## Negative index modes in surface plasmon waveguides: a study of the relations between lossless and lossy cases

Optics Express, Vol. 18, Issue 12, pp. 12213-12225 (2010)

http://dx.doi.org/10.1364/OE.18.012213

Acrobat PDF (3345 KB)

### Abstract

Surface plasmon modes in structures of metal-insulator-metal (MIM), insulator-insulator-metal (IIM) and insulator-metal-insulator (IMI) are studied theoretically for both lossless and lossy cases. Causality dictates which solutions of Maxwell’s equations we accept for these structures. We find that for both lossless and lossy cases, the negative index modes and positive index modes are independent and should be treated separately. For the lossless case, our results differ from some published papers. By studying in detail the lossy case, we demonstrate how the curves should look like.

© 2010 OSA

## 1. Introduction

## 2. Causality and equations

*d*, and the substrate and cover are infinite. Since negative refraction is caused by surface plasmon polaritons, and the surface wave is always bounded to interfaces, we can treat these structures as plasmon waveguides. Given the symmetry of the structure, the field distribution is the same in the

*x*and

*y*directions, thus we can confine our discussion to the

*x-z*plane, and consider surface modes of the TM polarization (i.e. only field components

*E*,

_{x}*E*and

_{z}*H*are nonzero).

_{y}13. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. **96**(7), 073907 (2006). [CrossRef] [PubMed]

14. M. I. Stockman, “Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality,” Phys. Rev. Lett. **98**(17), 177404 (2007). [CrossRef]

17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express **16**(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

18. E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,” Opt. Express **17**(4), 2465–2469 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2465. [CrossRef] [PubMed]

1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. **10**(4), 509–514 (1968). [CrossRef]

16. A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B **78**(15), 155405 (2008). [CrossRef]

17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express **16**(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

19. I. I. Smolyaninov, Y. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B **76**(20), 205424 (2007). [CrossRef]

*x*direction and we just need to study the

*ω*-

*k*dispersion curves, where

_{x}*ω*is the angular frequency of the incident wave and

*k*is the

_{x}*x*component of wave vector in the structure. We can find the dispersion relation for such structures from Eq. (1) [21]:

*k*(

_{i}*i*= 1, 2, 3) should fulfill:Where,

*k*

_{0}is the wave vector in vacuum, and

*k*is the transverse wave vector which is conserved throughout the three distinct regions.

_{x}*ε*is the permittivity in each area, and here we use

_{i}*ε*= 4 (such as Si

_{d}_{3}N

_{4}) as the permittivity of the dielectric layers. For the metal, we first use the lossy Drude model to describe the permittivity, which takes the form:

*ε*= 1,

_{∞}*ω*= 9eV, and

_{p}*Г*= 0.2687 eV, we approximately represent the permittivity of silver). We note that the value of

*Г*is a little larger than usual to make the dispersion curves clear but this does not affect the results. Taking a smaller value, the curves of the lossless and lossy cases are too close to each other to be able to distinguish them. The materials used here are non-magnetic, thus permeability

*μ*= 1 (

_{i}*i*= 1, 2, 3).

*S*› is also derived to study the energy transport in these layered structures. As we mainly study the bound surface modes, we just need to study the

*x*component of ‹

*S*› (represented by ‹

*S*›), since ‹

_{x}*S*› can be used to describe the propagation of energy flow in the structures. As in Eq. (4) ‹

_{x}*S*› is composed of three parts, the first one (‹

_{x}*S*›) represents the energy flow in layer 1, and the last two (‹

_{1x}*S*› and ‹

_{2x}*S*›) are for the substrate and cover, respectively:With the components: Where the coefficients have the following form:

_{3x}*ω-k*curves. The mode refractive index is decided by:

_{x}*k*indicates negative (positive) index in the structures. Generally, for a certain frequency, there is always an infinite number of solutions that solve Eq. (1) because of the periodic character of the formula and both the real and imaginary parts of

_{x}*k*can be positive or negative. Mathematically, these solutions all satisfy Eq. (1), thus we must choose the solutions which are physically correct. Ref. [17

_{x}17. J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express **16**(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

*k*, where

_{x}*imag*(

*k*) must always be positive, to decide which solution is reasonable. This is reliable when the metal is lossy and

_{x}*k*always has a non-zero imaginary part. In contrast, for the lossless case

_{x}*imag*(

*k*) of the propagating mode is always zero of course, in which case this approach fails leading to a contradiction between Fig. 1 and Figs. 2 to 4 in Ref. [17

_{x}**16**(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

*S*› to study the energy flow directions, which is more reliable and in compliance with causality. From this we can find out the relationship between lossless and lossy cases, and clearly see how negative refraction arises in these layered structures.

