Abstract

We examine a new class of volumetric metamaterials based on two-dimensional (2D) transmission-line layers that exhibit a negative refractive index (NRI). The dispersion characteristics of a single 2D layer are revealed through a periodic analysis, and the effective-medium response of the volumetric layered topology is predicted by an intuitive equivalent circuit model. Dispersion and transmission characteristics are also obtained for various designs by using full-wave finite-element method (FEM) simulations, including one design meeting the requirements of Veselago’s slab lens in free space, and suggest an isotropic NRI over bandwidths anywhere from 25% to 45%. Finally, the potential to implement these metamaterials from terahertz to near-infrared and optical frequencies is discussed.

© 2006 Optical Society of America

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  2. W. E. Kock, "Radio lenses," Bell Lab. Rec. 24, 177-216 (1946).
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    [CrossRef] [PubMed]
  8. D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, "3D wire mesh photonic crystals," Phys. Rev. Lett. 76, 2480-2483 (1996).
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  9. R. N. Bracewell, "Analogues of an ionized medium: applications to the ionosphere," Wirel. Eng. 31, 320-326 (1954).
  10. J. B. 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).
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  12. G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
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    [CrossRef]
  23. A. Grbic and G. V. Eleftheriades, "Periodic analysis of a 2-D negative refractive index transmission line structure," Special Issue on Metamaterials, IEEE Trans. Antennas Propag. 51, 2604-2611 (2003).
    [CrossRef]
  24. A. K. Iyer, K. G. Balmain, and G. V. Eleftheriades, "Dispersion analysis of resonance cone behaviour in magnetically anisotropic transmission-line metamaterials," in 2004 IEEE Antennas and Propagation Society International Symposium Digest (IEEE, 2004), pp. 3147-3150.
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    [CrossRef]
  26. A. K. Iyer and G. V. Eleftheriades, "Negative-refractive-index transmission-line metamaterials," in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G.V.Eleftheriades and K.G.Balmain, eds. (Wiley-IEEE, 2005), pp. 1-52.
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    [CrossRef]
  29. M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar, "Resonant frequencies of a split-ring resonator: analytical solutions and numerical simulations," Microwave Opt. Technol. Lett. 44, 133-136 (2005).
    [CrossRef]
  30. G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microw. Wirel. Compon. Lett. 13, 51-53 (2003).
    [CrossRef]
  31. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  32. A. Alù and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling, and transparency," Special Issue on Metamaterials, IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
    [CrossRef]
  33. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  34. A. Grbic and G. V. Eleftheriades, "Overcoming the diffraction limit with a planar left-handed transmission-line lens," Phys. Rev. Lett. 92, 117403 (2004).
    [CrossRef] [PubMed]
  35. A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades, "Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial," Opt. Express 11, 696-708 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696.
    [CrossRef] [PubMed]
  36. G. Shvets, "Photonic approach to making a material with a negative index of refraction," Phys. Rev. B 67, 035109 (2003).
    [CrossRef]
  37. S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).
    [CrossRef]

2005 (5)

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors," Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef] [PubMed]

A. Grbic and G. V. Eleftheriades, "An isotropic three-dimensional negative-refractive-index transmission-line metamaterial," J. Appl. Phys. 98, 043106 (2005).
[CrossRef]

F. Elek and G. V. Eleftheriades, "A two-dimensional uniplanar transmission-line metamaterial with a negative index of refraction," New J. Phys. 7, 163 (2005).
[CrossRef]

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar, "Resonant frequencies of a split-ring resonator: analytical solutions and numerical simulations," Microwave Opt. Technol. Lett. 44, 133-136 (2005).
[CrossRef]

2004 (3)

A. Grbic and G. V. Eleftheriades, "Overcoming the diffraction limit with a planar left-handed transmission-line lens," Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

A. Sanada, C. Caloz, and T. Itoh, "Characteristics of the composite right/left-handed transmission lines," IEEE Microw. Wirel. Compon. Lett. 14, 68-70 (2004).
[CrossRef]

A. Sanada, C. Caloz, and T. Itoh, "Planar distributed structures with negative refractive index," IEEE Trans. Microwave Theory Tech. 52, 1252-1263 (2004).
[CrossRef]

2003 (6)

