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Optics Express

Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 19, Iss. 7 — Mar. 28, 2011
  • pp: 6587–6598
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Interaction and spectral gaps of surface plasmon modes in gold nano-structures

Alexandre Kolomenskii, Siying Peng, Jeshurun Hembd, Andrei Kolomenski, John Noel, James Strohaber, Winfried Teizer, and Hans Schuessler  »View Author Affiliations


Optics Express, Vol. 19, Issue 7, pp. 6587-6598 (2011)
http://dx.doi.org/10.1364/OE.19.006587


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Abstract

The transmission of ultrashort (7 fs) broadband laser pulses through periodic gold nano-structures is studied. The distribution of the transmitted light intensity over wavelength and angle shows an efficient coupling of the incident p-polarized light to two counter-propagating surface plasmon (SP) modes. As a result of the mode interaction, the avoided crossing patterns exhibit energy and momentum gaps, which depend on the configuration of the nano-structure and the wavelength. Variations of the widths of the SP resonances and an abrupt change of the mode interaction in the vicinity of the avoided crossing region are observed. These features are explained by the model of two coupled modes and a coupling change due to switching from the higher frequency dark mode to the lower frequency bright mode for increasing wavelength of the excitation light.

© 2011 OSA

1. Introduction

The interaction of light with periodic metal structures, allowing for efficient coupling of light to surface plasmons (SPs) with their remarkable properties continues attracting considerable interest. In particular, sharp resonances [1

1. R. W. Wood, “On the remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–408 (1902).

4

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

], localization and enhancement of the electromagnetic field [5

5. M. Weber and D. L. Mills, “Interaction of electromagnetic waves with periodic gratings: enhanced fields and the reflectivity,” Phys. Rev. B 27, 2698–2709 (1983). [CrossRef]

,6

6. D. C. Skigin and R. A. Depine, “Resonant enhancement of the field within a single ground-plane cavity: comparison of different rectangular shapes,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59, 3661–3668 (1999). [CrossRef]

], and wave guiding along metal surfaces [7

7. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

] were demonstrated. Corrugated metal surfaces and arrays of nanowires find applications in sensing [8

8. C. J. Alleyne, A. G. Kirk, R. C. McPhedran, N. A. Nicorovici, and D. Maystre, “Enhanced SPR sensitivity using periodic metallic structures,” Opt. Express 15(13), 8163–8169 (2007). [CrossRef] [PubMed]

] and enhancing light absorption [9

9. T. J. Davis, S. C. Mayo, and B. A. Sexton, “Optical absorption by surface plasmons in deep sub-wavelength channels,” Opt. Commun. 267, 253–259 (2006). [CrossRef]

,10

10. Z. Chen, I. R. Hooper, and J. R. Sambles, “Strongly coupled surface plasmons on thin shallow metallic gratings,” Phys. Rev. B 77, 161405 (2008). [CrossRef]

]. The interaction of light with periodic structures is also of interest, because excited SPs exhibit gaps in the energy spectrum (ω-gaps) [3

3. R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968). [CrossRef]

,5

5. M. Weber and D. L. Mills, “Interaction of electromagnetic waves with periodic gratings: enhanced fields and the reflectivity,” Phys. Rev. B 27, 2698–2709 (1983). [CrossRef]

,7

7. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

,11

11. N. Kroo, Z. Szentirmay, and J. Felszerfalvi, “Dispersion anomalies of surface plasma oscillations in MOM tunnel structures,” Phys. Lett. 86A, 445–448 (1981).

,12

12. D. Heitmann, N. Kroo, C. Schulz, and Z. Szentirmay, “Dispersion anomalies of surface plasmons on corrugated metal-insulator interfaces,” Phys. Rev. B Condens. Matter 35(6), 2660–2666 (1987). [CrossRef] [PubMed]

], rendering these structures as simple plasmonic crystals. The gaps appear due to mode interaction near the crossing of their unperturbed dispersion curves and can be employed for wave guiding [13

13. S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001). [CrossRef] [PubMed]

], developing sensors [14

14. A. J. Benahmed and C.-M. Ho, “Bandgap-assisted surface-plasmon sensing,” Appl. Opt. 46(16), 3369–3375 (2007). [CrossRef] [PubMed]

] and photonic notch filters [15

15. D. Noordegraaf, L. Scolari, J. Laegsgaard, T. Tanggaard Alkeskjold, G. Tartarini, E. Borelli, P. Bassi, J. Li, and S. T. Wu, “Avoided-crossing-based liquid-crystal photonic-bandgap notch filter,” Opt. Lett. 33(9), 986–988 (2008). [CrossRef] [PubMed]

]. The existence of momentum gaps (or k-gaps) was also proposed and confirmed in experimental [11

11. N. Kroo, Z. Szentirmay, and J. Felszerfalvi, “Dispersion anomalies of surface plasma oscillations in MOM tunnel structures,” Phys. Lett. 86A, 445–448 (1981).

