Abstract

We analyze the transmission properties of double negative metamaterials (DNM). Numerical simulations, based on the transfer matrix algorithm, show that some portion of the electromagnetic wave changes its polarization inside the DNM structure. As the transmission properties depend strongly on the polarization, this complicates the interpretation of experimental and numerical data, both inside and outside of the pass band. From the transmission data, the effective permittivity, permeability and refractive index are calculated. In the pass band, we found that the real part of permeability and both the real and the imaginary part of the permittivity are negative. Transmission data for some new structures are also shown. Of particular interest is the structure with cut wires, which possesses two double negative pass bands.

© 2003 Optical Society of America

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References

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  1. J.B. Pendry, A.J. Holden, W.J. Stewart and I. Youngs, �??Extremely Low Frequency Plasmons in Metallic Mesostructures,�?? Phys. Rev. Lett. 76, 4773 (1996)
    [CrossRef] [PubMed]
  2. J.B. Pendry, A.J. Holden, D.J. Robbins and W.J. Stewart, (1999) �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. on Microwave Theory and Technol. 47, 2075 (1999)
    [CrossRef]
  3. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz �??A Composite medium with simultaneously negative permeability and permittivity,�?? Phys. Rev. Lett. 84, 4184 (2000)
    [CrossRef] [PubMed]
  4. D.R. Smith and N. Kroll, �??Negative Refractive Index in Left-Handed Materials,�?? Phys. Rev. Lett. 85, 2933 (2000)
    [CrossRef] [PubMed]
  5. R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, and S. Schultz, �??Microwave transmission through a two-dimensional, isotropic, left-handed meta material,�?? Appl. Phys. Lett. 78, 489 (2001)
    [CrossRef]
  6. P. Marko¡s and C.M. Soukoulis, �??Transmission Studies of the Left-handed Materials,�?? Phys. Rev. B 65 033401 (2002)
  7. P. Marko¡s and C.M. Soukoulis, �??Numerical Studies of Left-handed materials and Arrays of Split Ring Resonators,�?? Phys. Rev. E 65, 036622 (2002)
    [CrossRef]
  8. P. Marko¡s, I. Rousochatzakis, and C.M. Soukoulis, �??Transmission Losses in Left-handed Materials,�?? Phys. Rev. E 66, 045601 (2002)
    [CrossRef]
  9. C. D. Moss, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, �??Numerical Studies of Left-handed Metamaterials,�?? Prog. Electromagnetics Res. PIER 35, 315 (2002)
    [CrossRef]
  10. V.G. Veselago, �??The electrodynamics of substances with simultaneously negative values of permittivity and permeability,�?? Sov. Phys. Usp. 10, 509 (1968)
    [CrossRef]
  11. M.M. Sigalas, C.Y. Chan, K.M. Ho, and .M. Soukoulis, �??Metallic photonic band-gap materials,�?? Phys. Rev. B 52, 11744 (1999)
    [CrossRef]
  12. R.A. Shelby, D.R. Smith, and S. Schultz, �??Experimental veri.cation of a negative index of refraction,�?? Science 292, 77 (2001)
    [CrossRef] [PubMed]
  13. C. G. Parazzoli, R. B. Gregor, K. Li, B. E. C. Koltenbah, and M. Tanielian, �??Experimental Veri.cation and Simulation of Negative Index of Refraction Using Snell�??s Law,�?? Phys. Rev. Lett. 90 107401 (2003)
    [CrossRef] [PubMed]
  14. D.R. Smith, S. Shultz, P. Marko¡s, and C.M. Soukoulis, �??Determination of Effective Permittivity and Permeability of Metamaterials from Re.ection and Transmission Coefficient,�?? Phys. Rev. B 65 195104 (2002)
    [CrossRef]
  15. D. R. Smith and D. Schurig, �??Electromagnetic Wave propagation in Media with Inde.nite Permittivity and Permeability Tensors,�?? Phys. Rev. Lett. 90 077405 (2003)
    [CrossRef] [PubMed]
  16. S. O�??Brien, and J.B. Pendry, �??Photonic band gap e.ects and magnetic activity of dielectric composites,�?? J. Phys.: Condens. Matter 14 4035 (2002)
    [CrossRef]
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    [CrossRef] [PubMed]
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  19. P. Marko¡s and C. M. Soukoulis, �??Absorption losses in periodic arrays of thin metallic wires,�?? e-print cond-mat/0212343 to appear in Opt. Lett. (2003)
    [CrossRef]
  20. L.D. Landau, E.M. Lifshitz, and L.P. Pitaevski, �??Electrodynamics of Continuous Media,�?? Pergamon Press 1984
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    [CrossRef]
  22. J.D. Jackson, �??Classical Electrodynamic,�?? (3rd edition), J.Willey and Sons, 1999, p. 312
  23. M. Bayindir, K. Aydin, E. Ozbay, P. Marko¡s, and C.M. Soukoulis, �??Transmission Properties of Composite Metamaterials in Free Space,�?? Appl. Phys. Lett. 81, 120 (2002)
    [CrossRef]
  24. E. Ozbay, K. Aydin, E. Cubukcu, and M. Bayindir, �??Transmission and reflection properties of composite double negative metamaterials in free-space,�?? to be published (2003)
  25. P. Marko¡s, D.R. Smith, and C.M. Soukoulis, unpublished.
  26. R. Ruppin, �??Electromagnetic energy density in a dispersive and absorptive material,�?? Phys. Lett. A 299 309 (2002)
    [CrossRef]