A. Grbic and G. V. Eleftheriades, "Periodic analysis of a 2-D negative refractive index transmission line structure," Special Issue on Metamaterials, IEEE Trans. Antennas Propag. 51, 2604-2611 (2003).
[CrossRef]

A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades, "Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial," Opt. Express 11, 696-708 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696.
[CrossRef] [PubMed]

G. Shvets, "Photonic approach to making a material with a negative index of refraction," Phys. Rev. B 67, 035109 (2003).
[CrossRef]

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).
[CrossRef]

G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microw. Wirel. Compon. Lett. 13, 51-53 (2003).
[CrossRef]

A. Alù and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling, and transparency," Special Issue on Metamaterials, IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

2002 (1)

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000 (2)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

1999 (1)

J. B. 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]

1996 (2)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, "3D wire mesh photonic crystals," Phys. Rev. Lett. 76, 2480-2483 (1996).
[CrossRef] [PubMed]

1980 (1)

C. R. Brewitt-Taylor and P. B. Johns, "On the construction and numerical solution of transmission-line and lumped network models of Maxwell's equations," Int. J. Numer. Methods Eng. 15, 13-30 (1980).
[CrossRef]

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and µ," Sov. Phys. Usp. 10, 509-514 (1968) [translation based on the original Russian document, dated 1967].
[CrossRef]

1962 (1)

W. Rotman, "Plasma simulation by artificial dielectrics and parallel-plate media," IRE Trans. Antennas Propag. AP-10, 82-85 (1962).
[CrossRef]

1954 (1)

R. N. Bracewell, "Analogues of an ionized medium: applications to the ionosphere," Wirel. Eng. 31, 320-326 (1954).

1948 (1)

W. E. Kock, "Metallic delay lenses," Bell Syst. Tech. J. 27, 58-82 (1948).

1946 (1)

W. E. Kock, "Radio lenses," Bell Lab. Rec. 24, 177-216 (1946).

1944 (2)

G. Kron, "Equivalent circuit of the field equations of Maxwell," Proc. IRE 32, 289-299 (1944).
[CrossRef]

J. R. Whinnery and S. Ramo, "A new approach to the solution of high-frequency field problems," Proc. IRE 32, 284-288 (1944).
[CrossRef]

Alù, A.

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors," Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef] [PubMed]

A. Alù and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling, and transparency," Special Issue on Metamaterials, IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Atwater, H. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Baena, J. D.

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

Balmain, K. G.

A. K. Iyer, K. G. Balmain, and G. V. Eleftheriades, "Dispersion analysis of resonance cone behaviour in magnetically anisotropic transmission-line metamaterials," in 2004 IEEE Antennas and Propagation Society International Symposium Digest (IEEE, 2004), pp. 3147-3150.

Bonache, J.

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, "Analogues of an ionized medium: applications to the ionosphere," Wirel. Eng. 31, 320-326 (1954).

Brewitt-Taylor, C. R.

C. R. Brewitt-Taylor and P. B. Johns, "On the construction and numerical solution of transmission-line and lumped network models of Maxwell's equations," Int. J. Numer. Methods Eng. 15, 13-30 (1980).
[CrossRef]

Caloz, C.

A. Sanada, C. Caloz, and T. Itoh, "Planar distributed structures with negative refractive index," IEEE Trans. Microwave Theory Tech. 52, 1252-1263 (2004).
[CrossRef]

A. Sanada, C. Caloz, and T. Itoh, "Characteristics of the composite right/left-handed transmission lines," IEEE Microw. Wirel. Compon. Lett. 14, 68-70 (2004).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (Wiley-IEEE, 1990).
[CrossRef]

Eleftheriades, G. V.

F. Elek and G. V. Eleftheriades, "A two-dimensional uniplanar transmission-line metamaterial with a negative index of refraction," New J. Phys. 7, 163 (2005).
[CrossRef]

A. Grbic and G. V. Eleftheriades, "An isotropic three-dimensional negative-refractive-index transmission-line metamaterial," J. Appl. Phys. 98, 043106 (2005).
[CrossRef]

A. Grbic and G. V. Eleftheriades, "Overcoming the diffraction limit with a planar left-handed transmission-line lens," Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microw. Wirel. Compon. Lett. 13, 51-53 (2003).
[CrossRef]

A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades, "Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial," Opt. Express 11, 696-708 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696.
[CrossRef] [PubMed]

A. Grbic and G. V. Eleftheriades, "Periodic analysis of a 2-D negative refractive index transmission line structure," Special Issue on Metamaterials, IEEE Trans. Antennas Propag. 51, 2604-2611 (2003).
[CrossRef]

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

A. K. Iyer, K. G. Balmain, and G. V. Eleftheriades, "Dispersion analysis of resonance cone behaviour in magnetically anisotropic transmission-line metamaterials," in 2004 IEEE Antennas and Propagation Society International Symposium Digest (IEEE, 2004), pp. 3147-3150.

