## Surface-plasmon polariton solutions at a lossy slab in a symmetric surrounding |

Optics Express, Vol. 22, Issue 4, pp. 4628-4648 (2014)

http://dx.doi.org/10.1364/OE.22.004628

Acrobat PDF (1806 KB)

### 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

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

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

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]

## 2. Slab geometry

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

*α*,

*β*∈ {1, 2}) is the complex field amplitude and are the normalized polarization and wave vectors, respectively, with

**ê**

*being the unit vector in the positive*

_{y}*y*direction and

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. 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

*k*and

_{x}*x*and

*z*components. The tangential components of the wave vectors are continuous across the boundaries and, therefore,

*α*,

*β*∈ {1, 2}. Since also

*ε′*

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

*ε′*

_{r}_{1}> 0.

### 3.1. M1:
k 1 z ( 1 ) = k 1 z ( 2 )

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

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]

*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.

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

### 3.2. M2 and M3:
k 1 z ( 1 ) = − k 1 z ( 2 )

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

*r*

^{(1)}is the Fresnel reflection coefficient for the plane wave

*r*

^{(2)}likewise for

*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.

#### 3.2.1. M2:
k 2 z ( 1 ) = k 2 z ( 2 )

*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. 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]

**ê**

*and*

_{x}**ê**

*are the unit vectors in the positive*

_{z}*x*and

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

*m*

_{+}= 2

*n*and

*m*

_{−}= 2

*n*+ 1 with

*n*∈ ℤ. In contrast to the solutions of M1 given by Eq. (5), the wave-vector components of M2 in Eq. (10) stand for an infinite number of modes and depend on the film thickness. We note, however, that

*d*→ ∞, and thus all the M2 modes vanish in this limit. Further we observe that

*ε*

_{r}_{1}is complex) and fully independent of material parameters, while

*ε*

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

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

#### 3.2.2. M3:
k 2 z ( 1 ) = − k 2 z ( 2 )

*z*= 0, respectively. Equation (11) can further be split into with

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

*d*→ ∞, both Eq. (13) and Eq. (14) have solutions that converge towards common values for which

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]

*Principles of Nano-Optics*, 2nd ed. (Cambridge University, 2012). [CrossRef]

**38**, 1119–1121 (2013). [CrossRef] [PubMed]

*d*→ ∞ are therefore regarded (at any

*d*) as fundamental modes (FMs) [22

**15**, 183–197 (2007). [CrossRef] [PubMed]

*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 Similarly to M2, the corresponding fields vanish as

*d*becomes infinite [

*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*→ ∞.

*d*→ 0, Eq. (13) has a symmetric solution for which 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. P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. **1**, 484–588 (2009). [CrossRef]

## 4. Field propagation and energy flow

*k*=

_{x}*k′*+

_{x}*ik″*and

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

*ε′*

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

*ε′*

_{r}_{1}> 0.

### 4.1. Phase propagation and amplitude attenuation

*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.

*z*| ≥

*d*/2, the wave-vector components are connected as 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′*and

_{x}*k″*have the same (opposite) sign, Eq. (19) implies that

_{x}*z*| <

*d*/2, the wave-vector components behave as 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″*

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

### 4.2. Flux of energy

*Principles of Nano-Optics*, 2nd ed. (Cambridge University, 2012). [CrossRef]

**H**(

**r**) = (

*cμ*

_{0}

*k*

_{0})

^{−1}[

**k**×

**E**(

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

*c*is the speed of light, and

*μ*

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

*σ*≡

*k*

_{0}/(2

*cμ*

_{0}) and

*ε*

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

*z*| <

*d*/2, where after some calculations we find

*ε*

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

## 5. Bound modes and leaky modes

### 5.1. M1

**38**, 1119–1121 (2013). [CrossRef] [PubMed]

*x*axis, but along the

*z*axis they are opposite in the region |

*z*| ≥

*d*/2, respectively.

