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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 1 — Jan. 10, 2005
  • pp: 70–76
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Resonant transmission through a metallic film due to coupled modes

F. Marquier, J.-J. Greffet, S. Collin, F. Pardo, and J.L. Pelouard  »View Author Affiliations


Optics Express, Vol. 13, Issue 1, pp. 70-76 (2005)
http://dx.doi.org/10.1364/OPEX.13.000070


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Abstract

Enhanced transmission and absorption by a silver film with a periodic array of slits has been studied numerically. We find that transmission and absorption peaks coincide and can be attributed to resonances of the structure. We show that these modes can be viewed as a coupling between cavity modes and surface plasmon polaritons. A quantitative analysis shows that the coupled mode can have a cavity mode character or a surface plasmon character depending on the distance to the crossing point of their dispersion relation. Finally, we provide a simple model for the peak transmission value by introducing the concept of radiative yield.

© 2005 Optical Society of America

Since the publication six years ago of a paper [1

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

] demonstrating resonant transmission by a film with a periodic array of holes, over 100 papers have been devoted to analyse the mechanisms of this phenomena both experimentally and theoretically. Whereas the enhanced transmission due to a perfectly conducting structure has been known for some time and used to produce grid polarizers in the infrared or frequency selective surfaces in the microwave domain, the physics involved in the optical regime for noble metals is less simple. The presence of surface-plasmon polaritons which are excitations of a flat bare interface and the presence of losses introduce significant differences. Many papers have put forward the role of surface plasmons [2

2. J. A. Porto, F. J. Garcia Vidal, and J. B. Pendry, “Transmission Resonances on Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. 83, 2845 (1999). [CrossRef]

, 3

3. L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. 86, 1114 (2001). [CrossRef] [PubMed]

, 4

4. E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000). [CrossRef]

, 5

5. F. I. Baida and D. van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt.Commun. 209, 17 (2002). [CrossRef]

, 6

6. A. Krishnan, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolff, J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1 (2000). [CrossRef]

, 7

7. N. Bonod, S. Enoch, L. Li, E. Popov, and M. Neviere, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express 11, 482 (2003) [CrossRef] [PubMed]

]. More recently, it has been argued that excitation of surface-plasmon polaritons entails nearly zero transmission [8

8. Q. Cao and P. Lalanne, “Negative Role of Surface Plasmons in the Transmission of Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. 88, 057403 (2002) [CrossRef] [PubMed]

]. A very recent paper has shown that surface plasmons yield a peak of absorption that coexists with a peak of transmission in periodic arrays of subwavelength holes in a metal film [9

9. W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface Plasmon Polaritons and Their Role in the Enhanced Transmission of Light through Periodic Arrays of Subwavelength Holes in a Metal Film,” Phys. Rev. Lett. 92, 107401 (2004). [CrossRef] [PubMed]

]. Many authors have studied the enhanced transmission by a periodic array of slits. For p-polarization, surface waves can be excited but there is no cut-off frequency for the modes guided in the slits. It has been argued that the key mechanism is the excitation of resonances in the slits [10

10. S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A : Pure Appl. Opt. 4, S154 (2002). [CrossRef]

, 11

11. P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68, 125404 (2003). [CrossRef]

], a mechanism that cannot be invoked in the case of holes due to the presence of a cut-off frequency. It has also been argued that a resonant transmission is a manifestation of the excitation of a resonance of a diffracting structure independently of the presence of surface-plasmon polaritons [8

8. Q. Cao and P. Lalanne, “Negative Role of Surface Plasmons in the Transmission of Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. 88, 057403 (2002) [CrossRef] [PubMed]

, 12

12. M. M. J. Treacy, “Dynamical diffraction in metallic optical gratings,” Appl. Phys. Lett. 75, 606 (1999). [CrossRef]

]. This extremely rapid review of the litterature indicates that the resonant transmission is explained in terms of excitation of modes in the slits for the case of slits, in terms of excitation of surface plasmons for both slits and holes. The relative role of these excitations is not clearly established, the role of losses seems to have been clarified. Finally, we note that a problem is still pending. No simple explanation has been put forward to explain the large transmission. In this paper, we focus on the problem of propagation through a silver film with a periodic array of parallel slits. We will argue that the enhanced transmission is associated with the excitation of modes of the whole structure. We will show that these modes can be described as coupled modes mixing both cavity modes and surface modes. We will finally discuss the origin of the large value of the transmission.