A. Grbic and G. V. Eleftheriades, "Super-resolving negative-refractive-index transmission-line lenses," in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G.V.Eleftheriades and K.G.Balmain, eds. (Wiley-IEEE, 2005), pp. 93-170.
[CrossRef]

A. K. Iyer and G. V. Eleftheriades, "Negative-refractive-index transmission-line metamaterials," in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G.V.Eleftheriades and K.G.Balmain, eds. (Wiley-IEEE, 2005), pp. 1-52.
[CrossRef]

Elek, F.

F. Elek and G. V. Eleftheriades, "A two-dimensional uniplanar transmission-line metamaterial with a negative index of refraction," New J. Phys. 7, 163 (2005).
[CrossRef]

Engheta, N.

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors," Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef] [PubMed]

A. Alù and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling, and transparency," Special Issue on Metamaterials, IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Falcone, F.

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

García-García, J.

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

Gil, I.

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

Grbic, A.

A. Grbic and G. V. Eleftheriades, "An isotropic three-dimensional negative-refractive-index transmission-line metamaterial," J. Appl. Phys. 98, 043106 (2005).
[CrossRef]

A. Grbic and G. V. Eleftheriades, "Overcoming the diffraction limit with a planar left-handed transmission-line lens," Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

A. Grbic and G. V. Eleftheriades, "Periodic analysis of a 2-D negative refractive index transmission line structure," Special Issue on Metamaterials, IEEE Trans. Antennas Propag. 51, 2604-2611 (2003).
[CrossRef]

A. Grbic and G. V. Eleftheriades, "Super-resolving negative-refractive-index transmission-line lenses," in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G.V.Eleftheriades and K.G.Balmain, eds. (Wiley-IEEE, 2005), pp. 93-170.
[CrossRef]

Hoefer, W. J.

W. J. Hoefer, P. P. So, D. Thompson, and M. M. Tentzeris, "Topology and design of wideband 3D metamaterials made of periodically loaded transmission line arrays," presented at the IEEE Microwave Theory and Techniques Society International Microwave Symposium, Long Beach, Calif., June 12-17, 2005.

Holden, A. J.

J. B. 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. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Itoh, T.

A. Sanada, C. Caloz, and T. Itoh, "Planar distributed structures with negative refractive index," IEEE Trans. Microwave Theory Tech. 52, 1252-1263 (2004).
[CrossRef]

A. Sanada, C. Caloz, and T. Itoh, "Characteristics of the composite right/left-handed transmission lines," IEEE Microw. Wirel. Compon. Lett. 14, 68-70 (2004).
[CrossRef]

Iyer, A. K.

A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades, "Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial," Opt. Express 11, 696-708 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-696.
[CrossRef] [PubMed]

G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microw. Wirel. Compon. Lett. 13, 51-53 (2003).
[CrossRef]

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

A. K. Iyer and G. V. Eleftheriades, "Negative-refractive-index transmission-line metamaterials," in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G.V.Eleftheriades and K.G.Balmain, eds. (Wiley-IEEE, 2005), pp. 1-52.
[CrossRef]

A. K. Iyer, K. G. Balmain, and G. V. Eleftheriades, "Dispersion analysis of resonance cone behaviour in magnetically anisotropic transmission-line metamaterials," in 2004 IEEE Antennas and Propagation Society International Symposium Digest (IEEE, 2004), pp. 3147-3150.

Johns, P. B.

C. R. Brewitt-Taylor and P. B. Johns, "On the construction and numerical solution of transmission-line and lumped network models of Maxwell's equations," Int. J. Numer. Methods Eng. 15, 13-30 (1980).
[CrossRef]

Kalinin, V.

