Theory of light reflection and transmission by a plasmonic nanocomposite slab: Emergence of broadband perfect absorption

Efficient broadband optical absorption is required in diverse applications, in particular, in solar cells, thermal sources and bolometers. On the other hand, the incident light being almost completely absorbed by a structure is faintly reflected that is an advantage for anti-reflective coatings which are in high demand, for example, for contrast improvement in optical instruments due to the elimination of stray light. A traditional approach to the latter problem is exploiting of either refractive index matching coatings or multi-layer films which provide destructive interference in reflection [1]. In both cases the structure of the coating is dictated by the wavelength of the incident light and is therefore wavelength-specific.
Plasmonic metamaterials, which are artificial structures containing nano-sized metallic inclusions, suggest a novel solution in this field. Metal nanoparticles (NPs) support collective electron oscillations known as localized surface plasmons (LSPs) which have broad resonances in the optical spectral range. NPs can be arranged at a metal film, either regularly or randomly, being separated from it by a spacer layer (film-coupled nanoantennas) [2, 3, 4, 5, 6, 7, 8, 9, 10]. In such a case the gaps between NPs and the film support nanocavity plasmonic resonances which participate in the optical response of the structure as well. These resonances are dictated by the geometry and dimensions of the metamaterial unit cell and are responsible for narrow-band perfect absorption. A different metamaterial design which ensures a broadband efficient absorption contains a nanostructured metal film instead of patch nanoantennas [11, 12].
An alternative approach to plasmonic perfect absorbers is based on nanocomposite ultra-thin films in which metal NPs are distributed randomly throughout the host dielectric in the course of the fabrication process [13, 14, 15, 16]. Such structures provide absorption at the level of above 90% in a wide spectral range for a broad interval of angles of incidence. Besides that, their absorption spectra weakly depend on the angle of incidence and polarization of the illuminating light. In this paper, we focus ourselves on this kind of plasmonic perfect absorbers.
To explain their results, the authors of Ref. [13] assumed that the optical response of the Ag-Al₂O₃ nanocomposite film is the same as that of an Ag film of thickness equal to the Ag NP diameter and introduced an effective complex refractive index deduced from ellipsometric measurements with a silver film. This approach allowed one to reproduce large values of the absorption coefficient, however did not display its broadband character. The broadband perfect absorption observed in the nanocomposite structures containing Au and Cu NPs [14, 15] was attributed to both the coupling between the LSPs in metal NPs and propagating surface plasmon polaritons in the underlying metal film and to the resonances in the spacer layer between them, although no supporting calculations were provided. These papers noted, however, that reflections from the nanocomposite boundaries probably play a role in perfect absorption.
In order to consistently explain these observations and be able to engineer plasmonic perfect absorbers one needs a first-principles approach. Recently, we have developed a theory which rigorously treats the local field in a nanocomposite film of sub-wavelength thickness [17]. We have shown that the account of the field scattered by NPs in the nanocomposite slab and reflected by its boundaries leads to anisotropy and nonlocality of the nanocomposite optical response. It was speculated that this effect may play a role in perfect absorption and that the light scattering at surface roughness can contribute to it.
In this paper, basing on a proper account of the local field in a plasmonic nanocomposite slab, we develop the theory of light reflection and transmission by such a slab. We derive analytical expressions for the reflection and transmission coefficients which should be used for such a structure instead of the Fresnel equations. To match the experimental conditions we develop a model which takes into account surface roughness and investigate its impact on the reflection spectrum. A special attention is paid to the enhancement of the relaxation processes which disrupt the LSP oscillations in dense NP arrays - the mechanisms which, to the best of our knowledge, have not been discussed in the literature on the effective medium theory before.
The paper is organized as follows. In Sec. II we briefly review the local field approach which was first developed in Ref. [17] and extends the Maxwell Garnett approximation to optically thin nanocomposites. In Sec. III we derive the reflection and transmission coefficients for a slab for both TM and TE polarizations which take into account light scattering at surface roughness. Some numerical results which illustrate the theory as well as their comparison with the Maxwell Garnett and Bruggeman approximations are given in Sec. IV. Section V summarizes the main results of the paper.

