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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 21 — Oct. 13, 2008
  • pp: 16305–16313
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Plasmonic mediated interactions between waveguides and nanostructured metal surfaces

P. A. Porta and B. Corbett  »View Author Affiliations


Optics Express, Vol. 16, Issue 21, pp. 16305-16313 (2008)
http://dx.doi.org/10.1364/OE.16.016305


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Abstract

We present the polarization resolved, angular dependent, optical reflectance properties for single TE mode optical waveguides in contact with a nanostructured gold surface. A substantial angle dependent resonant decrease in the TE polarized surface reflectivity is measured which cannot be explained by a simple waveguide coupling due to surface roughness. Rather we show that the resonance is due to the excitation of a coupled waveguide-plasmonic surface mode created by the interaction between the metal nanostructure and the waveguide. A model based on coupled mode theory is introduced in order to explain the experimental data.

© 2008 Optical Society of America

1. Introduction

The surfaces of metals such as aluminum, copper, gold and silver are able to sustain the propagation of electro-magnetic (EM) waves via the oscillation of free electrons in the conduction band. The collective motion of these electrons generates a TM polarized plasma wave at the flat interface between the metal and the surrounding dielectric medium [1

1. M. Fox, Optical Properties of Solids (Oxford University Press, Oxford, 2001).

]. The properties of this surface plasma wave, usually referred to as a surface plasmon, are described in general using dispersion relations and are characterized by specific resonances of the collective motion of the electrons [2

2. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, (1986).

]. In the case of a metal nanoparticle electron oscillations at its surface generate a localized surface plasmon characterized by discrete resonance frequencies that can be TM or TE polarized [3

3. A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A 5, S16–S50 (2003). [CrossRef]

].

Thermal annealing of thin metal films increases their surface roughness [4

4. D. Porath, O. Millo, and J. I. Gersten, “Scanning tunneling microscopy studies and computer simulations of annealing of gold films,” J. Vac. Sci. Technol. B 14, 30–37 (1996). [CrossRef]

]. The process of heating a gold film at temperatures typically between 300° and 400° C induces the metal atoms close to the surface to aggregate and form protuberances with dimensions of the order of tens of nanometers. This process is a simple way to generate a nanostructure on the surface of the metal and thus to modify its optical properties. The nanostructured surface can provide the additional in-plane momentum to incident optical radiation and make possible its coupling into planar optical modes. Surface roughness on a metal has the beneficial effect of coupling optical radiation into surface plasmon modes [2

2. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, (1986).

] while it has a detrimental effect in dielectric waveguides by inducing losses via optical scattering [5

5. F. Ladouceur, “Roughness, inhomogeneity, and integrated optics,” J. Lightwave Technol. 15, 1020–1025 (1997). [CrossRef]

].

Fig. 1. (a). A schematic representation layer by layer of the metal-dielectric nanostructure fabricated. (b). Atomic force microscope image of the thermally annealed surface (380° C) of a gold film showing roughness of the order of 50 nm laterally.

In this paper, we describe the optical properties resulting from the deposition of thin dielectric films on top of a nanostructured gold surface formed on a flat silicon wafer. The films have thicknesses between 100 and 200 nm and form a single mode TE polarized optical waveguide. TM polarized waveguide modes are not supported. The angularly resolved reflectivity from the structure is measured showing a resonant absorption ascribed to coupling into the waveguide. A single grating-like coupling mechanism is not sufficient to explain the experimental data. An additional mechanism that modifies the optical properties of the guided modes is required.

Recent work has demonstrated that unconnected metal nanostructures can host TE polarized surface modes with properties similar and complementary to those of surface plasmons [6

6. R. Sainidou and F. J. G. de Abajo, “Plasmon guided modes in nanoparticle metamaterials,” Opt. Express 16, 4499–4506 (2008). [CrossRef] [PubMed]

]. In this paper we show how surface modes created by a nanostructured metal surface interact with a single mode optical waveguide fabricated on top of it. This interaction defines a new resonant coupled mode and provides an explanation of the particular behavior of our experimental data. The optical properties of these new surface modes are due to localized vibrations of electrons at the interface between the isolated metal nanoparticles and the dielectric surrounding them.

