## Abstract

A rigorous theoretical formulation based on electromagnetic plane waves is utilized to construct a unified framework and identification of all possible surface-plasmon polariton solutions at an absorptive slab in a symmetric, lossless dielectric surrounding. In addition to the modes reported in literature, sets of entirely new mode solutions are presented. The corresponding fields are classified into different categories and examined in terms of bound and leaky modes, as well as forward-and backward-propagating modes, both outside and inside the slab. The results could benefit plasmon based applications in thin-film nanophotonics.

© 2014 Optical Society of America

## 1. Introduction

Recent years have witnessed a rapid expansion of plasmonics (plasmon photonics) [1–6], which constitutes an exceptionally rich area of nano-optics [7]. Plasmonics is based on the utilization of surface plasmons, which are resonant charge-density oscillations that may exist at a metal-dielectric interface. The coupled electromagnetic surface field, known as the surface-plasmon polariton (SPP), propagates along the boundary in a wave-like fashion. The resonant plasmon excitations exhibit unique features when interacting with light. In particular, they can strongly enhance the electromagnetic energy density in the near-field zone, at subwavelength distances from the surface, and enable localization of light energy into nanoscale regions. Since the first theoretical description of surface plasmons [8], the progress in plasmon nanophotonics has been fast and a broad range of applications have emerged, among them biomedical and chemical sensors, subwavelength imaging and spectroscopy, emitters and detectors, waveguides, nanoscale photonics and processing, and plasmon holography [9–18].

The fundamental properties of SPPs at planar (and rough) interfaces have been extensively studied in the past [19]. Therefore it is rather surprising that new and unexpected features concerning SPPs at a single interface and at a metal slab have been reported even quite recently. For instance, it was demonstrated that the conventional results on plasmon propagation at a metal-dielectric boundary presented in standard literature can be highly inaccurate even in some situations in which they are supposed to hold [20]. The existence of a new type of backward-propagating SPP, similar to those fields met in metamaterials, was also predicted in the same context. Furthermore, in addition to the well-known SPP modes at metallic slabs [21], it was shown that Maxwell’s equations admit an infinite number of other modes which play an important role in matching the boundary conditions upon launching a SPP [22]. These findings suggest that new types of SPPs, possessing peculiar physical properties, can be encountered at planar interfaces.

In this work we consider surface-plasmon polariton solutions at an absorptive slab in a symmetric, lossless dielectric surrounding. By using a rigorous formulation based on plane waves, with one wave on each side of the slab, we construct a unified framework and classification for all possible mode solutions. In addition to the ones found in literature, we present sets of new SPPs not encountered earlier to our knowledge. Likewise, we show that some commonly accepted mode solutions are not admitted by Maxwell’s equations. Regarding field propagation and energy flow, previous works have focused only on the region outside the film; however, we investigate the properties of the various mode types also within the slab. It is revealed, in particular, that both forward- and backward-propagating waves are possible, depending on the mode type and the material parameters. Backward propagation appears not to have been observed in natural media. Instead of using approximate models for the material parameters (such as the Drude theory), throughout the work we employ experimental values of optical constants. Our results could impact thin-film plasmonics and nanophotonics with polaritonic materials.

The paper is organized as follows. In Sec. 2 we introduce the geometry and specify the representations of the electric fields. In Sec. 3 we present the mode solutions and classify them into three classes: M1, M2, and M3. Section 4 deals with general field propagation and energy flow, both outside as well as inside the slab. In Sec. 5 we discuss bound modes and leaky modes and their phase movement. Section 6 addresses forward and backward propagation of the different mode types. The main conclusions of the work are summarized in Sec. 7.

## 2. Slab geometry

We consider monochromatic electromagnetic plane waves of angular frequency *ω* at two planar interfaces between linear, homogeneous, isotropic, non-magnetic, stationary, and spatially non-dispersive media. The geometry, as illustrated in Fig. 1, consists of a lossy (metal) film characterized by a complex relative permittivity *ε _{r}*

_{1}=

*ε′*

_{r}_{1}+

*iε″*

_{r}_{1}in the region −

*d*/2 <

*z*<

*d*/2, with

*ε′*

_{r}_{1}< 0 and

*ε″*

_{r}_{1}> 0, surrounded on both sides by the same dielectric media possessing a real and positive permittivity

*ε*

_{r}_{2}. We have chosen the related (generally complex) wave vectors ${\mathbf{k}}_{\alpha}^{(\beta )}$, for which ${\mathbf{k}}_{\alpha}^{(\beta )}\cdot {\mathbf{k}}_{\alpha}^{(\beta )}={k}_{\alpha}^{2}={k}_{0}^{2}{\epsilon}_{r\alpha}$ with

*α*,

*β*∈ {1, 2} and where

*k*

_{0}is the vacuum wave number, to lie in the

*xz*plane and the boundaries to be in the

*xy*plane. In our notation the subscript

*α*identifies the medium (1 or 2), and the superscript (

*β*) refers to the wave-vector direction, as will become clear below.

The spatial part of the *p*-polarized electric field, which follows from the Helmholtz equation, can be expressed explicitly as

*α*,

*β*∈ {1, 2}) is the complex field amplitude and

**ê**

*being the unit vector in the positive*

_{y}*y*direction and $\left|{\mathbf{k}}_{\alpha}^{(\beta )}\right|={\left[{{\mathbf{k}}_{\alpha}^{(\beta )}}^{*}\cdot {\mathbf{k}}_{\alpha}^{(\beta )}\right]}^{1/2}$ (the asterisk * denotes complex conjugation). We note that ${\widehat{\mathbf{p}}}_{\alpha}^{(\beta )}$ and ${\widehat{\mathbf{k}}}_{\alpha}^{(\beta )}$ are not orthogonal if ${\mathbf{k}}_{\alpha}^{(\beta )}$ is complex, in which case also $\left|{\mathbf{k}}_{\alpha}^{(\beta )}\right|\ne {k}_{\alpha}$ [20, 23].

According to Eq. (1) we consider, as is customary, a system in which there are two plane waves within and one wave on each side of the slab. However, unlike in previous studies, we have so far placed no restrictions on the wave-vector directions inside or outside the slab. For instance, the two wave vectors within the slab may coincide, in which case there are altogether three plane waves. Likewise, the two wave vectors outside the slab may be identical. These situations allow a rich variety of entirely new slab-mode solutions.