We consider an infinite nanocomposite slab of thickness $d$ (medium 2) which is enclosed between semi-infinite media 1 and 3 with the dielectric functions $\epsilon_{1}$ and $\epsilon_{3}$, respectively (Fig. 1). We direct the $z$ coordinate axis from medium 1 to medium 3 along the normal to the slab boundaries and place its origin at the midpoint between them. We assume that the slab contains spherical metal nanoparticles of radius $R$ randomly distributed with the number density $N$, the dielectric function of the host material is $\epsilon_{h}$ and the dielectric function of NPs is described by the Drude model

$$\epsilon(\omega)=\epsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega(\omega+i\Gamma_{NP})},$$

(1)

where $\omega=2\pi c/\lambda$ is the frequency of the incident light with $c$ being the speed of light in vacuum and $\lambda$ being the wavelength, $\epsilon_{\infty}$ is the offset originating from the interband transitions, $\omega_{p}$ is the metal plasma frequency and $\Gamma_{NP}$ is the relaxation constant.
We follow Maxwell Garnett’s assumption [18] that the metal spheres having the radius much smaller than the wavelength of the incident light ($R\ll\lambda$) can be replaced by point dipoles with the dipole moments

$${\bf p}({\bf r},\omega)=\alpha(\omega){\bf E}^{\prime}({\bf r},\omega),$$

(2)

where

$$\alpha(\omega)=\epsilon_{h}R^{3}\frac{\epsilon(\omega)-\epsilon_{h}}{\epsilon(\omega)+2\epsilon_{h}}$$

(3)

is the sphere polarizability in the quasistatic approximation and ${\bf E}^{\prime}({\bf r},\omega)$ is the microscopic (local) field acting on the sphere. Let us note that the zero of the denominator in $\alpha(\omega)$ determines the frequency of the LSP at which the optical response of NPs is maximum. The dipole moment of NPs in a unit volume of the nanocomposite, or their polarization, equals ${\bf P}({\bf r},\omega)=N{\bf p}({\bf r},\omega)$.
The local field in turn can be written in the form of an integral equation [19, 17]

$${\bf E}^{\prime}({\bf r},\omega)={\bf E}({\bf r},\omega)+\int_{V^{\prime}}\bar{\bf F}({\bf r},{\bf r}^{\prime};\omega){\bf P}({\bf r}^{\prime},\omega)d{\bf r}^{\prime},$$

(4)

which can be understood as a general solution of the microscopic Maxwell’s equations represented as a sum of the general solution of the homogeneous Maxwell’s equations, ${\bf E}({\bf r},\omega)$, and the partial solution of the inhomogeneous equations with the source given by the polarization ${\bf P}({\bf r},\omega)$. Here $\bar{\bf F}({\bf r},{\bf r}^{\prime};\omega)$ is the so-called field susceptibility tensor [20, 21] which relates the electric field at the point ${\bf r}$ generated by a classical dipole, oscillating at frequency $\omega$, with the dipole moment itself, located at ${\bf r}^{\prime}$. This quantity can be expressed in terms of the Green’s function of the vector wave equation with appropriate boundary conditions [22]. The symbol $V^{\prime}$ in Eq. (4) denotes the slab volume after removal of a small volume around the NP under consideration that excludes the singularity associated with the self-action of NPs. It is a common procedure when one considers integral equations for the local field [19]. When writing Eq. (4) one formally treats the quantity ${\bf P}({\bf r},\omega)$ as a continuous function of ${\bf r}$ that is a good approximation when the average distance between NPs is much less than $\lambda$.
The quantity $\bar{\bf F}({\bf r},{\bf r}^{\prime};\omega)$ can be split into two contributions as

$$\bar{\bf F}({\bf r},{\bf r}^{\prime};\omega)=\bar{\bf F}^{0}({\bf r},{\bf r}^{\prime};\omega)+\bar{\bf F}^{R}({\bf r},{\bf r}^{\prime};\omega),$$