,12

12. D. Heitmann, N. Kroo, C. Schulz, and Z. Szentirmay, “Dispersion anomalies of surface plasmons on corrugated metal-insulator interfaces,” Phys. Rev. B Condens. Matter 35(6), 2660–2666 (1987). [CrossRef] [PubMed]

] and theoretical works [16

16. V. Celli V, P. Tran, A. A. Maradudin, and D. L. Mills, “k gaps for surface polaritons on gratings,” Phys. Rev. B Condens. Matter 37(15), 9089–9092 (1988). [CrossRef] [PubMed]

18

18. E. Popov, “Plasmon interactions in metallic gratings: w- and k-minigaps and their connection with poles and zeros,” Surf. Sci. 222, 517–527 (1989). [CrossRef]

]. However, it has also been suggested that a k-gap is an artifact, resulting from the mode over coupling [19

19. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996). [CrossRef] [PubMed]

]. Recently, after the discovery of the extraordinary transmission (EOT) of light through arrays of small holes [20

20. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]

] the investigation of the role of SPs in this phenomenon was extensively investigated, revealing effects of the diffraction, resonance excitations, and interplay between localized and delocalized surface plasmons on the enhanced light transmission (see recent reviews [21

21. J. Weiner, “The physics of light transmission through subwavelength apertures and aperture arrays,” Rep. Prog. Phys. 72(6), 064401 (2009). [CrossRef]

,22

22. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]

]).

In this paper we investigate the interaction of SP modes observed with the transmission far-field spectroscopy [23

23. E. Altewischer, X. Ma, M. P. van Exter, and J. P. Woerdman, “Resonant Bragg scatter of surface plasmons on nanohole arrays,” N. J. Phys. 8(57), 1–14 (2006). [CrossRef]

,24

24. P. Vasa, C. Ropers, R. Pomraenke, and C. Lienau, “Ultra-fast nano-optics,” Laser Photonics Rev. 3(6), 483–507 (2009). [CrossRef]

] of broadband laser pulses (transform limited duration 7 fs) through gold nano-structures with gratings. We study the interaction of light with gold gratings at parameters that are quite different from those used for the observation of the EOT. The latter is observed in optically thick (150-300nm) metal films with small holes (surface fraction of openings less than 20%). The thickness of the gold layer in our samples is about 50 nm, so that the film is partially transparent (the absorption length of light around 700 nm, calculated from the optical constants of gold [25

25. B. R. Cooper, H. Ehrenreich, and H. R. Philipp, “Optical properties of noble metals,” Phys. Rev. 138(2A), A494–A507 (1965). [CrossRef]

] is about 14 nm). If the slits (through or partial) are also taken into account the transmittance of our samples is much higher than in EOT observations. Consequently, at such conditions the positions of the SP resonances are close to the minima in the transmission [26

26. 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(18), 183901 (2003). [CrossRef] [PubMed]

,27

27. H. Lochbihler, “Surface polaritons on gold-wire gratings,” Phys. Rev. B Condens. Matter 50(7), 4795–4801 (1994). [CrossRef] [PubMed]

], while for typical conditions of the EOT the loci of the SP resonances correspond to the transmission enhancement [22

22. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]

]. We observe efficient coupling of the incident light to two counter-propagating SP modes, exhibiting avoided crossing with features that were not reported previously. The experimentally measured minima in the transmitted light form patterns, indicating the existence of ω- and k-gaps, confirmed also by calculations with the coupled mode model [28

28. H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972). [CrossRef]

,29

29. R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42(7), 4315–4324 (1990). [CrossRef] [PubMed]

].

2. Experimental section

We studied two types of metallic structures: a periodic array of gold nanowires on a glass substrate and a similar array with an additional gold sub-layer. The choice of the samples was aimed at observing the differences in the dispersion relations and the interaction of the SP modes in a sample with and without slits, since these configurations can be expected to have different radiative damping and mode coupling. The periods of the arrays were chosen to provide SP resonances at normal incidence close to the middle of the spectral range (from 650 to 850 nm) of the laser pulses. The configurations of the two samples produced by electron beam lithography and thermal vapor deposition are shown in Fig. 1(a)
Fig. 1 Measured samples: (a). Schematic of the sample profile and the geometry of the laser beam incidence (red arrows). The two samples had the following dimensions. Sample 1: d=705  nm, d1=390  nm,h=27  nm, h1=35  nm. Sample 2: d=722  nm, d1=395  nm, h=50  nm, h1=0. (b). SEM image of sample 1.
. Both samples had a gold grating structure with a profile close to rectangular. For the first sample the grating was thinner, and it was deposited onto an underlying gold film. The thickness of the films during the deposition was monitored with a quartz crystal monitor. The samples were also characterized with an atomic force microscope (AFM) and by scanning electron microscopy (SEM). A SEM image of the surface of the first sample is shown in Fig. 1(b).

Measurements of the transmitted light for a set of incidence angles were performed. A portion of the beam from a broadband pulsed laser (Rainbow, Femtolasers) was weakly focused and front-illuminated the sample, as shown in Fig. 1(a). The sample was mounted on a rotation stage allowing for variation of the incidence angle from −6° to + 6° with small steps and accuracy better than 0.05°. The light that interacted with the grating on the sample was selected by a small aperture and directed into an Ocean Optics spectrometer, registering the distribution of the spectral intensity.

3. Results

SPs can be efficiently excited when the incident light interacting with the grating produces a polarization wave propagating along the metal surface with the phase velocity of the SP wave.