Appl. Phys. Lett.

R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, and S. Schultz, �??Microwave transmission through a two-dimensional, isotropic, left-handed meta material,�?? Appl. Phys. Lett. 78, 489 (2001)
[CrossRef]

E.V. Ponizovskaya, M. Nieto-Vesperinas, and N. Garcia, �??Losses for microwave transmission metamaterials for producing left-handed materials: The strip wires,�?? Appl. Phys. Lett. 81, 4470 (2002)
[CrossRef]

M. Bayindir, K. Aydin, E. Ozbay, P. Marko¡s, and C.M. Soukoulis, �??Transmission Properties of Composite Metamaterials in Free Space,�?? Appl. Phys. Lett. 81, 120 (2002)
[CrossRef]

IEEE Trans. Microwave Theory Technol.

J.B. Pendry, A.J. Holden, D.J. Robbins and W.J. Stewart, (1999) �??Magnetism from conductors and enhanced nonlinear phenomena,�?? IEEE Trans. on Microwave Theory and Technol. 47, 2075 (1999)
[CrossRef]

J. Mod. Opt.

J.B. Pendry and A. MacKinnon, Phys. Rev. Lett. 69 2772 (1992). J.B. Pendry, �??Photonic band gap structures,�?? J. Mod. Opt. 41, 209 (1994). J.B. Pendry and P.M. Bell 1996, �??Transfer matrix techniques for electromagnetic waves,�?? in Photonic Band Gap Materials vol. 315 of NATO ASI Ser. E: Applied Sciences (1996), ed. by C.M. Soukoulis (Plenum, NY) p. 203
[CrossRef] [PubMed]

J. Phys.: Condens. Matter

S. O�??Brien, and J.B. Pendry, �??Photonic band gap e.ects and magnetic activity of dielectric composites,�?? J. Phys.: Condens. Matter 14 4035 (2002)
[CrossRef]

Phys. Lett. A

R. Ruppin, �??Electromagnetic energy density in a dispersive and absorptive material,�?? Phys. Lett. A 299 309 (2002)
[CrossRef]

Phys. Rev. B

D.R. Smith, S. Shultz, P. Marko¡s, and C.M. Soukoulis, �??Determination of Effective Permittivity and Permeability of Metamaterials from Re.ection and Transmission Coefficient,�?? Phys. Rev. B 65 195104 (2002)
[CrossRef]

M.M. Sigalas, C.Y. Chan, K.M. Ho, and .M. Soukoulis, �??Metallic photonic band-gap materials,�?? Phys. Rev. B 52, 11744 (1999)
[CrossRef]

P. Marko¡s and C.M. Soukoulis, �??Transmission Studies of the Left-handed Materials,�?? Phys. Rev. B 65 033401 (2002)

Phys. Rev. E

P. Marko¡s and C.M. Soukoulis, �??Numerical Studies of Left-handed materials and Arrays of Split Ring Resonators,�?? Phys. Rev. E 65, 036622 (2002)
[CrossRef]

P. Marko¡s, I. Rousochatzakis, and C.M. Soukoulis, �??Transmission Losses in Left-handed Materials,�?? Phys. Rev. E 66, 045601 (2002)
[CrossRef]

Phys. Rev. Lett.

