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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 14 — Jul. 5, 2010
  • pp: 14913–14925
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Plasmon dispersion diagram and localization effects in a three-cavity commensurate grating

A. Barbara, S. Collin, Ch. Sauvan, J. Le Perchec, C. Maxime, J-L. Pelouard, and P. Quémerais  »View Author Affiliations


Optics Express, Vol. 18, Issue 14, pp. 14913-14925 (2010)
http://dx.doi.org/10.1364/OE.18.014913


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Abstract

Commensurate gratings of deep-metallic grooves have highly localized cavity resonances which do not exist for purely periodic gratings. In this paper we present the experimental dispersion diagram of the resonances of a commensurate grating with three sub-wavelength cavities per period. We observe selective light localization within the cavities, transition from a localized to a delocalized mode and modifications of the coupling of modes with the external plane-wave that may lead to the generation of black modes. This unexpected complexity is analyzed via a theoretical study in full agreement with the experiments. These results open a way to the control of wavelength-dependent hot spot predicted in more complex commensurate gratings.

© 2010 Optical Society of America

1. Introduction

Within the vast framework of controlling light properties at a sub-wavelength scale one important issue is to understand and produce - in a reproducible manner - field localization and enhancements at the surface of metals, one reason being that those are linked to the Surface Enhanced Raman Scattering (SERS) effect[1

1. K. Kneipp, M. Moskovits, and H. Kneipp, ed., Surface-Enhanced Raman Scattering, Topics in Applied Physics, 103, (Springer, 2006). [CrossRef]

]. Beyond the fundamental interest of understanding SERS, many applications could benefit from the possibility of controlling the local field enhancement factors. They might actually be very useful for ultra-sensitive spectroscopic techniques which are widely used for identifying molecules in physics, chemistry, biology and pharmacology.

2. Experiments

2.1. Commensurate gratings and experimental set-up

Before we focus on the sample studied, let us briefly recall what the commensurate gratings we are dealing with are. Commensurate gratings are long-period gratings containing several identical, deep, rectangular and sub-wavelength aperture grooves per period. Within the period, two consecutive cavities may be separated either by a long distance L or a shorter one S, their order of appearance being such that no additional periodicity arises. As we increase the number of grooves within a period, we create more and more complex periodic gratings. The period tends to infinity and the arrangement of the grooves within a period tends, in a selfsimilar manner, to an aperiodic sequence[18

18. P. Quémerais, “Model of growth for long-range chemically ordered structures: application to quasicrystals,” J. Phys. I France 4, 1669 (1994). [CrossRef]

, 19

19. F. Ducastelle and P. Quémerais, “Chemical self-organization during crystal growth,” Phys. Rev. Lett. 78, 102 (1997). [CrossRef]

]. By this approach we can study the gradual apparition and modifications of light localization effects, starting from a simple-period grating to an almost aperiodic one[15

15. A. Barbara, J. Le Perchec, S. Collin, C. Sauvan, J-L. Pelouard, T. López-Ríos, and P. Quémerais, “Generation and control of hot spots on commensurate metallic gratings,” Opt. Exp. 16, 19127 (2008). [CrossRef]

]. We have previously theoretically predicted that critically wavelength-dependent hot spots could be generated and controlled from far-field considering such arrangements. Strong and sharp electromagnetic resonances appear. They are due to new cavity resonances which do not exist for gratings with one cavity per period. These states are more or less radiative depending on the coupling between the cavities[20

20. J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Controlling strong electromagnetic fields at subwavelength scales,” Phys. Rev. Lett. 97, 036405 (2006). [CrossRef] [PubMed]

] and will consequently exhibit different enhancement factors and coupling with the incident exciting wave. Their localization may also be altered by coupling with horizontal surface plasmons. The far-field signatures of these modes are dips observed in the reflectivity measurements of p-polarized light. They have been recently evidenced experimentally for the first time, in the infra-red region[15

15. A. Barbara, J. Le Perchec, S. Collin, C. Sauvan, J-L. Pelouard, T. López-Ríos, and P. Quémerais, “Generation and control of hot spots on commensurate metallic gratings,” Opt. Exp. 16, 19127 (2008). [CrossRef]

] and in the microwave regime[21

21. M. Navarro-Cía, D. Skigin, M. Beruete, and M. Sorolla, “Experimental observation of phase resonances in metallic compound gratings with subwavelength slits in the millimiter wave regime,” Appl. Phys. Lett. 94, 091107 (2009). [CrossRef]

] for gratings with two or three slits per period.

