, the boundary condition of Eq. (4) leads to coupling between the solution \(\psi_{n}\) and that of \(\psi_{n - 1}\), and \(\psi_{n + 1}\). For very small distance between the two metallic spheres, i.e., when \(\delta\) is much smaller than \(R\), one enters into convergence problems as increased number of summations terms are needed for accurate analysis and the calculations by recursion relations becomes extremely complicated. In the present work we follow the idea, that for treating such cases we can use certain approximations which will simplify the analysis and will be suitable for the treatment of hot spots in dimers. The EM potential of hot spots in spherical dimers is strongly localized at the small gap between the metallic spheres. Following the analysis for this case by bi-spherical coordinates it is possible to use special approximations analyzed as follows.

We use the relation

$$ - \left[ {\frac{\partial }{\partial \eta }\left( {\cosh \eta - \cos \alpha } \right)^{1/2} } \right]_{{\eta = \eta_{9} }} = \frac{{\left( {1/2} \right)\sinh \eta_{0} }}{{\left( {\cosh \eta_{0} - \cos \alpha } \right)^{1/2} }}. $$
(23)

The maximal value of this derivative is obtained for \(\cos \alpha = 1\). Then, under the approximation, \(\delta\) is much smaller than \(R\) we can use for hot spots the approximations:

$$ \sinh \eta_{0} \cong \eta_{0} ;\quad \left[ {\cosh \eta_{0} - 1} \right]^{1/2} \cong \frac{{\eta_{0} }}{\sqrt 2 }. $$
(24)

Substituting these approximations in Eq. (23), we get

$$ - \left[ {\frac{\partial }{\partial \eta }\left( {\cosh \eta - \cos \alpha } \right)^{1/2} } \right]_{{\eta = \eta_{9} }} \le \frac{1}{\sqrt 2 }, $$
(25)

where for general cases, it is even much smaller. On the other hand, under the condition \(\delta\) is much smaller than \(R\) there are many \(B_{n}\) and \(A_{n}\) coefficients multiplied by exponential terms with derivatives which are very large. Neglecting the derivative of Eq. (23), in Eq. (21), we get for very small \(\delta\):

$$ \begin{aligned} \varepsilon (\omega )\left[ {\frac{\partial }{\partial \eta }B_{n} \exp^{{ - \left( {n + \frac{1}{2}} \right)\eta }} } \right]_{{\eta = \eta_{0} }} & = - \varepsilon (\omega )\left( {n + \frac{1}{2}} \right)B_{n} \exp^{{ - \left( {n + \frac{1}{2}} \right)\eta_{0} }} \\ & = \varepsilon_{1} \left[ {\frac{\partial }{\partial \eta }\left\{ {A_{n} \sinh \left( {n + \frac{1}{2}} \right)\eta - Ea2^{1/2} \left( {2n + 1} \right)e^{{ - \eta \left( {n + 1/2} \right)}} } \right\}} \right]_{{\eta = \eta_{0} }} \\ & = \varepsilon_{1} \left[ {A_{n} \left( {n + \frac{1}{2}} \right)\cosh \left( {n + \frac{1}{2}} \right)\eta_{0} + Ea2^{1/2} \left( {2n + 1} \right)(n + 1/2)e^{{ - \left( {n + 1/2} \right)\eta_{0} }} } \right]. \\ \end{aligned} $$
(26)

Here, the functions \(P_{n} (\cos \alpha )\) were cancelled from the two sides of Eq. (21) due to the approximation of Eq. (25).

By substituting Eq. (20) into Eq. (26), we get

$$ \begin{gathered} - \varepsilon (\omega )\left( {n + \frac{1}{2}} \right)\left[ {A_{n} \sinh \left( {n + \frac{1}{2}} \right)\eta_{0} - 2^{1/2} Ea\left( {2n + 1} \right)e^{{ - (n + 1/2)n_{0} }} } \right] \hfill \\ = \varepsilon_{1} (n + 1/2)\left[ {A_{n} \cosh \left( {n + \frac{1}{2}} \right)\eta_{0} + 2^{1/2} Ea\left( {2n + 1} \right)e^{{ - \left( {n + 1/2} \right)\eta_{0} }} } \right]. \hfill \\ \end{gathered} $$
(27)