*z*| <

*d*/2, on the other hand, we obtain (for a derivation, see [20

**38**, 1119–1121 (2013). [CrossRef] [PubMed]

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

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

*z*axis). M1

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

*z*direction and the wavefronts advance only along the

*x*axis. We note that M1

_{I}and M1

_{III}are identical with the modes SPP I and SPP II introduced in [20

**38**, 1119–1121 (2013). [CrossRef] [PubMed]

*x*direction [

**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]

### 5.2. M2

*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,

*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.

*x*direction [

*z*| <

*d*/2 the fields are purely propagating in the

*z*direction and attenuating only along the

*x*axis.

### 5.3. M3

**33**, 5186–5201 (1986). [CrossRef]

*z*| <

*d*/2, we show that all of the cases in Eqs. (21)–(23) are possible for M3 and thus the behavior of

*z*axis, making the phase motion perpendicular to the interfaces ambiguous (cf. M2 in Sec. 5.2).

*x*direction, i.e.,

#### 5.3.1. Fundamental modes

*ε*

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

*d*for different vacuum wavelengths

*λ*

_{0}= 2

*π*/

*k*

_{0}. We observe that in all cases

*d*gets smaller, the behavior of the symmetric and antisymmetric fields starts to diverge. Particularly, the symmetric modes become LRSPPs with

*d*→ 0, in accordance with Eq. (17), while the antisymmetric modes acquire large values for

_{2}), and air, as a function of

*d*at

*λ*

_{0}= 632.8 nm. We see that increasing the refractive index of the surrounding medium leads to larger real and imaginary parts of

*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

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

*z*| <

*d*/2, we consider as an example symmetric FMs at a Ag film in an air surrounding. Fixing

*λ*

_{0}= 350 nm, corresponding to

*ε*

_{r}_{1}≈ −1.79 +

*i*0.60 [27], gives

*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.

*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]

#### 5.3.2. Higher-order modes

*d*

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

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

_{II}s can be either bound (

*d*<

*d*

_{c}), or leaky (

*d*>

*d*

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

*z*axis outside the slab (

*d*=

*d*

_{c}).

_{I}s (upper row) and HOM

_{II}s (lower row) having the lowest

*d*when

*λ*

_{0}= 632.8 nm. The figure confirms the result given by Eq. (16): when the slab thickness increases all of the solutions converge towards

*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

*x*axis, HOMs are generally needed to match the boundary conditions upon launching a SPP [22

**15**, 183–197 (2007). [CrossRef] [PubMed]

*d*becomes smaller the real parts of HOM

_{I}s slightly decrease before diverging as

*d*tends to zero. Regarding HOM

_{II}s, when

*d*is reduced

*d*

_{c}(which depends on the particular mode), the real part becomes zero. It is observed that the HOM

_{II}having the lowest (highest)

*d*further decreases (

*d*<

*d*

_{c}) the real part acquires a negative value and finally diverges when

*d*→ 0.

_{I}(upper row) and HOM

_{II}(lower row) having the lowest

*d*when

*λ*

_{0}is varied. The figure shows that, for both mode types, decreasing the wavelength gives a smaller

*λ*

_{0}results in a slight increase of the real part in the case of HOM

_{I}and for HOM

_{II}the critical thickness is shifted towards larger values:

*d*

_{c}≈ 66 nm (

*λ*

_{0}= 700 nm),

*d*

_{c}≈ 79 nm (

*λ*

_{0}= 600 nm),

*d*

_{c}≈ 90 nm (

*λ*

_{0}= 500 nm), and

*d*

_{c}≈ 109 nm (

*λ*

_{0}= 400 nm).

