INTRODUCTION

Surface plasmons allow one to restrict electromagnetic fields to a subwave scale, significantly exceeding the classical limit of optical diffraction. A new era of quantum plasmonics, which investigates the quantum behavior of surface plasmons and their interactions with a substance, has opened with a continuous decrease in the volume of optical modes in the depth of a subwave scale. This novel and intriguing field creates many new opportunities for pushing the boundaries of fundamental science and applied quantum technology [1].

In recent decades, the use of core–shell particles has gained huge popularity due to the possibility of manipulation of materials and sizes of both the core and the shell to control their plasmonic properties. This issue distinguishes them favorably from homogeneous particles. Due to impressive advances in material science, core–shell nanoparticles can now be synthesized with improved physicochemical properties and clearly determined sizes, shapes, and compositions [2]. The ability of core–shell nanoparticles for ultrahigh enhancement of the field and concentration of a subwave field has led to a wide variety of applications in many fields, such as energy accumulators and converters, optical sensors, and means for diagnostics and treatment of tumors [3–5]. The problem of the development and realization of a plasmonic nanolaser generates particular interest in nanoplasmonics. Plasmonic nanolasers based on the use of core–shell resonators have the advantages of their nanosize, low energy threshold, and ultrashort response time [6, 7].

In a number of practical applications, it is necessary to study the optical properties of core–shell nanoparticles located on a dielectric substrate. In this case, it is very important to strictly take the interaction between the nanoparticle and surface of the substrateinto account. In particular, this issue appears in solving the problem of synthesis of core–shell structures for provision of required spectral properties [8, 9]. When the thickness of plasmon coverage decreases to several nanometers, the interaction of electrons in the plasmonic metal should be taken into account on a different level. In this case, the layer thickness becomes comparable with the Fermi wavelength for electrons in metal (\(\sim\)5 nm for gold and silver), and so-called spatial dispersion occurs. In this case, the classical Maxwell theory turns out to be insufficient for description of the electromagnetic properties of a plasmonic metal [1].

There are various approaches for description of emerging spatial dispersion (SD) in a metal, starting from the complete quantum-mechanical model, within which the behavior of each electron inside the metal is modeled on the base of the Schr\(\ddot{o}\)dinger equation. It well suited to explain experimental results, but applicable merely to plasmonic particles with sizes of only several nanometers [10]. At present, quasi-classical approaches, which allows one to take emerging quantum effects within the Maxwell electromagnetic theory into account are most demanded. The Drude hydrodynamic model [11, 12] is one of these approaches, which considers the SD of a metal. It allows one to take longitudinal fields emerging in the metal into account, but requires correction of the quantum parameters of a metal when considering particles whose shape differs from a spherical one [13]. The next model, which is free of this restriction, called Generalized Nonlocal Optical Response (GNOR) [14]. Within the GNOR model, the diffusion of electrons in the metal is taken into account in the frame of the hydrodynamic theory. This enables one to study both nonspherical particles and their clusters. The GNOR theory allows one to consider both longitudinal fields that emerge inside the metal and additional boundary conditions, which are set on interfaces between the metal and dielectric [15, 16].

We use the GNOR model within the Discrete Source Method (DSM) [17]. The DSM is a strict numerically analytical surface oriented method. It is based on representation of fields in the form of a finite linear combination of distributed lower order multipoles [18], which satisfy semiclassical Maxwell equations, including longitudinal fields, inside the metal shell [19]. In the presence of a substrate, the Green tensor for a half-space [20] is used for representation of scattered fields out of the particle. Therefore, representations of fields in all regions satisfy the Maxwell equations, infinity conditions, and conditions of matching on an infinite substrate surface, in the analytical form. The corresponding Discrete Source (DS) amplitudes are determined from matching conditions, which are set on surfaces that bound the metal layer. An exceptional feature of DSM consists in the fact that it allows one to evaluate a real error of the obtained solution by calculation of the field discrepancy on the discontinuity surfaces of the characteristics of a medium of the core–shell particle. All these points allow one to use the DSM for analysis of plasmonic nanostructures with account for the SD effect, within the model of the generalized nonlinear response [17–20].

1 DIFFRACTION BOUNDARY PROBLEM FORMULATION

Let the space \({\textrm{R}}^{3}\) be split into two half-spaces: the upper half-space is \(D_{0}\): \((z>0)\) and the dielectric substrate is \(D_{1}:(z<0)\). We denote the flat interface \(\Sigma:\ (z=0)\). Let a spherical core–shell particle be entirely located in the upper half-space \(D_{0}\). The particle core we denote as \(D_{i}\), and the metal shell as \(D_{s}\). We will denote as \(\partial D_{i,s}\) the corresponding spherical boundaries. All media are assumed to be nonmagnetic, and we denote \(\varepsilon_{\nu}\), \(\nu=0,1,i,s\) as their dielectric permeabilities.

