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_r1 = ε′_r1 + iε″_r1 in the region −d/2 < z < d/2, with ε′_r1 < 0 and ε″_r1 > 0, surrounded on both sides by the same dielectric media possessing a real and positive permittivity ε_r2. 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 k₀ 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.

 

Fig. 1 Illustration of the geometry and notation related to electromagnetic plane waves at a lossy film of thickness d surrounded by non-absorptive dielectrics, possessing relative permittivities ε_r1 (complex) and ε_r2 (real), respectively. The quantities ${\widehat{\mathbf{k}}}_1^{(1)}$, ${\widehat{\mathbf{k}}}_1^{(2)}$, ${\widehat{\mathbf{k}}}_2^{(1)}$, and ${\widehat{\mathbf{k}}}_2^{(2)}$, and ${\widehat{\mathbf{p}}}_1^{(1)}$, ${\widehat{\mathbf{p}}}_1^{(2)}$, ${\widehat{\mathbf{p}}}_2^{(1)}$, and ${\widehat{\mathbf{p}}}_2^{(2)}$ in the xz plane are the unit wave and polarization vectors, respectively. The interfaces are at z = ±d/2.

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

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′_r1 < 0, we remark that the results presented in this section are also valid for ε′_r1 > 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

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

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

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

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]

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

4. Field propagation and energy flow

Let us write k_x = k′_x + ik″_x and $k_{\alpha z}^{(\beta )}={k_{\alpha z}^{(\beta )}}^{\prime }+i{k_{\alpha z}^{(\beta )}}^{″}$, with ′ and ″ denoting the real and imaginary parts, respectively. As with Sec. 3, even if we are interested in the case ε′_r1 < 0, the results in this section also hold for ε′_r1 > 0.

4.1. Phase propagation and amplitude attenuation

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

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

For |z| < d/2, the wave-vector components behave as

4.2. Flux of energy

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

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

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]

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

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

 

Fig. 2 Illustration of the possible directions of phase propagation (black arrows) and field attenuation (solid-red curves) for M1 decaying to the right. In the figure, (a) and (d) correspond to M1_I [Eq. (34)], (b) and (e) to M1_II [Eq. (35)], and (c) and (f) to M1_III [Eq. (36)], respectively. The graphs in the bottom row are mirror images of those in the top row.

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5.2. M2

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

Figure 3 illustrates the wavefront propagation and field attenuation for M2 decaying in the positive x direction [ ${k_x^{(\text{M}2)}}^{″}>0$]. 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.

 

Fig. 3 Illustration of the possible directions of phase propagation (black arrows) and field attenuation (solid-red curves) for M2 decaying to the right. Within the slab there are two fields whose wave-vector components perpendicular to the surfaces are purely real and opposite in sign. Graph (b) is a mirror image of graph (a).

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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., ${k_x^{(\text{M}3)}}^{″}>0$, 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 ${k_x^{(\text{M}3)}}^{″}/k_0$ (right column) for symmetric (upper row) and antisymmetric (lower row) FMs at a silver (Ag) slab surrounded by air (ε_r2 = 1), as a function of d for different vacuum wavelengths λ₀ = 2π/k₀. 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 ${k_x^{(\text{M}3)}}^{″}$. 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.

 

Fig. 4 Behavior of the real (left column) and imaginary (right column) parts of the tangential wave-vector component $k_x^{(\text{M}3)}$ for symmetric (upper row) and antisymmetric (lower row) FMs at a silver slab in an air surrounding, as a function of the thickness d. The solid (blue), dashed (green), dash-dotted (orange), and dotted (red) lines correspond to the relative permittivities ε_r1 = −3.77+i0.67 (λ₀ = 400 nm), ε_r1 = −8.50+i0.76 (λ₀ = 500 nm), ε_r1 = −13.91 + i0.93 (λ₀ = 600 nm), and ε_r1 = −20.44 + i1.29 (λ₀ = 700 nm) of silver [27], with λ₀ = 2π/k₀ and k₀ being the vacuum wavelength and wave number, respectively.