2. Device structure and optical properties

The general structure of the metal-dielectric nanostructures being considered is schematically illustrated in Fig. 1(a). A thin layer of gold (80 nm) on 10 nm of chrome is deposited by standard evaporation techniques onto a flat silicon substrate. The samples are annealed at temperatures between 370°C and 430°C for durations between 8 and 15 minutes, to produce a variety of disordered nanostructures on the metal surface.

Fig. 2. (a). TE polarized and (b) TM polarized reflectivity spectra of the annealed gold metal film coated with a single mode (TE) waveguide. The annealing process of the samples show in this figure lasted 8 minutes at 400° C. Figures (c) and (d) are optical images of the surface when viewed under TE and TM polarized white light at an angle of 70° degrees.

The atomic force microscope (AFM) image of the annealed gold surface is shown in Fig. 1(b) demonstrating a disordered metal nanostructure with lateral dimensions of the order of 50 nm. The height of the metal nanostructures varies between 10 and 30 nm. Using samples annealed at different conditions, thin layers of Polymethyl methacrylate (PMMA) with thickness varying between 100 and 200 nm were deposited by spin coating. For thicknesses, dwg, of the PMMA greater than 120 nm and less than 220 nm, a single mode optical waveguide is formed for TE polarized light in the visible range of the optical spectrum [7

7. A. Adams, J. Moreland, P. K. Hansma, and Z. Schlesinger, “Light-Emission from Surface-Plasmon and Waveguide Modes Excited by N-Atoms near a Silver Grating,” Phys. Rev. B 25, 3457–3461 (1982). [CrossRef]

].

The optical properties of the structures were characterized by measurement of the angle and polarization resolved reflectivity spectra from 400 nm to 800 nm. This experiment, as can be seen in Fig. 3, was performed by placing a polarizer in front of a halogen lamp whose emission was focused onto the surface of the structure. A multimode optical fiber with 62.5 µm core diameter was brought close to the surface at a reflectance angle equal to that of the incident radiation in order to collect the reflected light. Spectra from the optical fiber were recorded using an optical spectrum analyzer (ANDO AQ6313).

Fig. 3. Experimental set-up used during the optical characterization of the samples. Light from a light source S is focused by a lens L onto the sample surface. A polarizer P was inserted between the lens and the sample. The light was collected by an optical fiber OF and sent into the optical spectrum analyzer SA. The inset in the figure indicates how the angles were measured during the experiments.

Figures 3(a) and 3(b) show the polarization resolved reflectivity spectra as a function of incident angle between 20° and 70° of one of the first samples with a 200 nm thick waveguide. The TE polarized spectra show a dependence on incident angle that is absent for the TM polarized light. This angular dependence of the reflectivity results in a striking change of coloration of the gold surface when the surface is viewed with a digital camera at high incident angles when illuminated by polarized white light. Digital camera images of the surface, viewed at an angle of 70° and illuminated with TE polarized light and with TM polarized light, are shown in Figs. 3(c) and 3(d). The color contrast between these two figures is due to the decrease of reflected light in the red part of the optical spectrum so that the dominant component of the light reflected lies in the blue range. The TM polarized reflectivity spectra does not show this drop in reflectivity since the waveguide doesn’t support any TM polarized optical modes.

Fig. 4. (a). Experimental TE polarized reflectivity spectra of the bare annealed thin gold film. The inset shows the calculated reflectivity spectra of a gold surface. (b). Experimental TE polarized reflectivity spectra of an annealed thin gold film in contact with a single mode optical waveguide. The annealing process of these samples lasted 15 minutes at 380° C.