The essence of the SD lies in the fact that the connection between electric displacement \(\mathbf{D}(M)\) and electric field \(E(M)\) becomes nonlocal, i.e., the relation \(\mathbf{D}(M)=\varepsilon(M)\mathbf{E}(M)\) is replaced by \(\mathbf{D}(M)=\int\varepsilon(M-M^{\prime})\mathbf{E}(M^{\prime})dM^{\prime}\), which changes to a local one in the case of \(\varepsilon(M-M^{\prime})=\delta(M-M^{\prime})\) [14]. One of the SD manifestations is the appearance of longitudinal electromagnetic fields inside the metal. In this case, the internal electric field \(\mathbf{E}\) ceases to be purely transverse (\(\textrm{div}\mathbf{E}^{T}=0\)) due to a volume charge formation, and in order to adequately describe processes there it is necessary to additionally involve longitudinal fields (\(\textrm{curl}\mathbf{E}^{L}=0\)) [21]. In order to consider nonlocality, the Drude hydrodynamic theory and its generalization are used, i.e., the GNOR model [12]. Within the GNOR theory, the generalization of Ohm’s law for the conductivity current inside the metal is performed, that is a change of is carried out:

$$\mathbf{J}=\sigma\mathbf{E}\quad\Rightarrow\quad\xi^{2}\textrm{grad}(\textrm{div}\mathbf{J})+\mathbf{J}=\sigma\mathbf{E},$$

where \(\sigma\) is the conductivity of the metal and \(\xi\) is the nonlocality parameter [16]. As a result, the corresponding equation of the Maxwell system for a magnetic field is changed. In view of the above, the electric field is split into transverse and longitudinal fields inside the plasmonic metal: \(\mathbf{E}_{s}=\mathbf{E}_{s}^{T}+\mathbf{E}_{s}^{L}\), \(\textrm{div}\mathbf{E}_{s}^{T}=0\), \(\textrm{curl}\mathbf{E}_{s}^{L}=0\). It can be shown [22] that inside shell \(D_{s}\), these fields satisfy the Helmholtz equations

$$\Delta\mathbf{E}^{T}(M)+k_{T}^{2}\mathbf{E}^{T}(M)=0,$$
(1)
$$\Delta\mathbf{E}^{L}(M)+k_{L}^{2}\mathbf{E}^{L}(M)=0,$$
(2)

here \(k_{T}^{2}=k^{2}\varepsilon_{s}\), \(k_{L}^{2}=\varepsilon_{s}/\xi\) are the transverse and longitudinal wavenumbers, and \(k={\omega/c}\).

We change to the mathematical formulation of the scattering boundary problem for the Maxwell system taking GNOR into account. We denote \(\{\mathbf{E}^{0},\ \mathbf{H}^{0}\}\) as the field of a linearly polarized plane electromagnetic wave, propagating from the lower half-space in a half plane \(\phi=\pi\) and making the angle \(\theta_{0}\) with the 0z axis, directed from \(D_{1}\) to \(D_{0}\). The boundary problem formulation, taking SD into account, can then be written as follows:

$$\textrm{curl}\ \mathbf{H}_{\zeta}=jk\varepsilon_{\zeta}\mathbf{E}_{\zeta};\quad\textrm{curl}\ \mathbf{E}_{\zeta}=-jk\mathbf{H}_{\zeta}$$
$$\text{in }D_{\zeta},\ \zeta=0,1,i,$$
$$\textrm{curl}\ \mathbf{H}_{s}=jk(\varepsilon_{s}+\xi^{2}\textrm{grad}\ \textrm{div})\mathbf{E}_{s}(M);$$
$$\textrm{curl}\ \mathbf{E}_{s}=-jk\mathbf{H}_{s}\quad\text{in }D_{s},$$
$$\mathbf{n}_{i}\times(\mathbf{E}_{i}(P)-\mathbf{E}_{s}(P))=0,$$
$$\mathbf{n}_{i}\times(\mathbf{H}_{i}(P)-\mathbf{H}_{s}(P))=0,\quad P\in\partial D_{i};$$
$$\varepsilon_{i}\mathbf{n}_{i}\cdot\mathbf{E}_{i}(P)=\varepsilon_{L}\mathbf{n}_{l}\cdot\mathbf{E}_{s}(P),$$
$$\mathbf{e}_{z}\times(\mathbf{E}_{0}(Q)-\mathbf{E}_{1}(Q))=0,$$
$$\mathbf{e}_{z}\times(\mathbf{H}_{0}(Q)-\mathbf{H}_{1}(Q))=0,\quad Q\in\Sigma,$$
$$\mathbf{n}_{l}\times(\mathbf{E}_{s}(P)-\mathbf{E}_{0}^{s}(P))=\mathbf{n}_{s}\times\mathbf{E}_{0}^{0}(P),$$
$$\mathbf{n}_{l}\times(\mathbf{H}_{s}(P)-\mathbf{H}_{0}^{s}(P))=\mathbf{n}_{s}\times\mathbf{H}_{0}^{0}(P),\quad P\in\partial D_{s},$$
$$\varepsilon_{L}\mathbf{n}_{s}\cdot\mathbf{E}_{s}(P)=\varepsilon_{0}\mathbf{n}_{s}\cdot(\mathbf{E}_{0}^{0}(P)+\mathbf{E}_{0}^{s}(P)),$$
$$\mathop{\lim}\limits_{r\to\infty}r\cdot\left(\mathbf{H}_{\zeta}^{s}\times\frac{\mathbf{r}}{r}-\sqrt{\varepsilon_{\zeta}}\mathbf{E}_{\zeta}^{s}\right)=0,$$
$$r=|M|\to\infty,\quad\zeta=0,1,\quad z\neq 0;$$
$$\max\big{(}\big{|}\mathbf{H}_{\zeta}^{s}\big{|},\big{|}\mathbf{E}_{\zeta}^{s}\big{|}\big{)}=O\big{(}\rho^{-1/2}\big{)},$$
$$\rho=\sqrt{x^{2}+y^{2}},\quad\rho\to\infty,\quad z=\pm 0.$$
(3)