(5)

where $\bar{\bf F}^{0}({\bf r},{\bf r}^{\prime};\omega)$ originates from the dipole field of NPs and leads to the Lorentz local field [1]. The other term, $\bar{\bf F}^{R}({\bf r},{\bf r}^{\prime};\omega)$, results from the NPs dipole field reflected from the boundaries and is found in Ref. [21]. It was first considered in the context of the local field theory in Ref. [17]. The reflected contribution, $\bar{\bf F}^{R}({\bf r},{\bf r}^{\prime};\omega)$, decreases with the distance from the boundary and becomes insignificant at distances much larger than the wavelength. However, in a subwavelength slab its influence is substantial.
The field ${\bf E}({\bf r},\omega)$ in Eq. (4) corresponds to a solution for the slab without sources, i.e. with the NPs being removed. It can be therefore interpreted as the ”incident” field in the slab which is not yet disturbed by NPs. Such a consideration is typical for a scattering problem. In the bulk medium, where the reflected contribution can be neglected, this field equals the mean field due to an isotropic distribution of the NP dipoles. However, in the near field of the boundaries or inside a subwavelength slab this equivalence is violated.
Introducing the spatial Fourier transforms

$${\bf E}({\bf r},\omega)=\frac{1}{(2\pi)^{2}}\int{\bf E}({\bf k}_{\parallel},\omega;z)e^{i{\bf k}_{\parallel}\cdot{\bf r}_{\parallel}}d{\bf k}_{\parallel},$$

(6)

$${\bf P}({\bf r},\omega)=\frac{1}{(2\pi)^{2}}\int{\bf P}({\bf k}_{\parallel},\omega;z)e^{i{\bf k}_{\parallel}\cdot{\bf r}_{\parallel}}d{\bf k}_{\parallel}$$

(7)

and

$$\bar{\bf F}^{R}({\bf r},{\bf r}^{\prime};\omega)=\frac{1}{(2\pi)^{2}}\int\bar{\bf F}^{R}({\bf k}_{\parallel},\omega;z,z^{\prime})e^{i{\bf k}_{\parallel}\cdot({\bf r}_{\parallel}-{\bf r}^{\prime}_{\parallel})}d{\bf k}_{\parallel}$$

(8)

with ${\bf r}_{\parallel}$ and ${\bf k}_{\parallel}$ being the radius vector and wave vector components parallel to the slab boundaries, respectively, one obtains from Eqs. (2), (4) and (5)

	$\displaystyle{\bf P}(k_{\parallel},\omega;z)-\chi(\omega)\int_{-d/2}^{d/2}\bar{\bf F}^{R}(k_{\parallel},\omega;z,z^{\prime}){\bf P}(k_{\parallel},\omega;z^{\prime})dz^{\prime}$
	$\displaystyle=\chi(\omega){\bf E}(k_{\parallel},\omega;z).$

Here

$$\chi(\omega)=\frac{N\alpha(\omega)}{1-(4\pi/3\epsilon_{h})N\alpha(\omega)}$$

(10)

is the linear susceptibility of NPs [1] and we have taken into account the Lorentz relation

$$\int_{V^{\prime}}\bar{\bf F}^{0}({\bf r},{\bf r}^{\prime};\omega){\bf P}({\bf r}^{\prime},\omega)d{\bf r}^{\prime}=\frac{4\pi}{3\epsilon_{h}}{\bf P}({\bf r},\omega).$$

(11)

In what follows we assume that the slab is optically thin so that $d/\lambda\ll 1$. In this case one can approximate the kernel in Eq. (II) as

$$\bar{\bf F}^{R}(k_{\parallel},\omega;z,z^{\prime})\approx\bar{\bf F}^{R}(k_{\parallel},\omega;0,0)\equiv\bar{\bf F}_{0}^{R}(k_{\parallel},\omega).$$

(12)

As a result, Eq. (II) is reduced to

$\displaystyle{\bf P}^{m}(k_{\parallel},\omega)-d\chi(\omega)\bar{\bf F}_{0}^{R}(k_{\parallel},\omega){\bf P}^{m}(k_{\parallel},\omega)=\chi(\omega){\bf E}^{m}(k_{\parallel},\omega),$

where the superscript $m$ denotes the averaged quantities

$${\bf P}^{m}(k_{\parallel},\omega)=\frac{1}{d}\int_{-d/2}^{d/2}{\bf P}(k_{\parallel},\omega;z)dz$$

(14)

and

$${\bf E}^{m}(k_{\parallel},\omega)=\frac{1}{d}\int_{-d/2}^{d/2}{\bf E}(k_{\parallel},\omega;z)dz.$$

(15)

