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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 4 — Feb. 24, 2014
  • pp: 4628–4648
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Surface-plasmon polariton solutions at a lossy slab in a symmetric surrounding

Andreas Norrman, Tero Setälä, and Ari T. Friberg  »View Author Affiliations


Optics Express, Vol. 22, Issue 4, pp. 4628-4648 (2014)
http://dx.doi.org/10.1364/OE.22.004628


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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

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

6

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

], which constitutes an exceptionally rich area of nano-optics [7

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

]. 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

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

], 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

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

18

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

].

The fundamental properties of SPPs at planar (and rough) interfaces have been extensively studied in the past [19

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

]. 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

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]

]. 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

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]

], 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

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

]. These findings suggest that new types of SPPs, possessing peculiar physical properties, can be encountered at planar interfaces.

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

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 εr1 (complex) and εr2 (real), respectively. The quantities k^1(1), k^1(2), k^2(1), and k^2(2), and p^1(1), p^1(2), p^2(1), and p^2(2) in the xz plane are the unit wave and polarization vectors, respectively. The interfaces are at z = ±d/2.

The spatial part of the p-polarized electric field, which follows from the Helmholtz equation, can be expressed explicitly as
E(r)={E2(1)eik2(1)rp^2(1),zd/2,E1(1)eik1(1)rp^1(1)+E1(2)eik1(2)rp^1(2),|z|<d/2,E2(2)eik2(2)rp^2(2),zd/2,
(1)
where Eα(β) (α, β ∈ {1, 2}) is the complex field amplitude and
p^α(β)=k^α(β)×e^y,k^α(β)=kα(β)/|kα(β)|,α,β{1,2},
(2)
are the normalized polarization and wave vectors, respectively, with êy being the unit vector in the positive y direction and |kα(β)|=[kα(β)*kα(β)]1/2 (the asterisk * denotes complex conjugation). We note that p^α(β) and k^α(β) are not orthogonal if kα(β) is complex, in which case also |kα(β)|kα [20

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]

, 23

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]

].

3. Slab modes

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: k1z(1)=k1z(2)

In this case k1(1)=k1(2)k1(M1), whereupon also p^1(1)=p^1(2)p^1(M1) according to Eq. (2). The two parts of the field inside the slab [see Eq. (1)] can be combined and written as
E1(M1)(r)=E1(M1)eik1(M1)rp^1(M1),
(3)
where E1(M1)E1(1)+E1(2). Hence in this situation there is only one plane wave in each of the three regions. It now follows from Eqs. (1)(3), Maxwell’s equations, and the electromagnetic boundary conditions, that the fields have to satisfy
{εr2k1z(M1)=εr1k2z(1),z=d/2,εr2k1z(M1)=εr1k2z(2),z=d/2,
(4)
leading to k2z(1)=k2z(2)k2z(M1). In addition, from Eq. (4) one obtains
kx(M1)=k0εr1εr2εr1+εr2,kαz(M1)=k0εrαεr1+εr2,α{1,2},
(5)
which are exactly the same as the corresponding wave-vector components of SPPs at a single interface [7

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

, 20

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]

]. We note that Eq. (5) is completely independent of 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

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

]. 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: k1z(1)=k1z(2)

In this case the boundary conditions result in
r(1)e2ik1z(1)d=r(2),
(6)
where
r(β)εr2k1z(1)εr1k2z(β)εr2k1z(1)+εr1k2z(β),β{1,2}.
(7)
One can readily verify that k1z(1)=0 always satisfies Eq. (6), regardless of the film thickness 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 E1(1), and 1/r(2) likewise for E1(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., k2z(1)=±k2z(2).