In order to analyse the resonant transmission, we have extensively studied the transmission and the absorption of a free standing silver film in air with a periodic array of slits, period of 500 nm, thickness of 400 nm, slits of 50 nm-width for wavelengths in the range [0.5,2] µm. We use a Drude model for the dielectric constant ε(ω)=1-ωp2 /(ω 2+iγω) with ωp =1.29.1016 rad.s-1 and γ=1.14.1014 rad.s-1. The numerical results obtained with the method described in ref. [13

13. N. Chateau and J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321 (1994). [CrossRef]

] are plotted in Figure 1 in a (ω,k) plane.

Fig. 1. Transmission efficiency (a) and Absorption efficiency (b) in the plane (ω, k). Period of the grating : Λ=500 nm, Filling factor : F=0.9 and thickness h=400 nm. The brighter the region, the larger the transmission or absorption.

Large transmission peaks are observed with a transmission that is above 0.6–0.8. By looking at Fig. 1(b), we observe that the transmission peaks coincide with the absorption peaks. This result agrees with recent experimental data reported in ref. [9

9. W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface Plasmon Polaritons and Their Role in the Enhanced Transmission of Light through Periodic Arrays of Subwavelength Holes in a Metal Film,” Phys. Rev. Lett. 92, 107401 (2004). [CrossRef] [PubMed]

]. It indicates that both the enhanced transmission and the enhanced absorption can be attributed to the resonant excitation of a mode of the structure. It also appears that the transmission is limited by losses. We will come back to this point later.

We now study the resonances. It is clear from Fig. 1 (a,b) that the loci of maxima yields the dispersion relation of modes of the structure. The peaks follow closely the dispersion relation of the surface-plasmon polariton of a flat metallic surface in many parts of the (ω,k) space. This is particularly true close to the light line ω=ck and the corresponding folded branches. Yet, we also observe horizontal lines that do not match the usual dispersion relation for surface-plasmon polaritons. A closer look reveals that the position of the resonant frequencies of these horizontal modes are multiple of the same fundamental frequency. This suggests that organ-pipe type modes, hereafter called cavity modes, are excited in the slits. The frequency of these modes should be given by the condition k2d+2Φ=n2π where d stands for the thickness of the film, k for the wavector of the mode, Φ is the phase change upon the reflection at the end of the cavity and n is an integer. By assuming that the mode has a constant phase velocity v and neglecting the phase change at reflection [14

14. F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002) [CrossRef]

], we find ω=nπv/d. In order to further confirm the existence of such a cavity mode, we have studied the electromagnetic field in the structure. Fig. 2(a) shows the field at point B (see Fig. 1). It is clearly seen that there are cavity modes with two nodes excited in the cavity. It is also seen that the field is enhanced along the interfaces. The existence of cavity modes and their contribution to modes with flat dispersion relation has already been discussed in different contexts previously [15

15. T. Lopez-Rios, D. Mendoza, F. J. Garcia-Vidal, J. Sanchez-Dehesa, and B. Pannetier, “Surface Shape Resonances in Lamellar Metallic Gratings,” Phys. Rev. Lett. 81, 665 (1998). [CrossRef]

, 16

16. W. C. Tan, T. W. Preist, J. R. Sambles, and N. P. Wanstall, “Flat surface-plasmon-polariton bands and resonant optical absorption on short-pitch metal gratings,” Phys. Rev. B 59, 12661 (1999). [CrossRef]

, 17

17. F. J. Garcia-Vidal, J. Sanchez-Dehesa, A. Dechelette, E. Bustarret, T. Lopez-Rios, T. Fournier, and B. Pannetier, “Localized Surface Plasmons in Lamellar Metallic Gratings,” J. Lightwave Technol. 17, 2191 (1999) [CrossRef]

].