_{I}(upper row) and HOM

_{II}(lower row) of the lowest

_{2}, and air, as a function of

*d*at

*λ*

_{0}= 632.8 nm. We particularly observe that for both mode types varying the refractive index of the surrounding medium has a negligible effect on

*x*axis. This is in strong contrast to FMs in Fig. 5. For example, at

*d*= 100 nm the propagation length of the symmetric FM is about 230 times larger for Ag/air than in the case of Ag/GaP, while for HOMs the corresponding difference is practically zero. The real part, instead, is affected notably as

*ε*

_{r}_{2}is varied and

*d*is sufficiently small, i.e., increasing the refractive index outside the slab leads to a larger

_{I}and an increased critical thickness of HOM

_{II}[

*d*

_{c}≈ 73 nm (air),

*d*

_{c}≈ 97 nm (SiO

_{2}),

*d*

_{c}≈ 137 nm (C), and

*d*

_{c}≈ 164 nm (GaP)]. In the limit

*d*→ ∞, on the other hand, changing

*ε*

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

_{I}s, all the plots in Figs. 7 and 9 correspond to

*d*(as in the case with FMs), justifying the definition in Eq. (40). Consequently, the numerical results suggest that only bound HOM

_{I}s can exist. From the same figures one also finds that the real and imaginary parts of

_{I}s in the

*x*direction and outside the slab is similar to that of FMs [cf. Eq. (39)]. Concerning

_{I}s in the figures is as summarized in Fig. 6(c).

_{II}s, all the plots in Figs. 7 and 9 correspond to

*d*<

*d*

_{c}),

*d*=

*d*

_{c}), and

*d*>

*d*

_{c}), justifying our definition in Eq. (41). Further, Figs. 7 and 9 show that

*d*<

*d*

_{c}(

*d*>

*d*

_{c}). In addition, at the transition point

*d*=

*d*

_{c}, where

_{II}s obey

*z*direction for |

*z*| ≥

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

_{I}s [Eq. (42)]. Concerning

*x*direction (in both media). The last situation, Eq. (45), corresponds to (leaky) modes for which the behavior of

_{I}s, respectively.

_{II}s, one readily confirms from Eqs. (21), (43), and (44) that the real and imaginary parts always have the same sign for

*d*≤

*d*

_{c}. Although not analytically seen by using Eq. (45), according to Eq. (21) the same turns out to be true for all the numerical plots in Figs. 7 and 9 also when

*d*>

*d*

_{c}. Hence we may conclude that HOM

_{II}s satisfy

*d*, in contrast to HOM

_{I}s having

_{II}s discussed above. As

*d*<

*d*

_{c}(left figure), the field attenuation is similar to that of FMs and HOM

_{I}s in Fig. 6, but the phases advance in the opposite directions. In particular, the wavefront motion is in the negative

*x*direction even though the fields decay in the positive direction. When

*d*reaches the critical thickness (middle figure), the waves become purely evanescent with no phase movement at all along the

*x*axis (in both media) and strictly propagating away from the interfaces outside the slab. For |

*z*| <

*d*/2, the field is neither exclusively evanescent nor solely propagating in the

*z*direction and the phases of the two waves propagate perpendicular to the boundaries in opposite directions. Finally, as

*d*>

*d*

_{c}(right figure), the fields become leaky and the wavefronts advance away from the slab for |

*z*| ≥

*d*/2, in contrast to FMs and HOM

_{I}s in Fig. 6. On the other hand, inside the film the wave propagation is analogous to FMs and HOM

_{I}s and, in addition, HOM

_{II}s also decay in the same (positive) direction as the phases propagate along the

*x*axis in both media.

## 6. Forward- and backward-propagating modes

*z*| ≥

*d*/2 any further.

*x*direction, i.e.,

*k″*> 0, but the results on forward or backward propagation that are obtained also hold for

_{x}*k″*< 0.

_{x}### 6.1. M1

*x*axis, we have (as in [20

**38**, 1119–1121 (2013). [CrossRef] [PubMed]

*ε″*

_{r}_{1}. Since

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

_{I}–M1

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

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

**38**, 1119–1121 (2013). [CrossRef] [PubMed]

_{I}is a FPM while M1

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

_{II}, which is neither a FPM nor a BPM [since

*z*axis.