Fig. 1. (a) SEM image of the gold grating and (b) experimental set-up as described in [22]. The sample is mounted in the center of the motorized stage. For each θ/2θ position of the sample/detector, specular reflectivity as a function of the incident wavelength is measured. (c) shows the spectrum measured at θ = 20.5° corresponding to a scan along the blue line added in the experimental dispersion maps displayed in (d). The dispersion diagram is obtained from the normalized intensity of the specular reflectivity plotted as a function of the in-plane wave vector k// and the wave number 1/λ. The excited modes appear as dips in the specular reflectivity dips and as dark lines in the dispersion diagram. The hatched peak is due to the infra-red absorption of residual water in the experiment set-up.

In this paper we report on the measurements and analyze the dispersion of the modes of a gold commensurate grating whose unit cell is composed of three grooves positioned at x 1=0, x 2=0.7 mm and x 3=1.4 µm respectively and whose period is D=2.6 µm. These parameters were chosen in order to have resonances in the infra-red region. Figure 1a displays the SEM image of the grating which was fabricated using electron-beam lithography and a double lift-off technique described in ref.[15

15. A. Barbara, J. Le Perchec, S. Collin, C. Sauvan, J-L. Pelouard, T. López-Ríos, and P. Quémerais, “Generation and control of hot spots on commensurate metallic gratings,” Opt. Exp. 16, 19127 (2008). [CrossRef]

] and references therein. The grooves obtained are trapezoidal with a height h=0.59 µm and a width w=0.36 µm at the bottom and w=0.62 µm at the top. In the calculations we took w =0.62 µm.

The dispersion curves of the excited modes are obtained by measuring the specular reflectivity of the grating illuminated by a p-polarized light in the spectral range 1.5–16µm and at various incident angles θ ranging from 2.5–74° with a step of 0:5°. For each angle, the specular reflectivity measured as a function of the wave number of the incident light is normalized by that of a flat gold surface performed in the same spectral range and at θ = 2:5°. As an example, the reflectivity spectrum measured at θ = 20.5° is shown in figure 1c. The experimental set-up, whose geometry is sketched in figure 1b, is based on a commercial Fourier transform spectrometer and a home-made optical system permitting a spectral resolution of 10 cm−1 and a convergence angle Δθ = ±0.5°[22

22. C. Billaudeau, S. Collin, C. Sauvan, N. Bardou, F. Pardo, and J-L Pelouard, “Angle-resolved transmission measurements through anisotropic 2D plasmonic crystals,” Opt. Lett. 33, 165 (2008). [CrossRef] [PubMed]

]. The sample and the detector are mounted on two co-axial motorized stages permitting θ/2θ reflectivity measurements. The resulting dispersion diagram (figure 1d) is a gray scale intensity map of the normalized specular reflectivity plotted as a function of the normalized in-plane wave-vector k // = (2Dsinθ)/λ in abscissa and the wave-number 1/λ in ordinate.

2.2. Experimental results

As can be observed in figure 1c, the grating presents three cavity modes (CM) in the 0.4–0.55 µm−1 spectral range. In the dispersion diagram (Fig.1d) they appear as dark lines, the two darkest ones corresponding to the two narrow resonances. Two additional straight-line dispersions are also visible in the dispersion diagram. They correspond to the branches n = +1 and n = −2 of the well-known horizontal surface plasmon (SP) propagating along a metal/air interface of a periodic system as 1λ=±nD(ε(ε+1)±sinθ) . The branch n = −1, that should start at 1/λ=0.385 µm−1 at normal incidence, is not clearly observed since it forms a coupled mode with the broad cavity mode observed at 0.45 µm−1. Lastly, water condensation on the detector window gives rise to the non-dispersive water absorption bands hatched in the figure to distinguish them from the resonances of the grating.