Fig. 2. Near-field intensity of the electric field in logarithmic scale at point B (a), and at point C (b).

We now examine the near field structure of the field for the point C close to the light line (see Fig. 2(b)). It is seen that a cavity mode is still excited in the cavity, but the energy is lower than at point B. Conversely, surface modes are excited on both sides of the film. In order to further prove that surface modes of the surface polariton type are excited along the interface, we have studied the amplitude of the order -1 of the grating which is an evanescent mode. We have found that the amplitude of this mode as a function of frequency for a fixed wavevector has resonances (not shown) for the same frequencies than the transmission. This indicates that the surface wave is indeed excited when the transmission is enhanced.

Having established the character of cavity modes or surface modes of the structure modes depending on the position in the (ω,k) plane, we now propose to analyse the data using the concept of coupled mode. We plot in Fig. 3 a schematic view of the dispersion relation of a surface-plasmon polariton in the reduced zone scheme.

We have also added the dispersion relation of the cavity modes which are horizontal as we discussed above. Since the dispersion curves have crossing points, the modes must couple producing new modes with a higher and a lower frequency. This allow to draw schematically a dispersion relation that has the features observed in Fig. 1. We conclude that the enhanced transmission of a periodic array of slits can be understood by introducing the concept of coupled modes. We argue that away from the crossing point, the modes have either a well-defined cavity-mode or surface-mode character. Conversely, close to the crossing point, the modes are in the strong coupling regime and cannot be called either surface-modes nor cavity-modes. A more detailed analysis should take into account the fact that surface plasmons polaritons of both interfaces are coupled through the cavity producing symmetric and antisymmetric modes with slightly different frequencies.

In order to quantitatively check this concept, we introduce a measure of the surface-mode character. Let us denote by Us the energy of the surface mode component whereas Uc is the energy of the cavity-mode component of a given coupled mode. Uc is obtained by integrating the electromagnetic energy in the cavity (i.e. in the slit), Us is obtained by integrating the electromagnetic energy of all the evanescent waves, on both interfaces. We now define the surface plasmon character of a mode by a normalized quantity that we will denote as surface-cavity ratio SC=Us /(Us +Uc ).

Fig. 3. Dispersion relation of surface plasmons (long-dashed lines) and cavity modes (dotted lines). The plain lines shows the coupling between both modes.
Fig. 4. Branch B-C of the dispersion relation (a) and Surface-cavity ratio SC for this branch (b).

We have plotted in Fig. 4 the result of our computations. It is seen that we can clearly define a region where the mode has a well-defined cavity type and there are regions where the mode has a well-defined surface-wave type.

So far, we have argued that the resonant transmission can be attributed to the excitation of modes of the structure. We have proposed to analyse these modes in terms of the coupling between the modes of the bare surface and the modes of the cavity. Yet, we have only studied the transmission and absorption resonances. We now proceed to a direct analysis of the modes themselves. To this aim, we have searched the complex frequencies that are poles of the scattering matrix following the approach outlined in ref. [10

10. S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A : Pure Appl. Opt. 4, S154 (2002). [CrossRef]

]. The outcome is displayed in Fig. 5 (a).

Fig. 5. Exact dispersion relation of the structure (a), and associated quality factor (b).

We see that the exact dispersion relation is in close agreement with the loci of the enhanced transmission. An additional interesting information can be gained from this study. Since we look for complex eigenfrequencies of the structure, we obtain the decay rate γ=Im(ω) of each mode. Using this information, we have plotted the quality factor Re(ω)/2γ (ω) in Fig. 5(b). From this result, we learn that cavity modes usually have a low quality factor. Conversely, surface modes may have a large quality factor. By comparing Fig. 1 and Fig. 5(b), it is seen that the quality factor of the modes is the factor that governs the width of the transmission peaks, as discussed by Collin et al. in ref. [18

18. 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]

].