Rearranging the terms in Eq. (27), we get

$$ A_{n} \left\{ {\varepsilon_{1} \cosh \left( {n + \frac{1}{2}} \right)\eta_{0} + \varepsilon (\omega )\sinh \left( {n + \frac{1}{2}} \right)\eta_{0} } \right\} = \left( {\varepsilon (\omega ) - \varepsilon_{1} } \right)2^{1/2} Ea\left( {2n + 1} \right)e^{{ - (n + 1/2)n_{0} }} . $$
(28)

so that

$$ A_{n} = \frac{{2^{1/2} (2n + 1)(\varepsilon (\omega ) - \varepsilon_{1} )Eae^{{ - \left( {n + 1/2} \right)n_{0} }} }}{{\left\{ {\varepsilon_{1} \cosh \left( {n + \frac{1}{2}} \right)\eta_{0} + \varepsilon (\omega )\sinh \left( {n + \frac{1}{2}} \right)\eta_{0} } \right\}}};\quad {\text{for}}\;\delta \le \left( {1/10} \right)R. $$
(29)

We note that the calculation of the coefficients \(A_{n}\) by the use of Eq. (29) becomes quite simple as it is derived in a straightforward way by the use of the parameter \(\eta_{0}\) and the experimental parameters \(\varepsilon (\omega )\), and \(\varepsilon_{1}\). Once these coefficients are obtained the potential \(\psi^{(1)} (\eta ,\alpha )\) can be calculated using Eq. (15) with these coefficients. The number of coefficients \(A_{n}\) needed in the present analysis increases for lower values of \(\eta_{0}\) corresponding to lower values of \(\delta\) but their calculation using Eq. (29) is quite simple in comparison to the complicated calculations of these coefficients made by truncation of infinite number of linear equations used by other authors [31,32,33,34,35]. The use of the present approach is limited, however, by the validity of the approximation \(\delta\) is much smaller than \(R\) which is valid for hot spots. These results were derived in compact form by the use of bi-spherical coordinates.

5 The EM field described with bi-spherical coordinates at the hot spots and its transformation to the \(x,z\) coordinates

The bi-spherical radial EM field \(E_{\eta }\) outside the metallic spheres is given by

$$ - E_{\eta } = \left[ {\left( {\cosh \eta - \cos \alpha } \right)/a} \right]\left( {d\psi^{(1)} /d\eta } \right). $$
(30)

One should notice that the last term of Eq. (15) denotes the external EM potential \(V_{ext}\) given as \(V_{ext} = - Ez\), where \(E\) is given in short notation for \(E_{ext} \equiv E_{z}\) so that the incident field \(E\) might be obtained as \(E = - \frac{{dV_{ext} }}{dz}\) and this field is transmitted thorough this system. As the EM field at the hot spot \(E_{spot}\) is amplified by many orders of magnitudes relative to the incident field \(E\) we consider only the amplified EM field relative to the incident EM field. The amplified electric field is obtained by operating with \(E_{n}\) of Eq. (30) on other terms of Eq. (15). We get

$$ \begin{gathered} E_{\eta } = - \left[ {\left( {\cosh \eta - \cos \alpha } \right)/a} \right] \hfill \\ d\left[ {\left( {\cosh \eta - \cos \alpha } \right)^{1/2} \sum\limits_{n = 0}^{\infty } {A_{n} } \sinh \left[ {\left( {n + \frac{1}{2}} \right)\eta } \right]P_{n} \left( {\cos \alpha } \right)} \right]/d\eta . \hfill \\ \end{gathered} $$
(31)

Since the derivative of \(\left( {\cosh \eta - \cos \alpha } \right)^{1/2}\) relative to \(\eta\) is very small relative to the derivatives of the \(\sinh\) functions for \(\delta\) much smaller than \(R\) (where the number of coefficients \(A_{n}\) is very large) we neglect this derivative and get

$$ E_{\eta } = - \left[ {\left( {\cosh \eta - \cos \alpha } \right)^{3/2} /a} \right]\left[ {\sum\limits_{n = 0}^{n = \infty } {\left( {n + \frac{1}{2}} \right)\cosh \left[ {\left( {n + \frac{1}{2}} \right)\eta } \right]P_{n} \left( {\cos \alpha } \right)} } \right]. $$
(32)

Equation (32) gives the general solution for the radial EM field in bi-spherical coordinates for hot spots for which \(\delta\) is much smaller than \(R\) and for which the amplified electric field is much larger than the incident field \(E\) and where the coefficients \(A_{n}\) are given by Eq. (29).