The three cavity resonances result from the non-periodic arrangement of the cavities within the period of the grating which modifies their near-field coupling. This coupling splits the cavity resonance, of an individual cavity or of a simple-period grating[23

23. T. López-Ríos, D. Mendoza, FJ. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665 (1998). [CrossRef]

], into three resonances[15

15. A. Barbara, J. Le Perchec, S. Collin, C. Sauvan, J-L. Pelouard, T. López-Ríos, and P. Quémerais, “Generation and control of hot spots on commensurate metallic gratings,” Opt. Exp. 16, 19127 (2008). [CrossRef]

]. The striking point of these resonances is that two of them are spectrally very narrow i.e they are only weakly radiative and they have a large quality factor Q. The third resonance has a broad bandwidth. It is more radiative and has consequently a smaller Q factor and an enhancement factor. At first sight one can notice that the two narrow cavity modes exhibit a very different behaviour: the lower wave number branch vanishes at normal incidence and disperses very strongly when the incidence angle increases. Conversely the higher wave-number mode disperses very weakly, as it is expected for a cavity mode, and is strongly excited at normal incidence. However, its excitation strength unexpectedly diminishes down to extinction in the range k //=0.75 to 0.94 which corresponds to incident angles θ =29 to 37°. These features show that this commensurate grating with only three cavities per period already presents sophisticated localization effects that need to be thoroughly investigated.

3. Results analysis

3.1. Theoretical method

Hz(I)(x,y)=eik(γ0xβ0y)+Σm=m=+Rmeik(γmx+βmy),
(1)

with k = 2π/λ being the wave vector of the incident plane wave. The terms γm = sinθ+/D, β2m= 1 − γ 2 m and Rm are the normalized wave vectors and the amplitude of the mth order of reflection respectively.

In the grating, the field is the sum of the field inside each cavity. The expression of the field in the pth cavity is obtained by considering the vertical walls as perfectly conducting and the impedance condition at the bottom of the cavity. It writes as:

Hz,p(II)(x,y)=Σ=0+A,pcos[πw(xxp+w2)](eiμ(y+2h)+reiμy).
(2)

where Ap and μ=k1(λ2w)2 are the amplitude and the vertical component of the wave vector of the th mode in the pth cavity respectively. xp is the coordinate of the center of the pth cavity. The general expression of the series of xp of complex commensurate gratings can be found in ref.[15

15. A. Barbara, J. Le Perchec, S. Collin, C. Sauvan, J-L. Pelouard, T. López-Ríos, and P. Quémerais, “Generation and control of hot spots on commensurate metallic gratings,” Opt. Exp. 16, 19127 (2008). [CrossRef]

, 18

18. P. Quémerais, “Model of growth for long-range chemically ordered structures: application to quasicrystals,” J. Phys. I France 4, 1669 (1994). [CrossRef]

, 19

19. F. Ducastelle and P. Quémerais, “Chemical self-organization during crystal growth,” Phys. Rev. Lett. 78, 102 (1997). [CrossRef]

]. The coefficients r = (µ/k+η)/(µ/kη) are the reflection coefficients of the th mode at the bottom of the cavity, where η = 1/√ε and ε is the relative dielectric constant of the metal.

3.2. Eigenmodes of the grating

The determination of the eigenmodes of the system, i.e the resonances supported by the grating in the absence of excitation, permits to gain an insight in the origin and the nature of the three resonances observed. An eigenmode exists each time det(M)=0 (or real part: (det(M)=0) for a complex matrix). This condition is numerically found by diagonalizing M and finding the frequency and incident angle (related to the value of k //) for which at least one eigenvalue is null.