M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar, "Resonant frequencies of a split-ring resonator: analytical solutions and numerical simulations," Microwave Opt. Technol. Lett. 44, 133-136 (2005).
[CrossRef]

Kik, P. G.

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Kock, W. E.

W. E. Kock, "Metallic delay lenses," Bell Syst. Tech. J. 27, 58-82 (1948).

W. E. Kock, "Radio lenses," Bell Lab. Rec. 24, 177-216 (1946).

W. E. Kock, "Metal lens antennas," in Proceedings of IRE and Waves and Electrons (Institute of Radio Engineers, 1946), pp. 828-836.

Kremer, P. C.

Kron, G.

G. Kron, "Equivalent circuit of the field equations of Maxwell," Proc. IRE 32, 289-299 (1944).
[CrossRef]

Laso, M. A.

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

Lopetegi, T.

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

Maier, S. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Martin, F.

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

J. B. 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. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Portillo, M. F.

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

Ramo, S.

J. R. Whinnery and S. Ramo, "A new approach to the solution of high-frequency field problems," Proc. IRE 32, 284-288 (1944).
[CrossRef]

Robbins, D. J.

J. B. 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]

Rotman, W.

W. Rotman, "Plasma simulation by artificial dielectrics and parallel-plate media," IRE Trans. Antennas Propag. AP-10, 82-85 (1962).
[CrossRef]

Salandrino, A.

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors," Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef] [PubMed]

Sanada, A.

A. Sanada, C. Caloz, and T. Itoh, "Planar distributed structures with negative refractive index," IEEE Trans. Microwave Theory Tech. 52, 1252-1263 (2004).
[CrossRef]

A. Sanada, C. Caloz, and T. Itoh, "Characteristics of the composite right/left-handed transmission lines," IEEE Microw. Wirel. Compon. Lett. 14, 68-70 (2004).
[CrossRef]

Sarychev, A. K.

A. K. Sarychev and V. M. Shalaev, "Plasmonic nanowire materials," in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G.V.Eleftheriades and K.G.Balmain, eds. (Wiley-IEEE, 2005), pp. 313-338.
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Shalaev, V. M.

A. K. Sarychev and V. M. Shalaev, "Plasmonic nanowire materials," in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G.V.Eleftheriades and K.G.Balmain, eds. (Wiley-IEEE, 2005), pp. 313-338.
[CrossRef]

Shamonin, M.

M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar, "Resonant frequencies of a split-ring resonator: analytical solutions and numerical simulations," Microwave Opt. Technol. Lett. 44, 133-136 (2005).
[CrossRef]

Shamonina, E.

M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar, "Resonant frequencies of a split-ring resonator: analytical solutions and numerical simulations," Microwave Opt. Technol. Lett. 44, 133-136 (2005).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shvets, G.

G. Shvets, "Photonic approach to making a material with a negative index of refraction," Phys. Rev. B 67, 035109 (2003).
[CrossRef]

Sickmiller, M. E.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, "3D wire mesh photonic crystals," Phys. Rev. Lett. 76, 2480-2483 (1996).
[CrossRef] [PubMed]

Siddiqui, O.

G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microw. Wirel. Compon. Lett. 13, 51-53 (2003).
[CrossRef]

Sievenpiper, D. F.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, "3D wire mesh photonic crystals," Phys. Rev. Lett. 76, 2480-2483 (1996).
[CrossRef] [PubMed]

Sillero, R. Marqués

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

Simons, R.

R. Simons, Coplanar Waveguide Circuits, Components, and Systems (Wiley, 2001).
[CrossRef]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

So, P. P.

W. J. Hoefer, P. P. So, D. Thompson, and M. M. Tentzeris, "Topology and design of wideband 3D metamaterials made of periodically loaded transmission line arrays," presented at the IEEE Microwave Theory and Techniques Society International Microwave Symposium, Long Beach, Calif., June 12-17, 2005.

Solymar, L.

M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar, "Resonant frequencies of a split-ring resonator: analytical solutions and numerical simulations," Microwave Opt. Technol. Lett. 44, 133-136 (2005).
[CrossRef]

Sorolla, M.

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

Stewart, W. J.

J. B. 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. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Tentzeris, M. M.

