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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 25 — Dec. 5, 2011
  • pp: 25290–25297
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Surface impedance model for extraordinary transmission in 1D metallic and dielectric screens

V. Delgado and R. Marqués  »View Author Affiliations


Optics Express, Vol. 19, Issue 25, pp. 25290-25297 (2011)
http://dx.doi.org/10.1364/OE.19.025290


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Abstract

Extraordinary Optical Transmission of TM waves impinging at oblique incidence on metallic or high permittivity dielectric screens with a periodic distribution of 1D slits or any other kind of 1D defects is analyzed. Generalized waveguide theory altogether with the surface impedance concept are used for modeling such phenomena. A numerical analysis based on the mode matching technique proves to be an efficient tool for the characterization of these structures for any angle of incidence and slit or defect apertures.

© 2011 OSA

1. Introduction

The phenomenon of Extraordinary Optical Transmission (EOT) through opaque screens with a periodic array of subwavelength holes, first reported in [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, 667–669 (1998). [CrossRef]

], is a current topic of intensive research due to its potential applications in photonic circuits [2

2. E. Devaux, T. W. Ebbesen, J.-C. Weeber, and A. Dereux, “Launching and decoupling surface plasmons via micro-gratings,” Appl. Phys. Lett. 83, 4936 (2003). [CrossRef]

], optical sensing [3

3. A. A. Yanik, M. Huang, O. Kamohara, A. Artar, T. W. Geisbert, J. H. Connor, and H. Altug, “An optofluidic nanoplasmonic biosensor for direct detection of live viruses from biological media,” Nano Lett. 10, 4962–4969 (2010). [CrossRef]

] or fabrication of left-handed metamaterials. As nanofabrication techniques improve, efficient numerical models are desirable in order to provide fast characterization of the designs as well as physical insight into the phenomena. The physical background of the phenomena is an issue of controversy. Channeling of surface plasmons (SP) [4

4. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(15), 6779–6782 (1998). [CrossRef]

] through the holes was first proposed, but could not explain Extraordinary Transmission (ET) in metals at microwave frequencies [5

5. M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Martín-Moreno, J. Bravo-Abad, and F. J. García-Vidal, “Enhanced millimeter-wave transmission through subwavelength hole arrays,” Opt. Lett. 29(21), 2500–2502 (2004). [CrossRef] [PubMed]

] or in dielectric screens [6

6. M. Sarrazin and J. P. Vigneron, “Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes,” Phys. Rev. E 68, 016603 (2003). [CrossRef]

] where real part of permittivity is positive and SP do not exist. Subsequently, the surface plasmon concept was rescued to explain ET after considering that plasmon-like surface waves, or spoof plasmons, can be supported by structured metallic or dielectric surfaces [7

7. J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004). [CrossRef] [PubMed]

]. Recently, some of the authors presented an analytical theory of EOT through perfect conducting screens based on waveguide analysis [8

8. R. Marqués, F. Mesa, L. Jelinek, and F. Medina, “Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes,” Opt. Express 17(7), 5571–5579 (2009). [CrossRef] [PubMed]

]. In this paper the role of spoof plasmons in extraordinary transmission was also discussed (see e.g. Table 1 in [8

8. R. Marqués, F. Mesa, L. Jelinek, and F. Medina, “Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes,” Opt. Express 17(7), 5571–5579 (2009). [CrossRef] [PubMed]

]). This theory was later extended to screens made of realistic conductors such as metals at optical frequencies [9

9. V. Delgado, R. Marqués, and L. Jelinek, “Analytical theory of extraordinary optical transmission through realistic metallic screens,” Opt. Express 18(7), 6506–6515 (2010). [CrossRef] [PubMed]

] and 1D dielectric configurations [10

10. V. Delgado, R. Marqués, and L. Jelinek, “Extraordinary transmission through dielectric screens with 1D sub-wavelength metallic inclusions,” Opt. Express 19(14), 13612–13617 (2011). [CrossRef] [PubMed]

]. These generalizations were based on the surface impedance concept, which is widely used in classical electromagnetism for the analysis of skin effect in imperfect conductors [11

11. J. D. Jackson, Classical Electrodynamics, 3rd Ed. (John Wiley & Sons, Inc., 1998)

]. The model in [9

9. V. Delgado, R. Marqués, and L. Jelinek, “Analytical theory of extraordinary optical transmission through realistic metallic screens,” Opt. Express 18(7), 6506–6515 (2010). [CrossRef] [PubMed]

,10

10. V. Delgado, R. Marqués, and L. Jelinek, “Extraordinary transmission through dielectric screens with 1D sub-wavelength metallic inclusions,” Opt. Express 19(14), 13612–13617 (2011). [CrossRef] [PubMed]

] has some limitations as it is only valid in the case of normal incidence and for holes smaller than a quarter wavelength. In this letter, we present a unified analysis of extraordinary transmission at oblique incidence through periodic metallic or dielectric screens, loaded with transparent or opaque defects, including thin (sub-skin-depth) screens and wide (∼λ/2) slits or defects.