In our case of a grating with sub-wavelength aperture grooves, we know that the resonances are essentially contented in the fundamental mode of the cavities since it is the only propagative one[24

24. A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of subwavelength rectangular metallic gratings,”Eur. Phys. J. D. 23, 143–154 (2003). [CrossRef]

]. Considering only this first mode ( = 0 in eq. 2) reduces M to a (3×3) matrix presenting three eigenvalues ei=1,2,3 associated with three eigenvectors Ui=1,2,3. Since the eigenvalues are complex, the system presents an eigenmode each time that the real part of ei (noted (ei)) is null, the remaining imaginary part, ℑ(ei), being linked to the width of the resonance. The frequency for which the annulation occurs is the resonance frequency. The eigenvectors respectively associated with the eigenvalues at the resonance frequency also permit to characterize the eigenmode since they have three complex components proportional to the amplitude (module) and to the phase (argument) of the electromagnetic field inside the three cavities respectively. A field localization distribution can thus be associated with each eigenmode. In the present case, the annulation of the eigenvalues of M in the 0.4–0.55 µm−1 does highlight the three cavity resonances observed and on which we will focus our attention. According to the experiments, the numerical analysis shows that out of the three resonances two of them have thin widths, ℑ(e 1) ≃ −0.01 and ℑ(e 2) ≃ −0.03, and one is large ℑ(e 3) ≃ −4.5. These modes are associated with the eigenvectors U1≈ (1,0,−1), U2≈ (1,−2.3,1) and U3≈ (1,1,1) respectively. Let us note that these vectors do not form an orthogonal basis.

The grating considered in this paper can be seen as a three-groove system periodically reproduced. Its optical response is thus quite close to that of a unique three-groove system. Consequently we may deduce that the resonances we observe in the case of the grating are the following: the first mode, associated with U1, is an out-of phase resonance of the two external slits, which localize the same amount of electromagnetic energy, leading to the extinction of the field inside the central cavity. This anti-symmetrical mode will be denoted (→ 0 ←). The second mode, associated with U2 is an in-phase resonance of the two external slits, both out-of phase with respect to the central one. The latter localizes an electromagnetic energy four times larger than the one in the external slits (twice larger in amplitude). Due to the symmetrical coupling of the external cavities this mode will be called a pseudo-symmetric mode and denoted (→←→). Lastly, the third mode, associated with U3 and denoted (→→→) corresponds to the in-phase resonance of the three cavities which all localize the same electromagnetic energy. The latter is a purely symmetric mode. It is equivalent to the usual Fabry-Perot-like resonance of the simple-period grating for which all the cavities also resonate in-phase. The phase-difference obtained between the cavities are in full agreement with the phase-resonance description of similar modes observed in compound gratings[25

25. D. Skigin and R. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. lett. 95, 217402 (2005). [CrossRef] [PubMed]

]. The widths of the resonances of the gratings are essentially driven by the radiation losses which depend on the different phases of the resonating grooves. The broad mode U3 has a large radiative damping and will consequently weakly localize the electromagnetic energy. By contrast, the two thin modes U1 and U2 have less radiation losses and will present strong enhancement factors of the field intensities.

Fig. 2. Experimental (a) and calculated (b) dispersion diagram of the two thin cavity modes of the grating. The dashed line indicates the theoretical dispersion of the plasmon branch n = −1. (c) to (f): experimental (black) and calculated (gray) spectra measured at an incidence angle θ = 2.5° showing the weak excitation of the anti-symmetrical mode (→ 0 ←) (c), θ = 36° where the pseudo-symmetrical mode (→←→) is extinguished (d), θ = 18° where the pseudo-symmetrical mode should cross the horizontal surface branch (e) and θ = 70° where the anti-symmetrical mode couples with the surface plasmon branch (f). The hatched dip is due to water pollution in the experiment.
Fig. 3. (a) Projection as a function of the incidence angle of the eigenvectors U1 (squares) and U2 (dots) on the excitation vector V. (b–e) Intensity maps of the magnetic near-field intensity for the anti-symmetrical mode calculated at θ = 36°, 1/λ = 0.215µm−1 (b) and at θ =0°, 1/λ = 0.255µm−1 (c) and for the pseudo-symmetrical mode calculated at θ = 36°,1/λ = 0.296µm−1 (d) and θ = 0°,1/λ = 0.283µm−1 (e). The color bar gives the scale of the normalized magnetic field intensities.