3.2.1. M2: k2z(1)=k2z(2)

We first consider k2z(1)=k2z(2)k2z(M2) for which r(1) = r(2)r (the Fresnel reflection coefficients are not bounded in magnitude by unity [7

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

, 23

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]

]). Equation (6) then reduces to
eik1z(M2)d=±1,
(8)
with k1z(M2)k1z(1). Thus, depending on the sign of the right-hand side above, we obtain two different kinds of mode solutions inside the slab. According to Eqs. (1) and (2), Maxwell’s equations, and the boundary conditions, they can be expressed as
E1±(M2)(r)=E1(M2)eik1(M2)r|k1(M2)|{k1z(M2)[1re2ik1z(M2)z]e^x+kx(M2)[1±re2ik1z(M2)z]e^z},
(9)
where E1(M2)E1(1) and |k1(M2)||k1(1)|=|k1(2)|. The subscript ± corresponds to ± in Eq. (8), while êx and êz are the unit vectors in the positive x and z directions, respectively. Equation (8) further yields
kx(M2)=k0εr1(m±πk0d)2,k1z(M2)=m±(πd),k2z(M2)=k0εr2εr1+(m±πk0d)2,
(10)
where m+ = 2n and m = 2n + 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 k1z(M2)0 when d → ∞, and thus all the M2 modes vanish in this limit. Further we observe that k1z(M2) is real (even when εr1 is complex) and fully independent of material parameters, while kx(M2) does not depend on εr2, 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

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

].

3.2.2. M3: k2z(1)=k2z(2)

This last case corresponds to the conventionally studied situation. On writing k1z(1)k1z(M3), Eq. (6) becomes
r(1)eik1z(M3)d=±1,
(11)
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
E1±(M3)(r)=E1(M3)eik1(M3)r|k1(M3)|{k1z(M3)[1e2ik1z(M3)z]e^x+kx(M3)[1±e2ik1z(M3)z]e^z},
(12)
where E1(M3)E1(1) and |k1(M3)||k1(1)|=|k1(2)|. The modes E1+(M3)(r) and E1(M3)(r) represent electric fields whose normal components (or magnetic fields) are symmetric and antisymmetric with respect to z = 0, respectively. Equation (11) can further be split into
Symmetric(+):εr1εr2k2z(M3)k1z(M3)=tanh[12ik1z(M3)d],
(13)
Antisymmetric():εr1εr2k2z(M3)k1z(M3)=coth[12ik1z(M3)d],
(14)
with k2z(M3)k2z(1), which are transcendental equations for kx(M3) and generally require numerical methods to solve.

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

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

]. However, in marked contrast to Eq. (8), when d → ∞, both Eq. (13) and Eq. (14) have solutions that converge towards common values for which k1z(M3)0, namely [21

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

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

]
kx(M3)k0εr1εr2εr1+εr2,kαz(M3)k0εrαεr1+εr2,α{1,2}.
(15)
These degenerate wave-vector components are exactly those obtained for SPPs at a single interface [7

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

,20

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]

] [we also encountered the same solutions already in the context of M1, see Eq. (5) in Sec. 3.1]. The fields associated with the solutions of Eqs. (13) and (14) that approach the limit of Eq. (15) as d → ∞ are therefore regarded (at any d) as fundamental modes (FMs) [22

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

]. 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
kx(M3)k0εr1,k1z(M3)0,k2z(M3)k0εr2εr1.
(16)
Similarly to M2, the corresponding fields vanish as d becomes infinite [ k1z(M3)=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
kx(M3)k0εr2,k1z(M3)k0εr1εr2,k2z(M3)0,
(17)
representing a wave that is vertically polarized outside the slab and of infinite extent along the x axis, whereupon it is generally termed the long-range surface-plasmon polariton (LRSPP) [25

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

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

]. It can readily be verified from Eq. (14) that no such antisymmetric LRSPP exists.

4. Field propagation and energy flow

Let us write kx = k′x + ik″x and kαz(β)=kαz(β)+ikαz(β), 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
eikα(β)r=e[kxx+kαz(β)z]ei[kxx+kαz(β)z],α,β{1,2},
(18)
where the first term is responsible for the decay or growth of the field amplitude and the second is associated with the phase of the wave. With the time dependence exp(−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
kxkx=k2z(β)k2z(β),β{1,2},
(19)
indicating that if the directions of phase propagation and field attenuation are the same along one axis, then they will be opposite along the other axis. More precisely, if k′x and k″x have the same (opposite) sign, Eq. (19) implies that k2z(β)k2z(β)<0[k2z(β)k2z(β)>0] and, consequently, k2z(β) and k2z(β) have opposite (same) signs. Furthermore, it can be shown that
kx=0k2z(β)=0,k2z(β)=0kx=0,β{1,2},
(20)
demonstrating that a purely evanescent field along one axis is strictly propagating in the perpendicular direction (but not vice versa).