This condition, following also from the conservation of energy and the momentum component along the surface, can be realized by adjusting the wavelength, the incidence angle of the light wave, the period of the grating [30

30. I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D 9(17), 2423–2432 (1976). [CrossRef]

32

32. S. H. Zaidi, M. Yousaf, and S. R. Brueck, “Grating coupling to surface plasma waves. II. Interactions between first- and second-order coupling,” J. Opt. Soc. Am. B 8(6), 1348–1359 (1991). [CrossRef]

] or the refractive index of the dielectric medium adjacent to the metal [33

33. M. J. A. de Dood, E. F. C. Driessen, D. Stolwijk, and M. P. van Exter, “Observation of coupling between surface plasmons in index-matched hole arrays,” Phys. Rev. B 77, 115437 (2008). [CrossRef]

]. Thus, the resonant coupling takes place, when the wave vector of light after interaction with the grating has a component in the plane of the grating equal to the wave vector of the SP,
ksinθ+nkgr=ksp,
(1)
where k=2π/λ is the wave number of light, θ is the incidence angle, integer number n is the order of the diffraction interaction with the grating, kgr=2π/d, and ksp is the SP wave number.

Normalized transmission spectra for two samples are shown as density plots in Fig. 2(a,c)
Fig. 2 Transmitted light intensity for different incidence angles and wavelengths: (a,b) sample 1, (c,d) sample 2. Figures (a, c) show false color density plots with the dark (blue) color corresponding to the reduction of the light transmission due to the SP excitation. Figures (b,d) depict the wavelength dependences of the intensity for a set of angles, showing that while in (b) the minimum at shorter wavelength becomes narrower for smaller angles, in (d) the similar minimum becomes slightly broader; in (b,d) the curves for different wavelengths are equidistantly shifted vertically for better viewing.
. The intensity distribution over incidence angle and wavelength displays two diagonal valleys corresponding to n=±1orders of the SP resonance excitation, producing a decrease in the transmitted intensity. The higher frequency branch demonstrates a narrowing for sample 1 and a slight broadening for sample 2, when approaching the crossing region. This can be seen by comparing the shorter-wavelength dips of the spectral intensity profiles shown for a set of angles in Fig. 2(b,d), which exhibit asymmetric Fano-type resonances [34

34. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961). [CrossRef]

]. Such changes indicate damping variations of the SP modes. In particular, radiative damping can change due to a shift of the mode intensity distribution relative to the grooves of the grating, as the wavelength changes [35

35. C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, “Femtosecond light transmission and subradiant damping in plasmonic crystals,” Phys. Rev. Lett. 94(11), 113901 (2005). [CrossRef] [PubMed]

,36

36. C. Ropers, T. Elsaesser, G. Cerullo, M. Zavelani-Rossi, and C. Lienau, “Ultrafast optical excitations of metallic nanostructures: from light confinement to a novel electron source,” N. J. Phys. 9(397), 1–32 (2007). [CrossRef]

], and also the effect of the mode interaction due to Bragg scattering [23

23. E. Altewischer, X. Ma, M. P. van Exter, and J. P. Woerdman, “Resonant Bragg scatter of surface plasmons on nanohole arrays,” N. J. Phys. 8(57), 1–14 (2006). [CrossRef]

] increases closer to the normal incidence, corresponding in ksp-space to the boundary of the Brillouin zone.

We note that the coupling of the incident light to SPs at a grating surface is highly dependent on the polarization. The maximum coupling is achieved, when the projection of the electric field vector on the sample surface is perpendicular to the grooves. We measured SP coupling with light at normal incidence for varied polarizations, and the normalized data (Fig. 3
Fig. 3 Dependence of coupling on polarization angleϕwith ϕ=0°corresponding to polarization in the plane of incidence.
) suggests a cos2(ϕ) dependence on the polarization angle ϕ, i.e. only the component with the polarization perpendicular to the grooves couples to SPs.

For sample 2, the mode pattern in the band-gap region is qualitatively different for higher and lower frequency branches. For the higher frequency branch, the dispersion curves with n=±1 do not tend to merge when they approach the avoided crossing region, but rather form a gap in θ values near θ=0 (k-gap, Δk=1.91 × 103 cm- 1).

Figures 5(b,e) show the evolution of the transmission angular dependence for the two samples in the avoided crossing regions, indicated by dashed boxes in Fig. 5(a,d), as the wavelength is changed in small steps. With increasing wavelength the two side minima in the angular intensity distribution (Fig. 5(b), sample 1), which are seen clearly for λ=716nm, initially merge into a valley and then re-appear again. In Fig. 5(e) (sample 2) with increasing wavelength the two side minima stay separated until they disappear (four lower curves), and then two shallow minima in the central part of the plot are formed.

To describe the observed behavior we used a model of two coupled modes [28

28. H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972). [CrossRef]

,29

29. R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42(7), 4315–4324 (1990). [CrossRef] [PubMed]

,39

39. S. L. Chuang, Physics of Photonic Devices, (Wiley, 2009).

] with slowly varying complex amplitudes A1(z) (mode with n = + 1) and A2(z)(n = −1) and propagation constants (without accounting for interaction) ksp;1=ksp;1(ω) and ksp;2=ksp;2(ω)=ksp;1, counter-propagating along z-axis. The main mechanism of the interaction of these modes is their Bragg scattering with 2kgr [5

5. M. Weber and D. L. Mills, “Interaction of electromagnetic waves with periodic gratings: enhanced fields and the reflectivity,” Phys. Rev. B 27, 2698–2709 (1983). [CrossRef]

,19

19. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996). [CrossRef] [PubMed]

,23

23. E. Altewischer, X. Ma, M. P. van Exter, and J. P. Woerdman, “Resonant Bragg scatter of surface plasmons on nanohole arrays,” N. J. Phys. 8(57), 1–14 (2006). [CrossRef]