## 3. Slab modes

We let
${\mathbf{k}}_{\alpha}^{(\beta )}={k}_{x}{\widehat{\mathbf{e}}}_{x}+{k}_{\alpha z}^{(\beta )}{\widehat{\mathbf{e}}}_{z}$, where *k _{x}* and
${k}_{\alpha z}^{(\beta )}$ refer, respectively, to the Cartesian

*x*and

*z*components. The tangential components of the wave vectors are continuous across the boundaries and, therefore, ${k}_{\alpha x}^{(\beta )}={k}_{x}$ for all

*α*,

*β*∈ {1, 2}. Since also ${k}_{\alpha}^{2}={k}_{x}^{2}+{k}_{\alpha z}^{(\beta )2}$, one gets two options for the normal wave-vector components in each medium, namely ${k}_{\alpha z}^{(2)}=\pm {k}_{\alpha z}^{(1)}$. These possibilities lead to three different types of mode solutions, which we call M1, M2, and M3. We next analyze mathematically each type separately. The physical nature of these solutions will be addressed in later sections.

Although this work is focused on the situation where *ε′ _{r}*

_{1}< 0, we remark that the results presented in this section are also valid for

*ε′*

_{r}_{1}> 0.

#### 3.1. M1: ${k}_{1z}^{(1)}={k}_{1z}^{(2)}$

In this case ${\mathbf{k}}_{1}^{(1)}={\mathbf{k}}_{1}^{(2)}\equiv {\mathbf{k}}_{1}^{(\text{M}1)}$, whereupon also ${\widehat{\mathbf{p}}}_{1}^{(1)}={\widehat{\mathbf{p}}}_{1}^{(2)}\equiv {\widehat{\mathbf{p}}}_{1}^{(\text{M}1)}$ according to Eq. (2). The two parts of the field inside the slab [see Eq. (1)] can be combined and written as

*d*. Physically these results are intuitive as only one plane wave exists on each side of the two interfaces. Hence the same single-interface SPP condition can be met at both boundaries.

From a broader perspective, Eqs. (4) and (5) can be understood as generalizations of the standard Brewster angle for a propagating plane wave incident on an interface between two dielectric media producing no reflected field [24]. Thus the M1 modes may be interpreted as generalized Brewster fields for which the Fresnel reflection coefficient goes to zero outside as well as inside the film.

#### 3.2. M2 and M3: ${k}_{1z}^{(1)}=-{k}_{1z}^{(2)}$

In this case the boundary conditions result in

where*d*. Nevertheless, this solution is not interesting, since when substituted into Eq. (1) it leads to a field which vanishes everywhere.

Since *r*^{(1)} is the Fresnel reflection coefficient for the plane wave
${\mathbf{E}}_{1}^{(1)}$, and 1/*r*^{(2)} likewise for
${\mathbf{E}}_{1}^{(2)}$, Eq. (6) is at once recognized as a Fabry-Perot resonance condition for wave motion between the boundaries at *z* = ±*d*/2. This is an important result that is a consequence of the two transversally counter-propagating fields within the slab. In addition, if a ‘reflected’ field is introduced under the slab in Fig. 1, Eq. (6) corresponds to the zero of the film’s reflection coefficient.

In contrast to the situation of M1 in Sec. 3.1, in the present case we have three plane waves interacting at both interfaces and this allows two possibilities for the normal wave-vector components outside the slab, viz., ${k}_{2z}^{(1)}=\pm {k}_{2z}^{(2)}$.

### 3.2.1. M2: ${k}_{2z}^{(1)}={k}_{2z}^{(2)}$

We first consider
${k}_{2z}^{(1)}={k}_{2z}^{(2)}\equiv {k}_{2z}^{(\text{M}2)}$ for which *r*^{(1)} = *r*^{(2)} ≡ *r* (the Fresnel reflection coefficients are not bounded in magnitude by unity [7, 23]). Equation (6) then reduces to

**ê**

*and*

_{x}**ê**

*are the unit vectors in the positive*

_{z}*x*and

*z*directions, respectively. Equation (8) further yields

*m*

_{+}= 2

*n*and

*m*

_{−}= 2

*n*+ 1 with

*n*∈ ℤ. In contrast to the solutions of M1 given by Eq. (5), the wave-vector components of M2 in Eq. (10) stand for an infinite number of modes and depend on the film thickness. We note, however, that ${k}_{1z}^{(\text{M}2)}\to 0$ when

*d*→ ∞, and thus all the M2 modes vanish in this limit. Further we observe that ${k}_{1z}^{(\text{M}2)}$ is real (even when

*ε*

_{r}_{1}is complex) and fully independent of material parameters, while ${k}_{x}^{(\text{M}2)}$ does not depend on

*ε*

_{r}_{2}, also at variance with the M1 solutions.

It is of interest to note that the M2 modes above represent a general solution under the circumstances where the wave vectors in the regions on the two sides of the slab are identical and there are two perpendicularly counter-propagating waves within the slab (for M1 only one wave exists inside the slab). The mode solutions in Eq. (10) therefore are a generalization (to an absorptive film and more general field types) of the propagating plane waves that on interference at a dielectric plane-parallel plate produce no reflected field [24].

### 3.2.2. M3: ${k}_{2z}^{(1)}=-{k}_{2z}^{(2)}$

This last case corresponds to the conventionally studied situation. On writing ${k}_{1z}^{(1)}\equiv {k}_{1z}^{(\text{M}3)}$, Eq. (6) becomes

which, similarly to Eq. (8) in Sec. 3.2.1, gives rise to two different kinds of modes. This time, however, the field expression inside the film reads as*z*= 0, respectively. Equation (11) can further be split into

Similarly to the case discussed in Sec. 3.2.1, Eqs. (13) and (14) have an infinite number of solutions for any given media, thickness, and wavelength [22]. However, in marked contrast to Eq. (8), when *d* → ∞, both Eq. (13) and Eq. (14) have solutions that converge towards common values for which
${k}_{1z}^{(\text{M}3)}\ne 0$, namely [21, 22]

*d*→ ∞ are therefore regarded (at any

*d*) as fundamental modes (FMs) [22]. There are always two FMs, one symmetric and one antisymmetric, even when

*d*increases without limit. In addition to FMs, Eqs. (13) and (14) possess a second set of solutions such that for

*d*→ ∞ they reduce to

*d*becomes infinite [ ${k}_{1z}^{(\text{M}3)}=0$ amounts to a field which is zero everywhere]. We refer to these fields (at any

*d*) as higher-order modes (HOMs). We remark that the two sets of wave-vector components in Eqs. (15) and (16) above are the only solutions of Eqs. (13) and (14) when

*d*→ ∞.

In the opposite limit, *d* → 0, Eq. (13) has a symmetric solution for which

*x*axis, whereupon it is generally termed the long-range surface-plasmon polariton (LRSPP) [25, 26]. It can readily be verified from Eq. (14) that no such antisymmetric LRSPP exists.