We complete our discussion of the enhanced transmission by analysing the height of the transmission peaks. The question still open is to understand the origin of the large transmission factor. We first use a fundamental result derived many years ago [19

19. E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607 (1986) [CrossRef]

] showing that a non-lossy resonant structure has a zero reflection and therefore a unity transmission if it is symetric with respect to the plane (y,z) and if there is a single propagating order. Owing to this property, which is a fundamental consequence of symmetry, unitarity and reciprocity, one expects a transmission peak of 1. This is a property often observed in the microwave or far infrared regime where losses are practically negligible [20

20. A. Janhsen and V. Hansen, “Multiple dielectric loaded perforated screens as frequency selective surfaces,” IEE proceedings-H 138, 1 (1991).

, 21

21. Orta, Tascone, and Zien, “Arrays of finite or infinite extent in multilayered media for use as passive frequency-selective surfaces,” IEE proceedings 135, 75 (1988).

, 22

22. Pous and Pozar, “Frequency selective surface using aperture-coupled microstrip patches,” Electron. Lett. 25, 1136 (1989) [CrossRef]

]. When using noble metals in the optical spectrum, losses cannot be neglected and are responsible for the finite value of the transmission factor as discussed in ref. [9

9. W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface Plasmon Polaritons and Their Role in the Enhanced Transmission of Light through Periodic Arrays of Subwavelength Holes in a Metal Film,” Phys. Rev. Lett. 92, 107401 (2004). [CrossRef] [PubMed]

]. It is thus useful to introduce the radiative decay rate γR and the non-radiative decay rate γNR of the modes. The quantity η=γR /(γR +γNR ), analog to the quantum yield of a molecule, is the probability that the mode releases its energy by emitting a photon. We will refer to it as radiative yield. A high transmission is expected if η is large. It turns out that the transmission peak value can be shown to be η2. Indeed, owing to the existence of a resonance, the transmission factor can be cast in the form :

t(ω)=f(ω)ωω0+iγ
(1)

where f(ω) is an unknown analytic function. A similar form was introduced in ref. [23

23. W. C. Tan, T. W. Preist, and R. J. Sambles, “Resonant tunneling of light through thin metal films via strongly localized surface plasmons,” Phys. Rev. B 62, 11134 (1986) [CrossRef]

]. In this paper, we make no assumption regarding f(ω). In the presence of ohmic losses, γ=γR +γNR . We know from ref. [19

19. E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607 (1986) [CrossRef]

] that without losses (γNR =0), |t(ω 0)|2=1 so that |f(ω 0)|=γR . It follows that in the presence of losses, the transmission factor takes the peak value [γR /(γR +γNR )]2. In figure 6, we plot the value of the peak transmission derived from a scattering calculation [13

13. N. Chateau and J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321 (1994). [CrossRef]

] and η2 derived from the complex frequencies of the mode of the structure [10

10. S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A : Pure Appl. Opt. 4, S154 (2002). [CrossRef]

]. It is seen that η2 follows quantitatively the transmission peak.

Fig. 6. Zero-order transmission efficiency and square of the quantum yield η2 along the branch B-C of the dispersion relation.

References and links

1.

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

2.

J. A. Porto, F. J. Garcia Vidal, and J. B. Pendry, “Transmission Resonances on Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. 83, 2845 (1999). [CrossRef]

3.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. 86, 1114 (2001). [CrossRef] [PubMed]

4.

E. Popov, M. Neviere, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000). [CrossRef]

5.

F. I. Baida and D. van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt.Commun. 209, 17 (2002). [CrossRef]

6.

A. Krishnan, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolff, J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1 (2000). [CrossRef]

7.