]. The wave numbers of re-scattered waves can be shifted by 2kgr and create a polarization wave with a wave vector value ksp;2(s)=ksp;2+2kgr. Then ksp;1and ksp;2(s)considered as functions of the incident light frequency ω=2πc/λ cross at some frequency value ωc, such that ksp;1(ωc)ksp;2(s)(ωc)=0. The crossing of the modes means that they propagate with the same phase velocity and therefore their coupling can take place, modifying also the dispersion relations. To find these new relations we use a system of two coupled equations [39

39. S. L. Chuang, Physics of Photonic Devices, (Wiley, 2009).

]:
ddzA1(z)=iK1,2A2(z)exp[i(ksp;2+2kgrksp;1)],ddzA2(z)=iK2,1A1(z)exp[i(ksp;2+2kgrksp;1)],
(2)
where the coupling coefficients K1,2=K1,2(ω) and K2,1=K2,1(ω) are functions of frequency. Assuming dependences exp(ik'spz), where k'spis the new value of the propagation constant accounting for the mode interaction, we find the eigenvalues of Eqs. (2) that describe dispersion relations. To have the correct choice of signs we note that far from the crossing point new solutions k'sp;1,2 should transform into unperturbed values, ksp;1,2.Consequently, we find the new propagation constants as functions of ω:
k'sp;1,2=±(ksp;1+ksp;2+2kgr)/2q  for ω<ωc  andk'sp;1,2=±(ksp;1+ksp;2+2kgr)/2±q  for ω>ωc,
(3)
where q=Δ2+G, Δ=(ksp;1ksp;22kgr)/2 and G=K1,2K2,1. These equations show that for real values of ksp;1,ksp;2 and K1,2=K2,1*(typical for small losses and contra-directional coupling) the condition G<Δ2 is fulfilled for the interval of ω values (ω-gap), where q becomes imaginary, giving rise to the avoided crossing and a strong suppression (attenuation) of the SP modes within the gap. Thus, for G<0 the two SP modes form an ω-gap (conservative coupling), while for G>0 the gap appears in the k'sp values (dissipative coupling) [29

29. R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42(7), 4315–4324 (1990). [CrossRef] [PubMed]

]. When attenuation of the SP modes takes place or the coupling constants are complex, the values of k'sp;1,2 also become complex, so that the real parts Re[k'sp;1,2] determine the phase constants of the modes, and the imaginary parts Im[k'sp;1,2] determine their attenuation or amplification, depending on the sign.

The results of the dispersion relation calculations with the model of two coupled SP modes (Eq. (2)) are shown in Figs. 5(c,f) as λ(θ) dependences to enable direct comparison with the experimental data. The unperturbed dispersion relations ksp;1,2(ω) were calculated using the three layer model [40

40. E. Kretschmann, “The determination of the optical constants of metals by excitation of surface plasmons,” Z. Phys. 241, 313–324 (1971). [CrossRef]

], taking into account the grating as an additional gold layer with effective thicknesses heff,1(1)=15nm and heff,1(2)=35nm for samples 1 and 2, respectively. For sample 1, agreement with the experiment is obtained when the value of G has a negative real part and a relatively smaller imaginary part, namely G=(310137i)(meV)2, so that in the assumption |K1,2|=|K2,1|=K we obtain for the magnitude of the coupling constant K=18.4meV. A significant negative real part Re[G] is required for a large ω-gap with a strong attenuation of SP modes within the respective frequency interval. When Re[G] is positive, a k-gap (or a gap in θvalues) and two almost vertical portions of the dispersion curves appear. Note that the presence of the imaginary part Im[G]0 can also lead to the formation of a k-gap. This can be the reason for the appearance of a relatively narrow k-gap in calculated dispersion curves of Fig. 5(c). When this k-gap is small and damping is present, as in our case, the vertical portions of the dispersion curves can merge, forming a vertical line of minima in the transmission, similar to the observed in the experiment (Fig. 5(a), the data points shown by crosses). Thus, for Re[G]<0 and Im[G]0 features of both ω- and k-gaps can be present as in Fig. 5(c), where strong attenuation of the SP modes takes place for the vertical portions of the dispersion curves. This strong attenuation leads to the broadening of the central portion of the SP modes seen in Fig. 2(a).

For sample 2, the lower frequency branch looks similar to that of sample 1. However, the higher frequency branch shows the divergence of the dispersion curves for n=+1 and n=1 in the vicinity of the avoided crossing region (λ<λc=2πc/ωc=728nm), while in the region λ>λc the portions of the dispersion curve tend to merge as the wavelength is approaching the central wavelength λc. The different behavior in these two wavelength regions provides evidence that the sign of the real part of the product G changes, as the wavelength changes from λ<λctoλ>λc. The solid lines in Fig. 5(f) are calculated with G=289(meV)2 for λ<λc and with G=576(meV)2 for λ>λc and give good agreement with experimental points.