W. J. Hoefer, P. P. So, D. Thompson, and M. M. Tentzeris, "Topology and design of wideband 3D metamaterials made of periodically loaded transmission line arrays," presented at the IEEE Microwave Theory and Techniques Society International Microwave Symposium, Long Beach, Calif., June 12-17, 2005.

Thompson, D.

W. J. Hoefer, P. P. So, D. Thompson, and M. M. Tentzeris, "Topology and design of wideband 3D metamaterials made of periodically loaded transmission line arrays," presented at the IEEE Microwave Theory and Techniques Society International Microwave Symposium, Long Beach, Calif., June 12-17, 2005.

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and µ," Sov. Phys. Usp. 10, 509-514 (1968) [translation based on the original Russian document, dated 1967].
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Whinnery, J. R.

J. R. Whinnery and S. Ramo, "A new approach to the solution of high-frequency field problems," Proc. IRE 32, 284-288 (1944).
[CrossRef]

Yablonovitch, E.

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, "3D wire mesh photonic crystals," Phys. Rev. Lett. 76, 2480-2483 (1996).
[CrossRef] [PubMed]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Bell Lab. Rec. (1)

W. E. Kock, "Radio lenses," Bell Lab. Rec. 24, 177-216 (1946).

Bell Syst. Tech. J. (1)

W. E. Kock, "Metallic delay lenses," Bell Syst. Tech. J. 27, 58-82 (1948).

IEEE Microw. Wirel. Compon. Lett. (2)

A. Sanada, C. Caloz, and T. Itoh, "Characteristics of the composite right/left-handed transmission lines," IEEE Microw. Wirel. Compon. Lett. 14, 68-70 (2004).
[CrossRef]

G. V. Eleftheriades, O. Siddiqui, and A. K. Iyer, "Transmission line models for negative refractive index media and associated implementations without excess resonators," IEEE Microw. Wirel. Compon. Lett. 13, 51-53 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

A. Alù and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling, and transparency," Special Issue on Metamaterials, IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

A. Grbic and G. V. Eleftheriades, "Periodic analysis of a 2-D negative refractive index transmission line structure," Special Issue on Metamaterials, IEEE Trans. Antennas Propag. 51, 2604-2611 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (4)

J. D. Baena, J. Bonache, F. Martin, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, and M. Sorolla, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microwave Theory Tech. 53, 1451-1461 (2005).
[CrossRef]

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, "Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

A. Sanada, C. Caloz, and T. Itoh, "Planar distributed structures with negative refractive index," IEEE Trans. Microwave Theory Tech. 52, 1252-1263 (2004).
[CrossRef]

J. B. 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]

Int. J. Numer. Methods Eng. (1)

C. R. Brewitt-Taylor and P. B. Johns, "On the construction and numerical solution of transmission-line and lumped network models of Maxwell's equations," Int. J. Numer. Methods Eng. 15, 13-30 (1980).
[CrossRef]

IRE Trans. Antennas Propag. (1)

W. Rotman, "Plasma simulation by artificial dielectrics and parallel-plate media," IRE Trans. Antennas Propag. AP-10, 82-85 (1962).
[CrossRef]

J. Appl. Phys. (1)

A. Grbic and G. V. Eleftheriades, "An isotropic three-dimensional negative-refractive-index transmission-line metamaterial," J. Appl. Phys. 98, 043106 (2005).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

M. Shamonin, E. Shamonina, V. Kalinin, and L. Solymar, "Resonant frequencies of a split-ring resonator: analytical solutions and numerical simulations," Microwave Opt. Technol. Lett. 44, 133-136 (2005).
[CrossRef]

New J. Phys. (1)

F. Elek and G. V. Eleftheriades, "A two-dimensional uniplanar transmission-line metamaterial with a negative index of refraction," New J. Phys. 7, 163 (2005).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (2)

G. Shvets, "Photonic approach to making a material with a negative index of refraction," Phys. Rev. B 67, 035109 (2003).
[CrossRef]

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Phys. Rev. Lett. (6)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

A. Grbic and G. V. Eleftheriades, "Overcoming the diffraction limit with a planar left-handed transmission-line lens," Phys. Rev. Lett. 92, 117403 (2004).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
[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, 4773-4776 (1996).
[CrossRef] [PubMed]

D. F. Sievenpiper, M. E. Sickmiller, and E. Yablonovitch, "3D wire mesh photonic crystals," Phys. Rev. Lett. 76, 2480-2483 (1996).
[CrossRef] [PubMed]