Here, \(\{\mathbf{E}_{\zeta},\ \mathbf{H}_{\zeta}\}\) are the total fields in \(D_{\zeta}\), \(\zeta=0,1,i,s\), respectively, \(\mathbf{n}_{i,s}\) are the unit normals to the surfaces \(\partial D_{i,s}\), \(\mathbf{e}_{z}\) is the normal to the substrate, and the characteristics of the medium are chosen in such a way that \(\textrm{Im}\varepsilon_{0,1,i}=0\), \(\textrm{Im}\varepsilon_{s}\leq 0\), \(\textrm{Im}\varepsilon_{L}\leq 0\). It is assumed that the time dependence is chosen in the form of \(\exp\{j\omega t\}\).

We specify the remaining entities that enter the formulation of problem (3). The fields \(\{\mathbf{E}_{\zeta}^{0},\ \mathbf{H}_{\zeta}^{0}\}\), \(\zeta=0,1\) are the result of solving the problem of reflection and refraction of the field of plane wave \(\{\mathbf{E}^{0},\ \mathbf{H}^{0}\}\) on the surface of separation of the half-spaces \(\Sigma\). \(\{\mathbf{E}_{\zeta}^{s},\ \mathbf{H}_{\zeta}^{s}\}\), \(\zeta=0,1\) is the scattered field in each of the half-spaces, which is determined as \(\mathbf{E}_{\zeta}^{s}=\mathbf{E}_{\zeta}-\mathbf{E}_{\zeta}^{0}\), \(\mathbf{H}_{\zeta}^{s}=\mathbf{H}_{\zeta}-\mathbf{H}_{\zeta}^{0}\), \(\zeta=0,1\). In view of the construction of the external excitation field and boundary conditions on \(\Sigma\), the scattered field \(\{\mathbf{E}_{\zeta}^{s},\ \mathbf{H}_{\zeta}^{s}\}\), \(\zeta=0,1\) should also satisfy the matching conditions for tangential components on an infinite boundary \(\Sigma\).

Additional conditions for normal components of fields, which are necessary for a unique solution, are set on surfaces \(\partial D_{i,s}\) in addition to the classical matching conditions. These conditions physically correspond to conditions for the vanishing of the normal component of the conduction current on metal–dielectric interfaces \(\mathbf{n}\cdot\mathbf{J}=0\), which are subsequently transformed into conditions for normal components of fields [16]. The radiation conditions for problem (3) are formulated in such a way that the energy flux vanishes at infinity for the homogeneous problem (3) [23]. We will assume that the formulated boundary problem (3) has a unique classical solution.

The parameters \(\xi\) and \(\varepsilon_{L}\), which are related to the longitudinal field \(\mathbf{E}_{s}^{L}\), are determined as follows \(\varepsilon_{L}=\varepsilon_{s}-{\omega_{p}^{2}/\left(j\gamma\omega-\omega^{2}\right)}\), \(\xi^{2}=\) \({\varepsilon_{s}\left(\beta^{2}+D\left(\gamma+j\omega\right)\right)/\left(\omega^{2}-j\gamma\omega\right)}\). Here \(\omega_{p}\) is the plasmonic frequency of the metal, \(\gamma\) is the damping coefficient, \(\beta\) is the hydrodynamic velocity in the plasma, which is connected with the Fermi velocity \(v_{\textrm{F}}\) by relation \(\beta^{2}=3/5v_{\textrm{F}}^{2}\), and \(D\) is the diffusion constant of electrons [14].