As the hot spots are produced in dimers on (or near) the symmetric axis for which \(x = y = 0\) we obtain using Eq. (7) the following approximations on the symmetric \(z\)-axis:

$$ \begin{gathered} \sin \alpha = 0;\tanh \eta = \frac{\sinh \eta }{{\sqrt {1 + \sinh^{2} \eta } }} = \frac{2az}{{z^{2} + a^{2} }}; \hfill \\ \sinh \eta = \frac{2az}{{a^{2} - z^{2} }};\cosh \eta = \frac{{a^{2} + z^{2} }}{{a^{2} - z^{2} }};e^{ \pm \eta } = \cosh \eta \pm \sinh \eta = \frac{{\left( {a \pm z} \right)^{2} }}{{a^{2} - z^{2} }} = \frac{a \pm z}{{a \mp z}}. \hfill \\ \end{gathered} $$
(33)

At the focal points, for which: \(z = \pm a\), the functions \(\sinh \eta\),\(\cosh \eta\) and \(e^{\eta }\), are diverging to \(\infty\). Substituting the values of \(\sinh \eta\) and \(\cosh \eta\) from Eq. (33) into \(z\) of Eq. (6), we get

$$ z = \frac{a\sinh \eta }{{\cosh \eta - \cos \alpha }} = \frac{{2a^{2} z}}{{a^{2} + z^{2} - \left( {a^{2} - z^{2} } \right)cos\alpha }} \to \cos \alpha = - 1. $$
(34)

This result is demonstrated also in Fig. 2, where the curve for \(\alpha = \pi\) coincides with the \(z\) coordinate in the gap between the metallic surfaces. Therefore, the curve \(\alpha = \pi\) leads to the approximation \(\cos \alpha = - 1\) representing the central part of hot spots. This result leads to special values of the Legendre polynomials on the symmetric \(z\)-axis given by

$$ P_{n} (\cos (\alpha )) = P_{n} ( - 1) = ( - 1)^{n} = \left\{ \begin{gathered} 1\quad {\text{for}}\;\;{\text{any}}\;\;{\text{even}}\;\;{\text{number}}\;\;n \hfill \\ - 1\quad {\text{for}}\;\;{\text{any}}\;\;{\text{odd}}\;\;{\text{number}}\;\;n \hfill \\ \end{gathered} \right\} $$
(35)

By substituting the value \(\cos \alpha = - 1\) and Eq. (35) into Eq. (32) we obtain the result for the EM field in bi-spherical coordinates on the symmetric coordinate \(z\) including the hot spot:

$$ \begin{gathered} E_{spot} = \sum\limits_{n = 0}^{\infty } {E_{n} } \hfill \\ E_{n} = - \left[ {\left( {\cosh \eta + 1} \right)^{3/2} /a} \right]\left[ {(n + \frac{1}{2})\cosh \left[ {\left( {n + \frac{1}{2}} \right)\eta } \right]A_{n} \left\{ { \left( { - 1} \right)^{n} } \right\}} \right]. \hfill \\ \end{gathered} $$
(36)

We are interested in calculations of the total EM field intensity at the hot spot given by \(E_{spot}^{2}\) We notice that in the calculation of \(E_{spot}^{2}\) we have non-diagonal products \(E_{n} E_{n^{\prime}} (n \ne n^{\prime})\) with alternating signs so that their total contribution approximately vanishes. We take into account, therefore, only the diagonal incoherent elements. Then, for the electric field amplified factor \(\left| {\frac{{E_{spot} }}{E}} \right|^{2}\) and for the SERS measurements which are proportional to \(\left| {\frac{{E_{spot} }}{E}} \right|^{4}\) [4] we get

$$ \begin{gathered} \left| {\frac{{E_{spot} }}{E}} \right|^{2} = \left[ {\sum\limits_{n = 0}^{{n_{\max } }} {\left| {\frac{{E_{n} }}{E}} \right|^{2} } } \right]; \hfill \\ \left| {E_{n} } \right| = \left[ {\left( {\cosh \eta + 1} \right)^{3/2} /a} \right]\left[ {(n + \frac{1}{2})\cosh \left[ {\left( {n + \frac{1}{2}} \right)\eta } \right]A_{n} } \right]; \hfill \\ \left| {\frac{{E_{spot} }}{E}} \right|^{4} = \left[ {\sum\limits_{n = 0}^{{n_{\max } }} {\left| {\frac{{E_{n} }}{E}} \right|^{2} } } \right]^{2} . \hfill \\ \end{gathered} $$
(37)

