Abstract

We theoretically analyze the properties of metamaterials which have been designed by taking advantage of Babinet’s principle. It is shown that the complementary structure exhibits both a complementary spectral response and field distribution of the respective eigenmodes. For complementary split-ring resonators, we show that the spectral resonance features can be explained from two different perspectives. On one hand they can be explained as plasmon polariton resonances in dielectric nanostructures surrounded by metal, on the other hand they can be understood as guided mode resonances with vanishing propagation constant. The physical origin of these modes and differences to the conventional split-ring geometry are discussed.

© 2008 Optical Society of America

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  1. V. M. Shalaev, "Optical negative-index metamaterials," Nature Photonics 1,41-48 (2007).
    [CrossRef]
  2. C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelengths," Science 315,47-49 (2007).
    [CrossRef] [PubMed]
  3. J. D. JoannopoulosPhotonic crystals: Molding the flow of light (Princeton University Press, Princeton, New Jersey, 1995).
  4. D. R. Smith, S. Schultz, P. Marko¡s, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65,195104 (2002).
    [CrossRef]
  5. D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and Anne-Claude Tarot, "Effective parameters of resonant negative refractive index metamaterials: Interpretation and validity," J. Appl. Phys. 98,063505 (2005).
    [CrossRef]
  6. C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, "Investigating the transition from thin film to bulk properties of metamaterials" Phys. Rev. B 77,035126 (2008).
    [CrossRef]
  7. T. C. Choy, Effective Medium Theory, Principles and Applications, (Oxford University Press, 1999).
  8. R. D. Grober, R. J. Schoelkopf, and D. E. Prober, "Optical antenna: Towards a unity efficiency near-field optical probe," Appl. Phys. Lett. 70,1354-1356 (1997).
    [CrossRef]
  9. L. Lewin, "The electrical constants of a material loaded with spherical particles," Proc. Inst. Elec. Eng., Part 3,  94,65-68 (1947).
  10. C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, " Design of an Artificial Three-Dimensional Composite Metamaterial with Magnetic Resonances in the Visible Range of the Electromagnetic Spectrum," Phys. Rev. Lett. 99,017401 (2007).
    [CrossRef] [PubMed]
  11. J. P. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors, and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47,2075-2084 (1999).
    [CrossRef]
  12. F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marqués, F. Martín, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93,197401 (2004).
    [CrossRef] [PubMed]
  13. H.-T. Chen, J. F. O’Hara, A. J. Taylor, R. D. Averitt, C. Highstrete, M. Lee, and W. J. Padilla, "Complementary planar terahertz metamaterials," Opt. Express 15,1084-1095 (2007).
    [CrossRef] [PubMed]
  14. M. Born and E. Wolf, Principles of optics (Cambridge University Press, Cambridge, 1999).
  15. R. Ullrich, "Far-infrared properties of metallic mesh and its complementary structure," Infrared Phys. 7,37-55 (1967).
    [CrossRef]
  16. L. C. Botton, R. C. McPhedran, and G. W. Milton, "Perfectly conducting lamellar gratings: Babinet’s principle and circuit models," J. Mod. Opt. 42,2453-2473 (1995).
    [CrossRef]
  17. T. Zentgraf, C. Rockstuhl, T. P. Meyrath, A. Seidel, S. Kaiser, F. Lederer, and H. Giessen, "Babinet’s principle for optical frequency metamaterials and nanoantennas," Phys. Rev. B 76,033407 (2007).
    [CrossRef]
  18. C. Rockstuhl and F. Lederer, "Negative-index metamaterials from nanoapertures," Phys. Rev. B 76,125426 (2007).
    [CrossRef]
  19. J. A. Porto, F. J. Garca-Vidal, and J. B. Pendry, "Transmission Resonances on Metallic Gratings with Very Narrow Slits," Phys. Rev. Lett. 83,2845-2848 (1999).
    [CrossRef]
  20. X. Shi, L. Hesselink, and R. L. Thornton, "Ultrahigh light transmission through a C-shaped nanoaperture," Opt. Lett. 28,1320-1322 (2003).
    [CrossRef] [PubMed]
  21. C. Rockstuhl, F. Lederer, T. Zentgraf, and H. Giessen, "Enhanced transmission of periodic, quasi-periodic, and random nanoaperture arrays," Appl. Phys. Lett. 91,151109 (2007).
    [CrossRef]
  22. F. J. García de Abajo, R. G’omez-Medina, and J. J. S’aenz, "Full transmission through perfect-conductor subwavelength hole arrays," Phys. Rev. E 72,016608 (2005).
    [CrossRef]
  23. F. J. García de Abajo, "Light scattering by particle and hole arrays," Rev. Mod. Phys. 79,1267-1290 (2007).
    [CrossRef]
  24. P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6,4370-4379 (1972).
    [CrossRef]
  25. D. H. Dawes, R. C. McPhedran, and L. B. Whitbourn, "Thin capacitive meshes on a dielectric boundary: theory and experiment," Appl. Opt. 26,3498-3510 (1989).
    [CrossRef]
  26. L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14,2758-2767 (1997).
    [CrossRef]
  27. C. Rockstuhl, T. Zentgraf, C. Etrich, J. Kuhl, F. Lederer, and H. Giessen, "On the reinterpretation of resonances in split-ring-resonators at normal incidence," Opt. Express 14,8827-8836 (2006).
    [CrossRef] [PubMed]
  28. C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, "Resonances of split-ring resonator metamaterials in the near infrared," Appl. Phys. B. 84,219-227 (2006).
    [CrossRef]
  29. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31,2972-2974 (2006).
    [CrossRef] [PubMed]
  30. C. Rockstuhl, T. Zentgraf, E. Pshenay-Severin, J. Petschulat, A. Chipouline, J. Kuhl, H. Giessen, T. Pertsch, and F. Lederer, "The origin of magnetic polarizability in metamaterials at optical frequencies - an electrodynamic approach," Opt. Express 15,8871-8883 (2007).
    [CrossRef] [PubMed]
  31. Z. Zhu and T. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10,853-864 (2002).
    [PubMed]
  32. V. V. Varadan and A. R. Tellakula, "Effective properties of split-ring resonator metamaterials using measured scattering parameters: Effect of gap orientation," J. Appl. Phys. 100,034910 (2006).
    [CrossRef]
  33. M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7,S12-S22 (2005).
    [CrossRef]