N. Engheta, A. Salandrino, and A. Alù, "Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors," Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef] [PubMed]

Proc. IRE (2)

G. Kron, "Equivalent circuit of the field equations of Maxwell," Proc. IRE 32, 289-299 (1944).
[CrossRef]

J. R. Whinnery and S. Ramo, "A new approach to the solution of high-frequency field problems," Proc. IRE 32, 284-288 (1944).
[CrossRef]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and µ," Sov. Phys. Usp. 10, 509-514 (1968) [translation based on the original Russian document, dated 1967].
[CrossRef]

Wirel. Eng. (1)

R. N. Bracewell, "Analogues of an ionized medium: applications to the ionosphere," Wirel. Eng. 31, 320-326 (1954).

Other (8)

W. E. Kock, "Metal lens antennas," in Proceedings of IRE and Waves and Electrons (Institute of Radio Engineers, 1946), pp. 828-836.

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (Wiley-IEEE, 1990).
[CrossRef]

A. K. Sarychev and V. M. Shalaev, "Plasmonic nanowire materials," in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G.V.Eleftheriades and K.G.Balmain, eds. (Wiley-IEEE, 2005), pp. 313-338.
[CrossRef]

A. Grbic and G. V. Eleftheriades, "Super-resolving negative-refractive-index transmission-line lenses," in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G.V.Eleftheriades and K.G.Balmain, eds. (Wiley-IEEE, 2005), pp. 93-170.
[CrossRef]

W. J. Hoefer, P. P. So, D. Thompson, and M. M. Tentzeris, "Topology and design of wideband 3D metamaterials made of periodically loaded transmission line arrays," presented at the IEEE Microwave Theory and Techniques Society International Microwave Symposium, Long Beach, Calif., June 12-17, 2005.

A. K. Iyer, K. G. Balmain, and G. V. Eleftheriades, "Dispersion analysis of resonance cone behaviour in magnetically anisotropic transmission-line metamaterials," in 2004 IEEE Antennas and Propagation Society International Symposium Digest (IEEE, 2004), pp. 3147-3150.

A. K. Iyer and G. V. Eleftheriades, "Negative-refractive-index transmission-line metamaterials," in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G.V.Eleftheriades and K.G.Balmain, eds. (Wiley-IEEE, 2005), pp. 1-52.
[CrossRef]

R. Simons, Coplanar Waveguide Circuits, Components, and Systems (Wiley, 2001).
[CrossRef]

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Figures (23)

Fig. 1
Fig. 1

(a) Two-dimensional series TL unit cell. (b) Two-dimensional shunt TL unit cell.

Fig. 2
Fig. 2

Creating a volumetric medium by layering 2D planes.

Fig. 3
Fig. 3

Array of 2D series TL unit cells with generalized lumped loading. The lightly shaded region views the unit cell as the series interconnection of four two-wire lines, and the darkly shaded region views the unit cell as a loaded ring. The magnetic field of the impinging plane wave is normal to the plane of the page.

Fig. 4
Fig. 4

Definition of the ABCD transfer matrix for a two-port network.

Fig. 5
Fig. 5

Periodic analysis of the generalized series TL unit cell.

Fig. 6
Fig. 6

Specification of the constituents of the branches of the series TL unit cell depicted in Fig. 5.

Fig. 7
Fig. 7

Brillouin-zone boundary for a 2D rectangular lattice indicating high-symmetry points.

Fig. 8
Fig. 8

Series TL node array loaded in a dual configuration. The arrows indicate the current directions.

Fig. 9
Fig. 9

Axial propagation in the 2D series NRI-TL node array. S. C., short-circuit planes.

Fig. 10
Fig. 10

Representative series NRI-TL dispersion relations for axial propagation ( μ p = μ 0 , ϵ p = 3 ϵ 0 , g = 0.325 , C 0 = 1 pF , d = 5 mm ) : (a) L 0 = 3 nH , impedance-mismatched (open-stop-band) case; (b) L 0 = 10 nH , impedance-matched (closed-stop-band) case.

Fig. 11
Fig. 11

Series TL unit cells: (a) unloaded, (b) shunt inductors, (c) series capacitors, (d) composite series NRI-TL unit cell.