2. Theory

We will consider a metallic or high permittivity dielectric screen with a periodic distribution of slits. These configurations have been analyzed by many authors [12

12. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999). [CrossRef]

15

15. F. Medina, F. Mesa, and D. C. Skigin, “Extraordinary transmission through arrays of slits: a circuit theory model,” IEEE Trans. Microwave Theory Tech. 58(1), 105–115 (2010). [CrossRef]

] in order to simplify numerical problems and subsequently have taken on a live of its own. In this section such structure will be analyzed using an approximate mode-matching analysis which is simplified making use of the surface impedance concept.

Fig. 1 Front (a) and side (b) view of the screen with a periodic distribution of slits. Unit cell of the structure (c) with the incident TM wave. The impinging wave imposes periodic boundary conditions with a constant phase shift Δϕ = ky,0a along y.

Now, multiplying Eq. (6) by
cos[mπb(y+b/2)],m=1,..,M;
(10)
and Eq. (7) by
exp[i(ky,0+2nπa)y],n=N,,0,N;
(11)
and integrating in the corresponding ranges of y, we obtain a set of overdetermined equations which can be numerically solved using a least square algorithm.

3. Results

In Figs. 26 the transmission coefficients obtained with the reported model are compared with electromagnetic simulations using CST Microwave Studio. In order to obtain convergent results, the resolution of the higher modes inside the slits (M) and in the input and output regions (N) must be similar. Convergent results were obtained employing N = 60 modes in Eq. (6) and MN(b/a) modes in Eq. (7) with a CPU time of ∼ 0.5s vs CPU time of ∼ 10s with the electromagnetic solver.

Fig. 2 Transmission through an array of slits in a lossy copper screen (σ = 59.6×106S/m) at normal incidence and different sizes of the slits. Periodicity is a = 300μm and thickness of the screen is t = a/20. Continuous lines correspond to mode matching model, dashed lines to CST simulations and dot-dashed lines to our previous numerical model [9].
Fig. 3 Transmission through an array of slits in a lossy copper screen (σ = 59.6×106S/m) for different angles of incidence and Wood’s anomalies resonances. Periodicity is a = 300μm, size of slits is b = a/4 and thickness of the screen is t = a/20. Continuous lines correspond to mode matching model and dashed lines to CST simulations.
Fig. 4 Transmission through an array of slits in a silver screen (ωp = 2π × 2175THz and fc = 1.26 × fc = 1.26 × 2π × 4.35 THz) for different angles of incidence and Wood’s anomalies resonances. Periodicity is a = 1μm, size of slits is b = a/4 and thickness of the screen is t = a/20. Continuous lines correspond to mode matching model and dashed lines to CST simulations.
Fig. 5 Transmission through an array of slits in a zirconium-tin-titanate (ɛ = 92.7(1 + 0.005i) [17]) screen for different angles of incidence and Wood’s anomalies resonances. Periodicity is a = 3mm, size of slits is b = a/6 and thickness of the screen is t = a/25. Continuous lines correspond to mode matching model and dashed lines to CST simulations.
Fig. 6 Transmission through an array of PEC inclusions in a zirconium-tin-titanate (ɛ = 92.7(1 + 0.005i) [17]) screen for different angles of incidence and Wood’s anomalies resonances. Periodicity is a = 3mm, size of the inclusions is b = a/6 and thickness of the screen is t = a/25. Continuous lines correspond to mode matching model and dashed lines to CST simulations.

In Figs. 2 and 3 the metallic screen is modeled by a finite conductivity σ = 59.6 ×106 S/m (corresponding to copper) at frequencies around 1 THz:
ɛsiσωɛ0,
(12)
and in Fig. 4 by a Drude-Lorentz permittivity for silver [16

16. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr., and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–1119 (1983). [CrossRef] [PubMed]

] at frequencies around 300 THz
ɛsɛ0(1ωp2ω(ωifc)),
(13)
with plasma frequency ωp = 2π × 2175 THz and frequency of collision fc = 1.26 × fc = 1.26 × 2π × 4.35 THz, where a correction due to finite thickness of the screen has been taken into account [9