N. Bonod, S. Enoch, L. Li, E. Popov, and M. Neviere, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express 11, 482 (2003) [CrossRef] [PubMed]

8.

Q. Cao and P. Lalanne, “Negative Role of Surface Plasmons in the Transmission of Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. 88, 057403 (2002) [CrossRef] [PubMed]

9.

W. L. Barnes, W. A. Murray, J. Dintinger, E. Devaux, and T. W. Ebbesen, “Surface Plasmon Polaritons and Their Role in the Enhanced Transmission of Light through Periodic Arrays of Subwavelength Holes in a Metal Film,” Phys. Rev. Lett. 92, 107401 (2004). [CrossRef] [PubMed]

10.

S. Collin, F. Pardo, R. Teissier, and J. L. Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A : Pure Appl. Opt. 4, S154 (2002). [CrossRef]

11.

P. Lalanne, C. Sauvan, J. P. Hugonin, J. C. Rodier, and P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68, 125404 (2003). [CrossRef]

12.

M. M. J. Treacy, “Dynamical diffraction in metallic optical gratings,” Appl. Phys. Lett. 75, 606 (1999). [CrossRef]

13.

N. Chateau and J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321 (1994). [CrossRef]

14.

F. J. Garcia-Vidal and L. Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002) [CrossRef]

15.

T. Lopez-Rios, D. Mendoza, F. J. Garcia-Vidal, J. Sanchez-Dehesa, and B. Pannetier, “Surface Shape Resonances in Lamellar Metallic Gratings,” Phys. Rev. Lett. 81, 665 (1998). [CrossRef]

16.

W. C. Tan, T. W. Preist, J. R. Sambles, and N. P. Wanstall, “Flat surface-plasmon-polariton bands and resonant optical absorption on short-pitch metal gratings,” Phys. Rev. B 59, 12661 (1999). [CrossRef]

17.

F. J. Garcia-Vidal, J. Sanchez-Dehesa, A. Dechelette, E. Bustarret, T. Lopez-Rios, T. Fournier, and B. Pannetier, “Localized Surface Plasmons in Lamellar Metallic Gratings,” J. Lightwave Technol. 17, 2191 (1999) [CrossRef]

18.

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]

19.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607 (1986) [CrossRef]

20.

A. Janhsen and V. Hansen, “Multiple dielectric loaded perforated screens as frequency selective surfaces,” IEE proceedings-H 138, 1 (1991).

21.

Orta, Tascone, and Zien, “Arrays of finite or infinite extent in multilayered media for use as passive frequency-selective surfaces,” IEE proceedings 135, 75 (1988).

22.

Pous and Pozar, “Frequency selective surface using aperture-coupled microstrip patches,” Electron. Lett. 25, 1136 (1989) [CrossRef]

23.

W. C. Tan, T. W. Preist, and R. J. Sambles, “Resonant tunneling of light through thin metal films via strongly localized surface plasmons,” Phys. Rev. B 62, 11134 (1986) [CrossRef]

24.

K. Joulain, R. Carminati, J. P. Mulet, and J. J. Greffet, “Definition and measurement of the local density of electromagnetic states close to an interface,” Phys. Rev. B 68, 245405 (2004) [CrossRef]

25.

J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature (London) 416, 61(2002). [CrossRef]

26.

A. Shchegrov, K. Joulain, R. carminati, and J. J. Greffet, “Near-Field Spectral Effects due to Electromagnetic Surface Excitations,” Phys. Rev. Lett. 85, 1548 (2000). [CrossRef] [PubMed]

27.