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Figure 5 illustrates the behavior of ${k_x^{(\text{M}3)}}^{\prime }/k_0$ (left column) and ${k_x^{(\text{M}3)}}^{″}/k_0$ (right column) for symmetric (upper row) and antisymmetric (lower row) FMs at a Ag slab surrounded by gallium phosphide (GaP), diamond (C), fused silica (SiO₂), and air, as a function of d at λ₀ = 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 ${k_x^{(\text{M}3)}}^{″}$.

 

Fig. 5 Behavior of the real (left column) and imaginary (right column) parts of the tangential wave-vector component $k_x^{(\text{M}3)}$ for symmetric (upper row) and antisymmetric (lower row) FMs, as a function of the thickness d at a silver slab in different surroundings: gallium phosphide (solid lines), diamond (dashed lines), fused silica (dash-dotted lines), and air (dotted lines), having the relative permittivities ε_r2 = 11.01 [28], ε_r2 = 5.82 [29], ε_r2 = 2.12 [29], and ε_r2 = 1, respectively, at the vacuum wavelength λ₀ = 2π/k₀ = 632.8 nm (k₀ is the free-space wave number). The relative permittivity of silver is ε_r1 = −15.87 + i1.07 [27].

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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 ${k_{2z}^{(\text{M}3)}}^{″}>0$. Thus ${k_{2z}^{(\text{M}3)}}^{″}<0$ 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 ${k_x^{(\text{M}3)}}^{″}$ 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 ${k_{2z}^{(\text{M}3)}}^{″}>0$], we have

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 λ₀ = 350 nm, corresponding to ε_r1 ≈ −1.79 + i0.60 [27], gives $k_{1z}^{(\text{M}3)}\approx \pm k_0(0.031+i1.868)$, $k_{1z}^{(\text{M}3)}\approx \pm i1.893k_0$, 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].

 

Fig. 6 Illustration of the possible directions of phase propagation (black arrows) and field attenuation (solid-red curves) for FMs, (a)–(c), and HOM_Is, (c), decaying to the right. In the figure, (a) corresponds to Eq. (21), (b) to Eq. (22), and (c) to Eq. (23).

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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

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 ${k_x^{(\text{M}3)}}^{″}/k_0$ (right column) for the four symmetric HOM_Is (upper row) and HOM_IIs (lower row) having the lowest ${k_x^{(\text{M}3)}}^{″}$ at a Ag slab surrounded by air, as a function of d when λ₀ = 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 ${k_x^{(\text{M}3)}}^{″}$ 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 [ $1/{k_x^{(\text{M}3)}}^{″}$] 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_Is slightly decrease before diverging as d tends to zero. Regarding HOM_IIs, 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) ${k_x^{(\text{M}3)}}^{″}$ 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.

 

Fig. 7 Behavior of the real (left column) and imaginary (right column) parts of the tangential wave-vector component $k_x^{(\text{M}3)}$ for the four symmetric HOM_Is (upper row) and HOM_IIs (lower row) possessing the lowest ${k_x^{(\text{M}3)}}^{″}$ at a silver slab in an air surrounding, as a function of the thickness d at the vacuum wavelength λ₀ = 2π/k₀ = 632.8 nm (k₀ is the free-space wave number). The relative permittivity of silver is ε_r1 = −15.87 + i1.07 [27].

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Figure 8 illustrates the behavior of ${k_x^{(\text{M}3)}}^{\prime }/k_0$ (left column) and ${k_x^{(\text{M}3)}}^{″}/k_0$ (right column) for the symmetric HOM_I (upper row) and HOM_II (lower row) having the lowest ${k_x^{(\text{M}3)}}^{″}$ at a Ag slab surrounded by air, as a function of d when λ₀ is varied. The figure shows that, for both mode types, decreasing the wavelength gives a smaller ${k_x^{(\text{M}3)}}^{″}$ 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 λ₀ 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 (λ₀ = 700 nm), d_c ≈ 79 nm (λ₀ = 600 nm), d_c ≈ 90 nm (λ₀ = 500 nm), and d_c ≈ 109 nm (λ₀ = 400 nm).