## 4. Field propagation and energy flow

Let us write *k _{x}* =

*k′*+

_{x}*ik″*and ${k}_{\alpha z}^{(\beta )}={{k}_{\alpha z}^{(\beta )}}^{\prime}+i{{k}_{\alpha z}^{(\beta )}}^{\u2033}$, with

_{x}*′*and ″ denoting the real and imaginary parts, respectively. As with Sec. 3, even if we are interested in the case

*ε′*

_{r}_{1}< 0, the results in this section also hold for

*ε′*

_{r}_{1}> 0.

#### 4.1. Phase propagation and amplitude attenuation

According to the notation above, the exponential functions in Eq. (1) can be split into

*iωt*), the real parts of the wave-vector components thus give the wavefront propagation direction and the imaginary parts specify the direction in which the field amplitude decreases. The results established below for phase motion and amplitude variation are general and hold for all mode solutions M1–M3. They are eventually used to assess the relationship between the directions of phase and energy propagation of the various mode types.

In the regions outside the film, |*z*| ≥ *d*/2, the wave-vector components are connected as

*k′*and

_{x}*k″*have the same (opposite) sign, Eq. (19) implies that ${{k}_{2z}^{(\beta )}}^{\prime}{{k}_{2z}^{(\beta )}}^{\u2033}<0\left[{{k}_{2z}^{(\beta )}}^{\prime}{{k}_{2z}^{(\beta )}}^{\u2033}>0\right]$ and, consequently, ${{k}_{2z}^{(\beta )}}^{\prime}$ and ${{k}_{2z}^{(\beta )}}^{\u2033}$ have opposite (same) signs. Furthermore, it can be shown that

_{x}For |*z*| < *d*/2, the wave-vector components behave as

*z*axis, while in the case of Eq. (23) they are opposite. At the transition point, Eq. (22), the field is purely evanescent or strictly propagating along the

*z*axis (even if

*ε″*

_{r}_{1}> 0).

#### 4.2. Flux of energy

To analyze energy flow we make use of the time-averaged Poynting vector [7]

**H**(

**r**) = (

*cμ*

_{0}

*k*

_{0})

^{−1}[

**k**×

**E**(

**r**)] is the magnetic field,

*c*is the speed of light, and

*μ*

_{0}is the vacuum permeability. Outside the slab, for all mode types (M1–M3), the Poynting vector becomes

*σ*≡

*k*

_{0}/(2

*cμ*

_{0}) and $\left|{\mathbf{k}}_{2}\right|\equiv \left|{\mathbf{k}}_{2}^{(1)}\right|=\left|{\mathbf{k}}_{2}^{(2)}\right|$. Since

*ε*

_{r}_{2}> 0, the Poynting vector given in Eq. (25) always points in the same direction in which the wavefront propagates.

The situation is more involved for |*z*| < *d*/2, where after some calculations we find

*ε*

_{r}_{1}as well as on the position.

## 5. Bound modes and leaky modes

We now analyze field propagation of modes M1–M3 introduced in Sec. 3.

#### 5.1. M1

It was shown in Sec. 3.1 that ${k}_{2z}^{(1)}={k}_{2z}^{(2)}$. Consequently, according to Eq. (18), the directions of amplitude attenuation and phase propagation of M1 are the same above and below the slab. Thus we always have one exponentially decaying (bound wave) and one exponentially growing mode (leaky wave) outside the film when moving away from the interfaces. In addition, it can be shown that ${k}_{x}^{(\text{M}1)}$ and ${k}_{2z}^{(\text{M}1)}$ of Eq. (5) always satisfy [20]

*x*axis, but along the

*z*axis they are opposite in the region |

*z*| ≥

*d*/2, respectively.

For
${k}_{1z}^{(\text{M}1)}$ in the region |*z*| < *d*/2, on the other hand, we obtain (for a derivation, see [20])

_{I}corresponds to the phase moving in the same direction as the field decays, while for M1

_{III}the phase propagation is opposite to that of attenuation (along the

*z*axis). M1

_{II}stands for the case where the field is purely evanescent in the

*z*direction and the wavefronts advance only along the

*x*axis. We note that M1

_{I}and M1

_{III}are identical with the modes SPP I and SPP II introduced in [20] for a single interface.

Furthermore, by separating Eq. (4) into its real and imaginary contributions, it follows from the latter that the imaginary parts of the normal wave-vector components fulfill

i.e., they have opposite signs. Restricting to fields decaying in the positive*x*direction [ ${{k}_{x}^{(\text{M}1)}}^{\u2033}>0$], Eqs. (29)–(37) result altogether in six different combinations of phase movement and amplitude attenuation, which are illustrated in Fig. 2. We find, in particular, that in all of the situations one has bound waves at one interface (as in [20]), but leaky waves at the other surface. These solutions have practical significance as there always is a wave bound to the slab but also because even leaky waves are physical over limited distances [21].

#### 5.2. M2

In this case Eq. (10) implies ${{k}_{1z}^{(\text{M}2)}}^{\u2033}=0,$, and therefore, according to Eqs. (19) and (22), we again have [cf. Eq. (29)]

*x*axis and in the regions |

*z*| ≥

*d*/2 is similar to that of M1. However, the situation is different for |

*z*| <

*d*/2 since, as Eq. (10) shows, ${k}_{1z}^{(\text{M}2)}$ is purely real and therefore the field is neither exponentially decaying nor growing in the

*z*direction. In addition, for M2 there are two waves (of different amplitudes) within the slab propagating in opposite directions along the

*z*axis and, consequently, inside the slab we cannot identify the direction of the total-field phase movement perpendicular to the interfaces.

Figure 3 illustrates the wavefront propagation and field attenuation for M2 decaying in the positive *x* direction [
${{k}_{x}^{(\text{M}2)}}^{\u2033}>0$]. We see that outside the slab one has a bound mode and a leaky mode (as for M1 in Fig. 2), whereas for |*z*| < *d*/2 the fields are purely propagating in the *z* direction and attenuating only along the *x* axis.

#### 5.3. M3

The situation outside the slab in this case is different from that of M1 and M2, because for M3 the waves on each side are both either bound, or leaky, or strictly propagating in opposite directions perpendicular to the slab [ ${k}_{2z}^{(2)}=-{k}_{2z}^{(1)}$]. However, there is another fundamental difference between M3 and M1 and M2. Unlike frequently asserted [21], we emphasize that both signs in ${k}_{2z}^{(\text{M}3)}=\pm {\left[{k}_{2}^{(\text{M}3)2}-{k}_{x}^{(\text{M}3)2}\right]}^{1/2}$ are not allowed for a fixed ${k}_{x}^{(\text{M}3)}$, since Eqs. (13) and (14) are not invariant with respect to the change of sign of ${k}_{2z}^{(\text{M}3)}$. This implies that for a given bound (leaky) solution Eqs. (13) and (14) do not admit the corresponding leaky (bound) mode. In other words, for any particular tangential wave-vector component ${k}_{x}^{(\text{M}3)}$, Maxwell’s equations permit for M3 only a bound or a leaky mode, but not both. This property is at variance with M1 and M2 where both signs are simultaneously allowed for the transverse wave-vector component outside the slab.