_{x}*k*is always complex. A large imaginary part of

_{x}*k*means heavy damping when propagating along

_{x}*x*direction, thus we will mainly study those modes, the imaginary parts (

*imag*(

*k*)) of which are small or comparable with the real part (

_{x}*real*(

*k*)).

_{x}## 3. The MIM case

13. H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. **96**(7), 073907 (2006). [CrossRef] [PubMed]

*d*= 20nm, and

*ε*=

_{1}*ε*,

_{metal}*ε*=

_{2}*ε*= 4.

_{3}*H*for clarity) will be either symmetric or anti-symmetric along the

_{y}*z*direction. As the dispersion curves of these two situations are different, they will be discussed separately. We first consider the symmetric mode. The dispersion relations for both lossless and lossy cases are shown in Fig. 3 where (a) shows the real parts of

*k*; (b) shows the imaginary parts of

_{x}*k*; and (c) shows the corresponding time averaged Poynting vectors ‹

_{x}*S*› calculated by Eq. (6) (normalized for convenience). Note that the curves in each of these figures correspond to each other in color and line-style. The horizontal dotted lines indicate

_{x}*Г*= 0), we can plot the possible

*ω-k*curves of propagating modes (

*imag*(

*k*) = 0) as black solid and dashed curves in Fig. 3(a); in other words, both of the curves satisfy Eq. (1). It is clear that the term

_{x}*k*in Eq. (2) leads to ±

_{x}^{2}*k*being solutions. For these two cases, we have

_{x}*imag*(

*k*) = 0, which means we cannot pick out the correct one from

_{x}*imag*(

*k*)>0. The more efficient and reliable way is to study the value of ‹

_{x}*S*›. We need ‹

_{x}*S*› > 0 to avoid the energy accumulation at the interface mentioned above and which ensures that the energy is consistent. In Fig. 3(c), we can see that for the lossless case, ‹

_{x}*S*› > 0 only when

_{x}*real*(

*k*)>0 (black solid curve). In contrast, ‹

_{x}*S*› will be negative when

_{x}*real*(

*k*)<0, which means this set of solutions are non physical, and these pseudo solutions are plotted as dashed curves hereafter. The correct branch for the lossless case is then decided. As a general point, when loss in the Drude model increases, the curves for the lossless and lossy cases should not behave differently. When

_{x}*Г*= 0.2687 eV, we plot the dispersion curves in red in Fig. 3. We can see that curves for the lossy case are very close to the lossless case at lower frequencies, and follow the black curve as asymptotes. Moreover,

*imag*(

*k*) and ‹

_{x}*S*› are both positive, which is consistent with causality; the other set of solutions having

_{x}*imag*(

*k*)<0 and ‹

_{x}*S*›<0 is ignored. For higher frequencies, as there is no solution for the fundamental symmetric mode for the lossless case, the asymptote for the lossy case corresponds to the higher order mode of lossless case. This is indicated by the green dash-dot curve and we can see from Figs. 3(a) and 3(b) that both the real and imaginary parts of

_{x}*k*for the lossless and lossy cases are close and have similar trends at frequencies above

_{x}*ω*. For high order modes and lossless case, we have ‹

_{sp}*S*› = 0 which means this mode is non-propagating. For the lossy case,

_{x}*imag*(

*k*) is large comparing with

_{x}*real*(

*k*) at higher frequencies, thus the fields will decay rapidly along the

_{x}*x*direction; at the same time, ‹

*S*› is very small for higher frequencies but we do not show these high order curves here, as they are not the salient for this paper. It is these features that indicate the similarity between the lossless and lossy cases.

_{x}**16**(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

*ω = ω*asymptotically, as

_{sp}*real*(

*k*) increases, the black solid and dashed curves do not tend to it directly but first cross

_{x}*ω = ω*and then tend to the asymptote. The

_{sp}*ω*-

*k*and

_{x}*ω*-‹

*S*› curves are shown in Fig. 5(a) and 5(b), respectively. Based on causality, the physical solutions are plotted in red solid (for negative index property) and light green solid (for positive index property) curves, and the dashed curves are pseudo or non-physical solutions (corresponding negative ‹

_{x}*S*›) while the dash-dot curves are non-propagating modes like that in Fig. 4. This further indicates that negative index modes and positive modes should be treated separately, which also applies to the IIM and IMI cases.