D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz �??A Composite medium with simultaneously negative permeability and permittivity,�?? Phys. Rev. Lett. 84, 4184 (2000)
[CrossRef] [PubMed]

D.R. Smith and N. Kroll, �??Negative Refractive Index in Left-Handed Materials,�?? Phys. Rev. Lett. 85, 2933 (2000)
[CrossRef] [PubMed]

D. R. Smith and D. Schurig, �??Electromagnetic Wave propagation in Media with Inde.nite Permittivity and Permeability Tensors,�?? Phys. Rev. Lett. 90 077405 (2003)
[CrossRef] [PubMed]

J.B. Pendry, A.J. Holden, W.J. Stewart and I. Youngs, �??Extremely Low Frequency Plasmons in Metallic Mesostructures,�?? Phys. Rev. Lett. 76, 4773 (1996)
[CrossRef] [PubMed]

C. G. Parazzoli, R. B. Gregor, K. Li, B. E. C. Koltenbah, and M. Tanielian, �??Experimental Veri.cation and Simulation of Negative Index of Refraction Using Snell�??s Law,�?? Phys. Rev. Lett. 90 107401 (2003)
[CrossRef] [PubMed]

Prog. Electromagnetics Res. PIER

C. D. Moss, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, �??Numerical Studies of Left-handed Metamaterials,�?? Prog. Electromagnetics Res. PIER 35, 315 (2002)
[CrossRef]

Science

R.A. Shelby, D.R. Smith, and S. Schultz, �??Experimental veri.cation of a negative index of refraction,�?? Science 292, 77 (2001)
[CrossRef] [PubMed]

Sov. Phys. Usp.

V.G. Veselago, �??The electrodynamics of substances with simultaneously negative values of permittivity and permeability,�?? Sov. Phys. Usp. 10, 509 (1968)
[CrossRef]

Other

Photonic Band GapMaterials, ed. by C.M. Soukoulis (Kluwer, Dordrecht, 1996); Photonic Crystals and Light Localization in the 21st Century, ed. by C.M. Soukoulis (Kluwer, Dordrecht, 2001)

P. Marko¡s and C. M. Soukoulis, �??Absorption losses in periodic arrays of thin metallic wires,�?? e-print cond-mat/0212343 to appear in Opt. Lett. (2003)
[CrossRef]

L.D. Landau, E.M. Lifshitz, and L.P. Pitaevski, �??Electrodynamics of Continuous Media,�?? Pergamon Press 1984

E. Ozbay, K. Aydin, E. Cubukcu, and M. Bayindir, �??Transmission and reflection properties of composite double negative metamaterials in free-space,�?? to be published (2003)

P. Marko¡s, D.R. Smith, and C.M. Soukoulis, unpublished.

J.D. Jackson, �??Classical Electrodynamic,�?? (3rd edition), J.Willey and Sons, 1999, p. 312

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Figures (8)

Fig. 1.
Fig. 1.

Typical DNM structure consists of periodic array of split-ring resonators (SRRs), combined with periodic arrangement of thin metallic wires. The structure is symmetric in all lateral directions. Only one unit cell is shown. This structure possesses negative permittivity and permittivity only for a wave, propagating in the z direction and polarized with E⃗y [3].

Fig. 2.
Fig. 2.

Effective permittivity for an array of thin square metallic wires of the size of 200×200µm. Closed symbols shows the imaginary part ∊″eff, open symbols are for the real part ∊′eff. The period of the square lattice is a=5 mm. Three different discretizations of the unit cell were used, in which the wire is represented by 1×1 (circles), 2×2 (diamonds) and 4×4 (triangles) mesh points. Solid line is a fit of the data to the analytical formula (1). Data confirm the stability of the transfer matrix simulations, at least in the frequency interval f>6 GHz. Permittivity of the metal is ∊m=-2000+106 i. More detailed numerical analysis of the effective permittivity of array of metallic wires was performed in Ref. [19].