2 THE DISCRETE SOURCE METHOD FOR ACCOUNTING FOR THE PRISM AND GNOR MODEL

We will construct an approximate solution of problem (3) in accordance with scheme [20]. We restrict ourselves to the case of P-polarization because this scheme has the largest amplitude of the plasmonic resonance [17]. Since the particle is entirely located in the upper half-space \(D_{0}\), then the field of the refracted wave \(\{\mathbf{E}_{0}^{0},\ \mathbf{H}_{0}^{0}\}\) takes the form

$$\mathbf{E}_{0}^{0}(M)=T^{P}(-\mathbf{e}_{x}\cos\theta_{0}+\mathbf{e}_{z}\sin\theta_{0})$$
$${}\times\exp\{-jk_{0}(x\sin\theta_{0}+z\cos\theta_{0})\},$$
$$\mathbf{H}_{0}^{0}(M)=T^{P}n_{0}e_{y}\exp\{-jk_{0}(x\sin\theta_{0}+z\cos\theta_{0})\}.$$

Here, \(n_{0}=\sqrt{\varepsilon_{0}}\), \(T^{P}\) is the Fresnel refraction coefficient [24], and \(\mathbf{e}_{x},\mathbf{e}_{y},\mathbf{e}_{z}\) are unit vectors of Cartesian frame. In accordance with the Snell law, we have \(n_{0}\sin\theta_{0}=n_{1}\sin\theta_{1}\), \(n_{1}=\sqrt{\varepsilon_{1}}\). Therefore, the refracted angle \(\theta_{0}=\arcsin({n_{1}/n_{0}}{\textrm{sin}}\theta_{1})\) occur. In the case where a wave falls from a denser medium into a less dense medium \(n_{1}>n_{0}\), there is a critical angle \(\theta_{c}=\arcsin({n_{0}/n_{1}})\) (the total reflection angle) beyond which the wave does not pass into the upper half-space, since it is completely reflected from the surface \(\Sigma\). As well, the energy propagates along the surface of separation of the half-spaces and exponentially decays in the direction \(\mathbf{e}_{z}\). In this case \(\sin\theta_{0}>1\), and \(\cos\theta_{0}\) takes the value \(\cos\theta_{0}=-j\sqrt{\sin^{2}\theta_{0}-1}\). Therefore, the amplitude of the plane wave in \(D_{0}\) takes the form \(\exp\left\{-jk_{0}x\sin\theta_{0}\right\}\exp\left\{-k_{0}z\sqrt{\sin^{2}\theta_{0}-1}\right\}\).

We construct an approximate solution of problem (3) for the scattered field in \(D_{0}\) considering an axial symmetry and polarization that satisfies the quasi-classical system of Maxwell equations in all regions of constancy of medium parameters, conditions of radiation, and conditions of matching for fields on \(\Sigma\). We take the Fourier components of the Green tensor of the half-space as a basis for the scattered field representation. These can be written in the form of integral Weyl–Sommerfeld representations [20]:

$$G^{e,h}_{m}(\eta,z_{n})=\int\limits_{0}^{\infty}J_{m}(\lambda\rho)v^{e,h}_{11}(\lambda,z,z_{n})\lambda^{1+m}d\lambda,$$
$$g^{e,h}_{m}(\eta,z_{n})=\int\limits_{0}^{\infty}J_{m}(\lambda\rho)v^{e,h}_{31}(\lambda,z,z_{n})\lambda^{1+m}d\lambda,$$

here, \(J_{m}(\cdot)\) is the cylindrical Bessel function, the point \(\eta=(\rho,z)\) is located at half-plane \(\phi=\mathrm{const}\), and the points of localization of multipoles are distributed along the symmetry axis \(z_{n}\in OZ\), strictly inside \(D_{i}\cup D_{s}\). The spectral functions of the electric and magnetic types \(v_{11}^{e,h},v_{31}^{e,h}\) provide the fulfillment of the matching conditions at the boundary of the interface \(z=0\). In this case, the following expressions are valid for them:

$$v_{11}^{e,h}(\lambda,z,z_{n})$$
$${}=\begin{cases}\cfrac{\exp\left\{-\eta_{0}\left|z-z_{n}\right|\right\}}{\eta_{0}}\\ +A_{11}^{e,h}(\lambda)\cfrac{\exp\left\{-\eta_{0}(z+z_{n})\right\}}{\eta_{0}},\\ z_{n}>0,\quad z\geq 0\\ B_{11}^{e,h}(\lambda)\cfrac{\exp\left\{\eta_{1}z-\eta_{0}z_{n}\right\}}{\eta_{0}},\\ z_{n}>0,\quad z\leq 0,\end{cases}$$
$$v_{31}^{e,h}(\lambda,z,z_{n})$$
$${}=\begin{cases}A_{31}^{e,h}(\lambda)\exp\left\{-\eta_{0}(z+z_{n})\right\},\\ z_{n}>0,\quad z\geq 0\\ B_{31}^{e,h}(\lambda)\exp\left\{\eta_{1}z-\eta_{0}z_{0}\right\},\\ z_{0}>0,\quad z\leq 0.\end{cases}$$