3.3. Excited modes: dispersion, extinction and coupling with the horizontal SP

Let us now focus on the excitation of the two thin cavity modes. The description of the broad mode (→→→) is similar to that of the usual cavity resonance of a simple period grating and can be found in the literature[4

4. A. Wirgin and A. A. Maradudin, “Resonant response of a bare metallic grating to s-polarized light,” Prog. Surf. Sci. 22, 1 (1986). [CrossRef]

, 23

23. T. López-Ríos, D. Mendoza, FJ. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665 (1998). [CrossRef]

]. Figure 2 shows a detail of the experimental dispersion diagram and that of the calculated one, together with spectra obtained at several incidence angles. Just as for the experimental diagram, the calculated one is a gray-scale map of the specular reflectivity spectra obtained by calculating the coefficients R 0 R * 0 as a function of the wave-number for different angles. The gold dielectric constants were taken from ref.[26

26. E. D. Palik, Handbook of optical constants of solids, Academic Press.

] and eight modes inside the cavities were considered (=8 in eq.2). Figures 2a and 2b show that the calculation reproduces in a very convincing manner all the features of the experiments: the positions, the dispersion, the relative intensities and the regions of extinction of the two resonances. A more quantitative comparison also gives a very good agreement between experiments and calculations as can be seen on the spectra presented on figure 2c–2f. Beyond this quantitative comparison, these four spectra were chosen because they contain specific points of the dispersion diagram (points A to D on figure 2a). They were respectively measured and calculated at the following incidence angles: θ= 2.5° showing the quasi-extinction of the anti-symmetrical mode (point A); θ= 36° showing the extinction of the pseudo-symmetrical mode (point B); θ=70° showing the antisymmetrical coupled with the n = −1 SP branch (point D) and θ=18° showing the crossing point between the n = −1 SP branch and the pseudo-symmetrical mode (point C).

Fig. 4. Intensity maps of the magnetic field of the anti-symmetrical mode when excited at θ = 10°, 1/λ = 0.249µm−1 (a) and at θ = 66°, 1/λ = 0.188µm−1 (c). (b) shows the magnetic field intensity along the solid black line in (a) and the dotted line in (b). At θ = 10° the mode (→0←) is strongly confined inside the cavities. Its expansion outside the cavities increases at larger incident angles due to its coupling with the horizontal surface plasmon. (d) Intensity maps of the magnetic field of the pseudo-symmetrical mode (→←→) when excited at θ = 18°, 1/λ = 0:29µm−1.

All these results are supported by the intensity maps of the magnetic field in the near-field of the grating shown in figure 3b–e. At the angles of efficient coupling of the modes with the external field, i.e at θ=36° for the anti-symmetrical mode (→ 0 ←) and at normal incidence for the pseudo-symmetrical mode (→←→), the maps present strong enhancement factors indicating the large amount of electromagnetic energy localized inside the cavities. This localization is cavity-selective according to the distributions imposed by the symmetry of the modes: the anti-symmetrical mode associated with an eigenvector of the form (1,0–1) localizes a large amount of electromagnetic intensity in the two external cavities while the central one is almost extinguished (fig. 3b) and the pseudo-symmetrical mode whose eigenvector takes the form (1,≈ −2,1) localizes an electromagnetic field intensity in the central cavity about four times larger than the two external cavities (Fig. 3d).

When considering the maps calculated at the respective extinction angles of the modes, i.e at θ=0° for the anti-symmetrical mode (→ 0 ←) and at θ=36° for the pseudo-symmetrical mode (→←→), the maps hardly show any enhancement factor of the fields. Only a weak near-field subsists inside the cavities and its distribution is interestingly not that of the resonant mode but that of the non-resonant mode. The distribution of the pseudo-symmetrical mode (resp. anti-symmetrical mode) appears at the resonant frequency and angle of the extinction of the anti-symmetrical mode (resp. pseudo-symmetrical) as shown in figure 3c (resp. 3e). The resonant mode is thus fully extinguished and the non-resonant one is very weakly excited due to the spectral expansion of the modes which permits a slight overlap.