2008 (1)

C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, "Investigating the transition from thin film to bulk properties of metamaterials" Phys. Rev. B 77,035126 (2008).
[CrossRef]

2007 (9)

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photonics 1,41-48 (2007).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelengths," Science 315,47-49 (2007).
[CrossRef] [PubMed]

C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, " Design of an Artificial Three-Dimensional Composite Metamaterial with Magnetic Resonances in the Visible Range of the Electromagnetic Spectrum," Phys. Rev. Lett. 99,017401 (2007).
[CrossRef] [PubMed]

H.-T. Chen, J. F. O’Hara, A. J. Taylor, R. D. Averitt, C. Highstrete, M. Lee, and W. J. Padilla, "Complementary planar terahertz metamaterials," Opt. Express 15,1084-1095 (2007).
[CrossRef] [PubMed]

T. Zentgraf, C. Rockstuhl, T. P. Meyrath, A. Seidel, S. Kaiser, F. Lederer, and H. Giessen, "Babinet’s principle for optical frequency metamaterials and nanoantennas," Phys. Rev. B 76,033407 (2007).
[CrossRef]

C. Rockstuhl and F. Lederer, "Negative-index metamaterials from nanoapertures," Phys. Rev. B 76,125426 (2007).
[CrossRef]

C. Rockstuhl, F. Lederer, T. Zentgraf, and H. Giessen, "Enhanced transmission of periodic, quasi-periodic, and random nanoaperture arrays," Appl. Phys. Lett. 91,151109 (2007).
[CrossRef]