*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

*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

### 6.3. M3

*z*axis (see Sec. 5.3), we can only discuss forward or backward propagation along the

*x*axis. According to Eq. (28), in a manner similar to Eq. (50), one obtains We next analyze FMs and HOMs separately.

#### 6.3.1. Fundamental modes

**38**, 1119–1121 (2013). [CrossRef] [PubMed]

*λ*

_{0}= 325 nm (He-Cd laser), corresponding to

*ε*

_{r}_{1}≈ −0.08 +

*i*0.73 [27]. In this case Eqs. (13), (14), and (51) imply

*d*> 32.5 nm and

#### 6.3.2. Higher-order modes

_{I}solutions plotted in Figs. 7 and 8, as well as the dotted curves (air surrounding) in Fig. 9, obey

*d*< 143 nm (GaP),

*d*< 106 nm (C), and

*d*< 38 nm (SiO

_{2}), respectively, representing backward propagation. Since

_{I}s can be FPMs or BPMs depending on the direction of the energy flow, much as with FMs.

_{II}s, for

*d*<

*d*

_{c}Eqs. (43) and (51) give

*d*=

*d*

_{c}, Eqs. (44) and (51) likewise yield

*x*axis. We note, however, that

*x*direction [20

**38**, 1119–1121 (2013). [CrossRef] [PubMed]

*d*>

*d*

_{c}, all the HOM

_{II}solutions in Figs. 7 and 9 also imply

_{II}s, in a manner similar to FMs and HOM

_{I}s, with the difference that for HOM

_{II}s this behavior arises from the change of direction in the phase movement instead of the energy flow.

### 6.4. Summary

*z*| <

*d*/2. We find that along the

*x*axis both FPMs and BPMs are possible for all fields in M1 and M3, while those of M2 are exclusively FPMs. Further it is observed that in the

*z*direction M1

_{I}and M1

_{III}are, respectively, the only mode types for which FPMs and BPMs are found (and defined).

## 7. Conclusions

**33**, 5186–5201 (1986). [CrossRef]

**33**, 5186–5201 (1986). [CrossRef]

_{I}, M1

_{II}, and M1

_{III}, in analogy with the single-interface SPP [20

**38**, 1119–1121 (2013). [CrossRef] [PubMed]

**33**, 5186–5201 (1986). [CrossRef]

**15**, 183–197 (2007). [CrossRef] [PubMed]

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]

*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

**15**, 183–197 (2007). [CrossRef] [PubMed]

*d*becomes infinite. HOM

_{I}s are always bound, while HOM

_{II}s may be bound or leaky depending on

*d*. Concerning forward and backward propagation, all the modes in M3 can be either forward-propagating or backward-propagating waves. For FMs and HOM

_{I}s this feature was demonstrated to arise from the change of the energy-flow direction, whereas for HOM

_{II}s the phase movement changes direction.

**33**, 5186–5201 (1986). [CrossRef]

**33**, 5186–5201 (1986). [CrossRef]

**15**, 183–197 (2007). [CrossRef] [PubMed]

## Acknowledgments

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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. |

13. | S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. |

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 |

15. | E. Verhagen, M. Spacenovic, A. Polman, and L. K. Kuipers, “Nanowire plasmon excitation by adiabatic mode transformation,” Phys. Rev. Lett. |

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 |

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. |

18. | M. Ozaki, J.-I. Kato, and S. Kawata, “Surface-plasmon holography with white-light illumination,” Science |

19. | H. Raether, |

20. | A. Norrman, T. Setälä, and A. T. Friberg, “Exact surface-plasmon polariton solutions at a lossy interface,” Opt. Lett. |

21. | J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B |

22. | A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Surface plasmon polaritons on metallic surfaces,” Opt. Express |

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 |

24. | M. Born and E. Wolf, |

25. | F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B |

26. | P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. |

27. | E. D. Palik, ed., |

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 |

29. | M. Bass, C. DeCusatis, J. Enoch, V. Lakshminarayanan, G. Li, C. MacDonald, V. Mahajan, and E. V. Stryland, |

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 |

**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

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