The analysis of the explicit form of $F_{0,ij}^{R}$ (see the Appendix) reveals that $F_{0,xy}^{R}=F_{0,yx}^{R}=F_{0,yz}^{R}=F_{0,zy}^{R}=0$, while the tensorial components $F_{0,xz}^{R}$, $F_{0,zx}^{R}$ and $F_{0,zz}^{R}$ are proportional to $k_{\parallel}/\kappa$ and $(k_{\parallel}/\kappa)^{2}$, respectively, where

$$\kappa=\sqrt{\left(\frac{\omega}{c}\right)^{2}\epsilon_{h}-k_{\parallel}^{2}}.$$

(16)

and we assume that the vector ${\bf k}_{\parallel}$ is directed along the $x$ axis. Therefore in the case where $k_{\parallel}\ll\kappa$, only the components $F_{0,xx}^{R}$ and $F_{0,yy}^{R}$ contribute to Eq. (II) in the lowest order in $d/\lambda$. In the opposite limiting case, $k_{\parallel}\gg\kappa$, the dominant contributions originate from $F_{0,zz}^{R}$ and $F_{0,yy}^{R}$. For the sake of simplicity, we assume that the diagonal elements satisfactory describe the local field in the intermediate case as well.
In this mean-field approximation, the linear susceptibility tensor of the slab is given by

$$\eta^{m}_{jj}(k_{\parallel},\omega)=\frac{\chi(\omega)}{1-d\chi(\omega)F_{0,jj}^{R}(k_{\parallel},\omega)}.$$

(17)

Correspondingly, the mean-field dielectric tensor, which determines the nonlocal and anisotropic optical response of the slab, takes the form

$$\epsilon^{m}_{jj}(k_{\parallel},\omega)=\epsilon_{h}+4\pi\eta^{m}_{jj}(k_{\parallel},\omega).$$

(18)

Let us assume now that a plane monochromatic electromagnetic wave of the frequency $\omega$ falls at the nanocomposite slab from medium 1 so that its wave vector ${\bf k}_{1}$ makes an angle $\theta$ with the normal to the boundary (Fig. 1). We assume also that the boundary surface has a small random subwavelength roughness which leads to the scattering of the incident light. Two sources of scattering can contribute here: (i) surface roughness due to the surface morphology and (ii) ”volume” roughness due to inhomogeneities in the bulk originating from the distribution of nanoparticles [23].
The scattering cross-section is proportional to the power spectrum of the rough surface, $\mathcal{P}({\bf k}_{\parallel}-{\bf k}_{1\parallel})$, which is determined as the Fourier transform of the roughness correlation function, $C({\bf r}_{\parallel}^{\prime}-{\bf r}_{\parallel})$ [23, 24]. For an isotropic surface one usually assumes

$$C(\mid{\bf r}_{\parallel}^{\prime}-{\bf r}_{\parallel}\mid)=\delta^{2}\exp\left(-\frac{\mid{\bf r}_{\parallel}^{\prime}-{\bf r}_{\parallel}\mid^{2}}{\sigma^{2}}\right),$$

(19)

where $\delta$ is the root-mean-square height of the surface and $\sigma$ is the correlation length. Accordingly,

	$\displaystyle\mathcal{P}(\mid{\bf k}_{\parallel}-{\bf k}_{1\parallel}\mid)\sim G(\mid{\bf k}_{\parallel}-{\bf k}_{1\parallel}\mid)$
	$\displaystyle\equiv\frac{\sigma^{2}}{4\pi}\exp\left(-\frac{\sigma^{2}}{4}\mid{\bf k}_{\parallel}-{\bf k}_{1\parallel}\mid^{2}\right).$		(20)

We model therefore the spatial Fourier spectrum of the incident field amplitude by the Gaussian function

$$A_{1}^{i}({\bf k}_{\parallel},\omega)=A_{0}\exp\left(-\frac{\sigma^{2}}{8}\mid{\bf k}_{\parallel}-{\bf k}_{1\parallel}\mid^{2}\right),$$

(21)

so that the corresponding intensity is proportional to the power spectrum, Eq. (III).
We consider separately two cases: (i) when the electric field vector of the incident wave lies in the plane of incidence (TM polarization) and (ii) when it is perpendicular to the plane of incidence (TE polarization). It should be noted that generally the scattering at a rough surface is accompanied by light depolarization. However, for scattering at small roughness in the plane of incidence the polarization of light is not changed [24]. In what follows we discuss the latter case which implies that the vector ${\bf k}_{\parallel}$ for both the reflected and transmitted waves lies in the plane of incidence.