In Figs. 4(a) and 4(b) we compare the TE polarization reflectivity of a bare nanostructured metal surface to one with a 150 nm thick waveguide deposited on it. The reflectivity spectra of the structure with a waveguide show resonant absorption between 420 and 500 nm for all the angles of observations. The resonant absorption in Fig 4(b) is stronger than that in Fig. 2(a) for the different annealing process and the waveguide dimensions. The annealing process, performed at 380°C for 15 min, generated a nanostructured metal surface with larger average dimension and longer correlation length with higher wavevectors. The thinner waveguide (150 nm) produced a more localized near field interaction with the nanostructured surface. The TE reflectivity spectra from the uncoated metal surface in Fig. 4(a) show the expected behavior for a planar gold film simulated using the known dielectric indices for gold (Fig. 4(a) inset). These data demonstrate that the nanostructured surface by itself does not alter the far-field reflectivity properties. A comparison of the TE polarized reflectivity spectra in Figs. 4(a) and 4(b) provides a figure of merit of the efficiency of this coupling process. For the wavelength of 420 nm and the angle of incidence equal to 70°, a decrease of the reflectivity from 0.7 to 0.05 is measured suggesting that more than 90% of incident light is absorbed or coupled into planar waveguide modes.

3. Role of the optical waveguide

Coupling of EM radiation into an optical waveguide occurs when the k-vector of the waveguide, kwg, is equal to the momentum of the incident photons. This happens only if a surface nanostructure (roughness or a grating) supplies the extra-momentum, kns, to the incident light to satisfy the conservation of the in-plane momentum [8

8. R. G. Hunsperger, Integrated Optics: theory and technology (Springer-Verlag, Berlin, 1991).

]:

kinc+kns=kwg
(1)

where k‖inc is the in-plane component of the light incident onto the sample surface. The wavevector kns is given by 2π/Λ where Λ is the periodicity of the surface nanostructure in the case of a grating or the correlation length in the case of a disordered nanostructure.

Fig. 5. Calculated TE dispersion relations for a slab optical waveguide with thicknesses varying from 100 nm to 300 nm. Square points show the extracted dispersion relations obtained from the experimental reflectivity spectra of a nanostructured surface with a 150 nm thick waveguide, top curve, and with 200 nm thick waveguide, bottom curve. The dashed line shows the light line for air and a medium with a refractive index of 1.5.

The k-vectors of the optical waveguides defined by the polymer coating deposited onto the gold surface were determined by calculating the dispersion relationship of a three layer structure formed by air, the polymer PMMA (refractive index n=1.49) and a gold substrate [8

8. R. G. Hunsperger, Integrated Optics: theory and technology (Springer-Verlag, Berlin, 1991).

] and shown in Fig. 5 for waveguides of thickness from 100 nm up to 300 nm, where the structure supports only a TE polarized single mode in the visible part of the spectrum. Plotted in the same figure are the experimental resonant k-vectors corresponding to the minima in the reflectivity spectrum, for two waveguides with different thickness, 150 and 200 nm, when the angle of incidence varies between 20° and 70°. The experimental data relative to the 200 nm waveguide in Fig. 5 are different from those plotted in Fig. 2(a) because of the different thermal annealing process.

The difference kwg - k‖inc should be constant and equal to kns from (1) being the contribution to the in-plane momentum from the surface roughness. Table I compares the experimental data with the calculations in Fig. 5 with kwg - k‖inc varying between 8 and 4.3 µm-1. The fact that this is not constant means that the experimental resonances cannot be explained solely by incident light coupled into a waveguide mode mediated by the surface nanostructure, but that an additional mechanism is required.

Table I. Values of the k-vectors for a 150 nm thick waveguide.

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4. Plasmonic surface modes

The nanostructured metal surface can support surface modes of plasmonic nature caused by oscillations of free electrons at the interface between the metal and the adjacent dielectric. The irregularity of the interface between the metal and the dielectric allows free electrons to vibrate both orthogonal and parallel to the waveguide, as shown schematically in Fig. 6(a). An electric field propagating in the waveguide with TE polarization state induces vibrations of electrons in the in-plane directions. These electron oscillations generate localized TE polarized electric fields E i inside each cavity of the surface nanostructure. These electric field are usually referred to as Lorentz fields and can be calculated following the procedure described by Kittel [9

9. C. Kittel, Introduction to Solid State Physics (John Wiley and Sons, Hoboken, NJ, 2005).

]. The collective contribution of each of these localized electric fields E i is represented by an average evanescent mode along the metal surface that can be defined as a plasmonic surface mode (PSM).

Fig. 6. (a). A schematic representation of the PSM generated by the collective oscillations of localized dipoles in the single metal nanostructures. (b) Description of the coupling mechanism between the waveguide mode and the PSM.