Let us now comment on the dispersion of the modes and their coupling with the horizontal surface plasmons. A cavity mode is expected to very weakly disperse since it is a localized mode whose resonance frequency is mainly driven by the height of the cavity[23

23. T. López-Ríos, D. Mendoza, FJ. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665 (1998). [CrossRef]

]. This is what we observe for the pseudo-symmetrical mode. However, the anti-symmetrical mode disperses very strongly. To understand this point we recall that the grating has a period D = 2.6µm and should consequently support an horizontal surface plasmon in this spectral range, at a position highlighted in figure 2 by the added gray dotted line. In simple-period gratings, when a SP branch crosses a cavity mode branch, an energy splitting occurs: two hybrid modes separated by an energy gap centered on the crossing point appear[24

24. A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of subwavelength rectangular metallic gratings,”Eur. Phys. J. D. 23, 143–154 (2003). [CrossRef]

, 27

27. S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63, 033107 (2001). [CrossRef]

]. Here each mode has a very different behaviour. On the one hand, the pseudo-symmetrical one has no coupling with the surface plasmon branch and takes over the surface plasmon which completely vanishes. This can be seen on the map displayed in figure 4d which shows that the near-field distribution is of the type of the (→←→) resonance since the enhancement of the fields is mainly localized in the central groove and since no enhancement is observed at the horizontal interface. On the other hand the anti-symmetrical mode (→ 0 ←)) couples with the surface plasmon and forms a unique mode. Consequently the cavity mode disperses in a manner similar to that of the surface plasmon and the mode presents a transition from a very localized state, when excited at small incidence angles, to a partly delocalized state, when excited at higher incidence angles. These features are described in figure 4: the map of the magnetic field intensity calculated considering an incidence angle θ = 10° and at the frequency resonance of the anti-symmetrical mode (fig.4a) clearly evidences the signature of the (→ 0 ←)) mode since a strong enhancement of the fields occurs in the two external cavities while the central one is almost extinguished. The coupling of this mode with the surface plasmon of the horizontal interface appears in figure 4c, calculated for an incidence angle θ = 66°. In that case, the field is localized inside the external cavities (1 and and 3) but also above the cavities. Lastly, figure 4b shows the magnetic field intensity along the vertical direction in the middle of the third cavity. This highlights the variation of localization of the mode through its extension in the free space above the grating. A simple variation of the incidence angle can thus drive a transition from a localized to a partially delocalized state.

4. Conclusion

Appendix

  • Hz(I)(x,0+)y+ikηHz(I)(x,0+)=Hz(II)(x,0)y+ikηHz(II)(x,0)
  • H (I) z (x,0+) = H (II) z (x,0-).

The first equality is valid over the whole interface and is projected onto the set of basis vectors e(ikγmx) which leads to the following expression:

Rm=β0ηβ0+ηδm,0+wDΣp=1QΣ=0+ApSmeikγmxp(e2iμh1)(μk+ηβm+η),
(3)

with Sm±=1ww2+w2e±ikγmxcos[πw(x+w2)]dx.

The second boundary condition is valid at the mouth of each groove i.e for x ∊ [xp−w/2,xp+w/2] and p = 1,2..Q for Q cavities per period. It is projected onto the set of basis vectors cos[πw(xxp+w2)] to obtain:

A,p=(21+δ,0)1e2iμh+rΣm=+Sm+eikγmxp(δm,0+Rm).
(4)

Replacing the expression of the Rm (equation (3)) in equation (4) leads to the matrix system of the form: Σp=1QΣ=0+M,,p,pA,p=V,p

with M,,p,p=(e2iμh+r)δ,δp,pF,Σm=+Sm+Smβm+ηeikγm(xpxp)

and F,=(21+δ,0)wD(e2iμh1)(μk+η).

and with

V,p=(21+δ,0)2β0β0+ηS0+eikγ0xp.
(5)

The terms of the matrix M and that of V may be numerically calculated by truncation of the sums. In the unimodal approach we take ℓ = ℓ′ = 0 while typically mmax = 200 reflecting orders are considered. This permits to determine the values of the coefficients Aℓ,p. From the latter, the coefficients Rm may be obtained and the electromagnetic field can be reconstructed in the whole space.

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F. Ducastelle and P. Quémerais, “Chemical self-organization during crystal growth,” Phys. Rev. Lett. 78, 102 (1997). [CrossRef]

20.

J. Le Perchec, P. Quémerais, A. Barbara, and T. López-Ríos, “Controlling strong electromagnetic fields at subwavelength scales,” Phys. Rev. Lett. 97, 036405 (2006). [CrossRef] [PubMed]

21.