The spectral coefficients \(A,B\) are determined from a one-dimensional problem with matching conditions at \(z=0\), whence it is easy to obtain that

$$A_{11}^{e,h}(\lambda)=\frac{\chi_{0}^{e,h}-\chi_{1}^{e,h}}{\chi_{0}^{e,h}+\chi_{1}^{e,h}};\quad B_{11}^{e,h}(\lambda)=\frac{2\chi_{0}^{e,h}}{\chi_{0}^{e,h}+\chi_{1}^{e,h}};$$
$$A_{31}^{e,h}(\lambda)=\frac{2\delta}{\left(\chi_{0}^{e}+\chi_{1}^{e}\right)\left(\chi_{0}^{h}+\chi_{1}^{h}\right)};\quad\delta={1/\varepsilon_{0}}-{1/\varepsilon_{1}},$$
$$B_{31}^{e,h}(\lambda,z_{0})=\left(1,\frac{\varepsilon_{1}}{\varepsilon_{0}}\right)\frac{2\delta}{\left(\chi_{0}^{e}+\chi_{1}^{e}\right)\left(\chi_{0}^{h}+\chi_{1}^{h}\right)},$$

where \(\eta_{\zeta}=\sqrt{\lambda^{2}-k_{\zeta}^{2}}\), \(\chi_{\zeta}^{e}=\eta_{\zeta}\), \(\chi_{\zeta}^{h}={\eta_{\zeta}/\varepsilon_{\zeta}}\), \(\zeta=0,1\).

In order to construct an approximate solution for the scattered field in \(D_{0}\), vector potentials are used. They are written in the cylindrical frame as

$$\mathbf{A}_{mn}^{(e)0}=\{G_{m}^{e}(\eta,z_{n}^{e})\cos(m+1)\varphi;$$
$$-G_{m}^{e}(\eta,z_{n}^{e})\sin(m+1)\varphi;$$
$$-g_{m}^{e}(\eta,z_{n}^{e})\cos(m+1)\varphi\},$$
$$\mathbf{A}_{mn}^{(h)0}=\{G_{m}^{h}(\eta,z_{n}^{e})\sin(m+1)\varphi;$$
$$\phantom{-}G_{m}^{h}(\eta,z_{n}^{e})\cos(m+1)\varphi;$$
$$-g_{m+1}^{h}(\eta,z_{n}^{e})\sin(m+1)\varphi\},$$
$$\mathbf{A}_{0n}^{(e)0}=\{0;\ 0;\ G_{0}^{h}(\eta,z_{n}^{e})\}.$$
(4)

In order to construct fields inside regions \(D_{i,s}\), the following potentials will be used:

$$\mathbf{A}_{mn}^{(e)\nu}=\{Y_{m}^{\nu}(\eta,z_{n}^{\nu})\cos(m+1)\varphi;$$
$$-Y_{m}^{\nu}(\eta,z_{n}^{\nu})\sin(m+1)\varphi;\ 0\},\quad\nu=i,s\pm;$$
$$\mathbf{A}_{mn}^{(h)\nu}=\{Y_{m}^{\nu}(\eta,z_{n}^{\nu})\sin(m+1)\varphi;$$
$$Y_{m}^{\nu}(\eta,z_{n}^{\nu})\cos(m+1)\varphi;\ 0\},$$
$$\mathbf{A}_{n}^{(e)\nu}=\{0;\ 0;\ Y_{0}^{\nu}(\eta,z_{n}^{\nu})\}.$$
(5)

Here, \(Y_{m}^{i}(\eta,z_{n}^{i})=j_{m}(k_{i}r_{\eta{z}_{n}^{i}})({\rho/r_{\eta{z}_{n}^{i}}})^{m}\), \(j_{m}(\cdot)\) is the spherical Bessel function, \(Y_{m}^{s\pm}(\eta,z_{n}^{s})=h_{m}^{(2,1)}(k_{s}r_{\eta{z}_{n}^{s}}{)(}{\rho/r_{\eta{z}_{n}^{s}}})^{m}\), \(h_{m}^{(2,1)}(\cdot)\) are the spherical Hankel function, corresponding to outgoing and incoming waves, \(r_{\eta{z}_{n}}^{2}=\rho^{2}+\left(z-z_{n}\right)^{2}\), \(\eta=(\rho,z)\), \(k_{i,s}=k\sqrt{\varepsilon_{i,s}}\), and \(z_{n}^{i,s}\) are coordinates of discrete sources (DS). It should be noted that functions that are taken as a basis for the vector potentials (4), (5), satisfy the Helmholtz equation (1).

For the case of P-polarization, a longitudinal field is constructed, based on the following scalar potentials [19]:

$$\Psi_{mn}^{s\pm}(M)$$
$${}=h_{m+1}^{(2,1)}(k_{L}R_{\eta z_{n}^{s}})P_{m+1}^{m+1}(\cos\theta_{z_{n}^{s}})\cos(m+1)\varphi,$$
$$\Psi_{n}^{s\pm}(M)=h_{0}^{(2,1)}(k_{L}R_{\xi z_{n}^{s}}),$$

which satisfy the Helmholtz equation (2). Then, an approximate solution for the total field inside the particle and scattered into \(D_{0}\), corresponding to P-polarization takes the following form:

$$\mathbf{E}_{\nu}^{TN}=\sum_{m=0}^{M}\sum_{n=1}^{N_{\nu}^{m}}\Bigg{\{}p_{mn}^{\nu}\frac{j}{k\varepsilon_{\nu}}\textrm{curl}\ \textrm{curl}\ \mathbf{A}_{mn}^{(e)\nu}$$
$${}+q_{mn}^{\nu}\frac{1}{\varepsilon_{\nu}}\textrm{curl}\ \mathbf{A}_{mn}^{(h)\nu}\Bigg{\}}+\sum_{n=1}^{N_{v}^{0}}r_{n}^{\nu}\frac{j}{k\varepsilon_{\nu}}\textrm{curl}\ \textrm{curl}\ \mathbf{A}_{n}^{(e)\nu};$$
$$\mathbf{E}_{s\pm}^{LN}=\sum_{m=0}^{M}\sum_{n=1}^{N_{s}^{m}}\bar{p}_{mn}^{s\pm}\textrm{grad}\Psi_{mn}^{s\pm}+\sum_{n=1}^{N_{s}^{m}}\bar{r}_{n}^{s\pm}\textrm{grad}\Psi_{n}^{s\pm};$$
$$\mathbf{H}_{\nu}^{N}=\frac{j}{k}\textrm{curl}\ \mathbf{E}_{\nu}^{N},\quad\nu=0,i,s\pm.$$
(6)

We note that inside the shell \(D_{s}\), the electromagnetic field is constructed as a sum of outgoing and incoming waves, that is

$$\mathbf{E}_{s}^{N}=\mathbf{E}_{s+}^{TN}+\mathbf{E}_{s-}^{TN}+\mathbf{E}_{s+}^{LN}+E_{s-}^{LN},$$
$$\textrm{div}\mathbf{E}_{s\pm}^{TN}=0,\quad\textrm{curl}\ \mathbf{E}_{s\pm}^{LN}=0.$$

The constructed fields (6) satisfy the quasi-classical system of Maxwell equations of boundary problem (3) and matching conditions on an infinite substrate surface \(\Sigma\). As well, the unknown amplitudes of DS \(\left\{p_{mn}^{\nu},q_{mn}^{\nu},r_{n}^{\nu};\bar{p}_{mn}^{s\pm},\bar{r}_{n}^{s\pm}\right\}\) are determined from matching conditions on surfaces \(\partial D_{i,s}\). The numerical algorithm is constructed according to the standard scheme, as explained in [17], considering the features of the behavior of longitudinal and transverse wave numbers [19].

In order to calculate the characteristics of the far-field zone scattering we need the direction diagram for the scattered field \(\mathbf{F}(\theta,\phi)\), which is determined in the upper and lower half-spaces as

$${\mathbf{E}_{0,1}^{s}(r)/\left|\mathbf{E}^{0}(z=0)\right|}$$
$${}=\frac{\exp\{-jk_{0,1}r\}}{r}\mathbf{F}^{(0,1)}(\theta,\phi)+O({1/r^{2}}),\quad r\to\infty.$$

Then, for P-polarization, the (\(\theta,\varphi\)) components of the diagram on the unit upper hemisphere \(\Omega^{+}=\left\{0\leq\theta\leq\pi/2;0\leq\phi\leq 2\pi\right\}\) take the form

$$F_{\theta}^{P(0)}(\theta,\phi)=jk_{0}\sum_{m=0}^{M}\cos\left(\left(m+1\right)\phi\right)(j\sin\theta)^{m}$$
$${}\times\sum_{n=1}^{N_{0}^{m}}\{p_{nm}^{0}[\overline{G}_{n}^{e(0)}\cos\theta+jk_{0}\overline{g}_{n}^{e(0)}\sin^{2}\theta\,]$$
$${}+q_{nm}^{0}\overline{G}_{n}^{h(0)}\}-j\frac{k_{0}}{\varepsilon_{0}}\sin\theta\sum_{n=1}^{N_{0}^{0}}r_{n}^{0}\overline{G}_{n}^{h(0)},$$
$$F_{\phi}^{P(0)}(\theta,\phi)=-jk_{0}\sum_{m=0}^{M}\sin\left((m+1)\phi\right)(j\sin\theta)^{m}$$
$${}\times\sum_{n=1}^{N_{0}^{m}}\{p_{nm}^{0}\overline{G}_{n}^{e(0)}+q_{nm}^{0}[\overline{G}_{n}^{h(0)}\cos\theta$$
$${}+jk_{0}\overline{g}_{n}^{h(0)}\sin^{2}\theta]\},$$
(7)

where the corresponding spectral functions \(\overline{G}_{n}^{e,h}\), \(\overline{g}_{n}^{h}\) can be represented in the form

$$\overline{G}_{n}^{e,h(0)}(\theta)=\exp\left\{jk_{0}z_{n}\cos\theta\right\}$$
$${}+{\textrm{A}}_{11}^{e,h}(k_{0}\sin\theta)\exp\left\{-jk_{0}z_{n}\cos\theta\right\},\quad z_{n}>0;$$
$$\overline{g}_{n}^{e,h(0)}(\theta)=jk_{0}\cos\theta{\textrm{A}}_{31}^{e,h}(k_{0}\sin\theta)$$
$${}\times\exp\left\{-jk_{0}z_{n}\cos\theta\right\}.$$