We should take into account that \(\left| {\frac{{E_{spot} }}{E}} \right|^{2}\) gives the electric field squared at the hot spot, where products of \(E_{n}\) with \(E_{n^{\prime}}\) (\(n \ne n^{\prime}\)) vanish due to the approximation made after Eq. (35). We should take into account also that SERS measurements depend on \(\left| {\frac{{E_{spot} }}{E}} \right|^{4}\), so it is obtained by the square of the sum of Eq. (37) (as demonstrated later in the numerical calculations and discussed in Sect. 7).

The general solution for the electric field in our system including its dependence on \(x\) and \(y\) coordinates is given by Eq. (32). This equation which includes complicated Legendre polynomials can give numerical results by extensive calculations for parameters \(\eta\) and \(\alpha\), where the coefficients \(A_{n}\) are given by Eq. (29). We are interested in large enhancement fields (\(EF\)) [43,44,45,46] discussed later in Sect. 7. Large SERS enhancement fields (\(EF\)) occurs on the \(z\)-axis, where on this axis, we have \(\alpha = \pi\) as follows from Eq. (35) and also from Fig. 2. Under this approximation, we have still sum of many terms but each term is obtained by simple function of \(\eta\).

We inserted in Eq. (37), the maximal value \(n_{\max }\) which guarantees summation convergence. We transform Eq. (37) as a function of the \(z\) coordinate using the following relation from Eq. (33):

\(\cosh \left( \eta \right) = \frac{1}{2}\left[ {\frac{{\left( {a + z} \right)}}{a - z} + \frac{a - z}{{\left( {a + z} \right)}}} \right].\) (38}.

Using this relation, in Eq. (37), we get

$$ \begin{gathered} \left| {E_{n} } \right| = \left( \frac{1}{a} \right)\left\{ {\frac{1}{2}\left[ {\frac{{\left( {a + z} \right)}}{a - z} + \frac{a - z}{{\left( {a + z} \right)}}} \right] + 1} \right\}^{3/2} \hfill \\ \left[ {\left( {n + \frac{1}{n}} \right)\left( \frac{1}{2} \right)\left\{ {\left[ {\frac{a + z}{{a - z}}} \right]^{n + 1/2} + \left[ {\frac{a - z}{{a + z}}} \right]^{{\left( {n + 1/2} \right)}} } \right\}A_{n} } \right]. \hfill \\ \end{gathered} $$
(39)

One should take into account that we presented \(\cosh \left( \eta \right)\) in Eq. (39) in a symmetric form so that the inversion \(z \to - z\) does not change this function. We obtain the high EM fields at the hot spots by substituting Eq. (29), into Eq. (39), as demonstrated by numerical calculations in the next section. The number of coefficients \(A_{n}\) needed in the present analysis increases for lower values of \(\eta_{0}\) (corresponding to lower values of \(\delta\)), but their calculation using Eq. (29) is quite simple in comparison to the complicated calculations of these coefficient made by truncation of very large number of linear equations used by other authors [31,32,33,34,35]. In the center of the coordinate system Eq. (39) is reduced to simpler form given by

$$ \left| {E_{n} } \right| = \left( \frac{1}{a} \right)2^{3/2} \left( {n + \frac{1}{2}} \right)A_{n} . $$
(40)

6 Numerical calculations

In the present work, we developed solutions of Laplace-equation for the EM field in hot spots, in a system composed of two metallic spheres with the same radius \(R\) and incident homogenous EM field, in the symmetric \(z\) direction. The use of our general results is demonstrated here by numerical calculations in Tables 1 and . In Table 1, we use the parameters:

$$ \begin{gathered} \delta = \frac{1}{10}R;D = R + \delta ;a = \sqrt {D^{2} - R^{2} } = 0.4583R;\sinh \eta_{0} = \frac{a}{R} = 0.4583; \hfill \\ \eta_{0} = 0.4436;\cosh \eta_{0} = 1.1000;D = a\coth \eta_{0} . \hfill \\ \end{gathered} $$
(41)
Table 1 Calculations are made for two metallic spheres, with radius R, with dielectric constant \(\varepsilon (\omega ) = - 9\) and surrounding medium with dielectric constants \(\varepsilon_{1} = 1\) and with \(2\delta = 0.2R\)
Full size table