4. Discussion

We also would like to compare our results with those obtained in the observation of the EOT effect, which stimulated an extensive discussion of the role of SPs in this phenomenon. For the normal incidence the transmission minima take place when the period of the grating is close to a multiple of the wavelength of the surface plasmon wave [21

21. J. Weiner, “The physics of light transmission through subwavelength apertures and aperture arrays,” Rep. Prog. Phys. 72(6), 064401 (2009). [CrossRef]

]. This is consistent with our results, since at the normal incidence we observe a strong minimum in the transmission at the θ=0, which corresponds to ksp=kgr, i.e. λsp=d (d is the grating period). We determine the position of the SP mode from the transmission measurements. The transmitted intensity results from the interference of the direct wave, which is slowly varying, and contributions from scattered SPs that are re-emitted as bulk waves [35

35. C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, “Femtosecond light transmission and subradiant damping in plasmonic crystals,” Phys. Rev. Lett. 94(11), 113901 (2005). [CrossRef] [PubMed]

,36

36. C. Ropers, T. Elsaesser, G. Cerullo, M. Zavelani-Rossi, and C. Lienau, “Ultrafast optical excitations of metallic nanostructures: from light confinement to a novel electron source,” N. J. Phys. 9(397), 1–32 (2007). [CrossRef]

]. Since at resonance SPs and the waves they re-emit experience a sharp change of phase, SP resonances are often exhibited in the transmission as Fano-type profiles [24

24. P. Vasa, C. Ropers, R. Pomraenke, and C. Lienau, “Ultra-fast nano-optics,” Laser Photonics Rev. 3(6), 483–507 (2009). [CrossRef]

]. The effect of the EOT is observed in optically thick metal films with small holes. Recent theoretical calculations have shown that for such conditions the positions of the surface plasmon resonances for a plane metal film correspond to minima in the transmission [42

42. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002). [CrossRef] [PubMed]

,43

43. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts,” Opt. Express 14(14), 6400–6413 (2006). [CrossRef] [PubMed]

], and transmission maxima match the positions of SP resonances, taking into account their displacement due to collective and localized interactions with the holes [44

44. P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nanohole arrays,” J. Opt. A, Pure Appl. Opt. 7(8), 422–426 (2005). [CrossRef]

].

A k-gap in our experiment was observed for a dark higher frequency mode, which corresponds to the maxima of the field on the metal ridges [35

35. C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, “Femtosecond light transmission and subradiant damping in plasmonic crystals,” Phys. Rev. Lett. 94(11), 113901 (2005). [CrossRef] [PubMed]

,36

36. C. Ropers, T. Elsaesser, G. Cerullo, M. Zavelani-Rossi, and C. Lienau, “Ultrafast optical excitations of metallic nanostructures: from light confinement to a novel electron source,” N. J. Phys. 9(397), 1–32 (2007). [CrossRef]

]. For such a spatial configuration of the mode the scattering should be reduced and the coupling is also reduced accordingly. Consequently, also for this reason over coupling cannot be responsible for the observed k-gap effect. This is consistent with the results of Ref. 18

18. E. Popov, “Plasmon interactions in metallic gratings: w- and k-minigaps and their connection with poles and zeros,” Surf. Sci. 222, 517–527 (1989). [CrossRef]

, showing that a k-gap appears for a small direct coupling of the SP modes. From calculations based on the Rayleigh expansion and transfer-matrix theory it was shown [46

46. P. Tran, V. Celli, and A. A. Maradudin, “Conditions for the occurrence of k gaps for surface polaritons on gratings,” Opt. Lett. 13(6), 530–532 (1988). [CrossRef] [PubMed]

] that k-gaps exist only up to a certain value of the ratio of the real and imaginary parts of the coupling constant. The presence of a significant imaginary part of the coupling constant is also a prerequisite of appearance of a k-gap according to presented in our paper model of two coupled modes. We note that recent theoretical analysis of a grating-coupled multimode waveguide based on a study of the trajectories of the poles and zeroes of the reflection and transmission coefficients also revealed the existence of significant k-gaps [47

47. A.-L. Fehrembach, S. Hernandez, and A. Sentenac, “k gaps for multimode waveguide gratings,” Phys. Rev. B 73(23), 233405 (2006). [CrossRef]

].

5. Conclusion

In conclusion, we experimentally investigated the interaction of light with periodic gold nano-structures on glass substrates and observed qualitatively different behavior near the avoided crossing region, depending on the configuration of the sample. The sample with a grating deposited on top of a gold film exhibited an energy (or ω-) gap between the two branches of the dispersion relation. The valley in the intensity distribution of the transmitted light in the vicinity of normal incidence and the calculation with the coupled mode model indicate possible presence of a small momentum (or k-) gap as well. Additionally, the higher frequency branch experienced narrowing of the width of the transmission minimum and a strong reduction of coupling to the incident light, which indicate a decrease of radiative damping in the avoided crossing region. For the sample with only a gold grating deposited on a glass substrate the two branches of the dispersion relation displayed qualitatively different patterns: while the lower frequency branch exhibited behavior typical for an ω-gap, the higher frequency branch was split into two portions showing a significant k-gap. Therefore, our observations indicate that interacting modes in one avoided crossing region can exhibit features characteristic of both a k-gap and an ω-gap; this behavior we explain by switching from the higher frequency dark mode to the lower frequency bright mode as the wavelength increases. The coupled mode model shows that ω-gaps are formed, when the coupling constants are predominantly real, and k-gaps appear for predominantly imaginary coupling constants. Thus, when the latter have both real and imaginary parts, features characteristic for gaps of both types are possible.

Acknowledgments

We thank Wonmuk Hwang for providing the AFM for sample profile measurements and Norbert Kroo for a stimulating discussion. This work was partially supported by the Robert A. Welch Foundation (grants Nos. A1546 and A1585), the National Science Foundation (NSF) (grants Nos. 0722800 and 0555568), the Qatar National Research Fund (grant NPRP30-6-7-35), and the United States Air Force Office of Scientific Research (USAFOSR) (grant FA9550-07-1-0069).