R. Carminati and J. J. Greffet, “Near-Field Effects in Spatial Coherence of Thermal Sources,” Phys. Rev. Lett. 82, 1660 (1999) [CrossRef]

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(230.3990) Optical devices : Micro-optical devices
(240.6680) Optics at surfaces : Surface plasmons
(310.2790) Thin films : Guided waves

ToC Category:
Research Papers

History
Original Manuscript: November 18, 2004
Revised Manuscript: December 20, 2004
Manuscript Accepted: December 20, 2004
Published: January 10, 2005

Citation
F. Marquier, J.-J. Greffet, S. Collin, F. Pardo, and J.L. Pelouard, "Resonant transmission through a metallic film due to coupled modes," Opt. Express 13, 70-76 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-1-70


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References

  1. T. W.  Ebbesen, H. J.  Lezec, H. F.  Ghaemi, T.  Thio, P. A.  Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667 (1998). [CrossRef]
  2. J. A.  Porto, F. J.  Garcia Vidal, J. B.  Pendry, “Transmission Resonances on Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. 83, 2845 (1999). [CrossRef]
  3. L.  Martin-Moreno, F. J.  Garcia-Vidal, H. J.  Lezec, K. M.  Pellerin, T.  Thio, J. B.  Pendry, T. W.  Ebbesen, “Theory of Extraordinary Optical Transmission through Subwavelength Hole Arrays,” Phys. Rev. Lett. 86, 1114 (2001). [CrossRef] [PubMed]
  4. E.  Popov, M.  Neviere, S.  Enoch, R.  Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000). [CrossRef]
  5. F. I.  Baida, D.  van Labeke, “Light transmission by subwavelength annular aperture arrays in metallic films,” Opt.Commun. 209, 17 (2002). [CrossRef]
  6. A.  Krishnan, T.  Thio, T. J.  Kim, H. J.  Lezec, T. W.  Ebbesen, P. A.  Wolff, J. B.  Pendry, L.  Martin-Moreno, F. J.  Garcia-Vidal, “Evanescently coupled resonance in surface plasmon enhanced transmission,” Opt. Commun. 200, 1 (2000). [CrossRef]
  7. N.  Bonod, S.  Enoch, L.  Li, E.  Popov, M.  Neviere, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express 11, 482 (2003) [CrossRef] [PubMed]
  8. Q.  Cao, P.  Lalanne, “Negative Role of Surface Plasmons in the Transmission of Metallic Gratings with Very Narrow Slits,” Phys. Rev. Lett. 88, 057403 (2002) [CrossRef] [PubMed]
  9. W. L.  Barnes, W. A.  Murray, J.  Dintinger, E.  Devaux, T. W.  Ebbesen, “Surface Plasmon Polaritons and Their Role in the Enhanced Transmission of Light through Periodic Arrays of Subwavelength Holes in a Metal Film,” Phys. Rev. Lett. 92, 107401 (2004). [CrossRef] [PubMed]
  10. S.  Collin, F.  Pardo, R.  Teissier, J. L.  Pelouard, “Horizontal and vertical surface resonances in transmission metallic gratings,” J. Opt. A : Pure Appl. Opt. 4, S154 (2002). [CrossRef]
  11. P.  Lalanne, C.  Sauvan, J. P.  Hugonin, J. C.  Rodier, P.  Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68, 125404 (2003). [CrossRef]
  12. M. M. J.  Treacy, “Dynamical diffraction in metallic optical gratings,” Appl. Phys. Lett. 75, 606 (1999). [CrossRef]
  13. N.  Chateau, J. P.  Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321 (1994). [CrossRef]
  14. F. J.  Garcia-Vidal, L.  Martin-Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B 66, 155412 (2002) [CrossRef]
  15. T.  Lopez-Rios, D.  Mendoza, F. J.  Garcia-Vidal, J.  Sanchez-Dehesa, B.  Pannetier, “Surface Shape Resonances in Lamellar Metallic Gratings,” Phys. Rev. Lett. 81, 665 (1998). [CrossRef]
  16. W. C.  Tan, T. W.  Preist, J. R.  Sambles, N. P.  Wanstall, “Flat surface-plasmon-polariton bands and resonant optical absorption on short-pitch metal gratings,” Phys. Rev. B 59, 12661 (1999). [CrossRef]
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