_{x}## 4. The IIM and IMI cases

**16**(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001. [CrossRef]

*ε*= 1) and

_{2}*ε*= 4,

_{1}*ε*=

_{3}*ε*, and let

_{metal}*d*= 10nm since the negative index property will be more evident when

*d*is thin. For the lossless case, the

*ω-k*curves are shown in Fig. 6(a) and for clarity, we do not show the curves in same figure when

_{x}*k*is large, instead, we will show these details in Fig. 7 . The red (blue) solid curves illustrate negative (positive) propagating mode directions; the dashed curves are for pseudo-solutions. We can obtain two pairs of non-propagating solutions (the dash-dot curves) for higher and lower frequencies (see the insets for details), which are axially symmetric and have large

_{x}*imag*(

*k*) and ‹

_{x}*S*› = 0. Similar to the anti-symmetric mode case in the MIM structure, these non-propagating curves will be asymptotes for curves of the lossy case and the curves for different modes also lie on different sides of the dash-dot curves. Based on the value of

_{x}*imag*(

*k*), we find that the IIM structure shows positive index properties at lower frequencies, both positive and negative index properties over the middle frequency range (as

_{x}*imag*(

*k*) is smaller in this frequency range), while decaying at higher frequencies. For propagating modes, the

_{x}*ω-k*curves are no longer continuous, which means the negative index mode and the positive index mode are never coincident and therefore should be treated separately. For the lossy case, we find physical solutions based on causality and plot them in light green (negative index mode) and light blue (positive index mode). We can see that the curves for the lossy case have those of the lossless case as asymptotes. The orange curve is a high order solution of the lossy case to further prove that the dash-dot curves are asymptotes of the lossy case. At these higher frequencies, there is another high order non-propagating mode of the lossless case which will be asymptotic to the orange curve, but we do not plot them here as it is not pertinent to this paper.

_{x}*k*and the lossless case, details are shown in Fig. 7. The situation is similar to the anti-symmetric mode of the MIM structure: the curves also tend asymptotically to

_{x}*ω = ω*, but first dip down, cross and then rise up to the asymptote. The right branches are consistent with causality in Fig. 7(b) and here we just plot the curves with ‹

_{sp}*S*›>0 and do not show the curves with negative value of ‹

_{x}*S*› which correspond to the dashed curves and are axially symmetric with the solid curves in (b).

_{x}*H*for definition as in MIM structure). The anti-symmetric mode shows no negative index property and

_{y}*imag*(

*k*) is always large and so we just study the symmetric mode here. We set

_{x}*ε*=

_{1}*ε*,

_{metal}*ε*= 4 and

_{2}= ε_{3}*d*= 10 nm. The

*ω-k*curves of the IMI system are very similar to those of IIM structure, and we just plot the physical curves in Fig. 8 in which the branches are picked based on consistency with causality and the case for large

_{x}*k*is shown in the upper left inset. Similarly, the negative and positive index modes should be treated separately for both lossless and lossy cases. For the lossless case, the red curve indicates the propagating negative index mode and the blue one is for positive; the dash-dot curves represent the non-propagating modes. For the lossy case, the light green curve represents the negative index mode and light blue represents positive and since the curves are very close to each other, the upper right inset shows more details. Considering the imaginary parts of

_{x}*k*for which a smaller value corresponds to a longer propagation length, we can see that the IMI structure shows negative and positive index properties simultaneously in certain frequency ranges.

_{x}## 5. The case of actual metal data

*real*(

*k*) decides the negative (

_{x}*real*(

*k*)<0) or positive (

_{x}*real*(

*k*)>0) indices of these modes. For the lossy case, the red (blue) solid curves represent the negative (positive) index modes, which have the lossless curves as asymptotes. We can see that the curves in Fig. 9 are very similar to those cases based on the Drude model in previous sections, which further supports the validity of our discussion.

_{x}## 6. Conclusion

*ω-k*and

_{x}*ω-*‹

*S*› curves for both lossless and lossy cases. We initially used the lossy Drude model for the permittivity of metal layers, and then we used real data for silver for verification. The results are in agreement for both cases. Using the principle of causality and in particular that the energy flow is along the positive

_{x}*x*direction, we were able to pick out the correct branches corresponding to solutions with physical meaning from the multiple mathematical solutions of Maxwell’s equations for these structures. We showed the relationship between the dispersion curves for the lossless and lossy cases, and explained how the curves vary as the loss increases. Based on this, for both the lossless and lossy cases, we see that the branches of negative and positive index modes are independent of each other, and should be treated separately. In particular for the lossless case, our results are different from previously published results.