For |z| < d/2, the wave-vector components behave as
kxkx<12k02εr1k1z(β)k1z(β)>0,β{1,2},
(21)
kxkx=12k02εr1k1z(β)k1z(β)=0,β{1,2},
(22)
kxkx>12k02εr1k1z(β)k1z(β)<0,β{1,2}.
(23)
Equation (21) corresponds to the situation in which the wavefront propagation and amplitude attenuation directions are the same along the 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″r1 > 0).

4.2. Flux of energy

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

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

]
S(r)12[E(r)×H*(r)],
(24)
where H(r) = (0k0)−1[k × E(r)] is the magnetic field, c is the speed of light, and μ0 is the vacuum permeability. Outside the slab, for all mode types (M1–M3), the Poynting vector becomes
S2(β)(r)=σεr2|E2(β)|2|k2|2e2k2(β)rk2(β),β{1,2},
(25)
where σk0/(20) and |k2||k2(1)|=|k2(2)|. Since εr2 > 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
S1(M1)(r)=σ|E1(M1)|2|k1(M1)|2e2k1(M1)r{[εr1kx(M1)+εr1kx(M1)]e^x+[εr1k1z(M1)+εr1k1z(M1)]e^z},
(26)
S1±(M2)(r)=σ|E1(M2)|2|k1(M2)|2e2kx(M2)x(εr1*{kx(M2)[1+|r|2±r*e2ik1z(M2)z±re2ik1z(M2)z]e^x+k1z(M2)[1|r|2±r*e2ik1z(M2)zre2ik1z(M2)z]e^z}),
(27)
S1±(M3)(r)=2σ|E1(M3)|2|k1(M3)|2e2kx(M3)x[εr1*(kx(M3){cosh[2k1z(M3)z]±cos[2k1z(M3)z]}e^xk1z(M3){sinh[2k1z(M3)z]isin[2k1z(M3)z]}e^z)].
(28)
As is evident from Eqs. (26)(28), the direction of the Poynting vector inside the slab does not necessarily coincide with that of phase propagation, but instead it depends on a combination of the real and imaginary parts of the wave vectors and εr1 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 k2z(1)=k2z(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 kx(M1) and k2z(M1) of Eq. (5) always satisfy [20

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]

]
kx(M1)kx(M1)>0,k2z(M1)k2z(M1)<0,
(29)
indicating that the directions of phase propagation and amplitude attenuation are the same along the x axis, but along the z axis they are opposite in the region |z| ≥ d/2, respectively.

For k1z(M1) in the region |z| < d/2, on the other hand, we obtain (for a derivation, see [20

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]

])
εr2<εr1k1z(M1)k1z(M1)>0,
(30)
(εr1+εr2)2>εr22εr12k1z(M1)k1z(M1)>0,
(31)
(εr1+εr2)2=εr22εr2k1z(M1)=0,
(32)
(εr1+εr2)2<εr22εr12k1z(M1)k1z(M1)<0.
(33)
Motivated by Eqs. (30)(33) we classify three different types of M1s as
M1I:=k1z(M1)k1z(M1)>0,
(34)
M1II:=k1z(M1)=0,
(35)
M1III:=k1z(M1)k1z(M1)<0,
(36)
where M1I corresponds to the phase moving in the same direction as the field decays, while for M1III the phase propagation is opposite to that of attenuation (along the z axis). M1II 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 M1I and M1III are identical with the modes SPP I and SPP II introduced in [20

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]

] 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
k1z(M1)k2z(M1)<0,
(37)
i.e., they have opposite signs. Restricting to fields decaying in the positive x direction [ kx(M1)>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

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]

]), 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

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]

].

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 M1I [Eq. (34)], (b) and (e) to M1II [Eq. (35)], and (c) and (f) to M1III [Eq. (36)], respectively. The graphs in the bottom row are mirror images of those in the top row.