We illustrate the above theory with some numerical calculations carried out for a model system which simulates the structure experimentally investigated in Ref. [14]. Namely, we assume that Au NPs of radius $R=2$ nm are randomly distributed in the dielectric host medium with $\epsilon_{h}=1.5^{2}$, the light beam is incident from vacuum ($\epsilon_{1}=1$) and the nanocomposite slab is supported by an optically thick gold film so that the transmission through the structure is zero. In such a case, assuming that the light scattered in the slab is completely absorbed as well, the absorptivity, $\mathcal{A}^{\alpha}$, can be expressed in terms of the reflectivity as $\mathcal{A}^{\alpha}=1-\mathcal{R}^{\alpha}$. The dielectric function of the gold film, $\epsilon_{3}(\omega)$, is taken in the Drude model, Eq. (1), with the parameters $\epsilon_{\infty}=9$, $\omega_{p}=13.8\times 10^{15}$ s^-1 and the relaxation constant $\Gamma=0.11\times 10^{15}$ s^-1 [25]. For gold NPs $\Gamma$ should be replaced by $\Gamma_{NP}$ which results not only from the electron collision rate given by $\Gamma$, but also from the electron collisions with the NP boundary, so that $\Gamma_{NP}\approx\Gamma+v_{F}/R$ with $v_{F}$ being the Fermi velocity [25]. For Au NPs of radius $R=2$ nm this gives $\Gamma_{NP}\approx 0.81\times 10^{15}$ s^-1.
For dense NP arrays which were realized in Ref. [14] we envisage, however, additional channels of LSP relaxation. A hint for that can be found, for example, in the optical absorbance spectrum of diamond-like nanocomposite films containing Ag NPs which demonstrates broadening with the increase of the NP concentration [26]. One can propose two mechanisms responsible for such behavior as follows.
In the quasistatic approximation, which is valid for NPs of subwavelength diameters, the LSP frequencies of different spherical NPs coincide with each other. For small enough distances between NPs this can lead to ”jumps” of LSP excitations from one NP to another which disrupt the collective electron oscillations in a single NP and reduce the LSP relaxation time [27].
Another possible reason for the enhancement of the LSP decay rate is the cooperative radiation of NPs. In the model under discussion the nanocomposite slab cab be regarded as an ensemble of point dipole emitters which is confined within a subwavelength layer. Such a system is known to radiate coherently with a decay rate enhanced by the factor $\eta=Nd(\lambda/2\pi)^{2}$ (collective spontaneous emission or Dicke superradiance) [28]. The radiative damping rate of a single spherical NP can be estimated using the formula [27, 29]

$$\Gamma_{rad}=\frac{2}{9}\frac{\sqrt{\epsilon_{h}}}{2\epsilon_{h}+1}\frac{\omega_{p}^{4}R^{3}}{c^{3}}.$$

(54)

Taking here $R=2$ nm, $f=0.1$, $d=50$ nm and $\lambda=500$ nm one obtains for gold NPs $\eta\Gamma_{rad}\approx 0.61\times 10^{15}$ s^-1 that is comparable with the estimate for $\Gamma_{NP}$ given above. The contributions from different relaxation channels are added together resulting in the total relaxation rate. Therefore in what follows we consider the quantity $\Gamma_{NP}$ as an empirical parameter and investigate its influence on the optical spectra.