We use the term “plasmonic” to indicate that these electric fields are produced by oscillation of free electrons in each metal nanostructure on the device surface. These electric fields at the surface are localized on each individual cavity. Any absorption resonances, as in the case of separated metal nanoparticles or metal islands [10

10. D. N. Jarrett and L. Ward, “Optical-Properties of Discontinuous Gold-Films,” J. Phys. D Appl. Phys. 9, 1515–1527 (1976). [CrossRef]

], are averaged out as can be seen from the TE polarized reflectivity spectra shown in Fig. 4(a) where no unaccounted-for features are measured.

The combination of nanostructured metal surface and dielectric waveguide has strong analogies with an optical device described by Christ et al. [11

11. A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: Strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91, (2003).

]. The authors of that work produced a metal-dielectric nanostructure formed by an indium tin oxide (ITO) waveguide with a metal grating fabricated on top of it. They found an anomalous behavior of the extinction spectra as the angle of observation was varied. They explained this behavior by considering the coupling between TM polarized plasmonic resonances in the metal nanowires and the ITO waveguide mode with the same polarization.

The presence of the PSM provides the explanation for the apparent breaking of the inplane momentum conservation. In the situation where the optical waveguide is in contact with the nanostructured metal, the evanescent field of the PSM overlaps substantially with the intensity profile of the waveguide mode resulting in their coupling.

5. Coupled mode theory

We use a model based on the general coupled mode theory to explain the experimental data [12

12. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New Jersey, 1984).

]. The k-vector for the waveguide, kwg(ω), is calculated from the dispersion relations for the optical waveguide while the k-vector for the PSM, kPSM(ω), requires knowledge of its refractive index, n(ω) (=√ε(ω)), which is calculated from the effective medium theory [13

13. R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, “Optical Properties of Granular Silver and Gold Films,” Phys. Rev. B 8, 3689 (1973). [CrossRef]

], [6

6. R. Sainidou and F. J. G. de Abajo, “Plasmon guided modes in nanoparticle metamaterials,” Opt. Express 16, 4499–4506 (2008). [CrossRef] [PubMed]

]. The effective dielectric constant, ε(ω), is a combination of the metal dielectric constant, εM(ω), of the waveguide dielectric constant, εD(ω), and of the volume fraction x occupied by the dielectric around the metal-dielectric interface. It is calculated from the Maxwell-Garnett equation [13

13. R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, “Optical Properties of Granular Silver and Gold Films,” Phys. Rev. B 8, 3689 (1973). [CrossRef]

]:

ε(ω)εD(ω)ε(ω)+2εD(ω)=(1x)εM(ω)εD(ω)εM(ω)+2εD(ω)
(2)

The quantity x is referred to as the filling factor and plays a crucial role in representing optical properties at an irregular metal dielectric interface. Different average dimensions and the correlation length of the nanostructured metal surface obtained by varying the thermal annealing process can be described by changing this filling factor. The nanostructured gold surface performs two functions that are shown schematically in Fig. 6(b). The first provides in-plane momentum via scattering of the incident EM radiation to allow light coupling into planar modes. The second creates a PSM that couples with the waveguide mode to modify the optical properties of the light injected into the nanostructure. The first function is modeled by adding extra-momentum to the incident light while the second is taken into account using Eq. (2). A system of coupled equations to describe the amplitude variation of the two optical modes present in the nanostructure [12

12. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New Jersey, 1984).

] can be written:

ddza1=ikwg(ω)a1+κ12(ω)a2
ddza2=ikPSM(ω)a2+κ12*(ω)a1
(3)

In Eqs. (3) a 1 represents the waveguide mode amplitude while a 2 represents the PSM amplitude. The coupling coefficient κ12 is a complex number that describes the transfer of energy between the two optical modes. This process is mediated by a polarization vector P that appears on the nanostructured metal surface as a result of the incident light [9

9. C. Kittel, Introduction to Solid State Physics (John Wiley and Sons, Hoboken, NJ, 2005).

]. Its expression is obtained using the effective medium model. The coefficient κ12 is set to be proportional to the scalar product between P and the optical field E W in the waveguide and as a consequence it is proportional to the effective dielectric constant ε(ω) that appears in the expression for P [12

12. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New Jersey, 1984).