5.2. M2

In this case Eq. (10) implies k1z(M2)=0,, and therefore, according to Eqs. (19) and (22), we again have [cf. Eq. (29)]
kx(M2)kx(M2)>0,k2z(M2)k2z(M2)<0.
(38)
Hence the propagation of M2 along the 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, k1z(M2) 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 [ kx(M2)>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).

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 [ k2z(2)=k2z(1)]. However, there is another fundamental difference between M3 and M1 and M2. Unlike frequently asserted [21

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]

], we emphasize that both signs in k2z(M3)=±[k2(M3)2kx(M3)2]1/2 are not allowed for a fixed kx(M3), since Eqs. (13) and (14) are not invariant with respect to the change of sign of k2z(M3). 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 kx(M3), 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 k1z(M3) for |z| < d/2, we show that all of the cases in Eqs. (21)(23) are possible for M3 and thus the behavior of k1z(M3) to some extent resembles that of k1z(M1) 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).

5.3.1. Fundamental modes

Let us first discuss FMs. Figure 4 illustrates the behavior of kx(M3)/k0 (left column) and kx(M3)/k0 (right column) for symmetric (upper row) and antisymmetric (lower row) FMs at a silver (Ag) slab surrounded by air (εr2 = 1), as a function of d for different vacuum wavelengths λ0 = 2π/k0. We observe that in all cases kx(M3) 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 kx(M3)k0 in the limit d → 0, in accordance with Eq. (17), while the antisymmetric modes acquire large values for kx(M3) and kx(M3). In addition, one finds that for both field types the real and imaginary parts of kx(M3) 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 kx(M3) 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 εr1 = −3.77+i0.67 (λ0 = 400 nm), εr1 = −8.50+i0.76 (λ0 = 500 nm), εr1 = −13.91 + i0.93 (λ0 = 600 nm), and εr1 = −20.44 + i1.29 (λ0 = 700 nm) of silver [27], with λ0 = 2π/k0 and k0 being the vacuum wavelength and wave number, respectively.

Figure 5 illustrates the behavior of kx(M3)/k0 (left column) and kx(M3)/k0 (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 kx(M3) 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 kx(M3) and kx(M3).

Fig. 5 Behavior of the real (left column) and imaginary (right column) parts of the tangential wave-vector component kx(M3) 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 εr2 = 11.01 [28], εr2 = 5.82 [29], εr2 = 2.12 [29], and εr2 = 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 εr1 = −15.87 + i1.07 [27].

We recall that Eqs. (13) and (14) permit only one sign for k2z(M3) at any given kx(M3) (see the beginning of Sec. 5.3). All the solutions plotted in Figs. 4 and 5 correspond to k2z(M3)>0. Thus k2z(M3)<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 kx(M3) and kx(M3) 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 k2z(M3)>0], we have
kx(M3)kx(M3)>0,k2z(M3)<0,
(39)
indicating that the directions of phase movement and amplitude attenuation of FMs are the same along the x axis (similarly to M1 and M2) and the wavefronts propagate towards the interfaces outside the slab.

Concerning k1z(M3) 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 εr1 ≈ −1.79 + i0.60 [27

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

], gives k1z(M3)±k0(0.031+i1.868), k1z(M3)±i1.893k0, and k1z(M3)±k0(0.002+i1.895) 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 k1(β), 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

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]

].

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

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
HOMI:=k2z(M3)>0,d,
(40)
HOMII:={k2z(M3)>0,d<dc,k2z(M3)=0,d=dc,k2z(M3)<0,d>dc,
(41)
where dc is a critical thickness that depends on the particular mode, media, and wavelength. Thus HOMIs are bound fields regardless of the film thickness, whereas HOMIIs can be either bound (d < dc), or leaky (d > dc), or strictly propagating in opposite directions along the z axis outside the slab (d = dc).