In this paper, we have developed the theory of the optical properties of an optically thin nanocomposite slab which contains randomly distributed metal nanoparticles. We have derived the analytical formulas for the reflection and transmission coefficients in both polarizations, Eqs. (32), (33), (43) and (44), which take into account the anisotropy and nonlocality of the slab optical response. In the developed theory the effective linear susceptibility of the nanocomposite slab is determined by the number density of NPs, while their positions are unimportant. This means that the approach under discussion is equally applicable to plasmonic metamaterials with a regular arrangement of spherical metal NPs.
We have examined the impact of different factors, which are beyond the scope of the Maxwell Garnett approximation, on the reflection spectrum of a plasmonic nanocomposite slab placed on a metal substrate. We have concluded that the broadband perfect absorption in such structures is associated with the LSP absorption band and originates from both reflections from the slab boundaries, which trap light within the slab, and enhanced decay channels for the LSP excitations in a dense NP array. In contrast to this, the omnidirectional character of perfect absorption and its polarization insensitivity stem from the light scattering and depolarization, respectively, at surface roughness or volume irregularities in the slab.
The reflection spectra calculated with the use of the developed approach satisfactory agree with the available experimental data, while those obtained in both Maxwell Garnett and Bruggeman approximations do not. This conclusion reveals the importance of a proper account of the local field in the slab and adequate modeling of the LSP relaxation in dense NP arrays.
These findings open up new possibilities for engineering broadband perfect absorption in plasmonic nanocomposites and can stimulate experimental investigations in this direction. The LSP relaxation rate is determined by both the NP filling factor and NP sizes, while the light scattering stems from surface morphology - the factors which can be well controlled in an experiment. The perfect absorption band can be extended to longer wavelengths when using metal nanorods with large aspect ratios as NPs in the nanocomposite [31].
Another important aspect which is highlighted by the present research is the mechanisms of the LSP relaxation in dense NP arrays which have not been discussed before. Both quantum jumps of LSP excitations between NPs and their cooperative radiation are processes which are inherent to quantum emitters. The experimental investigation of these mechanisms would therefore contribute to the rapidly developing field of quantum plasmonics [32].
Finally, it is necessary to stress that the approach developed in this paper is a fully analytical one which has been derived from first principles. Although it does not substitute the numerical methods in all respects, it can serve as a reliable test for them. The numerical calculations should always be accompanied by a careful analysis of adopted approximations and possible numerical errors, not saying about a necessary compromise between the accuracy and the required computational memory and time. They have therefore their own range of applicability. The usual practice is testing of a computational package against known analytical solutions [33]. In this context, any analytical solution derived from first principles is extremely important.

This work was carried out under a programme of, and funded by, the European Space Agency (ESA Contract No. 4000130078/20/NL/SC). The author is grateful to Prof. Sigitas Tamulevičius and Prof. Tomas Tamulevičius for useful discussions.

The components of the tensor $\bar{F}_{0}^{R}(k_{\parallel},\omega)$, Eq. (12), have the following form:

	$\displaystyle F^{R}_{0,xx}=-\frac{2\pi i\kappa}{\epsilon_{h}}\frac{r_{1}^{p}(1-r_{2}^{p})+r_{2}^{p}(1-r_{1}^{p})}{1-r_{1}^{p}r_{2}^{p}},$		(57)
	$\displaystyle F^{R}_{0,yy}=\frac{2\pi i\omega^{2}}{\kappa c^{2}}\frac{r_{1}^{s}+r_{2}^{s}+2r_{1}^{s}r_{2}^{s}}{1-r_{1}^{s}r_{2}^{s}},$		(58)
	$\displaystyle F^{R}_{0,zz}=\frac{2\pi ik_{\parallel}^{2}}{\epsilon_{h}\kappa}\frac{r_{1}^{p}(1+r_{2}^{p})+r_{2}^{p}(1+r_{1}^{p})}{1-r_{1}^{p}r_{2}^{p}},$		(59)
	$\displaystyle F^{R}_{0,xz}=-F^{R}_{0,zx}=\frac{2\pi ik_{\parallel}}{\epsilon_{h}}\frac{r_{2}^{p}-r_{1}^{p}}{1-r_{1}^{p}r_{2}^{p}},$		(60)
	$\displaystyle F^{R}_{0,xy}=F^{R}_{0,yx}=F^{R}_{0,yz}=F^{R}_{0,zy}=0.$		(61)

Here

	$\displaystyle r_{1}^{p}=\frac{\epsilon_{1}\kappa-\epsilon_{h}k_{1z}}{\epsilon_{1}\kappa+\epsilon_{h}k_{1z}},$		(62)
	$\displaystyle r_{1}^{s}=\frac{\kappa-k_{1z}}{\kappa+k_{1z}},$		(63)
	$\displaystyle r_{2}^{p}=\frac{\epsilon_{3}\kappa-\epsilon_{h}k_{3z}}{\epsilon_{3}\kappa+\epsilon_{h}k_{3z}},$		(64)
	$\displaystyle r_{2}^{s}=\frac{\kappa-k_{3z}}{\kappa+k_{3z}}$		(65)

are the Fresnel reflection coefficients for $p$ and $s$ polarizations at the medium 1/medium 2 and medium 2/medium 3 interfaces, respectively, in the approximation of small NP volume fraction.