Regarding
${k}_{1z}^{(\text{M}3)}$ for |*z*| < *d*/2, we show that all of the cases in Eqs. (21)–(23) are possible for M3 and thus the behavior of
${k}_{1z}^{(\text{M}3)}$ to some extent resembles that of
${k}_{1z}^{(\text{M}1)}$ discussed in Sec. 5.1. Nevertheless, M3 consists of two waves within the slab propagating and attenuating oppositely along the *z* axis, making the phase motion perpendicular to the interfaces ambiguous (cf. M2 in Sec. 5.2).

Besides the symmetric and antisymmetric modes, the M3 fields introduced in Sec. 3.2.2 were further divided into FMs [Eq. (15)] and HOMs [Eq. (16)] whose field propagation we next analyze separately. Due to the transcendental nature of Eqs. (13) and (14), one has to apply numerical techniques to find
${k}_{x}^{(\text{M}3)}$ for each chosen media, wavelength, and slab thickness. In the examples that follow we only consider modes decaying in the positive *x* direction, i.e.,
${{k}_{x}^{(\text{M}3)}}^{\u2033}>0$, but the main conclusions also concern fields attenuating in the opposite direction.

### 5.3.1. Fundamental modes

Let us first discuss FMs. Figure 4 illustrates the behavior of
${{k}_{x}^{(\text{M}3)}}^{\prime}/{k}_{0}$ (left column) and
${{k}_{x}^{(\text{M}3)}}^{\u2033}/{k}_{0}$ (right column) for symmetric (upper row) and antisymmetric (lower row) FMs at a silver (Ag) slab surrounded by air (*ε _{r}*

_{2}= 1), as a function of

*d*for different vacuum wavelengths

*λ*

_{0}= 2

*π*/

*k*

_{0}. We observe that in all cases ${k}_{x}^{(\text{M}3)}$ approaches the value given by Eq. (15) as the film thickness increases. This is precisely the property that identifies FMs. However, as

*d*gets smaller, the behavior of the symmetric and antisymmetric fields starts to diverge. Particularly, the symmetric modes become LRSPPs with ${k}_{x}^{(\text{M}3)}\to {k}_{0}$ in the limit

*d*→ 0, in accordance with Eq. (17), while the antisymmetric modes acquire large values for ${{k}_{x}^{(\text{M}3)}}^{\prime}$ and ${{k}_{x}^{(\text{M}3)}}^{\u2033}$. In addition, one finds that for both field types the real and imaginary parts of ${k}_{x}^{(\text{M}3)}$ increase as the wavelength is reduced.

Figure 5 illustrates the behavior of
${{k}_{x}^{(\text{M}3)}}^{\prime}/{k}_{0}$ (left column) and
${{k}_{x}^{(\text{M}3)}}^{\u2033}/{k}_{0}$ (right column) for symmetric (upper row) and antisymmetric (lower row) FMs at a Ag slab surrounded by gallium phosphide (GaP), diamond (C), fused silica (SiO_{2}), and air, as a function of *d* at *λ*_{0} = 632.8 nm. We see that increasing the refractive index of the surrounding medium leads to larger real and imaginary parts of
${k}_{x}^{(\text{M}3)}$ for both mode types. The variation of *d* in Fig. 5 has the same effect as in Fig. 4, namely, the solutions converge towards that of Eq. (15) when *d* increases, while for small slab thicknesses the symmetric fields become LRSPPs and the antisymmetric modes obtain large values for
${{k}_{x}^{(\text{M}3)}}^{\prime}$ and
${{k}_{x}^{(\text{M}3)}}^{\u2033}$.

We recall that Eqs. (13) and (14) permit only one sign for ${k}_{2z}^{(\text{M}3)}$ at any given ${k}_{x}^{(\text{M}3)}$ (see the beginning of Sec. 5.3). All the solutions plotted in Figs. 4 and 5 correspond to ${{k}_{2z}^{(\text{M}3)}}^{\u2033}>0$. Thus ${{k}_{2z}^{(\text{M}3)}}^{\u2033}<0$ is not possible and the numerical analysis suggests that only bound FMs are allowed by Maxwell’s equations. Furthermore, it can be observed from the plots in Figs. 4 and 5 that ${{k}_{x}^{(\text{M}3)}}^{\prime}$ and ${{k}_{x}^{(\text{M}3)}}^{\u2033}$ always have the same sign. Though not shown, the same features are also found when silver is replaced, for example, by gold (Au). Consequently, and according to Eq. (19) [and since ${{k}_{2z}^{(\text{M}3)}}^{\u2033}>0$], we have

*x*axis (similarly to M1 and M2) and the wavefronts propagate towards the interfaces outside the slab.

Concerning
${k}_{1z}^{(\text{M}3)}$ for |*z*| < *d*/2, we consider as an example symmetric FMs at a Ag film in an air surrounding. Fixing *λ*_{0} = 350 nm, corresponding to *ε _{r}*

_{1}≈ −1.79 +

*i*0.60 [27], gives ${k}_{1z}^{(\text{M}3)}\approx \pm {k}_{0}(0.031+i1.868)$, ${k}_{1z}^{(\text{M}3)}\approx \pm i1.893{k}_{0}$, and ${k}_{1z}^{(\text{M}3)}\approx \pm {k}_{0}\left(-0.002+i1.895\right)$ for

*d*= 100 nm,

*d*= 175 nm, and

*d*= 250 nm, respectively. The same kind of results are found for antisymmetric fields by modifying the wavelength and/or the film thickness. Thus each of the three cases in Eqs. (21)–(23) is possible for FMs.

Summarizing the results above leads to three different combinations of phase movement and field attenuation for FMs, which are illustrated in Fig. 6. The black arrows and red curves within the slab in Figs. 6(a) and 6(c) are merely intended to illustrate the propagation and attenuation of the two waves having
${\mathbf{k}}_{1}^{(\beta )}$, *β* ∈ {1, 2}, as given in Eq. (21) [Fig. 6(a)] and Eq. (23) [Fig. 6(c)]. In Fig. 6(b) [Eq. (22)], on the other hand, the fields are purely evanescent along the *z* axis for |*z*| < *d*/2 and the phases advance in the positive *x* direction. Consequently, in that case the black arrows represent the actual directions of wavefront motion. Note the absence of leaky FMs in Fig. 6. Dispersion, propagation, and localization of FMs have been studied elsewhere [30].