Fig. 3.
Fig. 3.

Left: Transmission |tyy |2, |txx |2, |txy |2, and |tyx |2 for a DNM structure shown in Fig. 1. Periodic boundary conditions are considered in the x and y directions. The length of the system is 20 unit cells along the z direction. The transmission tyy clearly determines the resonance interval (9.8–11 GHz), where we expect the permittivity and permeability to be both negative. Note that the “off-diagonal” transmissions txy and tyx are much higher than tyy outside the resonance interval. Right: the same for an array of SRR only.

Fig. 4.
Fig. 4.

Real part of the transmission tyy for a system on the left side (f=9.7 GHz) of the DNM pass band. Open symbols show the transmission tyy , closed symbols show the transmission multiplied by factor of 100 for a longer system length. Data show that the amplitude of the transmission does not decrease after passing more than 12 unit lengths. The spatial dependence of tyy is given by n=1, in agreement with Eq. 3

Fig. 5.
Fig. 5.

Effective permittivity, permeability, impedance and index of refraction for the DNM structure shown in the Fig.1. Solid (open) circles denote real (imaginary) part, respectively. Dashed area shows the left-handed frequency band. Note that not only ∊′eff and µ′eff but also imaginary part of the permittivity, ∊″eff is negative. Outside the pass band and for frequencies f<9.8 GHz, we did not succeeded to find the real part of the refractive index. The spatial oscillations of the transmission are very fast indicating that |neff|>4. Then, the wavelength of the EM wave becomes comparable with the size of the unit cell and the proper value of neff is not accessible.

Fig. 6.
Fig. 6.

Dependence of the transmission peak on the shape of the unit cell. In this simulations, the wire is deposited on the opposite side of the dielectric board. The width of the transmission band increases as the width of the unit cell decreases. Calculation of the effective refraction index (data not shown) confirmed that the band is left-handed with n eff<0.

Fig. 7.
Fig. 7.

Left: Transmission and the refraction index (bottom) for a one-dimensional DNM in which the wire is positioned along the SRR as shown in the top figure. The size of the unit cell is 1.764×3.35×3.84 mm. The size of the SRR is 3×3 mm. Azimuthal gap g=0.176 mm, radial gaps and width of rings is 0.35 mm. The wire width are 0.53 mm. The permittivity of the metallic components ∊m=-2000+106 i. Data for the refraction index confirm the left-handedness of the pass band. Right panel shows the same structure with one additional wire located on the opposite side of the dielectric board.

Fig. 8.
Fig. 8.

Transmission (top) and the refraction index (bottom) for a one-dimensional DNM with cut wires. The unit cell is shown on the right. The size of the unit cell is 0.789×3.31×4.26 mm. The size of the SRR is 3×3 mm with azimuthal gap 0.157 mm. The wire width is 0.979 mm with a cut of 0.157 mm. Dashed line on the top panel shows the transmission for an array of wires. For the DNM, we found three transmission peaks. Note that the first transmission peak is right-handed. The two other peaks are left-handed, as is proved by the sign of neff in the bottom panel.

Equations (16)

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eff = 1 f e 2 f 2 + i γ e f
μ eff = 1 F f 2 f 2 f m 2 + i γ m f ,
t yy 0 L = t yy ( 0 ) 0 L + t yy ( 1 )
t yy ( 1 ) = z , z t yx 0 z t xx z z t xy ( z L ) +
t 1 = [ cos ( nkL ) i 2 ( z + 1 z ) sin ( nkL ) ] ,
r t = i 2 ( z 1 z ) sin ( nkL ) ,
= n z and μ = n z .
z = ± ( 1 + r ) 2 t 2 ( 1 r ) 2 t 2
z > 0 ,
cos ( nkL ) = X = 1 2 t ( 1 r 2 + t 2 ) .
e n kL [ cos ( n k L ) + i sin ( n k L ) ] = Y = X ± 1 X 2 .
n > 0 ,
t xy t yy .
n 2 n 2 = μ μ
2 n n = μ + μ
Q = 1 2 π dωω H 2 × 2 n ( ω ) z ( ω )

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