F. J. García de Abajo, "Light scattering by particle and hole arrays," Rev. Mod. Phys. 79,1267-1290 (2007).
[CrossRef]

C. Rockstuhl, T. Zentgraf, E. Pshenay-Severin, J. Petschulat, A. Chipouline, J. Kuhl, H. Giessen, T. Pertsch, and F. Lederer, "The origin of magnetic polarizability in metamaterials at optical frequencies - an electrodynamic approach," Opt. Express 15,8871-8883 (2007).
[CrossRef] [PubMed]

2006 (4)

C. Rockstuhl, T. Zentgraf, C. Etrich, J. Kuhl, F. Lederer, and H. Giessen, "On the reinterpretation of resonances in split-ring-resonators at normal incidence," Opt. Express 14,8827-8836 (2006).
[CrossRef] [PubMed]

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, "Resonances of split-ring resonator metamaterials in the near infrared," Appl. Phys. B. 84,219-227 (2006).
[CrossRef]

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31,2972-2974 (2006).
[CrossRef] [PubMed]

V. V. Varadan and A. R. Tellakula, "Effective properties of split-ring resonator metamaterials using measured scattering parameters: Effect of gap orientation," J. Appl. Phys. 100,034910 (2006).
[CrossRef]

2005 (3)

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7,S12-S22 (2005).
[CrossRef]

F. J. García de Abajo, R. G’omez-Medina, and J. J. S’aenz, "Full transmission through perfect-conductor subwavelength hole arrays," Phys. Rev. E 72,016608 (2005).
[CrossRef]

D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and Anne-Claude Tarot, "Effective parameters of resonant negative refractive index metamaterials: Interpretation and validity," J. Appl. Phys. 98,063505 (2005).
[CrossRef]

2004 (1)

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marqués, F. Martín, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93,197401 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (2)

D. R. Smith, S. Schultz, P. Marko¡s, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65,195104 (2002).
[CrossRef]

Z. Zhu and T. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10,853-864 (2002).
[PubMed]

1999 (2)

J. P. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors, and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47,2075-2084 (1999).
[CrossRef]

J. A. Porto, F. J. Garca-Vidal, and J. B. Pendry, "Transmission Resonances on Metallic Gratings with Very Narrow Slits," Phys. Rev. Lett. 83,2845-2848 (1999).
[CrossRef]

1997 (2)

R. D. Grober, R. J. Schoelkopf, and D. E. Prober, "Optical antenna: Towards a unity efficiency near-field optical probe," Appl. Phys. Lett. 70,1354-1356 (1997).
[CrossRef]

L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14,2758-2767 (1997).
[CrossRef]

1995 (1)

L. C. Botton, R. C. McPhedran, and G. W. Milton, "Perfectly conducting lamellar gratings: Babinet’s principle and circuit models," J. Mod. Opt. 42,2453-2473 (1995).
[CrossRef]

1989 (1)

D. H. Dawes, R. C. McPhedran, and L. B. Whitbourn, "Thin capacitive meshes on a dielectric boundary: theory and experiment," Appl. Opt. 26,3498-3510 (1989).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6,4370-4379 (1972).
[CrossRef]

1967 (1)

R. Ullrich, "Far-infrared properties of metallic mesh and its complementary structure," Infrared Phys. 7,37-55 (1967).
[CrossRef]

1947 (1)

L. Lewin, "The electrical constants of a material loaded with spherical particles," Proc. Inst. Elec. Eng., Part 3,  94,65-68 (1947).