References and links

1.

R. W. Wood, “On the remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–408 (1902).

2.

U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld's waves),” J. Opt. Soc. Am. 31, 213–222 (1941). [CrossRef]

3.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968). [CrossRef]

4.

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

5.

M. Weber and D. L. Mills, “Interaction of electromagnetic waves with periodic gratings: enhanced fields and the reflectivity,” Phys. Rev. B 27, 2698–2709 (1983). [CrossRef]

6.

D. C. Skigin and R. A. Depine, “Resonant enhancement of the field within a single ground-plane cavity: comparison of different rectangular shapes,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59, 3661–3668 (1999). [CrossRef]

7.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]

8.

C. J. Alleyne, A. G. Kirk, R. C. McPhedran, N. A. Nicorovici, and D. Maystre, “Enhanced SPR sensitivity using periodic metallic structures,” Opt. Express 15(13), 8163–8169 (2007). [CrossRef] [PubMed]

9.

T. J. Davis, S. C. Mayo, and B. A. Sexton, “Optical absorption by surface plasmons in deep sub-wavelength channels,” Opt. Commun. 267, 253–259 (2006). [CrossRef]

10.

Z. Chen, I. R. Hooper, and J. R. Sambles, “Strongly coupled surface plasmons on thin shallow metallic gratings,” Phys. Rev. B 77, 161405 (2008). [CrossRef]

11.

N. Kroo, Z. Szentirmay, and J. Felszerfalvi, “Dispersion anomalies of surface plasma oscillations in MOM tunnel structures,” Phys. Lett. 86A, 445–448 (1981).

12.

D. Heitmann, N. Kroo, C. Schulz, and Z. Szentirmay, “Dispersion anomalies of surface plasmons on corrugated metal-insulator interfaces,” Phys. Rev. B Condens. Matter 35(6), 2660–2666 (1987). [CrossRef] [PubMed]

13.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001). [CrossRef] [PubMed]

14.

A. J. Benahmed and C.-M. Ho, “Bandgap-assisted surface-plasmon sensing,” Appl. Opt. 46(16), 3369–3375 (2007). [CrossRef] [PubMed]

15.

D. Noordegraaf, L. Scolari, J. Laegsgaard, T. Tanggaard Alkeskjold, G. Tartarini, E. Borelli, P. Bassi, J. Li, and S. T. Wu, “Avoided-crossing-based liquid-crystal photonic-bandgap notch filter,” Opt. Lett. 33(9), 986–988 (2008). [CrossRef] [PubMed]

16.

V. Celli V, P. Tran, A. A. Maradudin, and D. L. Mills, “k gaps for surface polaritons on gratings,” Phys. Rev. B Condens. Matter 37(15), 9089–9092 (1988). [CrossRef] [PubMed]

17.

P. Halevi and O. Mata-Méndez, “Electromagnetic modes of corrugated thin films and surfaces with a transition layer. II. Minigaps,” Phys. Rev. B Condens. Matter 39(9), 5694–5705 (1989). [CrossRef] [PubMed]

18.

E. Popov, “Plasmon interactions in metallic gratings: w- and k-minigaps and their connection with poles and zeros,” Surf. Sci. 222, 517–527 (1989). [CrossRef]

19.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996). [CrossRef] [PubMed]

20.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]

21.

J. Weiner, “The physics of light transmission through subwavelength apertures and aperture arrays,” Rep. Prog. Phys. 72(6), 064401 (2009). [CrossRef]

22.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]

23.

E. Altewischer, X. Ma, M. P. van Exter, and J. P. Woerdman, “Resonant Bragg scatter of surface plasmons on nanohole arrays,” N. J. Phys. 8(57), 1–14 (2006). [CrossRef]

24.

P. Vasa, C. Ropers, R. Pomraenke, and C. Lienau, “Ultra-fast nano-optics,” Laser Photonics Rev. 3(6), 483–507 (2009). [CrossRef]

25.

B. R. Cooper, H. Ehrenreich, and H. R. Philipp, “Optical properties of noble metals,” Phys. Rev. 138(2A), A494–A507 (1965). [CrossRef]

26.

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(18), 183901 (2003). [CrossRef] [PubMed]

27.

H. Lochbihler, “Surface polaritons on gold-wire gratings,” Phys. Rev. B Condens. Matter 50(7), 4795–4801 (1994). [CrossRef] [PubMed]

28.

H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972). [CrossRef]

29.

R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42(7), 4315–4324 (1990). [CrossRef] [PubMed]

30.

I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D 9(17), 2423–2432 (1976). [CrossRef]

31.

S. H. Zaidi, M. Yousaf, and S. R. Brueck, “Grating coupling to surface plasma waves. I. First-order coupling,” J. Opt. Soc. Am. B 8, 770 (1991). [CrossRef]

32.

S. H. Zaidi, M. Yousaf, and S. R. Brueck, “Grating coupling to surface plasma waves. II. Interactions between first- and second-order coupling,” J. Opt. Soc. Am. B 8(6), 1348–1359 (1991). [CrossRef]

33.

M. J. A. de Dood, E. F. C. Driessen, D. Stolwijk, and M. P. van Exter, “Observation of coupling between surface plasmons in index-matched hole arrays,” Phys. Rev. B 77, 115437 (2008). [CrossRef]

34.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961). [CrossRef]

35.