 

Fig. 8 Behavior of the real (left column) and imaginary (right column) parts of the tangential wave-vector component $k_x^{(\text{M}3)}$ for the symmetric HOM_I (upper row) and HOM_II (lower row) having the lowest ${k_x^{(\text{M}3)}}^{″}$, at a silver slab in an air surrounding as a function of the thickness d. The solid (blue), dashed (green), dash-dotted (orange), and dotted (red) lines correspond to the relative permittivities ε_r1 = −3.77 + i0.67 (λ₀ = 400 nm), ε_r1 = −8.50 + i0.76 (λ₀ = 500 nm), ε_r1 = −13.91 + i0.93 (λ₀ = 600 nm), and ε_r1 = −20.44 + i1.29 (λ₀ = 700 nm) of silver [27], with λ₀ = 2π/k₀ and k₀ being the wavelength and wave number in vacuum, respectively.

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As a final example, we have in Fig. 9 plotted ${k_x^{(\text{M}3)}}^{\prime }/k_0$ (left column) and ${k_x^{(\text{M}3)}}^{″}/k_0$ (right column) for the symmetric HOM_I (upper row) and HOM_II (lower row) of the lowest ${k_x^{(\text{M}3)}}^{″}$ at a Ag slab surrounded by GaP, diamond, SiO₂, and air, as a function of d at λ₀ = 632.8 nm. We particularly observe that for both mode types varying the refractive index of the surrounding medium has a negligible effect on ${k_x^{(\text{M}3)}}^{″}$ 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_r2 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 (SiO₂), d_c ≈ 137 nm (C), and d_c ≈ 164 nm (GaP)]. In the limit d → ∞, on the other hand, changing ε_r2 has no influence on the real part, as is also evident from Eq. (16).

 

Fig. 9 Behavior of the real (left column) and imaginary (right column) parts of the tangential wave-vector component $k_x^{(\text{M}3)}$ for the symmetric HOM_I (upper row) and HOM_II (lower row) having the lowest ${k_x^{(\text{M}3)}}^{″}$, as a function of the thickness d at a silver slab in different surroundings: gallium phosphide (solid lines), diamond (dashed lines), fused silica (dash-dotted lines), and air (dotted lines), possessing the relative permittivities ε_r2 = 11.01 [28], ε_r2 = 5.82 [29], ε_r2 = 2.12 [29], and ε_r2 = 1, respectively, at the vacuum wavelength λ₀ = 2π/k₀ = 632.8 nm (k₀ is the free-space wave number). The relative permittivity of silver is ε_r1 = −15.87 + i1.07 [27].

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Regarding HOM_Is, all the plots in Figs. 7 and 9 correspond to ${k_{2z}^{(\text{M}3)}}^{″}>0$ for any d (as in the case with FMs), justifying the definition in Eq. (40). Consequently, the numerical results suggest that only bound HOM_Is 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_Is 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., ${k_{1z}^{(\text{M}3)}}^{\prime }{k_{1z}^{(\text{M}3)}}^{″}<0$, and thus the total field-propagation behavior of HOM_Is in the figures is as summarized in Fig. 6(c).

As for HOM_IIs, all the plots in Figs. 7 and 9 correspond to ${k_{2z}^{(\text{M}3)}}^{″}>0$ (d < d_c), ${k_{2z}^{(\text{M}3)}}^{″}=0$ (d = d_c), and ${k_{2z}^{(\text{M}3)}}^{″}<0$ (d > d_c), justifying our definition in Eq. (41). Further, Figs. 7 and 9 show that ${k_x^{(\text{M}3)}}^{\prime }$ and ${k_x^{(\text{M}3)}}^{″}$ 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