9. V. Delgado, R. Marqués, and L. Jelinek, “Analytical theory of extraordinary optical transmission through realistic metallic screens,” Opt. Express 18(7), 6506–6515 (2010). [CrossRef] [PubMed]

]. In the abscissas, frequencies are normalized to the Wood’s anomalies frequencies for each angle of incidence
fw,n={nca(1sin(θ))forn>0|n|ca(1+sin(θ))forn<0
(14)
corresponding to the divergence of the TM2n modes excited in the generalized waveguide. In the case of metallic screens, the numerical model works quite well for any angle of incidence and different Wood’s anomalies resonances (n = −1, n = 1 and n = −2). At a given angle of incidence (θ = 19.47°) the Wood’s anomaly frequency corresponding to the divergence of the scattered TM2 and TM−4 modes is the same. For higher angles of incidence the peaks corresponding to the divergence of TM2 mode vanish (Fig. 2(b)) and those corresponding to the divergence of TM−4 grow.

In order to gain physical insight into the phenomenon, the longitudinal real part of the Poynting vector (ℜ(Sz)) at the frequencies of some of the peaks, computed with the electromagnetic simulator are plotted in Fig. 7 and 8. In Fig. 7 the screens are modeled as lossy copper. The graphics are complex, with vortexes of power bouncing forward and backward along the structure; a fact which express the important role played by evanescent modes near the screen. A high concentration of power appears at the edges of the slits, where strong currents are excited. Figure 7(a) corresponds to the divergence of the TM−2 mode and Fig. 7(b) to the divergence of the TM−4 mode excited in the screen for an angle of incidence θ = 20°. In both cases, the power flow pattern corresponding to each resonance can be clearly appreciated. In Fig. 8 the screens are modeled as zirconium-tin-titanate (ɛ = 92.7(1 + 0.005i)). In Fig. 8(a) the slits are empty and in Fig. 8(b) the slits are filled with PEC. In both cases the patterns correspond to the divergence of the TM−2 mode for an angle of incidence θ = 20°; and show that most power is transmitted through the dielectric screens, thus suggesting that EOT in these structures is obtained as a consequence of the periodic distribution of defects, and no necessarily associated to the presence of transparent slits. Even more, in Fig. 8(a) the empty slits seem to act as a drain of power, which is first transmitted through the dielectric from left to right and then partially put back in the left space through the slits.

Fig. 7 Propagating component of the real part of the Poynting vector at the frequencies of the EOT peaks for the lossy copper screen analyzed in Fig.2. In (a) EOT is associated to the divergence of the TM−2 mode and in (b) to the divergence of the TM−4 mode. Angle of incidence is θ = 20° in both cases.
Fig. 8 Propagating component of the real part of the Poynting vector at the frequencies of the EOT peaks for the zirconium-tin-titanate screen analyzed in Figs. 5 and 6. In (a) the slits are empty and in (b) filled with PEC. Angle of incidence is θ = 20° and EOT is associated to the divergence of the TM−2 mode in both cases.

4. Conclusion

Acknowledgments

This work has been supported by the Spanish Ministerio de Educación y Ciencia and European Union FEDER funds (project No. CSD2008-00066 and TEC2010-16948). The authors are grateful to Lukas Jelinek for many fruitful discussions.

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, 667–669 (1998). [CrossRef]

2.

E. Devaux, T. W. Ebbesen, J.-C. Weeber, and A. Dereux, “Launching and decoupling surface plasmons via micro-gratings,” Appl. Phys. Lett. 83, 4936 (2003). [CrossRef]

3.

A. A. Yanik, M. Huang, O. Kamohara, A. Artar, T. W. Geisbert, J. H. Connor, and H. Altug, “An optofluidic nanoplasmonic biosensor for direct detection of live viruses from biological media,” Nano Lett. 10, 4962–4969 (2010). [CrossRef]

4.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58(15), 6779–6782 (1998). [CrossRef]

5.

M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Martín-Moreno, J. Bravo-Abad, and F. J. García-Vidal, “Enhanced millimeter-wave transmission through subwavelength hole arrays,” Opt. Lett. 29(21), 2500–2502 (2004). [CrossRef] [PubMed]

6.

M. Sarrazin and J. P. Vigneron, “Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes,” Phys. Rev. E 68, 016603 (2003). [CrossRef]

7.

J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004). [CrossRef] [PubMed]

8.

R. Marqués, F. Mesa, L. Jelinek, and F. Medina, “Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes,” Opt. Express 17(7), 5571–5579 (2009). [CrossRef] [PubMed]

9.

V. Delgado, R. Marqués, and L. Jelinek, “Analytical theory of extraordinary optical transmission through realistic metallic screens,” Opt. Express 18(7), 6506–6515 (2010). [CrossRef] [PubMed]

10.