C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, “Femtosecond light transmission and subradiant damping in plasmonic crystals,” Phys. Rev. Lett. 94(11), 113901 (2005). [CrossRef] [PubMed]

36.

C. Ropers, T. Elsaesser, G. Cerullo, M. Zavelani-Rossi, and C. Lienau, “Ultrafast optical excitations of metallic nanostructures: from light confinement to a novel electron source,” N. J. Phys. 9(397), 1–32 (2007). [CrossRef]

37.

M. G. Weber and D. L. Mills, “Symmetry and the reflectivity of diffraction gratings at normal incidence,” Phys. Rev. B Condens. Matter 31(4), 2510–2513 (1985). [CrossRef] [PubMed]

38.

J. W. Lee, T. H. Park, P. Nordlander, and D. M. Mittleman, “Antibonding plasmon mode coupling of an individual hole in a thin metallic film,” Phys. Rev. B 80(20), 205417 (2009). [CrossRef]

39.

S. L. Chuang, Physics of Photonic Devices, (Wiley, 2009).

40.

E. Kretschmann, “The determination of the optical constants of metals by excitation of surface plasmons,” Z. Phys. 241, 313–324 (1971). [CrossRef]

41.

K. G. Lee and Q. H. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. 95(10), 103902 (2005). [CrossRef] [PubMed]

42.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002). [CrossRef] [PubMed]

43.

Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts,” Opt. Express 14(14), 6400–6413 (2006). [CrossRef] [PubMed]

44.

P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nanohole arrays,” J. Opt. A, Pure Appl. Opt. 7(8), 422–426 (2005). [CrossRef]

45.

H. Lochbihler, “Surface polaritons on metallic wire gratings studied via power losses,” Phys. Rev. B Condens. Matter 53(15), 10289–10295 (1996). [CrossRef] [PubMed]

46.

P. Tran, V. Celli, and A. A. Maradudin, “Conditions for the occurrence of k gaps for surface polaritons on gratings,” Opt. Lett. 13(6), 530–532 (1988). [CrossRef] [PubMed]

47.

A.-L. Fehrembach, S. Hernandez, and A. Sentenac, “k gaps for multimode waveguide gratings,” Phys. Rev. B 73(23), 233405 (2006). [CrossRef]

OCIS Codes
(240.0240) Optics at surfaces : Optics at surfaces
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: November 1, 2010
Revised Manuscript: January 17, 2011
Manuscript Accepted: February 4, 2011
Published: February 23, 2011

Citation
Alexandre Kolomenskii, Siying Peng, Jeshurun Hembd, Andrei Kolomenski, John Noel, James Strohaber, Winfried Teizer, and Hans Schuessler, "Interaction and spectral gaps of surface plasmon modes in gold nano-structures," Opt. Express 19, 6587-6598 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6587