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 kx(M3)/k0 (left column) and kx(M3)/k0 (right column) for the four symmetric HOMIs (upper row) and HOMIIs (lower row) having the lowest kx(M3) 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 kx(M3)=k0εr1k0(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 kx(M3) 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/kx(M3)] 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

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

]. As for kx(M3), when d becomes smaller the real parts of HOMIs slightly decrease before diverging as d tends to zero. Regarding HOMIIs, when d is reduced kx(M3) gets smaller and, at the critical thickness dc (which depends on the particular mode), the real part becomes zero. It is observed that the HOMII having the lowest (highest) kx(M3) possesses the smallest (largest) critical thickness. As d further decreases (d < dc) 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 kx(M3) for the four symmetric HOMIs (upper row) and HOMIIs (lower row) possessing the lowest kx(M3) 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 εr1 = −15.87 + i1.07 [27].

Figure 8 illustrates the behavior of kx(M3)/k0 (left column) and kx(M3)/k0 (right column) for the symmetric HOMI (upper row) and HOMII (lower row) having the lowest kx(M3) 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 kx(M3) and, consequently, larger propagation lengths along the interfaces. This feature is opposite to that of the corresponding FMs depicted in Fig. 4. Regarding kx(M3), reducing λ0 results in a slight increase of the real part in the case of HOMI and for HOMII the critical thickness is shifted towards larger values: dc ≈ 66 nm (λ0 = 700 nm), dc ≈ 79 nm (λ0 = 600 nm), dc ≈ 90 nm (λ0 = 500 nm), and dc ≈ 109 nm (λ0 = 400 nm).

Fig. 8 Behavior of the real (left column) and imaginary (right column) parts of the tangential wave-vector component kx(M3) for the symmetric HOMI (upper row) and HOMII (lower row) having the lowest kx(M3), 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 εr1 = −3.77 + i0.67 (λ0 = 400 nm), εr1 = −8.50 + i0.76 (λ0 = 500 nm), εr1 = −13.91 + i0.93 (λ0 = 600 nm), and εr1 = −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.

As a final example, we have in Fig. 9 plotted kx(M3)/k0 (left column) and kx(M3)/k0 (right column) for the symmetric HOMI (upper row) and HOMII (lower row) of the lowest kx(M3) 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 kx(M3) 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 εr2 is varied and d is sufficiently small, i.e., increasing the refractive index outside the slab leads to a larger kx(M3) for HOMI and an increased critical thickness of HOMII [dc ≈ 73 nm (air), dc ≈ 97 nm (SiO2), dc ≈ 137 nm (C), and dc ≈ 164 nm (GaP)]. In the limit d → ∞, on the other hand, changing εr2 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 kx(M3) for the symmetric HOMI (upper row) and HOMII (lower row) having the lowest kx(M3), 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 εr2 = 11.01 [28], εr2 = 5.82 [29], εr2 = 2.12 [29], and εr2 = 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 εr1 = −15.87 + i1.07 [27].

Regarding HOMIs, all the plots in Figs. 7 and 9 correspond to k2z(M3)>0 for any d (as in the case with FMs), justifying the definition in Eq. (40). Consequently, the numerical results suggest that only bound HOMIs can exist. From the same figures one also finds that the real and imaginary parts of kx(M3) have the same sign. Hence, and according to Eq. (19), we obtain
kx(M3)kx(M3)>0,k2z(M3)<0,
(42)

indicating that the wave propagation of HOMIs in the x direction and outside the slab is similar to that of FMs [cf. Eq. (39)]. Concerning k1z(M3), all the solutions plotted in Figs. 7 and 9 obey Eq. (23), i.e., k1z(M3)k1z(M3)<0, and thus the total field-propagation behavior of HOMIs in the figures is as summarized in Fig. 6(c).

As for HOMIIs, all the plots in Figs. 7 and 9 correspond to k2z(M3)>0 (d < dc), k2z(M3)=0 (d = dc), and k2z(M3)<0 (d > dc), justifying our definition in Eq. (41). Further, Figs. 7 and 9 show that kx(M3) and kx(M3) have the opposite (same) sign for d < dc (d > dc). In addition, at the transition point d = dc, where kx(M3)=0, only those solutions with k2z(M3)>0 are allowed by Eq. (13). Thus, and according to Eq. (19), the numerical analysis implies that
d<dc[k2z(M3)>0]kx(M3)kx(M3)<0,k2z(M3)>0,
(43)
d=dc[k2z(M3)=0]kx(M3)=0,k2z(M3)>0,
(44)
d>dc[k2z(M3)<0]kx(M3)kx(M3)>0,k2z(M3)>0.
(45)
The equations above particularly show that HOMIIs obey k2z(M3)>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 HOMIs [Eq. (42)]. Concerning kx(M3), 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 kx(M3) is analogous to that in Eqs. (39) and (42) of FMs and HOMIs, respectively.