In the lower half-space \(\Omega^{-}=\{\pi/2\leq\theta\leq\pi;0\leq\phi\leq 2\pi\}\), the components of the diagram have the form

$$F_{\theta}^{P(1)}(\theta,\phi)=jk\left|\cos\theta\right|\sum_{m=0}^{M}(jk_{1}\sin\theta)^{m}$$
$${}\times\cos(m+1)\varphi\sum_{n=1}^{N_{m}}\{p_{nm}^{0}[\bar{G}_{n}^{e(1)}\cos\theta$$
$${}+jk_{1}\sin^{2}\theta\bar{g}_{n}^{e(1)}]+q_{nm}^{0}\bar{G}_{n}^{h(1)}\}$$
$${}-jk\sin\theta\left|\cos\theta\right|\sum_{n=1}^{N_{0}}r_{n}^{0}\bar{G}_{n}^{e(1)},$$
$$F_{\phi}^{P(1)}(\theta,\phi)=-jk\left|\cos\theta\right|\sum_{m=0}^{M}(jk_{1}\sin\theta)^{m}$$
$${}\times\sin(m+1)\varphi\sum_{n=1}^{N_{m}}\{p_{nm}^{0}\bar{G}_{n}^{e(1)}+q_{nm}^{0}[\bar{G}_{n}^{h(1)}\cos\theta$$
$${}+jk_{1}\sin^{2}\theta\bar{g}_{n}^{h(1)}]\},$$
(8)

where the spectral functions \(\overline{G}_{n}^{e,h(1)},\overline{g}_{n}^{e,h(1)}\) are written as follows

$$\bar{G}_{n}^{e,h(1)}(\theta)=(k_{1},jk)B_{11}^{e,h}(k_{1}\sin\theta)$$
$${}\times\exp\left\{-jk\left(\sqrt{1-\varepsilon_{1}\sin^{2}\theta}\right)z_{n}\right\},$$
$$\bar{g}_{n}^{e,h(1)}(\theta)=(k_{1},jk)B_{31}^{e}(k_{1}\sin\theta)$$
$${}\times\exp\left\{-jk\left(\sqrt{1-\varepsilon_{1}\sin^{2}\theta}\right)z_{n}\right\}.$$

Having determined the DS amplitudes for the scattered field, one can easy calculate the components of the direction diagram (7), (8) everywhere on the unit sphere, as well as the field (6) in the immediate vicinity of the particle. It should be emphasized that the scattering diagram in the entire space is calculated based on the same DS amplitudes \(\left\{p_{nm}^{0},q_{nm}^{0},r_{n}^{0}\right\}\), which is a consequence of using the Green tensor that leads to a unified representation for the scattered field everywhere in \(D_{0,1}\).

3 RESULTS AND DISCUSSION

We consider a core–shell spherical particle with a fixed core diameter \(D=\) 16 nm, consisting of SiO\({}_{2}\) with refraction index \(n_{i}=1.46\) and a golden shell, whose thickness we denote as \(d\). Let the particle be located on the glassy substrate \({\textrm{SF}}11\) with refraction index \(n_{1}=1.78\) in water with index \(n_{0}=1.33\). In this case, the critical angle is \(\theta_{A}=48.35^{\circ}\), and the corresponding region of evanescent waves begins immediately beyond this angle. We mainly use the angle of incidence \(\theta_{0}=50^{\circ}\). The frequency dispersion of gold is taken into account in the calculations in accordance with the experimental results [25].

Fig. 1
figure 1

Comparative results of the intensity of scattering \(\sigma_{\text{sc}}^{0,1}(\theta_{0}=50^{\circ},\lambda)\) to the upper half-space and to the prism.

Full size image
Fig. 2
figure 2

The behavior of the absorption cross-section \(\sigma_{\text{abs}}(\theta_{0})\) depending on the angle of the wave incidence for the thicknesses \(d\) = 2, 4 nm.

Full size image

We are interested in the integral scattering cross-section

$$\sigma_{\text{sc}}^{0,1}(\theta_{0},\lambda)=\int\limits_{\Omega^{\pm}}\left|F_{\theta}^{(0,1)}(\theta_{0},\theta,\phi)\right|^{2}$$
$${}+\left|F_{\phi}^{(0,1)}(\theta_{0},\theta,\phi)\right|^{2}d\omega,$$
(9)

the absorption cross-section, which has the form

$$\sigma_{\text{abs}}(\theta_{0},\lambda)=-\textrm{Re}\int\limits_{\partial D_{s}}\left(\mathbf{E}_{0}^{N}+\mathbf{E}_{0}^{0}\right)$$
$${}\times\left(\mathbf{H}_{0}^{N}+\mathbf{H}_{0}^{0}\right)^{*}d\sigma,$$
(10)

as well as the integral field intensity enhancement coefficient

$$E(\theta_{0},\lambda)$$
$${}=\int\limits_{\partial D_{s}}\left|\mathbf{E}_{0}^{N}+\mathbf{E}_{0}^{0}\right|^{2}d\sigma\bigg{/}\int\limits_{\partial D_{s}}\left|\mathbf{E}_{0}^{0}\right|^{2}d\sigma.$$
(11)

The dimensions of the intensity and absorption cross-section are \(\mu\)m\({}^{2}\).