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References

  1. R. W. Wood, “On the remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–408 (1902).
  2. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld's waves),” J. Opt. Soc. Am. 31, 213–222 (1941). [CrossRef]
  3. R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968). [CrossRef]
  4. H. Raether, in Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).
  5. M. Weber and D. L. Mills, “Interaction of electromagnetic waves with periodic gratings: enhanced fields and the reflectivity,” Phys. Rev. B 27, 2698–2709 (1983). [CrossRef]
  6. D. C. Skigin and R. A. Depine, “Resonant enhancement of the field within a single ground-plane cavity: comparison of different rectangular shapes,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59, 3661–3668 (1999). [CrossRef]
  7. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
  8. C. J. Alleyne, A. G. Kirk, R. C. McPhedran, N. A. Nicorovici, and D. Maystre, “Enhanced SPR sensitivity using periodic metallic structures,” Opt. Express 15(13), 8163–8169 (2007). [CrossRef] [PubMed]
  9. T. J. Davis, S. C. Mayo, and B. A. Sexton, “Optical absorption by surface plasmons in deep sub-wavelength channels,” Opt. Commun. 267, 253–259 (2006). [CrossRef]
  10. Z. Chen, I. R. Hooper, and J. R. Sambles, “Strongly coupled surface plasmons on thin shallow metallic gratings,” Phys. Rev. B 77, 161405 (2008). [CrossRef]
  11. N. Kroo, Z. Szentirmay, and J. Felszerfalvi, “Dispersion anomalies of surface plasma oscillations in MOM tunnel structures,” Phys. Lett. 86A, 445–448 (1981).
  12. D. Heitmann, N. Kroo, C. Schulz, and Z. Szentirmay, “Dispersion anomalies of surface plasmons on corrugated metal-insulator interfaces,” Phys. Rev. B Condens. Matter 35(6), 2660–2666 (1987). [CrossRef] [PubMed]
  13. S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001). [CrossRef] [PubMed]
  14. A. J. Benahmed and C.-M. Ho, “Bandgap-assisted surface-plasmon sensing,” Appl. Opt. 46(16), 3369–3375 (2007). [CrossRef] [PubMed]
  15. D. Noordegraaf, L. Scolari, J. Laegsgaard, T. Tanggaard Alkeskjold, G. Tartarini, E. Borelli, P. Bassi, J. Li, and S. T. Wu, “Avoided-crossing-based liquid-crystal photonic-bandgap notch filter,” Opt. Lett. 33(9), 986–988 (2008). [CrossRef] [PubMed]
  16. V. Celli, P. Tran, A. A. Maradudin, and D. L. Mills, “k gaps for surface polaritons on gratings,” Phys. Rev. B Condens. Matter 37(15), 9089–9092 (1988). [CrossRef] [PubMed]
  17. P. Halevi and O. Mata-Méndez, “Electromagnetic modes of corrugated thin films and surfaces with a transition layer. II. Minigaps,” Phys. Rev. B Condens. Matter 39(9), 5694–5705 (1989). [CrossRef] [PubMed]
  18. E. Popov, “Plasmon interactions in metallic gratings: w- and k-minigaps and their connection with poles and zeros,” Surf. Sci. 222, 517–527 (1989). [CrossRef]
  19. W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996). [CrossRef] [PubMed]
  20. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391(6668), 667–669 (1998). [CrossRef]
  21. J. Weiner, “The physics of light transmission through subwavelength apertures and aperture arrays,” Rep. Prog. Phys. 72(6), 064401 (2009). [CrossRef]
  22. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010). [CrossRef]
  23. E. Altewischer, X. Ma, M. P. van Exter, and J. P. Woerdman, “Resonant Bragg scatter of surface plasmons on nanohole arrays,” N. J. Phys. 8(57), 1–14 (2006). [CrossRef]
  24. P. Vasa, C. Ropers, R. Pomraenke, and C. Lienau, “Ultra-fast nano-optics,” Laser Photonics Rev. 3(6), 483–507 (2009). [CrossRef]
  25. B. R. Cooper, H. Ehrenreich, and H. R. Philipp, “Optical properties of noble metals,” Phys. Rev. 138(2A), A494–A507 (1965). [CrossRef]
  26. 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(18), 183901 (2003). [CrossRef] [PubMed]
  27. H. Lochbihler, “Surface polaritons on gold-wire gratings,” Phys. Rev. B Condens. Matter 50(7), 4795–4801 (1994). [CrossRef] [PubMed]
  28. H. Kogelnik and C. V. Shank, “Coupled wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972). [CrossRef]
  29. R. J. C. Spreeuw, R. C. Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42(7), 4315–4324 (1990). [CrossRef] [PubMed]
  30. I. Pockrand, “Resonance anomalies in the light intensity reflected at silver gratings with dielectric coatings,” J. Phys. D 9(17), 2423–2432 (1976). [CrossRef]
  31. S. H. Zaidi, M. Yousaf, and S. R. Brueck, “Grating coupling to surface plasma waves. I. First-order coupling,” J. Opt. Soc. Am. B 8, 770 (1991). [CrossRef]
  32. S. H. Zaidi, M. Yousaf, and S. R. Brueck, “Grating coupling to surface plasma waves. II. Interactions between first- and second-order coupling,” J. Opt. Soc. Am. B 8(6), 1348–1359 (1991). [CrossRef]
  33. M. J. A. de Dood, E. F. C. Driessen, D. Stolwijk, and M. P. van Exter, “Observation of coupling between surface plasmons in index-matched hole arrays,” Phys. Rev. B 77, 115437 (2008). [CrossRef]
  34. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961). [CrossRef]
  35. C. Ropers, D. J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D. S. Kim, and C. Lienau, “Femtosecond light transmission and subradiant damping in plasmonic crystals,” Phys. Rev. Lett. 94(11), 113901 (2005). [CrossRef] [PubMed]
  36. C. Ropers, T. Elsaesser, G. Cerullo, M. Zavelani-Rossi, and C. Lienau, “Ultrafast optical excitations of metallic nanostructures: from light confinement to a novel electron source,” N. J. Phys. 9(397), 1–32 (2007). [CrossRef]
  37. M. G. Weber and D. L. Mills, “Symmetry and the reflectivity of diffraction gratings at normal incidence,” Phys. Rev. B Condens. Matter 31(4), 2510–2513 (1985). [CrossRef] [PubMed]
  38. J. W. Lee, T. H. Park, P. Nordlander, and D. M. Mittleman, “Antibonding plasmon mode coupling of an individual hole in a thin metallic film,” Phys. Rev. B 80(20), 205417 (2009). [CrossRef]
  39. S. L. Chuang, Physics of Photonic Devices, (Wiley, 2009).
  40. E. Kretschmann, “The determination of the optical constants of metals by excitation of surface plasmons,” Z. Phys. 241, 313–324 (1971). [CrossRef]
  41. K. G. Lee and Q. H. Park, “Coupling of surface plasmon polaritons and light in metallic nanoslits,” Phys. Rev. Lett. 95(10), 103902 (2005). [CrossRef] [PubMed]
  42. Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88(5), 057403 (2002). [CrossRef] [PubMed]
  43. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts,” Opt. Express 14(14), 6400–6413 (2006). [CrossRef] [PubMed]
  44. P. Lalanne, J. C. Rodier, and J. P. Hugonin, “Surface plasmons of metallic surfaces perforated by nanohole arrays,” J. Opt. A, Pure Appl. Opt. 7(8), 422–426 (2005). [CrossRef]
  45. H. Lochbihler, “Surface polaritons on metallic wire gratings studied via power losses,” Phys. Rev. B Condens. Matter 53(15), 10289–10295 (1996). [CrossRef] [PubMed]
  46. P. Tran, V. Celli, and A. A. Maradudin, “Conditions for the occurrence of k gaps for surface polaritons on gratings,” Opt. Lett. 13(6), 530–532 (1988). [CrossRef] [PubMed]
  47. A.-L. Fehrembach, S. Hernandez, and A. Sentenac, “k gaps for multimode waveguide gratings,” Phys. Rev. B 73(23), 233405 (2006). [CrossRef]

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