Regarding k1z(M3) of HOMIIs, one readily confirms from Eqs. (21), (43), and (44) that the real and imaginary parts always have the same sign for ddc. 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 > dc. Hence we may conclude that HOMIIs satisfy k1z(M3)k1z(M3)>0 regardless of d, in contrast to HOMIs having k1z(M3)k1z(M3)<0 at any slab thickness.

We have in Fig. 10 summarized the (three) different combinations of phase movement and amplitude attenuation for HOMIIs discussed above. As d < dc (left figure), the field attenuation is similar to that of FMs and HOMIs 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 > dc (right figure), the fields become leaky and the wavefronts advance away from the slab for |z| ≥ d/2, in contrast to FMs and HOMIs in Fig. 6. On the other hand, inside the film the wave propagation is analogous to FMs and HOMIs and, in addition, HOMIIs also decay in the same (positive) direction as the phases propagate along the x axis in both media.

Fig. 10 Illustration of the possible directions of phase propagation (black arrows) and field attenuation (solid-red curves) for HOMIIs decaying to the right, when the slab thickness d is smaller than (left), equal to (middle), and larger than (right) the critical thickness dc.

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, S2(β)(r) is parallel to k2(β) and decays exponentially at twice the rate of the field in the direction determined by k2(β). 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″x < 0.

6.1. M1

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

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]

])
εr1>12εr2S1x(M1)>0,
(46)
εr1=12εr2S1x(M1)=0,
(47)
εr1<12εr2S1x(M1)<0.
(48)
Hence the sign of S1x(M1) is totally independent of ε″r1. Since kx(M1)>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 M1I–M1III 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

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]

]
k1z(M1)0S1z(M1)<0,
(49)
where the upper or lower symbols are taken in both expressions, indicating that energy flows in the same direction as the field attenuates. Thus, from Eqs. (34), (36), and (49), we conclude that M1I is a FPM while M1III is a BPM perpendicular to the interfaces. Regarding M1II, which is neither a FPM nor a BPM [since k1z(M1)=0], Eq. (26) shows that S1z(M1)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 k1z(M2) is real], we find that the sign of S1x(M2) is identical to that of the real part of εr1*kx(M2), i.e.,
sgn[S1x(M2)]=sgn[εr1kx(M2)+εr1kx(M2)],
(50)
where sgn is the sign function. Further, by using Eqs. (10), (22), and (38) [the last one implies kx(M2)>0], one gets sgn[S1x(M2)]=sgn{εr1+2[kx(M2)/k0]2}. This result, together with Eq. (10), leads to S1x(M2)>0, which according to Eq. (38) shows that M2 is a FPM. As we will see, M2 is the only type of mode which is always a FPM.

6.3. M3

6.3.1. Fundamental modes

For all the solutions plotted in Figs. 4 and 5 Eq. (51) implies S1x(M3)<0 and, since kx(M3)>0 [cf. Eq. (39)], backward propagation. Nevertheless, also the situations of S1x(M3)=0 and S1x(M3)>0 are possible. For instance, if the slab thickness is sufficiently large, kx(M3)kx(M1) [see Eqs. (5) and (15)] and the behavior of S1x(M3) is analogous to that of S1x(M1) governed by Eqs. (46)(48). The energy flow then is also similar to that of the single-interface SPP [20

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]

]. As another example, we may consider a Ag slab surrounded by air when λ0 = 325 nm (He-Cd laser), corresponding to εr1 ≈ −0.08 + i0.73 [27