M. Navarro-Cía, D. Skigin, M. Beruete, and M. Sorolla, “Experimental observation of phase resonances in metallic compound gratings with subwavelength slits in the millimiter wave regime,” Appl. Phys. Lett. 94, 091107 (2009). [CrossRef]

22.

C. Billaudeau, S. Collin, C. Sauvan, N. Bardou, F. Pardo, and J-L Pelouard, “Angle-resolved transmission measurements through anisotropic 2D plasmonic crystals,” Opt. Lett. 33, 165 (2008). [CrossRef] [PubMed]

23.

T. López-Ríos, D. Mendoza, FJ. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665 (1998). [CrossRef]

24.

A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of subwavelength rectangular metallic gratings,”Eur. Phys. J. D. 23, 143–154 (2003). [CrossRef]

25.

D. Skigin and R. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. lett. 95, 217402 (2005). [CrossRef] [PubMed]

26.

E. D. Palik, Handbook of optical constants of solids, Academic Press.

27.

S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63, 033107 (2001). [CrossRef]

OCIS Codes
(050.2230) Diffraction and gratings : Fabry-Perot
(240.6680) Optics at surfaces : Surface plasmons
(240.6695) Optics at surfaces : Surface-enhanced Raman scattering

ToC Category:
Optics at Surfaces

History
Original Manuscript: May 25, 2010
Revised Manuscript: June 22, 2010
Manuscript Accepted: June 23, 2010
Published: June 28, 2010

Citation
Aude Barbara, Stéphane Collin, Christophe Sauvan, Jérôme Le Perchec, Camille Maxime, Jean-Luc Pelouard, and Pascal Quémerais, "Plasmon dispersion diagram and localization effects in a three-cavity commensurate grating," Opt. Express 18, 14913-14925 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14913


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References

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  15. A. Barbara, J. Le Perchec, S. Collin, C. Sauvan, J-L. Pelouard, T. López-Ríos, and P. Quémerais, “Generation and control of hot spots on commensurate metallic gratings,” Opt. Express 16, 19127 (2008). [CrossRef]
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  19. F. Ducastelle and P. Quémerais, “Chemical self-organization during crystal growth,” Phys. Rev. Lett. 78, 102 (1997). [CrossRef]
  20. J. Le Perchec, P. Quémerais, A. Barbara and T. López-Ríos, “Controlling strong electromagnetic fields at subwavelength scales,” Phys. Rev. Lett. 97, 036405 (2006). [CrossRef] [PubMed]
  21. M. Navarro-Cía, D. Skigin, M. Beruete, and M. Sorolla, “Experimental observation of phase resonances in metallic compound gratings with subwavelength slits in the millimiter wave regime,” Appl. Phys. Lett. 94, 091107 (2009). [CrossRef]
  22. C. Billaudeau, S. Collin, C. Sauvan, N. Bardou, F. Pardo, and J-L Pelouard, “Angle-resolved transmission measurements through anisotropic 2D plasmonic crystals,” Opt. Lett. 33, 165 (2008). [CrossRef] [PubMed]
  23. T. López-Ríos, D. Mendoza, F. J. García-Vidal, J. Sánchez-Dehesa, and B. Pannetier, “Surface shape resonances in lamellar metallic gratings,” Phys. Rev. Lett. 81, 665 (1998). [CrossRef]
  24. A. Barbara, P. Quémerais, E. Bustarret, T. López-Ríos, and T. Fournier, “Electromagnetic resonances of subwavelength rectangular metallic gratings,” Eur. Phys. J. D. 23, 143-154 (2003). [CrossRef]
  25. D. Skigin and R. Depine, “Transmission resonances of metallic compound gratings with subwavelength slits,” Phys. Rev. lett. 95, 217402 (2005) and references therein. [CrossRef] [PubMed]
  26. E. D. Palik, Handbook of optical constants of solids, Academic Press.
  27. S. Collin, F. Pardo, R. Teissier, and J.-L. Pelouard, “Strong discontinuities in the complex photonic band structure of transmission metallic gratings,” Phys. Rev. B 63, 033107 (2001). [CrossRef]

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