We notice that the circle corresponding to \(\eta_{0} = 0.4436\) has radius which is a little smaller from that of the circle with \(\eta_{0} = 0.4\) described in Fig. 2. We follow here electrostatic approach in which the dielectric constants of metallic spheres are known as empirical values at certain frequency \(\omega\) corresponding to a certain model (e.g., [39, 40]). The coefficients \(A_{n}\) of Eq. (29) are proportional to \(\varepsilon (\omega ) - \varepsilon_{1}\), so that consequently, the amplification of the EM at the hot spot has this simple dependence in addition to the dependence in the denominator of Eq. (29) on the dielectric constants. For illustration, in this example, we assumed for the two spheres \(\varepsilon (\omega ) \cong - 9\) (e.g., silver spheres, where the imaginary part \(\varepsilon_{2} (\omega )\) is very small), and for the surrounding medium, \(\varepsilon_{1} = 1\). The coefficients \(A_{n}\) were calculated by Eq. (29) and represented in Table 1 as \(\frac{{A_{n} }}{Ea}\), where \(E\) is the incident EM field. The electric field \(\left| {E_{n} (z)/E} \right|\) was calculated by Eq. (39) using Eq. (29) and presented in Table 1 for \(z = 0,0.08R\) and \(0.1R\). The electric field amplified factor is given by \(\left| {\frac{{E_{spot} }}{E}} \right|^{2} = \left[ {\sum\limits_{n = 0}^{{n_{\max } }} {\left| {\frac{{E_{n} }}{E}} \right|^{2} } } \right]\). The amplification factors \(\left| {E_{spot} } \right|^{2} = \sum\limits_{n = 0}^{{n_{\max } }} {\left| {E_{n} } \right|^{2} }\) representing the incident field squared \(E^{2}\) were calculated on the symmetric \(z\)-axis by the sum of the terms squared for the center of the coordinate system, and on the \(z\)-axis for the distance \(\delta = 0.08R\) and \(\delta = 0.1R\) from its center. They are presented in Table 1. We find that \(\left| {E_{spot} } \right|^{2}\) is amplified very much for molecules absorbed on the metallic surface on the symmetric z-axis. Table 1 demonstrates this effect.

We should take into account that SERS enhancement factor (\(EF\)) [43,44,45,46] is given by \(\left| {E_{spot} } \right|^{4}\) as given in Eq. (37).

In Table 2, we made the calculation for \(\sum\limits_{n} {\left| {E_{n} /E} \right|^{2} }\) and its square representing SERS \(EF\) at the center of the coordinates system using Eqs. (37) and (40). The distance between the two metallic spheres is assumed to be very small given by: \(2\delta /R = 0.01\). Straight forward calculation by Eq. (5) gives for this case: \(a = 0.1R\). Then by Eq. (9) we get approximately \(\sinh \eta_{0} = \frac{a}{R} = 0.1 = \eta_{0}\).We assumed in this table the same dielectric constants that are given in Table 1. Table 2 is given as follows:

Table 2 Values of \(\sum\limits_{n} {\left| {E_{n} /E} \right|^{2} }\) and its square representing SERS \(EF\) at the center of the coordinates system are calculated for the present dimer system for two different distances between the two metallic spheres
Full size table

Assuming, for example, that the radius of the metallic spheres is of order \(100\;{\text{nm}}\) which is smaller than the optical wavelength then the minimal length of the gap between the two spheres with \(2\delta /R = 0.01\) is order of 1 \({\text{nm}}\) which is still larger than the size of molecule (see e. g [44]). In such cases the SERS enhancement factor (EF) calculated by Eq. (37) becomes extremely large. To obtain the convergence of the summation of \(\sum\limits_{n = 0}^{{n_{\max } }} {\left| {E_{n} /E} \right|^{2} }\) for \(2\delta /R = 0.01\) the calculation of 36 terms (\(n_{\max } = 36\)) were needed. Such calculations were made in straightforward way but many lengthy calculations were needed.