### 5.3.2. Higher-order modes

We now turn our attention to HOMs. As will later be demonstrated, it turns out that there exists two different classes of HOMs, both including an infinite number of modes, which we already at this point introduce and define as

*d*

_{c}is a critical thickness that depends on the particular mode, media, and wavelength. Thus HOM

_{I}s are bound fields regardless of the film thickness, whereas HOM

_{II}s can be either bound (

*d*<

*d*

_{c}), or leaky (

*d*>

*d*

_{c}), or strictly propagating in opposite directions along the

*z*axis outside the slab (

*d*=

*d*

_{c}).

From now on we only consider symmetric HOMs, but remark that the results which follow are also valid for antisymmetric HOMs. Figure 7 illustrates the behavior of
${{k}_{x}^{(\text{M}3)}}^{\prime}/{k}_{0}$ (left column) and
${{k}_{x}^{(\text{M}3)}}^{\u2033}/{k}_{0}$ (right column) for the four symmetric HOM_{I}s (upper row) and HOM_{II}s (lower row) having the lowest
${{k}_{x}^{(\text{M}3)}}^{\u2033}$ at a Ag slab surrounded by air, as a function of *d* when *λ*_{0} = 632.8 nm. The figure confirms the result given by Eq. (16): when the slab thickness increases all of the solutions converge towards
${k}_{x}^{(\text{M}3)}={k}_{0}\sqrt{{\epsilon}_{r1}}\approx {k}_{0}(0.135+i3.987)$. As *d* decreases the imaginary parts, which are nearly identical for the two mode types, grow and diverge in the limit *d* → 0. Particularly we observe from Fig. 7 that
${{k}_{x}^{(\text{M}3)}}^{\u2033}$ of HOMs are considerably larger than those of FMs plotted in Fig. 5 (right column, dotted lines) and, consequently, HOMs suffer from a strong absorption and have much smaller propagation lengths [
$1/{{k}_{x}^{(\text{M}3)}}^{\u2033}$] along the interfaces than FMs do. However, despite their rapid decay along the *x* axis, HOMs are generally needed to match the boundary conditions upon launching a SPP [22]. As for
${{k}_{x}^{(\text{M}3)}}^{\prime}$, when *d* becomes smaller the real parts of HOM_{I}s slightly decrease before diverging as *d* tends to zero. Regarding HOM_{II}s, when *d* is reduced
${{k}_{x}^{(\text{M}3)}}^{\prime}$ gets smaller and, at the critical thickness *d*_{c} (which depends on the particular mode), the real part becomes zero. It is observed that the HOM_{II} having the lowest (highest)
${{k}_{x}^{(\text{M}3)}}^{\u2033}$ possesses the smallest (largest) critical thickness. As *d* further decreases (*d* < *d*_{c}) the real part acquires a negative value and finally diverges when *d* → 0.

Figure 8 illustrates the behavior of
${{k}_{x}^{(\text{M}3)}}^{\prime}/{k}_{0}$ (left column) and
${{k}_{x}^{(\text{M}3)}}^{\u2033}/{k}_{0}$ (right column) for the symmetric HOM_{I} (upper row) and HOM_{II} (lower row) having the lowest
${{k}_{x}^{(\text{M}3)}}^{\u2033}$ at a Ag slab surrounded by air, as a function of *d* when *λ*_{0} is varied. The figure shows that, for both mode types, decreasing the wavelength gives a smaller
${{k}_{x}^{(\text{M}3)}}^{\u2033}$ and, consequently, larger propagation lengths along the interfaces. This feature is opposite to that of the corresponding FMs depicted in Fig. 4. Regarding
${{k}_{x}^{(\text{M}3)}}^{\prime}$, reducing *λ*_{0} results in a slight increase of the real part in the case of HOM_{I} and for HOM_{II} the critical thickness is shifted towards larger values: *d*_{c} ≈ 66 nm (*λ*_{0} = 700 nm), *d*_{c} ≈ 79 nm (*λ*_{0} = 600 nm), *d*_{c} ≈ 90 nm (*λ*_{0} = 500 nm), and *d*_{c} ≈ 109 nm (*λ*_{0} = 400 nm).

As a final example, we have in Fig. 9 plotted
${{k}_{x}^{(\text{M}3)}}^{\prime}/{k}_{0}$ (left column) and
${{k}_{x}^{(\text{M}3)}}^{\u2033}/{k}_{0}$ (right column) for the symmetric HOM_{I} (upper row) and HOM_{II} (lower row) of the lowest
${{k}_{x}^{(\text{M}3)}}^{\u2033}$ at a Ag slab surrounded by GaP, diamond, SiO_{2}, and air, as a function of *d* at *λ*_{0} = 632.8 nm. We particularly observe that for both mode types varying the refractive index of the surrounding medium has a negligible effect on
${{k}_{x}^{(\text{M}3)}}^{\u2033}$ and, therefore, on the propagation lengths along the *x* axis. This is in strong contrast to FMs in Fig. 5. For example, at *d* = 100 nm the propagation length of the symmetric FM is about 230 times larger for Ag/air than in the case of Ag/GaP, while for HOMs the corresponding difference is practically zero. The real part, instead, is affected notably as *ε _{r}*

_{2}is varied and

*d*is sufficiently small, i.e., increasing the refractive index outside the slab leads to a larger ${{k}_{x}^{(\text{M}3)}}^{\prime}$ for HOM

_{I}and an increased critical thickness of HOM

_{II}[

*d*

_{c}≈ 73 nm (air),

*d*

_{c}≈ 97 nm (SiO

_{2}),

*d*

_{c}≈ 137 nm (C), and

*d*

_{c}≈ 164 nm (GaP)]. In the limit

*d*→ ∞, on the other hand, changing

*ε*

_{r}_{2}has no influence on the real part, as is also evident from Eq. (16).

Regarding HOM_{I}s, all the plots in Figs. 7 and 9 correspond to
${{k}_{2z}^{(\text{M}3)}}^{\u2033}>0$ for any *d* (as in the case with FMs), justifying the definition in Eq. (40). Consequently, the numerical results suggest that only bound HOM_{I}s can exist. From the same figures one also finds that the real and imaginary parts of
${k}_{x}^{(\text{M}3)}$ have the same sign. Hence, and according to Eq. (19), we obtain

indicating that the wave propagation of HOM_{I}s in the *x* direction and outside the slab is similar to that of FMs [cf. Eq. (39)]. Concerning
${k}_{1z}^{(\text{M}3)}$, all the solutions plotted in Figs. 7 and 9 obey Eq. (23), i.e.,
${{k}_{1z}^{(\text{M}3)}}^{\prime}{{k}_{1z}^{(\text{M}3)}}^{\u2033}<0$, and thus the total field-propagation behavior of HOM_{I}s in the figures is as summarized in Fig. 6(c).