Appl. Opt. (1)

D. H. Dawes, R. C. McPhedran, and L. B. Whitbourn, "Thin capacitive meshes on a dielectric boundary: theory and experiment," Appl. Opt. 26,3498-3510 (1989).
[CrossRef]

Appl. Phys. B. (1)

C. Rockstuhl, T. Zentgraf, H. Guo, N. Liu, C. Etrich, I. Loa, K. Syassen, J. Kuhl, F. Lederer, and H. Giessen, "Resonances of split-ring resonator metamaterials in the near infrared," Appl. Phys. B. 84,219-227 (2006).
[CrossRef]

Appl. Phys. Lett. (2)

C. Rockstuhl, F. Lederer, T. Zentgraf, and H. Giessen, "Enhanced transmission of periodic, quasi-periodic, and random nanoaperture arrays," Appl. Phys. Lett. 91,151109 (2007).
[CrossRef]

R. D. Grober, R. J. Schoelkopf, and D. E. Prober, "Optical antenna: Towards a unity efficiency near-field optical probe," Appl. Phys. Lett. 70,1354-1356 (1997).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. P. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors, and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47,2075-2084 (1999).
[CrossRef]

Infrared Phys. (1)

R. Ullrich, "Far-infrared properties of metallic mesh and its complementary structure," Infrared Phys. 7,37-55 (1967).
[CrossRef]

J. Appl. Phys. (2)

D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and Anne-Claude Tarot, "Effective parameters of resonant negative refractive index metamaterials: Interpretation and validity," J. Appl. Phys. 98,063505 (2005).
[CrossRef]

V. V. Varadan and A. R. Tellakula, "Effective properties of split-ring resonator metamaterials using measured scattering parameters: Effect of gap orientation," J. Appl. Phys. 100,034910 (2006).
[CrossRef]

J. Mod. Opt. (1)

L. C. Botton, R. C. McPhedran, and G. W. Milton, "Perfectly conducting lamellar gratings: Babinet’s principle and circuit models," J. Mod. Opt. 42,2453-2473 (1995).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7,S12-S22 (2005).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nature Photonics (1)

V. M. Shalaev, "Optical negative-index metamaterials," Nature Photonics 1,41-48 (2007).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. B (5)

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6,4370-4379 (1972).
[CrossRef]

T. Zentgraf, C. Rockstuhl, T. P. Meyrath, A. Seidel, S. Kaiser, F. Lederer, and H. Giessen, "Babinet’s principle for optical frequency metamaterials and nanoantennas," Phys. Rev. B 76,033407 (2007).
[CrossRef]

C. Rockstuhl and F. Lederer, "Negative-index metamaterials from nanoapertures," Phys. Rev. B 76,125426 (2007).
[CrossRef]

C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, "Investigating the transition from thin film to bulk properties of metamaterials" Phys. Rev. B 77,035126 (2008).
[CrossRef]

D. R. Smith, S. Schultz, P. Marko¡s, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65,195104 (2002).
[CrossRef]

Phys. Rev. E (1)

F. J. García de Abajo, R. G’omez-Medina, and J. J. S’aenz, "Full transmission through perfect-conductor subwavelength hole arrays," Phys. Rev. E 72,016608 (2005).
[CrossRef]

Phys. Rev. Lett. (3)

C. Rockstuhl, F. Lederer, C. Etrich, T. Pertsch, and T. Scharf, " Design of an Artificial Three-Dimensional Composite Metamaterial with Magnetic Resonances in the Visible Range of the Electromagnetic Spectrum," Phys. Rev. Lett. 99,017401 (2007).
[CrossRef] [PubMed]

J. A. Porto, F. J. Garca-Vidal, and J. B. Pendry, "Transmission Resonances on Metallic Gratings with Very Narrow Slits," Phys. Rev. Lett. 83,2845-2848 (1999).
[CrossRef]

F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marqués, F. Martín, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Phys. Rev. Lett. 93,197401 (2004).
[CrossRef] [PubMed]

Proc. Inst. Elec. Eng (1)

L. Lewin, "The electrical constants of a material loaded with spherical particles," Proc. Inst. Elec. Eng., Part 3,  94,65-68 (1947).