We change to the analysis of the SD influence on the scattering cross-section (9), absorption cross-section (10), and enhancement coefficient (11). For gold, the corresponding quantum parameters, which are necessary for the calculation of the nonlocal quantities \(\varepsilon_{L}\) and \(k_{L}\), are chosen in accordance with [26], that is

$$\hbar\omega_{p}=9.03\text{eV},\quad\hbar\gamma=0.053\text{eV},$$
$$v_{F}=1.40\times 10^{12}{\mu}\text{m/s},\quad D=8.62\times 10^{8}{\mu}\text{m}^{2}/\text{s}.$$

Giving the wavelength of the external excitation \(\lambda\) and calculating the corresponding value of \(\omega\), it is easy to determine the values of the nonlocal parameters \(\varepsilon_{L}\) and \(k_{L}\).

Figure 1 presents the comparative results of the intensity of scattering \(\sigma_{\text{sc}}^{0,1}(\theta_{0}=50^{\circ},\lambda)\) into the upper half-space and into a prism for the film thickness \(d=4\) nm at the angle of the wave incidence \(\theta_{0}=50^{\circ}\). The results show that under this excitation the major portion of the energy is scattered back into the prism. In addition, it can be seen that accounting for SD (NLE) reduces the intensity at the maximum by 25% at a small shift to a short-wave region. Similar results also occur at the film thickness \(d=2\) nm.

Figure 2 is devoted to consideration of behavior of the absorption cross-section \(\sigma_{\text{abs}}(\theta_{0})\) depending on the angle of the wave incidence for thicknesses \(d=2,4\) nm. The curves correspond to the wavelengths of the plasmonic resonance. It can be seen that maxima are reached near the total internal reflection angle \(\theta_{A}=48.35^{\circ}\). In this connection, the choice of \(\theta_{0}=50^{\circ}\) seems to be justified.

The integral enhancement coefficient \(E(\theta_{0}=50^{\circ},\lambda)\), which depends on the wavelength, and the comparison of the local (LRA) and nonlocal cases (NLE) can be seen in Fig. 3. It is seen that decreasing the film thickness leads to a significant increase of the field enhancement and the shift to a long-wavelength region. As before, taking SD into account leads to reduction of the amplitudes.

We change to the study of the influence of the asymmetry of the geometry of a core–shell particle on the characteristics of the near field, and we restrict ourselves to the nonlocal case. We consider the characteristic case where the centers of the spheres are shifted with respect to each other [27]. Figure 4 presents the values \(\sigma_{\text{abs}}(\theta_{0}=50^{\circ},\lambda)\) for \(d=2\) nm and three different configurations of the spheres: when the centers coincide, a shift of the center of the internal sphere up by 1 nm or down by 1 nm occurs. Therefore, upon an upward shift, the film thickness is 3 nm at the bottom and 1 nm at the top, and conversely, upon a downward shift, the film thickness is 1 nm at the bottom and 3 nm at the top. From these results a significant shift of the position of plasmonic resonance should be noted towards the long-wavelength region by 30 nm upon a simultaneous monotonic decrease in the amplitude by 25%. Figure 5 gives similar results for the enhancement coefficient \(E(\theta_{0}=50^{\circ},\lambda)\). In this case, we note that upon a shift by the same 30 nm, the maxima of the amplitudes change nonmonotonically.

Fig. 3
figure 3

The integral enhancement coefficient \(E(\theta_{0}=50^{\circ},\lambda)\) depending on the wavelength, comparison of the local (LRA) and nonlocal (NLE) cases.

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Fig. 4
figure 4

The values of \(\sigma_{\text{abs}}(\theta_{0}=50^{\circ},\lambda)\) for \(d=2\) nm and three different configurations of the spheres.

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Fig. 5
figure 5

The enhancement coefficients \(E(\theta_{0}=50^{\circ},\lambda)\) for \(d=2\) nm and three different configurations of the spheres.

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CONCLUSIONS

The discrete sources method was adapted for analysis of the scattering properties of core–shell particles, which are located on a substrate in a field on evanescent waves, considering the spatial dispersion in the metal and the asymmetry of the geometry. The numerical study showed that in the case of excitation by the evanescent wave, the major portion of the scattered energy returns to the substrate. It was established that the asymmetry of the geometry significantly affects the near field intensity; namely, the maximum of the absorption cross-section and the enhancement coefficient shifts towards a long-wavelength part of the spectrum by 30 nm and the change of the amplitude of plasmonic resonance reaches 25%.