As for HOM_{II}s, all the plots in Figs. 7 and 9 correspond to
${{k}_{2z}^{(\text{M}3)}}^{\u2033}>0$ (*d* < *d*_{c}),
${{k}_{2z}^{(\text{M}3)}}^{\u2033}=0$ (*d* = *d*_{c}), and
${{k}_{2z}^{(\text{M}3)}}^{\u2033}<0$ (*d* > *d*_{c}), justifying our definition in Eq. (41). Further, Figs. 7 and 9 show that
${{k}_{x}^{(\text{M}3)}}^{\prime}$ and
${{k}_{x}^{(\text{M}3)}}^{\u2033}$ have the opposite (same) sign for *d* < *d*_{c} (*d* > *d*_{c}). In addition, at the transition point *d* = *d*_{c}, where
${{k}_{x}^{(\text{M}3)}}^{\prime}=0$, only those solutions with
${{k}_{2z}^{(\text{M}3)}}^{\prime}>0$ are allowed by Eq. (13). Thus, and according to Eq. (19), the numerical analysis implies that

_{II}s obey ${{k}_{2z}^{(\text{M}3)}}^{\prime}>0$, indicating that the wavefronts advance away from the slab in the

*z*direction for |

*z*| ≥

*d*/2, in contrast to FMs [Eq. (39)] and HOM

_{I}s [Eq. (42)]. Concerning ${k}_{x}^{(\text{M}3)}$, the first case, Eq. (43), a situation not encountered before, represents (bound) fields for which the phase motion and amplitude attenuation are opposite along the surfaces. The second one, Eq. (44), not met earlier either, stands for waves that are purely evanescent in the

*x*direction (in both media). The last situation, Eq. (45), corresponds to (leaky) modes for which the behavior of ${k}_{x}^{(\text{M}3)}$ is analogous to that in Eqs. (39) and (42) of FMs and HOM

_{I}s, respectively.

Regarding
${k}_{1z}^{(\text{M}3)}$ of HOM_{II}s, one readily confirms from Eqs. (21), (43), and (44) that the real and imaginary parts always have the same sign for *d* ≤ *d*_{c}. Although not analytically seen by using Eq. (45), according to Eq. (21) the same turns out to be true for all the numerical plots in Figs. 7 and 9 also when *d* > *d*_{c}. Hence we may conclude that HOM_{II}s satisfy
${{k}_{1z}^{(\text{M}3)}}^{\prime}{{k}_{1z}^{(\text{M}3)}}^{\u2033}>0$ regardless of *d*, in contrast to HOM_{I}s having
${{k}_{1z}^{(\text{M}3)}}^{\prime}{{k}_{1z}^{(\text{M}3)}}^{\u2033}<0$ at any slab thickness.

We have in Fig. 10 summarized the (three) different combinations of phase movement and amplitude attenuation for HOM_{II}s discussed above. As *d* < *d*_{c} (left figure), the field attenuation is similar to that of FMs and HOM_{I}s in Fig. 6, but the phases advance in the opposite directions. In particular, the wavefront motion is in the negative *x* direction even though the fields decay in the positive direction. When *d* reaches the critical thickness (middle figure), the waves become purely evanescent with no phase movement at all along the *x* axis (in both media) and strictly propagating away from the interfaces outside the slab. For |*z*| < *d*/2, the field is neither exclusively evanescent nor solely propagating in the *z* direction and the phases of the two waves propagate perpendicular to the boundaries in opposite directions. Finally, as *d* > *d*_{c} (right figure), the fields become leaky and the wavefronts advance away from the slab for |*z*| ≥ *d*/2, in contrast to FMs and HOM_{I}s in Fig. 6. On the other hand, inside the film the wave propagation is analogous to FMs and HOM_{I}s and, in addition, HOM_{II}s also decay in the same (positive) direction as the phases propagate along the *x* axis in both media.

## 6. Forward- and backward-propagating modes

Finally we analyze energy flow of modes M1–M3 in terms of the Poynting vector introduced in Sec. 4.2. Fields whose wavefronts move in the direction parallel to the Poynting vector are regarded as forward-propagating modes (FPMs), while those waves for which the phase motion is antiparallel to the energy transfer are referred to as backward-propagating modes (BPMs).

According to Eq. (25), regardless of the mode type,
${\mathbf{S}}_{2}^{(\beta )}(\mathbf{r})$ is parallel to
${{\mathbf{k}}_{2}^{(\beta )}}^{\prime}$ and decays exponentially at twice the rate of the field in the direction determined by
${{\mathbf{k}}_{2}^{(\beta )}}^{\u2033}$. Hence all modes M1–M3 are FPMs outside the slab, and their energy flow behavior (in that region) is illustrated by the black arrows and solid-red curves in Figs. 2, 3, 6, and 10. These results are characteristic of electromagnetic plane waves and we will not discuss the energy transfer for |*z*| ≥ *d*/2 any further.

As Eqs. (26)–(28) show, the situation is more complex inside the slab and thus each mode type is assessed separately in the subsections below. For brevity we only consider fields decaying in the positive *x* direction, i.e., *k″ _{x}* > 0, but the results on forward or backward propagation that are obtained also hold for

*k″*< 0.

_{x}#### 6.1. M1

Equations (5) and (26) imply that, along the *x* axis, we have (as in [20])

*ε″*

_{r}_{1}. Since ${{k}_{x}^{(\text{M}1)}}^{\prime}>0$ according to Eq. (29), the first case above represents a FPM and the last a BPM. At the transition point, Eq. (47), no energy is transfered along the surfaces. By using Eqs. (30)–(36) and (46)–(48) it is straightforward to show that, in the

*x*direction, depending on the material parameters, each of M1

_{I}–M1

_{III}can be either a FPM, a BPM, or a field with no energy flow at all.

In the *z* direction, it can be numerically verified from Eqs. (5) and (26) that [20]

_{I}is a FPM while M1

_{III}is a BPM perpendicular to the interfaces. Regarding M1

_{II}, which is neither a FPM nor a BPM [since ${{k}_{1z}^{(\text{M}1)}}^{\prime}=0$], Eq. (26) shows that ${S}_{1z}^{(\text{M}1)}\ne 0$ due to losses, even if the field is purely evanescent along the

*z*axis.