Fig. 12
Fig. 12

(Color online) Volumetric NRI-TL unit cell for an infinite array excited by a TE z -polarized plane wave propagating in the y direction.

Fig. 13
Fig. 13

Interactions between a normally incident TE-polarized plane wave and the volumetric layered NRI-TL medium: (a) schematic representation of the unit cell, (b) equivalent circuit model of the unit cell.

Fig. 14
Fig. 14

Axial dispersion relations corresponding to the series unit-cell topologies of Figs. 11a, 11b, 11c, 11d and values reported in Table 3, obtained using the HFSS finite-element solver (dots) and equivalent circuit model (dashed curves): (a) unloaded, (b) discrete lumped shunt inductors, (c) discrete lumped series capacitors, (d) composite series NRI-TL unit cell.

Fig. 15
Fig. 15

Axial dispersion relation for a volumetric layered NRI-TL structure based on the design in Table 3 but with L 0 increased from 5.6 nH to approximately 10 nH to meet the impedance-matched condition of Eq. (29); obtained using the HFSS finite-element solver (dots) and equivalent circuit model (dashed curves).

Fig. 16
Fig. 16

Fully printed composite series NRI-TL unit cell employing interdigitated capacitors and strip inductors.

Fig. 17
Fig. 17

Dispersion relations corresponding to the series unit-cell topologies of Figs. 11a, 11b, 11c, 11d employing printed elements instead of discrete elements, obtained using the HFSS finite-element solver (dots): (a) unloaded, (b) printed lumped shunt inductors, (c) printed lumped series capacitors, (d) composite fully printed series NRI-TL unit cell.

Fig. 18
Fig. 18

Two-dimensional reduced Brillouin zone for the NRI band obtained using HFSS. The dispersion contours indicate that isotropy is achieved in the approach to the Γ point (the homogeneous limit).

Fig. 19
Fig. 19

(Color online) (a) Rectangular unit cell for a triangular series NRI-TL medium. (b) Axial dispersion relation for volumetric layered medium based on the unit cell in (a), taken in the direction of its long axis.

Fig. 20
Fig. 20

(Color online) (a) Volumetric layered NRI-TL slab of thickness three cells. (b) Representation of infinite slab for illumination by a TE plane wave at normal incidence using electric and magnetic walls.

Fig. 21
Fig. 21

(Color online) Printed lumped element design: (a) HFSS transmission phase for the slab arrangement of Fig. 20a (black dots) compared with the dispersion of the infinite structure (gray dots); (b) HFSS transmission ( S 21 —black curve) and reflection ( S 11 —gray curve) magnitudes.

Fig. 22
Fig. 22

(Color online) Discrete lumped element design matched to free space: (a) HFSS transmission phase for the slab arrangement of Fig. 20a (black dots) compared with the dispersion of the infinite structure (gray dots) and equivalent circuit mode (dashed curve); (b) HFSS transmission ( S 21 —black curve) and reflection ( S 11 —gray curve) magnitudes.

Fig. 23
Fig. 23

Possible optical implementations of the series NRI-TL array by using plasmonic nanoparticles for (a) square unit cells and (b) triangular unit cells.

Tables (4)

Tables Icon

Table 1 Description of the Symmetry Points of the 2D Brillouin Zone for a Rectangular Lattice a

Tables Icon

Table 2 Dispersion Relations for Isotropic and Anisotropic Cases a

Tables Icon

Table 3 Design Parameters Employed for a Volumetric Layered NRI-TL Medium by Using Discrete Lumped Elements

Tables Icon

Table 4 Design Parameters for Interdigitated Capacitors and Strip Inductors Employed in the Fully Printed NRI-TL Planar Unit Cell Depicted in Fig. 16

Equations (58)

Equations on this page are rendered with MathJax. Learn more.