The present article is based on classical model but when the gap length is of a truly atomic scale quantum effects become important. Such quantum effects, including tunneling between the nanoparticles which reduces the EF, were treated, for example, in Refs. [48, 49] added now to the article.

7 Comparison between the present analytical model for obtaining maximal enhancement factor (EF) for single molecule in hot spots of spherical dimers and other authors methods

Quite often the computational methods concentrate on the calculations of the absorbance and scattering of the nanoparticles, where such particles act as sensors (e.g., in [42]). We are interested, however, in the methods for obtaining extremely large enhancement of the EM field so that it can be used for SERS enhancement factor (\(EF)\) on the one molecule level stronger than other nonlinear effects [43,44,45,46]. One can find in the literature anything from \(10^{4}\) to \(10^{14}\), quoted as SERS enhancements [26, 45]. Such quotations led authors to claim: "the hot spot phenomenon is still an allusive and feebly understood subject" [44], also we find in the abstract of Ref. [7]: "\(\cdot \cdot \cdot\) the dimers provide a well-defined system for studying the hot spot phenomenon \(\cdot \cdot \cdot\) an extremely important but poorly understood subject \(\cdot \cdot \cdot\) ". However, this phenomenon can be understood by taking into account the different experimental conditions.

The common definition of EM enhancement field (\(EF\)) is given by [26, 43,44,45,46]

$$ {\text{EF}} = \frac{{I_{{{\text{SERS}}}} /N_{{{\text{SERS}}}} }}{{I_{{{\text{NRS}}}} /N_{{{\text{NRS}}}} }}, $$
(42)

where \(I_{{{\text{SERS}}}}\) and \(I_{{{\text{NRS}}}}\) are the SERS and normal Raman scattering (NRS) intensities, respectively, and where \(N_{{{\text{SERS}}}}\) and \(N_{{{\text{NRS}}}}\) are the number of molecules contributing to the scattering intensity of SERS and NRS, respectively. Equation (42) defines the one molecule enhancement factor (EF) depending on the numbers assumed for \(N_{{{\text{SERS}}}}\) and \(N_{{{\text{NRS}}}}\). It was shown by considering: "Basic electromagnetic theory of SERS" [47] that Local Field Intensity Enhancement Factor (LFIEF) is given by \(\frac{{\left| {E(\vec{r},\omega )} \right|^{2} }}{{\left| {E_{0} } \right|^{2} }}\) and SERS EF is given approximately by \(\frac{{\left| {E(\vec{r},\omega )} \right|^{4} }}{{\left| {E_{0} } \right|^{4} }}\) in the hot spot. Corresponding calculations were demonstrated by in Tables 1 and 2 using the bi-spherical model as a summation of \(n\) terms (\(n = n_{\max }\)) for the spherical dimer system at two different distances between the metallic spheres.

We would like to compare our results for \(EF\) with those of other computational models and for this purpose we use the results summarized in Table 3 for spherical dimers. An important parameter which follows from the use of the bi-spherical coordinates is given by the ratio \(\frac{d}{R}\), where \(d \equiv 2\delta\) is the gap length (i.e., the shortest distance between the two spheres in our analysis) and \(R\) is the radius of the two equal spheres (\(R_{1} = R_{2}\)). Maximal \(EF\) is obtained in our analysis under the condition that the electric field is along the inter-particle axis (parallel polarization). Although the general results for this case depend on two bi-spherical coordinates \(\eta\) and \(\alpha\) at the hot spot we have \(\cos \alpha = - 1\) and we remain with a dependence only on the radial coordinate \(\eta\) (see Fig. 2 for visualization these relation). Each spherical symmetric dimer is described in Fig. 2 by a parameter \(\eta_{0}\), where we have the relation \(\sinh \eta_{0} = \frac{a}{R}\) and at the hot spots \(\sinh \eta_{0}\) is nearly equal \(\eta_{0}\) (see Eq. (41) for realization of such calculations). The conclusion from this discussion is that the EF in spherical dimers depends on the combined parameter \(\frac{d}{R}\) and not on separate two parameters. The physical explanation for such relation is that not only shorter gaps increase \(EF\) but also it is increased by a larger curvature of the nano-particles at the hot spots. Although we showed this criterion for spherical dimers I believe that this ratio will play important role in more general cases of hot spots, where \(R\) will be exchanged by the curvature at the hot spot. I include the parameter \(\frac{d}{R}\) in Fig. 3, where I compare my results with other authors as the EF is found to be function of this parameter, while both parameters were given by other authors.