We note that the appearance of both forward and backward propagation along the *x* axis is a consequence of the change in the energy-flow direction, while along the *z* axis this behavior arises from the change of direction in the phase movement.

#### 6.2. M2

In this case there are two waves within the slab advancing in opposite directions perpendicular to the interfaces and, consequently, we cannot discuss forward or backward propagation along the *z* axis (see Sec. 5.2). Nevertheless, along the *x* axis the situation is different because the two fields propagate in the same direction. We thus consider the *x* component of Eq. (27). Noting that the factor in front and the expression inside the bracket are non-negative real numbers [since
${k}_{1z}^{(\text{M}2)}$ is real], we find that the sign of
${S}_{1x}^{(\text{M}2)}$ is identical to that of the real part of
${\epsilon}_{r1}^{*}{k}_{x}^{(\text{M}2)}$, i.e.,

#### 6.3. M3

Since M3 also contains two waves within the slab propagating in opposite directions along the *z* axis (see Sec. 5.3), we can only discuss forward or backward propagation along the *x* axis. According to Eq. (28), in a manner similar to Eq. (50), one obtains

### 6.3.1. Fundamental modes

For all the solutions plotted in Figs. 4 and 5 Eq. (51) implies
${S}_{1x}^{(\text{M}3)}<0$ and, since
${{k}_{x}^{(\text{M}3)}}^{\prime}>0$ [cf. Eq. (39)], backward propagation. Nevertheless, also the situations of
${S}_{1x}^{(\text{M}3)}=0$ and
${S}_{1x}^{(\text{M}3)}>0$ are possible. For instance, if the slab thickness is sufficiently large,
${k}_{x}^{(\text{M}3)}\approx {k}_{x}^{(\text{M}1)}$ [see Eqs. (5) and (15)] and the behavior of
${S}_{1x}^{(\text{M}3)}$ is analogous to that of
${S}_{1x}^{(\text{M}1)}$ governed by Eqs. (46)–(48). The energy flow then is also similar to that of the single-interface SPP [20]. As another example, we may consider a Ag slab surrounded by air when *λ*_{0} = 325 nm (He-Cd laser), corresponding to *ε _{r}*

_{1}≈ −0.08 +

*i*0.73 [27]. In this case Eqs. (13), (14), and (51) imply ${S}_{1x}^{(\text{M}1)}>0$ for the symmetric FM as

*d*> 32.5 nm and ${S}_{1x}^{(\text{M}1)}>0$ for the antisymmetric FM regardless of the slab thickness. Thus one concludes that FMs can be either FPMs or BPMs. Because ${{k}_{x}^{(\text{M}3)}}^{\prime}$ is positive, the appearance of both FPMs and BPMs is a consequence of the change in the energy-flow direction.

### 6.3.2. Higher-order modes

According to Eq. (51), all the HOM_{I} solutions plotted in Figs. 7 and 8, as well as the dotted curves (air surrounding) in Fig. 9, obey
${S}_{1x}^{(\text{M}3)}>0$, which corresponds to forward propagation [cf. Eq. (42)]. On the other hand, for the solid, dashed, and dash-dotted lines in Fig. 9 one has
${S}_{1x}^{(\text{M}3)}<0$ when *d* < 143 nm (GaP), *d* < 106 nm (C), and *d* < 38 nm (SiO_{2}), respectively, representing backward propagation. Since
${{k}_{x}^{(\text{M}3)}}^{\prime}>0$, HOM_{I}s can be FPMs or BPMs depending on the direction of the energy flow, much as with FMs.

Concerning HOM_{II}s, for *d* < *d*_{c} Eqs. (43) and (51) give
${S}_{1x}^{(\text{M}3)}>0$, corresponding to BPMs. As *d* = *d*_{c}, Eqs. (44) and (51) likewise yield
${S}_{1x}^{(\text{M}3)}>0$ but, since in this case
${{k}_{x}^{(\text{M}3)}}^{\prime}=0$, we cannot discuss forward or backward propagation even along the *x* axis. We note, however, that
${S}_{1x}^{(\text{M}3)}\ne 0$ because of absorption, in spite of the fact that the field is purely evanescent in the *x* direction [20]. When *d* > *d*_{c}, all the HOM_{II} solutions in Figs. 7 and 9 also imply
${S}_{1x}^{(\text{M}3)}>0$ which, according to Eq. (45), represents FPMs. Consequently, both forward and backward propagation occur for HOM_{II}s, in a manner similar to FMs and HOM_{I}s, with the difference that for HOM_{II}s this behavior arises from the change of direction in the phase movement instead of the energy flow.

#### 6.4. Summary

Table 1 summarizes the results on forward and backward propagation for M1–M3 in the region |*z*| < *d*/2. We find that along the *x* axis both FPMs and BPMs are possible for all fields in M1 and M3, while those of M2 are exclusively FPMs. Further it is observed that in the *z* direction M1_{I} and M1_{III} are, respectively, the only mode types for which FPMs and BPMs are found (and defined).

## 7. Conclusions

In summary, we have investigated surface-plasmon polariton solutions at a lossy slab in a non-absorptive dielectric surrounding. The specific aim was to identify all possible plane-wave mode solutions, classify the corresponding fields into different categories, and examine their field propagation and energy flow characteristics. We have analyzed bound modes and leaky modes, as well as forward- and backward-propagating modes. While previous studies have focused on the region outside the slab where only forward propagation can exist, in this work we investigated the field properties within the slab as well. The various surface-plasmon polaritons were divided into three main classes: M1, M2, and M3.

The first class, M1, which can be viewed as generalized Brewster fields, represents to our knowledge field modes of a new kind, not met before. They always contain bound waves at one interface and leaky waves at the other. Solutions of this type were discussed in [21] in the context of asymmetric structures, but for slabs in a symmetric environment such modes are possible only in the case when just one plane wave exists within the slab (two waves propagating in opposite directions orthogonal to the surfaces were exclusively employed in [21]). This led us to further classify M1 into subcategories M1_{I}, M1_{II}, and M1_{III}, in analogy with the single-interface SPP [20]. It was revealed that, inside the slab, depending on the mode type and the material parameters, both forward- and backward-propagating waves are possible. Along the film, the occurrence of forward and backward propagation was shown to originate from the change of direction in the energy flow, while perpendicularly to the slab this behavior is instead a consequence of the change of the phase-propagation direction.