*Real*(

*k*) of the propagating modes, where negative refraction occurs, will be negative and for IIM and IMI structures, the curves will not be continuous anymore; this is illustrated by the red curves in Fig. 2 for clarity. This result explains previously published dispersion curves for propagating modes were confused with non-physical modes. To the best of our knowledge, there are no publications showing such physically reasonable and self-consistent results regarding the dispersion relations of MIM, IIM, and IMI structures for both the lossy and lossless cases when negative refraction occurs. The results presented here show evident differences from those previously published papers which are ambiguous and/or self-contradictory.

_{x}## Acknowledgments

## References and links

1. | V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. |

2. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

3. | S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. |

4. | V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. |

5. | G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. |

6. | G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science |

7. | J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature |

8. | M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small |

9. | A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. |

10. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science |

11. | Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science |

12. | I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science |

13. | H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. |

14. | M. I. Stockman, “Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality,” Phys. Rev. Lett. |

15. | H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science |

16. | A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B |

17. | J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express |

18. | E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,” Opt. Express |

19. | I. I. Smolyaninov, Y. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B |

20. | H. Reather, |

21. | S. A. Maier, |

22. | E. D. Palik, |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(350.3618) Other areas of optics : Left-handed materials

(160.3918) Materials : Metamaterials

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: November 23, 2009

Revised Manuscript: March 18, 2010

Manuscript Accepted: March 18, 2010

Published: May 25, 2010

**Citation**

Yuan Zhang, Xuejin Zhang, Ting Mei, and Michael Fiddy, "Negative index modes in surface plasmon waveguides: a study of the relations between lossless and lossy cases," Opt. Express **18**, 12213-12225 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-12-12213

Sort: Year | Journal | Reset

### References

- V. G. Veselago, “Electrodynamics of substances with simultaneously negative values of sigma and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]
- R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
- S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005). [CrossRef] [PubMed]
- V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005). [CrossRef]
- G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007). [CrossRef]
- G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006). [CrossRef] [PubMed]
- J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef] [PubMed]
- M. C. Gwinner, E. Koroknay, L. Fu, P. Patoka, W. Kandulski, M. Giersig, and H. Giessen, “Periodic large-area metallic split-ring resonator metamaterial fabrication based on shadow nanosphere lithography,” Small 5(3), 400–406 (2009). [CrossRef] [PubMed]
- A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007). [CrossRef] [PubMed]
- N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005). [CrossRef] [PubMed]
- Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]
- I. I. Smolyaninov, Y. J. Hung, and C. C. Davis, “Magnifying superlens in the visible frequency range,” Science 315(5819), 1699–1701 (2007). [CrossRef] [PubMed]
- H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96(7), 073907 (2006). [CrossRef] [PubMed]
- M. I. Stockman, “Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality,” Phys. Rev. Lett. 98(17), 177404 (2007). [CrossRef]
- H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316(5823), 430–432 (2007). [CrossRef] [PubMed]
- A. Hohenau, A. Drezet, M. Weißenbacher, F. R. Aussenegg, and J. R. Krenn, “Effects of damping on surface-plasmon pulse propagation and refraction,” Phys. Rev. B 78(15), 155405 (2008). [CrossRef]
- J. A. Dionne, E. Verhagen, A. Polman, and H. A. Atwater, “Are negative index materials achievable with surface plasmon waveguides? A case study of three plasmonic geometries,” Opt. Express 16(23), 19001–19017 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-23-19001 . [CrossRef]
- E. Feigenbaum and M. Orenstein, “Backward propagating slow light in inverted plasmonic taper,” Opt. Express 17(4), 2465–2469 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-4-2465 . [CrossRef] [PubMed]
- I. I. Smolyaninov, Y. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B 76(20), 205424 (2007). [CrossRef]
- H. Reather, Surface plasmon (Springer, Berlin, 1988), Chap.2.
- S. A. Maier, Plasmonics: fundamentals and applications (Springer, 2007), Chap. 2.
- E. D. Palik, Handbook of Optical Constants in Solids (Boston, MA: Academic, 1991).

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