]. This approximates a regime where the EM radiation is well confined in the optical waveguide and the overlap between the two EM waves does not change substantially with the frequency. This approximation has the beneficial effect of simplifying the calculation to obtain the k-vector of the coupled mode. This k-vector is obtained by calculating the eigen-values of the matrix representing the system of Eqs. (3) and has the following form:

k=kwg+kPSM2±(kwgkPSM2)2κ1,22
(4)
Fig. 7. Comparison between dispersion relations data obtained from experimental reflectivity spectra (triangular dots) and calculations from coupled mode theory (square dots). The black dotted line represents the light line in air.

In order to concentrate on modeling the coupling strength between the two modes we ignore loss and consider only the real part of kwg(ω) and kPSM(ω). Thus Eq. (4) provides two complex conjugates and their real part gives the wavevector of the new coupled mode. The dispersion relations, for waveguides of thickness equal to 150 and 200 nm, are plotted in Fig. 7 with square dots where a filling factor x of 0.1 was used. The filling factor was varied from 0.5 to 0.01 with the value 0.1 best fitting the experimental data. These coupled modes lie below the light line in air because they are still waveguide modes. An extra-momentum given by the surface nanostructure is needed to transform them into radiative modes.

The experimental resonant k-vectors can be compared visually with the calculated values from the model in Fig. 7. The table II reports more precisely the values of the calculated k-vectors and the values of the k-vectors from the experimental data in the case of the 150 nm optical waveguide. Now the difference between these k-vectors, that represent the contribution of the surface nanostructure to the in-plane momentum, varies in a well defined numerical range and it is equal to an average value of 7.22±0.62 µm-1. Thus we have recovered, from a qualitative point of view, the law of momentum conservation in Eq. (1).

The error present in the data of Table II can be attributed to the approximation introduced in calculating the coupling coefficient κ12 where its dependence on the overlap integral between the electric field E in the waveguide and the polarization vector P on the nanostructured metal surface has been neglected [14

14. A. Yariv, “Coupled-Mode Theory for Guided-Wave Optics,” IEEE J. Quantum Elect. QE 9, 919–933 (1973).

]. The decay profile of the PSM varies substantially as the frequency of the incident light changes and this variation has to be included to obtain a better agreement in the entire wavevector range. Further calculations are needed to confirm this assumption.

Thus the incident light coupled by the nanostructured metal surface into the waveguide sees a mode that is not simply coincident to the waveguide but is the result of the coupling between this mode and the PSM at the metal dielectric interface.

Table II. Values of the k-vector for a 150 nm thick waveguide.

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6. Conclusions

We have analyzed the resonant effect of an optical waveguide in contact with a nanostructured gold surface that possesses plasmonic surface modes. Light impinging on the surface of this structure is coupled into a planar mode that is not simply a waveguide mode but is the result of the strong coupling between the optical waveguide mode and the plasmonic surface mode present on the nanostructured metal surface. The plasmonic resonance produced by the oscillation of free electrons at the metal-dielectric interface could also be used to couple light to underlying waveguide structures, if such a structure was fabricated. A promising application for these structures is in the development of new surface sensors capable of analyzing the properties of transparent materials deposited on its surface. The substance to be analyzed could be deposited directly onto the nanostructured gold surface to form an optical waveguide and its presence can be detected by measuring the variation of the reflectivity at a fixed angle.

Acknowledgments

We acknowledge John Pike, Sergei Romanov and Gaetan Leveque for useful discussions, Dan O’Connell for the fabrication of the metal-dielectric nanostructures and John Justice for the AFM images. This study was supported by the Science Foundation Ireland through the National Access Programme. One of the authors (P.A.P.) would like to acknowledge the Marie Curie Transfer of Knowledge scheme for supporting his activity at the Tyndall National Institute.

References and links

1.

M. Fox, Optical Properties of Solids (Oxford University Press, Oxford, 2001).

2.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, (1986).

3.

A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A 5, S16–S50 (2003). [CrossRef]

4.

D. Porath, O. Millo, and J. I. Gersten, “Scanning tunneling microscopy studies and computer simulations of annealing of gold films,” J. Vac. Sci. Technol. B 14, 30–37 (1996). [CrossRef]

5.