V. Delgado, R. Marqués, and L. Jelinek, “Extraordinary transmission through dielectric screens with 1D sub-wavelength metallic inclusions,” Opt. Express 19(14), 13612–13617 (2011). [CrossRef] [PubMed]

11.

J. D. Jackson, Classical Electrodynamics, 3rd Ed. (John Wiley & Sons, Inc., 1998)

12.

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999). [CrossRef]

13.

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

14.

M. M. J. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B 66, 195105 (2002). [CrossRef]

15.

F. Medina, F. Mesa, and D. C. Skigin, “Extraordinary transmission through arrays of slits: a circuit theory model,” IEEE Trans. Microwave Theory Tech. 58(1), 105–115 (2010). [CrossRef]

16.

M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr., and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–1119 (1983). [CrossRef] [PubMed]

17.

P. H. Bolivar, J. G. Rivas, R. Gonzalo, I. Ederra, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 51(4), 1062–1066 (2003). [CrossRef]

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(050.1960) Diffraction and gratings : Diffraction theory

ToC Category:
Diffraction and Gratings

History
Original Manuscript: July 28, 2011
Revised Manuscript: September 23, 2011
Manuscript Accepted: October 20, 2011
Published: November 23, 2011

Citation
V. Delgado and R. Marqués, "Surface impedance model for extraordinary transmission in 1D metallic and dielectric screens," Opt. Express 19, 25290-25297 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25290


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References

  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, 667–669 (1998). [CrossRef]
  2. E. Devaux, T. W. Ebbesen, J.-C. Weeber, and A. Dereux, “Launching and decoupling surface plasmons via micro-gratings,” Appl. Phys. Lett.83, 4936 (2003). [CrossRef]
  3. A. A. Yanik, M. Huang, O. Kamohara, A. Artar, T. W. Geisbert, J. H. Connor, and H. Altug, “An optofluidic nanoplasmonic biosensor for direct detection of live viruses from biological media,” Nano Lett.10, 4962–4969 (2010). [CrossRef]
  4. H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B58(15), 6779–6782 (1998). [CrossRef]
  5. M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Martín-Moreno, J. Bravo-Abad, and F. J. García-Vidal, “Enhanced millimeter-wave transmission through subwavelength hole arrays,” Opt. Lett.29(21), 2500–2502 (2004). [CrossRef] [PubMed]
  6. M. Sarrazin and J. P. Vigneron, “Optical properties of tungsten thin films perforated with a bidimensional array of subwavelength holes,” Phys. Rev. E68, 016603 (2003). [CrossRef]
  7. J. B. Pendry, L. Martín-Moreno, and F. J. García-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305, 847–848 (2004). [CrossRef] [PubMed]
  8. R. Marqués, F. Mesa, L. Jelinek, and F. Medina, “Analytical theory of extraordinary transmission through metallic diffraction screens perforated by small holes,” Opt. Express17(7), 5571–5579 (2009). [CrossRef] [PubMed]
  9. V. Delgado, R. Marqués, and L. Jelinek, “Analytical theory of extraordinary optical transmission through realistic metallic screens,” Opt. Express18(7), 6506–6515 (2010). [CrossRef] [PubMed]
  10. V. Delgado, R. Marqués, and L. Jelinek, “Extraordinary transmission through dielectric screens with 1D sub-wavelength metallic inclusions,” Opt. Express19(14), 13612–13617 (2011). [CrossRef] [PubMed]
  11. J. D. Jackson, Classical Electrodynamics, 3rd Ed. (John Wiley & Sons, Inc., 1998)
  12. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett.83, 2845–2848 (1999). [CrossRef]
  13. F. J. García-Vidal and L. Martín Moreno, “Transmission and focusing of light in one-dimensional periodically nanostructured metals,” Phys. Rev. B66, 155412 (2002). [CrossRef]
  14. M. M. J. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B66, 195105 (2002). [CrossRef]
  15. F. Medina, F. Mesa, and D. C. Skigin, “Extraordinary transmission through arrays of slits: a circuit theory model,” IEEE Trans. Microwave Theory Tech.58(1), 105–115 (2010). [CrossRef]
  16. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt.22(7), 1099–1119 (1983). [CrossRef] [PubMed]
  17. P. H. Bolivar, J. G. Rivas, R. Gonzalo, I. Ederra, A. L. Reynolds, M. Holker, and P. de Maagt, “Measurement of the dielectric constant and loss tangent of high dielectric-constant materials at terahertz frequencies,” IEEE Trans. Microwave Theory Tech.51(4), 1062–1066 (2003). [CrossRef]

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