The second class, M2, which includes two different kinds of submodes and has an analogue in dielectric thin-film interference of plane waves for which the reflection coefficient vanishes, is as far as we know likewise entirely new in plasmonics. We demonstrated that for M2 the behavior of the fields outside the slab is similar to that of modes in M1, whereas the field inside the slab is neither decaying nor growing transversally to the interfaces. Furthermore, it was shown that along the slab M2 includes solely forward-propagating waves, the only mode type with this property.

The last class, M3, consists of fields which are either symmetric or antisymmetric with respect to the center of the slab. This class was further divided into fundamental modes (FMs) and higher-order modes (HOMs). The well-known surface-plasmon polariton [21,22,30], which reduces to the single-interface case when the thickness *d* increases, in our case FM, was shown always to be bound to the film. The corresponding leaky wave is not admitted by Maxwell’s equations. It is not obvious why just one (symmetric and antisymmetric) FM exists, when HOMs, previously encountered only in [22], on the other hand are infinite in number. HOMs consist of two classes and vanish when *d* becomes infinite. HOM_{I}s are always bound, while HOM_{II}s may be bound or leaky depending on *d*. Concerning forward and backward propagation, all the modes in M3 can be either forward-propagating or backward-propagating waves. For FMs and HOM_{I}s this feature was demonstrated to arise from the change of the energy-flow direction, whereas for HOM_{II}s the phase movement changes direction.

Finally, we briefly comment on the physical interpretation and significance of the solutions derived and analyzed in this paper. Leaky waves are traditionally rejected as unphysical because their amplitudes and energy densities grow without limit when moving away from the boundaries outside the slab. Nevertheless, leaky waves can be practically meaningful in a transient sense over limited regions of space [21]; hence these fields have not been excluded in this work. All the various solutions (which followed directly from Maxwell’s equations) were formulated with just one wave on each side of the slab; hence the solutions can be regarded as resonance modes of the slab structure, as evidenced explicitly by the generalized Brewster condition for M1 in Sec. 3.1 and the Fabry-Perot resonance condition for M2 and M3 in Sec. 3.2. The various plasmon mode solutions are generally characterized by complex-valued wave vectors, which raises the question of whether or not the modes can be generated in a practical situation. Employing the traditional Otto or Kretschmann configuration [19] enables approximate excitation, by matching the real part of the tangential wave-vector component (or phase) of an external beam with that of the plasmon field. An alternative excitation approach is end-fire coupling [21,22]. A more rigorous resonance matching would also involve the imaginary parts of the wave vectors, necessitating modifications to the usual configurations. In the end, it is the interaction (or resonance) between an incident light and the plasmon modes which is physical in real experiments. Thus we may conclude that the various solutions and their realization describe different aspects of plasmon resonance.

## Acknowledgments

This work was partly funded by the Academy of Finland (projects 272414 and 268480). Andreas Norrman is thankful for support from the Emil Aaltonen Foundation, the Finnish Foundation for Technology Promotion, and the Jenny and Antti Wihuri Foundation.

## References and links

**1. **S. Kawata, *Near-Field Optics and Surface Plasmon Polaritons* (Springer, 2001). [CrossRef]

**2. **W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**, 824–830 (2003). [CrossRef] [PubMed]

**3. **A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. **408**, 131–314 (2005). [CrossRef]

**4. **J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. **70**, 1–87 (2007). [CrossRef]

**5. **D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics **4**, 83–91 (2010). [CrossRef]

**6. **M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express **19**, 22029–22106 (2011). [CrossRef] [PubMed]

**7. **L. Novotny and B. Hecht, *Principles of Nano-Optics*, 2nd ed. (Cambridge University, 2012). [CrossRef]

**8. **R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. **106**, 874–881 (1957). [CrossRef]

**9. **E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science **311**, 189–193 (2006). [CrossRef] [PubMed]

**10. **S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nature Photon. **1**, 641–648 (2007). [CrossRef]

**11. **C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature **445**, 39–46 (2007). [CrossRef] [PubMed]

**12. **D. M. Koller, A. Hohenau, H. Ditlbacher, N. Galler, F. Reil, F. R. Aussenegg, A. Leitner, E. J. W. List, and J. R. Krenn, “Organic plasmon-emitting diode,” Nature Photon. **2**, 684–687 (2008). [CrossRef]

**13. **S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. **101**, 047401 (2008). [CrossRef] [PubMed]

**14. **M. A. Noginov, G. Shu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature **460**, 1110–1112 (2009). [CrossRef] [PubMed]

**15. **E. Verhagen, M. Spacenovic, A. Polman, and L. K. Kuipers, “Nanowire plasmon excitation by adiabatic mode transformation,” Phys. Rev. Lett. **102**, 203904 (2009). [CrossRef] [PubMed]

**16. **R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature **461**, 629–632 (2009). [CrossRef] [PubMed]

**17. **M. Fukuda, T. Aihara, K. Yamaguchi, Y. Y. Ling, K. Miyaji, and M. Tohyama, “Light detection enhanced by surface plasmon resonance in metal film,” Appl. Phys. Lett. **96**, 153107 (2010). [CrossRef]

**18. **M. Ozaki, J.-I. Kato, and S. Kawata, “Surface-plasmon holography with white-light illumination,” Science **332**, 218–220 (2011). [CrossRef] [PubMed]

**19. **H. Raether, *Surface Plasmons on Smooth and Rough Surfaces and on Gratings* (Springer, 1988).

**20. **A. Norrman, T. Setälä, and A. T. Friberg, “Exact surface-plasmon polariton solutions at a lossy interface,” Opt. Lett. **38**, 1119–1121 (2013). [CrossRef] [PubMed]

**21. **J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B **33**, 5186–5201 (1986). [CrossRef]

**22. **A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Surface plasmon polaritons on metallic surfaces,” Opt. Express **15**, 183–197 (2007). [CrossRef] [PubMed]

**23. **A. Norrman, T. Setälä, and A. T. Friberg, “Partial spatial coherence and partial polarization in random evanescent fields on lossless interfaces,” J. Opt. Soc. Am. A **28**, 391–400 (2011). [CrossRef]

**24. **M. Born and E. Wolf, *Principles of Optics*, 7th ed. (Cambridge University, 1999). [CrossRef]

**25. **F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B **44**, 5855–5872 (1991). [CrossRef]

**26. **P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. **1**, 484–588 (2009). [CrossRef]

**27. **E. D. Palik, ed., *Handbook of Optical Constants of Solids* (Academic, 1998).

**28. **D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B **27**, 985–1009 (1983). [CrossRef]

**29. **M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. V. Stryland, *Handbook of Optics*, 3rd ed., Vol. 4. (McGraw-Hill, 2009).

**30. **J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Planar metal plasmon waveguides: frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B **72**, 075405 (2005). [CrossRef]