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

]. In this case Eqs. (13), (14), and (51) imply S1x(M1)>0 for the symmetric FM as d > 32.5 nm and S1x(M1)>0 for the antisymmetric FM regardless of the slab thickness. Thus one concludes that FMs can be either FPMs or BPMs. Because kx(M3) 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 HOMI solutions plotted in Figs. 7 and 8, as well as the dotted curves (air surrounding) in Fig. 9, obey S1x(M3)>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 S1x(M3)<0 when d < 143 nm (GaP), d < 106 nm (C), and d < 38 nm (SiO2), respectively, representing backward propagation. Since kx(M3)>0, HOMIs can be FPMs or BPMs depending on the direction of the energy flow, much as with FMs.

Concerning HOMIIs, for d < dc Eqs. (43) and (51) give S1x(M3)>0, corresponding to BPMs. As d = dc, Eqs. (44) and (51) likewise yield S1x(M3)>0 but, since in this case kx(M3)=0, we cannot discuss forward or backward propagation even along the x axis. We note, however, that S1x(M3)0 because of absorption, in spite of the fact that the field is purely evanescent in the x direction [20

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]

]. When d > dc, all the HOMII solutions in Figs. 7 and 9 also imply S1x(M3)>0 which, according to Eq. (45), represents FPMs. Consequently, both forward and backward propagation occur for HOMIIs, in a manner similar to FMs and HOMIs, with the difference that for HOMIIs 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 M1I and M1III are, respectively, the only mode types for which FPMs and BPMs are found (and defined).

Table 1. Summary of FPMs and BPMs for mode types M1–M3 inside the slab. The grey-shaded areas represent situations in which forward or backward propagation is not defined.

table-icon
View This Table

7. Conclusions

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

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]

] 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

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]

]). This led us to further classify M1 into subcategories M1I, M1II, and M1III, in analogy with the single-interface SPP [20

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]

]. 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

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

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

,30

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]

], 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

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

], on the other hand are infinite in number. HOMs consist of two classes and vanish when d becomes infinite. HOMIs are always bound, while HOMIIs 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 HOMIs this feature was demonstrated to arise from the change of the energy-flow direction, whereas for HOMIIs 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

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]

]; 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

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

] 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

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

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

]. 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]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(260.2110) Physical optics : Electromagnetic optics
(310.0310) Thin films : Thin films

ToC Category:
Plasmonics

History
Original Manuscript: December 17, 2013
Revised Manuscript: January 25, 2014
Manuscript Accepted: January 25, 2014
Published: February 20, 2014

Citation
Andreas Norrman, Tero Setälä, and Ari T. Friberg, "Surface-plasmon polariton solutions at a lossy slab in a symmetric surrounding," Opt. Express 22, 4628-4648 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4628


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References

  1. S. Kawata, Near-Field Optics and Surface Plasmon Polaritons (Springer, 2001). [CrossRef]
  2. W. L. Barnes, A. Dereux, T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). [CrossRef] [PubMed]
  3. A. V. Zayats, I. I. Smolyaninov, 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, P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70, 1–87 (2007). [CrossRef]
  5. D. K. Gramotnev, 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, 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, N. J. Halas, “Nano-optics from sensing to waveguiding,” Nature Photon. 1, 641–648 (2007). [CrossRef]
  11. C. Genet, 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, J. R. Krenn, “Organic plasmon-emitting diode,” Nature Photon. 2, 684–687 (2008). [CrossRef]
  13. S. Zhang, D. A. Genov, Y. Wang, M. Liu, 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, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009). [CrossRef] [PubMed]
  15. E. Verhagen, M. Spacenovic, A. Polman, 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, 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, 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, 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, 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, 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, M. Mansuripur, “Surface plasmon polaritons on metallic surfaces,” Opt. Express 15, 183–197 (2007). [CrossRef] [PubMed]
  23. A. Norrman, T. Setälä, 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, E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999). [CrossRef]
  25. F. Yang, J. R. Sambles, 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, 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, E. V. Stryland, Handbook of Optics, 3rd ed., Vol. 4. (McGraw-Hill, 2009).
  30. J. A. Dionne, L. A. Sweatlock, H. A. Atwater, 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]

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