[ V in I in ] = [ A B C D ] [ V out I out ] .
[ V 1 I 1 ] = [ V x I x ] ,
[ V 2 I 2 ] = [ V y I y ] ,
[ V 3 I 3 ] = [ V x I x ] exp ( j β x d ) ,
[ V 4 I 4 ] = [ V y I y ] exp ( j β y d ) .
A i = cos θ 2 ,
B i = Z i 4 cos θ 2 + j Z 0 sin θ 2 ,
C i = Y i 2 cos θ 2 + j Y 0 sin θ 2 ,
D i = ( 1 + Z i Y i 8 ) cos θ 2 + j 2 ( Z 2 Z 0 + Y Y 0 ) sin θ 2 ,
[ A 1 B 1 C 1 D 1 ] = [ A 3 B 3 C 3 D 3 ] = [ A x B x C x D x ] ,
[ A 2 B 2 C 2 D 2 ] = [ A 4 B 4 C 4 D 4 ] = [ A y B y C y D y ] .
{ A y C y [ ( 1 cos β x d ) + 2 B x C x ] + A x C x [ ( 1 cos β y d ) + 2 B y C y ] } = 0 .
Z B x = V x I x = j A x C x tan β x d 2 ,
Z B y = V y I y = j A y C y tan β y d 2 .
cos ( β x d ) 1 ( β x d ) 2 2 ,
cos ( β y d ) 1 ( β y d ) 2 2 .
( β d ) 2 = 4 C x C y { B x + B y } C x sin 2 ϕ + C y cos 2 ϕ .
( β d ) 2 = ( 4 sin θ 2 j Z Z 0 cos θ 2 ) ( 2 sin θ 2 j Y Y 0 cos θ 2 ) .
tan θ 2 = j Z 4 Z 0 ,
tan θ 2 = j Y 2 Y 0 .
Z B x = j A C β x d 2 = j A β d 2 C cos ϕ ,
Z B y = j A C β y d 2 = j A β d 2 C sin ϕ ,
Z B = ± Z B x 2 + Z B y 2 = ± j A β d 2 C ,
Z B x = Z B cos ϕ ,
Z B y = Z B sin ϕ .
β 2 = ω 2 L eff C eff ,
Z B 2 = L eff C eff L eff = β Z B ω C eff = β ω Z B .
L eff = 4 A B j ω d ,
C eff = 2 C j ω A d .
L eff = cos 2 θ 2 ( 4 Z 0 ω d tan θ 2 j Z ω d ) ,
C eff = ( 2 Y 0 ω d tan θ 2 j Y ω d ) .
L eff = 2 Z 0 θ ω d j Z ω d ,
C eff = Y 0 θ ω d j Y ω d .
θ = ω ϵ p μ p d ,
Z 0 = Y 0 1 = g p μ p ϵ p ,
L eff = 2 μ p g p j Z ω d ,
C eff = ϵ p g p j Y ω d ,
μ eff ( ω ) = L eff g eff = 2 μ p g p g eff p j Z g eff ω d ,
ϵ eff ( ω ) = C eff g eff = ϵ p g eff g p j Y g eff ω d .
μ eff ( ω ) = 2 μ p j Z g ω d ,
ϵ eff ( ω ) = ϵ p j Y g ω d .
μ eff ( ω ) = 2 μ p j ( j ω L 0 ) g ω d = 2 μ p + L 0 g d ,
ϵ eff ( ω ) = ϵ p j ( j ω C 0 ) g ω d = ϵ p + C 0 g d .
μ eff ( ω ) = 2 μ p j g ( j ω C 0 ) ω d = 2 μ p 1 ω 2 g C 0 d ,
ϵ eff ( ω ) = ϵ p j g ( j ω L 0 ) ω d = ϵ p g ω 2 L 0 d .
ω C , 1 = 1 2 μ p g C 0 d [ μ eff ( ω C , 1 ) = 0 ] ,
ω C , 2 = g ϵ p L 0 d [ ϵ eff ( ω C , 2 ) = 0 ] .
ω C , 1 = ω C , 2 L 0 C 0 = g 2 μ p ϵ p = 2 Z 0 = Z 0 , 2 D ,
Z 0 = η 0 ϵ r K ( k ) K ( k ) ,
k = [ 1 ( S S + 2 W ) 2 ] 1 2 ,
k = S S + 2 W ,
g = Z 0 ( η 0 ϵ r ) = K ( k ) K ( k ) .
μ v ( ω ) = j Z ω g wg ,
ϵ v ( ω ) = j Y g wg ω .
ω 0 = 1 L r C 0 ,
ω mp = 1 L r C 0 = 1 ( L r F 2 L wg ) C 0 ,
L 0 C 0 = L r C wg + C x d ,
ω mp = 1 L r C 0 = 1 L 0 ( C wg + C x d ) = ω x .

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