Internal Reflection of the Surface of a Plasmonic Substrate Covered by Active Nanoparticles

Abstract

Self-consistent analytical approach was developed to describe the optical properties of an assembly of evenly distributed nanoparticles on a plasmonic substrate. The approach is based on the effective susceptibility concept in the frame of the Green function method with the account of the local-field effects in the system. The analytical expressions for the effective susceptibility (polarizability) of a single cylinder-like particle as well as for an ensemble of such particles on plasmonic substrate were derived. In the frame of developed approach, the analytical expressions for reflection coefficients were obtained. On the base of these coefficients, the influence of geometrical parameters of the particle and their surface concentration on reflectance spectrum of p-polarized electromagnetic radiation in the Kretschmann configuration was studied. The strong coupling of surface and localized modes with their hybridization and Rabi splitting at high nanoparticle concentrations was demonstrated.

Appendices

Appendices

1.1 Appendix A

1.1.1 Susceptibility of a Nanoparticle on the Surface in the Near-Field Approximation

Let us consider a cylindrical nanoparticle placed on the surface of an isotropic active plasmonic substrate (metal or n-doped semiconductor) with the dielectric constant ε ₂ (ω) illuminated by long-range electrical field E ⁽⁰⁾(R,ω) as it is illustrated in Fig. 17.2. As linear dimension of the particle is supposed less than the wavelength, all electrodynamic interactions between the particle and the substrate as well as with a surrounding medium can be taken into account by using the Green function in near-field approximation [16, 17]. Then the local field E(R,ω) at any point R in the system can be calculated by the equation

(17.A1)

where V _δ is the so-called exclusion volume which is used to remove the singularity of at R = R′ and is the source dyadic accounting depolarizing properties of V _δ , which depends solely on the shape of the exclusion volume [36, 37].

Choosing the origin of a Cartesian coordinate system on the surface of the plasmonic film with the z-axis directed along the axis of the cylindrical particle, as shown in Fig. 17.2, for our case has the next simple diagonal form:

(17.A2)

where and with the transversal unit tensor e _i (i = x, y) and e _z denote unit vectors in a Cartesian xyz-coordinate system.

According to the method developed in [26], we suppose that similar to Eq. (17.2) for the local current density j(R,ω), one can introduce the relation connecting j(R,ω) with the external field E ⁽⁰⁾(R,ω) illuminating the system:

(17.A3)

where is the tensor of linear response of the particle on the surface to the external field E ⁽⁰⁾(R,ω). Then using the constitutive relation Eq. (17.2) in the form

(17.A4)

and Eq. (17.A3), the external field can be expressed as

(17.A5)

The reverse matrix exists because there is dissipation in the system.

Substituting Eq. (17.A4) into Eq. (17.A1), one obtains analogously to [26]

(17.A6)

Since Eq. (17.A6) fulfills for all points of the system including points within the volume V - V _δ , we can act on Eq. (17.A6) by the operator . Then, supposing that local field can be represented in the form [26]

(17.A7)

after interchanging of the dummy variables and, correspondingly, the order of integration, one obtains from Eq. (17.A6)

(17.A8)

Using the fact that exponents form the complete set of orthonormal functions, we obtain from Eq. (17.A8)

(17.A9)

Using Eqs. (17.A3), (17.A4), and (17.A9), one can receive the relation between the local and external fields existing within the volume V – V _δ :

(17.A10)

(17.A11)

The electric dipole moment of the current density distribution j(R,ω)

(17.A10)

Taking into account Eq. (17.A4), it can be rewritten in the form

(17.A11)

Replacing the local field E(R,ω) in Eq. (17.A11) by the external field E ⁽⁰⁾(R,ω), we obtain

(17.A12)

Since we consider nanoparticles with sizes less than the wavelength of the external illuminating field, one can omit the variation of the external field across the particle, i.e., in Eq. (17.A12), one can replace E ⁽⁰⁾(R,ω) by E ⁽⁰⁾(R_c,ω), then

(17.A13)

Evaluating the integral in square brackets of Eq. (17.A13) as

(17.A14)

we can introduce the dyadic dipole-dipole polarizability of the particle via

(17.A15)

The tensor for the considered case has the diagonal form

(17.A16)

with the components of the polarizability written as

(17.A17)

where

(17.A18)

(17.A19)

(17.A20)

ε₃ is a dielectric function of surrounding medium.

1.2 Calculation of the Susceptibility of a Nanoparticle Layer

Let us suppose that the tensor of linear response of the cylindrical particle on the surface of an active medium is known; then the electric field in the medium “3” where the particles are located can be written in the form similar to Eq. (17.5):

(17.B1)

where is the external long-range electrical field acting on the particle in the layer

(17.B2)

Let us assume that the Green function of the considered system is known, then

(17.B3)

Substituting Eq. (17.B3) into Eq. (17.B1), one can write

(17.B4)

Taking Eq. (17.B4) at the point z = z _c and using Eq. (17.B3), we obtain the simple relation between the local field E ⁽³⁾ and the external field E ⁽⁰³⁾ at the point z = z _c :

(17.B5)

(17.B6)

The tensor is the dimensionless effective susceptibility of the layer of cylindrical nanoparticles on a surface [16, 17], which accounts both near- and far-field electromagnetic interactions within the layer and with the substrate.