Table 3 Present theoretical results for \(EF\) given in Tables 1 and 2 are compared with \(EF\) calculations made by other authors
Full size table

For studying dimers with gaps lengths of order \(1\;{\text{nm}}\), it was found that is necessary to go beyond the accuracy of grid-based methods, such as DDA and FDTD and perform certain analytical results [52[. Electrodynamic calculations using FEM and analytical expressions were developed to determine the electric fields EF [50,51,52] with gap lengths of the \(1\;{\text{nm}}\) scale. We find also that in spite of the fact that the present analytical method is different from the analytical methods developed by the other authors given in Table 3, there is a fair correlation between the EF maximal values given by these different methods as function of the parameter \(\frac{d}{R}\) (Deviation from such correlations occurs in Ref. [7], where SERS was correlated with SEM imaging and it seems that such correlation is not successful). An important conclusion is obtained from Table 3 by which extremely large EF can be obtained around \(10^{12}\) for dimers with a gap length around 1 \({\text{nm}}\). One should take into account that for dimers with very short gaps which are on atomic scale quantum effects become important. A comment about this problem was given in the article on page 23.

8 Summary and conclusions

We developed in the present article Laplace equation solutions for two metallic spheres interacting with the external EM incident in the symmetric \(z\)-axis as described in Fig. 1, where this system has cylindrical symmetry around the \(z\)-axis. The electrostatic potential \(\psi_{1} (\eta ,\alpha )\) representing Laplace equation solution at the hot spot was developed in Eq. (15) as function of the bi-spherical coordinates \(\eta ,\alpha\), where for bi-spherical coordinates \(\eta\) represents the distance from the center and \(\alpha\) represents an angle from the reference direction. The coordinates \(\eta ,\alpha\) are described, therefore, as bi-spherical polar coordinates in the \(x,z\) plane of Fig. 2, and these coordinates are not changed by rotation around the \(z\)-axis. Such geometry leads to localized surface plasmons of opposite charge in the small gap between them. The potential \(\psi_{1} (\eta ,\alpha )\) in the small gap is proportional to summation of Legendre polynomials \(P_{n} (\alpha )\) with proportionality coefficients \(A_{n}\) and is also proportional to \(\sinh\) function. Using boundary conditions we obtained after some calculations and certain approximations (including the condition \(\delta \le (1/10)R\)) a general equation for the coefficients \(A_{n}\) in Eq. (29). These coefficients are proportional to the external EM field, are proportional to the difference in the dielectric constants \(\varepsilon (\omega ) - \varepsilon_{1}\) and functions of \(\varepsilon (\omega )\) and \(\varepsilon_{1}\) in the denominator of Eq. (29) and are given as a certain function of the parameter \(\eta_{0}\) (defined in Eq. (9)). General solution for the EM field in the bi-spherical radial direction \(\eta\) is derived in Eq. (32). The amplified EM field is found to be proportional to the sum of products of the coefficients \(A_{n}\) with Legendre polynomial \(P_{n} (\cos \alpha )\) and with \(\cosh\) function.

As the hot spots in dimers are produced on (or near) the symmetric \(z\)-axis (\(x = y = 0\)) we simplified the calculations using this condition. The relation: \(\cos \alpha = - 1\) was derived by following Eqs. (6–7) and is confirmed by the diagram of Fig. 2. The Legendre polynomials are simplified using Eq. (35) and the final result in bi-spherical coordinates for the EM field on the symmetric \(z\)-axis is given by Eq. (37). Taking into account symmetry considerations the fourth power of the electric field at hot spots for SERS and TPI-PL measurements is given by the incoherent summation of Eq. (37). Transforming Eq. (37) to its values on the \(z\) coordinate we obtain the simple Eq. (39) with a very simple form in the center of the coordinates system given by Eq. (40). We demonstrated our final results for two different distances between the metallic spheres in Tables 1, 2.

A comparison between the present work and other computational methods for getting large EF for spherical dimers with gaps lengths around \(1\;{\text{nm}}\) was made using Table 3. Although a very special spherical dimer system has been analyzed analogous properties are obtained for dimers which have different structures [44].