F. Ladouceur, “Roughness, inhomogeneity, and integrated optics,” J. Lightwave Technol. 15, 1020–1025 (1997). [CrossRef]

6.

R. Sainidou and F. J. G. de Abajo, “Plasmon guided modes in nanoparticle metamaterials,” Opt. Express 16, 4499–4506 (2008). [CrossRef] [PubMed]

7.

A. Adams, J. Moreland, P. K. Hansma, and Z. Schlesinger, “Light-Emission from Surface-Plasmon and Waveguide Modes Excited by N-Atoms near a Silver Grating,” Phys. Rev. B 25, 3457–3461 (1982). [CrossRef]

8.

R. G. Hunsperger, Integrated Optics: theory and technology (Springer-Verlag, Berlin, 1991).

9.

C. Kittel, Introduction to Solid State Physics (John Wiley and Sons, Hoboken, NJ, 2005).

10.

D. N. Jarrett and L. Ward, “Optical-Properties of Discontinuous Gold-Films,” J. Phys. D Appl. Phys. 9, 1515–1527 (1976). [CrossRef]

11.

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: Strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91, (2003).

12.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New Jersey, 1984).

13.

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, “Optical Properties of Granular Silver and Gold Films,” Phys. Rev. B 8, 3689 (1973). [CrossRef]

14.

A. Yariv, “Coupled-Mode Theory for Guided-Wave Optics,” IEEE J. Quantum Elect. QE 9, 919–933 (1973).

OCIS Codes
(250.5460) Optoelectronics : Polymer waveguides
(250.5403) Optoelectronics : Plasmonics
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Optics at Surfaces

History
Original Manuscript: May 23, 2008
Revised Manuscript: July 3, 2008
Manuscript Accepted: July 4, 2008
Published: September 29, 2008

Citation
P. A. Porta and B. Corbett, "Plasmonic mediated interactions between waveguides and nanostructured metal surfaces," Opt. Express 16, 16305-16313 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16305


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References

  1. M. Fox, Optical Properties of Solids (Oxford University Press, Oxford, 2001).
  2. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, (1986).
  3. A. V. Zayats, and I. I. Smolyaninov, "Near-field photonics: surface plasmon polaritons and localized surface plasmons," J. Opt. A 5, S16-S50 (2003). [CrossRef]
  4. D. Porath, O. Millo, and J. I. Gersten, "Scanning tunneling microscopy studies and computer simulations of annealing of gold films," J. Vac. Sci. Technol. B 14, 30-37 (1996). [CrossRef]
  5. F. Ladouceur, "Roughness, inhomogeneity, and integrated optics," J. Lightwave Technol. 15, 1020-1025 (1997). [CrossRef]
  6. R. Sainidou, and F. J. G. de Abajo, "Plasmon guided modes in nanoparticle metamaterials," Opt. Express 16, 4499-4506 (2008). [CrossRef] [PubMed]
  7. A. Adams, J. Moreland, P. K. Hansma, and Z. Schlesinger, "Light-Emission from Surface-Plasmon and Waveguide Modes Excited by N-Atoms near a Silver Grating," Phys. Rev. B 25, 3457-3461 (1982). [CrossRef]
  8. R. G. Hunsperger, Integrated Optics: theory and technology (Springer-Verlag, Berlin, 1991).
  9. C. Kittel, Introduction to Solid State Physics (John Wiley and Sons, Hoboken, NJ, 2005).
  10. D. N. Jarrett, and L. Ward, "Optical-Properties of Discontinuous Gold-Films," J. Phys. D Appl. Phys. 9, 1515-1527 (1976). [CrossRef]
  11. A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, "Waveguide-plasmon polaritons: Strong coupling of photonic and electronic resonances in a metallic photonic crystal slab," Phys. Rev. Lett. 91, (2003).
  12. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New Jersey, 1984).
  13. R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, "Optical Properties of Granular Silver and Gold Films," Phys. Rev. B 8, 3689 (1973). [CrossRef]
  14. A. Yariv, "Coupled-Mode Theory for Guided-Wave Optics," IEEE J. Quantum Elect. QE 9, 919-933 (1973).

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