Rev. Mod. Phys. (1)

F. J. García de Abajo, "Light scattering by particle and hole arrays," Rev. Mod. Phys. 79,1267-1290 (2007).
[CrossRef]

Science (1)

C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelengths," Science 315,47-49 (2007).
[CrossRef] [PubMed]

Other (3)

J. D. JoannopoulosPhotonic crystals: Molding the flow of light (Princeton University Press, Princeton, New Jersey, 1995).

T. C. Choy, Effective Medium Theory, Principles and Applications, (Oxford University Press, 1999).

M. Born and E. Wolf, Principles of optics (Cambridge University Press, Cambridge, 1999).

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Figures (8)

Fig. 1.
Fig. 1.

Normalized reflected (dashed red lines) and transmitted intensity (solid blue lines) for SRRs (a, b) and c-SRRs (c, d), as shown in the geometry on top. The structures are illuminated by a plane wave at normal incidence with its electric field component polarized parallel (a,c) or perpendicularly (b,d) to the gap. The numbers indicate the order of the resonance which can be deduced from the amplitude nodes of the electric/magnetic field component normal to the SRR/c-SRR plane. The SRR/c-SRR is characterized by an arm length of l = 300 nm, a width of w = 40 nm, and a height for the SRR of h = 15 nm which is the thickness of the metal film for the c-SRR. They are assumed to be arranged in a periodic lattice with the period being Λ = 400 nm.

Fig. 2.
Fig. 2.

Amplitude of the magnetic field component normal to the c-SRR surface at the spectral positions of the first three lowest-order eigenmodes. The electric polarization of the incident plane wave is normal to the c-SRR gap for odd (a, c) and parallel for even eigenmodes (b).

Fig. 3.
Fig. 3.

Normalized transmitted intensity through the c-SRR as a function of metal thickness for different polarizations of the incident electric field. The lowest order Fabry-Pérot resonance (kzd≈0) of the four lowest order guided modes are indicated by the dashed lines. For an increasing thickness higher order Fabry-Pérot resonances of these guided modes move asymptotically towards the spectral position of the lowest order resonance.

Fig. 4.
Fig. 4.

Ex and Ey component of the first-(a,b) and second-order (c,d) guided mode. The mode profiles were calculated by solving the eigenvalue equation in an infinite structure normal to the c-SRR surface for the eigenvalues (kz ≈0). These mode fields are qualitatively identical to those of plasmon resonances obtained by calculating the transmission of a h = 15 nm film by using the Fourier Modal Method.

Fig. 5.
Fig. 5.

Effective refractive index of the first three eigenmodes of the c-SRR nanoparture separated in the real (a) and the imaginary (b) part.

Fig. 6.
Fig. 6.

(a) Normalized reflected (red dotted line) and transmitted intensity (blue bold line) for an in-plane illuminated c-SRR with its gap at the front side. The amplitude of the magnetic field component perpendicular to the c-SRR and 20 nm above its surface at the resonance frequency of ν ¯ 2 c =7000 cm 1 is shown in (b). The amplitude of the x- and y-component of the electric field amplitude at the same frequency are displayed in (c) and (d), respectively.

Fig. 7.
Fig. 7.

Reflected and transmitted normalized intensity for an in-plane illuminated c-SRR with parallel orientation (a). The amplitude of the magnetic field component perpendicular to the c-SRR and 20 nm above its surface at the resonance frequency of ν ¯ 1 c =3500 cm 1 are shown in (b) and at the resonance frequency of ν ¯ 3 c =8000 cm 1 are shown in (c).

Fig. 8.
Fig. 8.

Conceptual sketch of the electric field in the nanoresonators that form the MM for the first three resonances. At a fixed time, the arrows denote the electric field, where their directions resolve the phase relation, and their thickness resolve the excitation strength. Resonances in the SRR are shown in the top row, resonances in the c-SRR are shown in the bottom row.

Equations (1)

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T = t 2 e i k z h 1 + r 2 e i 2 k z h ,

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