Abstract

The increasing interest in metamaterials with negative refractive index has been prompted by a variety of promising optical and microwave applications. Often, the resulting electromagnetic problems to be solve are not analytically derivable; therefore, numerical modeling must be employed and the Transmission Line Modeling (TLM) method constitutes a possible choice. After having greatly simplified the existing TLM techniques for the modeling of metamaterials, we propose in this paper to carry out a frequency study of cloaking structure.

© 2008 Optical Society of America

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  1. P. B. Johns and R. L. Beurle, "Numerical solution of 2-dimensional scattering problems using a transmission-line matrix," Proc. Inst. Elec. Eng. 118, 1203-1208 (1971).
    [CrossRef]
  2. C. Christopoulos, The Transmission-Line Modeling Method, The Institute of Electrical and Electronic Engineers (New York and Oxford University Press, Oxford, 1995).
    [CrossRef]
  3. C. Blanchard, J. A. Portí, J. A. Morente, A. Salinas, and E. A. Navarro, "Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method," J. Appl. Phys. 102, 064101 (2007).
    [CrossRef]
  4. P. P. M. So, H. Du, and W. J. R. Hoefer, "Modeling of metamaterials with negative refractive index using 2-D shunt and 3-D SCN TLM networks," IEEE Trans. Microwave Theory Tech. 53, 1496-1505 (2005).
    [CrossRef]
  5. C. Blanchard, J. Portí, B-I Wu, J. A. Morente, A. Salinas, and J. A. Kong, "Time domain simulation of electromagnetic cloaking structure with TLM method," Opt. Express 16, 6461-6470 (2008).
    [CrossRef] [PubMed]
  6. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
    [CrossRef] [PubMed]
  7. U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
    [CrossRef] [PubMed]
  8. 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]
  9. Y. Huang, Y. Feng, and T. Jiang, "Electromagnetic cloaking by layered structure of homogeneous isotropic materials," Opt. Express 15, 11133 (2007).
    [CrossRef] [PubMed]
  10. J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, and C. Blanchard, "On the circuit description of TLM nodes," Int. J. Electron. 93, 479-491 (2006).
    [CrossRef]
  11. J. A. Portí, J. A. Morente, and M. C. Carrión, "Simple derivation of scattering matrix for TLM nodes," Electron. Lett. 34, 1763-1764 (1998).
    [CrossRef]
  12. P. B. Johns, "A symmetrical condensed node for the TLM method," IEEE Trans. Microwave Theory Tech. 35, 370-377 (1987).
    [CrossRef]
  13. Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
    [CrossRef]

2008

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

C. Blanchard, J. Portí, B-I Wu, J. A. Morente, A. Salinas, and J. A. Kong, "Time domain simulation of electromagnetic cloaking structure with TLM method," Opt. Express 16, 6461-6470 (2008).
[CrossRef] [PubMed]

2007

Y. Huang, Y. Feng, and T. Jiang, "Electromagnetic cloaking by layered structure of homogeneous isotropic materials," Opt. Express 15, 11133 (2007).
[CrossRef] [PubMed]

C. Blanchard, J. A. Portí, J. A. Morente, A. Salinas, and E. A. Navarro, "Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method," J. Appl. Phys. 102, 064101 (2007).
[CrossRef]

2006

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, and C. Blanchard, "On the circuit description of TLM nodes," Int. J. Electron. 93, 479-491 (2006).
[CrossRef]

2005

P. P. M. So, H. Du, and W. J. R. Hoefer, "Modeling of metamaterials with negative refractive index using 2-D shunt and 3-D SCN TLM networks," IEEE Trans. Microwave Theory Tech. 53, 1496-1505 (2005).
[CrossRef]

2002

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]

1998

J. A. Portí, J. A. Morente, and M. C. Carrión, "Simple derivation of scattering matrix for TLM nodes," Electron. Lett. 34, 1763-1764 (1998).
[CrossRef]

1987

P. B. Johns, "A symmetrical condensed node for the TLM method," IEEE Trans. Microwave Theory Tech. 35, 370-377 (1987).
[CrossRef]

1971

P. B. Johns and R. L. Beurle, "Numerical solution of 2-dimensional scattering problems using a transmission-line matrix," Proc. Inst. Elec. Eng. 118, 1203-1208 (1971).
[CrossRef]

Beurle, R. L.

P. B. Johns and R. L. Beurle, "Numerical solution of 2-dimensional scattering problems using a transmission-line matrix," Proc. Inst. Elec. Eng. 118, 1203-1208 (1971).
[CrossRef]

Blanchard, C.

C. Blanchard, J. Portí, B-I Wu, J. A. Morente, A. Salinas, and J. A. Kong, "Time domain simulation of electromagnetic cloaking structure with TLM method," Opt. Express 16, 6461-6470 (2008).
[CrossRef] [PubMed]

C. Blanchard, J. A. Portí, J. A. Morente, A. Salinas, and E. A. Navarro, "Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method," J. Appl. Phys. 102, 064101 (2007).
[CrossRef]

J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, and C. Blanchard, "On the circuit description of TLM nodes," Int. J. Electron. 93, 479-491 (2006).
[CrossRef]

Carrión, M. C.

J. A. Portí, J. A. Morente, and M. C. Carrión, "Simple derivation of scattering matrix for TLM nodes," Electron. Lett. 34, 1763-1764 (1998).
[CrossRef]

Du, H.

P. P. M. So, H. Du, and W. J. R. Hoefer, "Modeling of metamaterials with negative refractive index using 2-D shunt and 3-D SCN TLM networks," IEEE Trans. Microwave Theory Tech. 53, 1496-1505 (2005).
[CrossRef]

Eleftheriades, G. V.

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]

Feng, Y.

Hoefer, W. J. R.

P. P. M. So, H. Du, and W. J. R. Hoefer, "Modeling of metamaterials with negative refractive index using 2-D shunt and 3-D SCN TLM networks," IEEE Trans. Microwave Theory Tech. 53, 1496-1505 (2005).
[CrossRef]

Huang, Y.

Iyer, A. K.

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]

Jiang, T.

Jiang, X.

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Johns, P. B.

P. B. Johns, "A symmetrical condensed node for the TLM method," IEEE Trans. Microwave Theory Tech. 35, 370-377 (1987).
[CrossRef]

P. B. Johns and R. L. Beurle, "Numerical solution of 2-dimensional scattering problems using a transmission-line matrix," Proc. Inst. Elec. Eng. 118, 1203-1208 (1971).
[CrossRef]

Kong, J. A.

Kremer, P. C.

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]

Leonhardt, U.

U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

Liang, Z.

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Morente, J. A.

C. Blanchard, J. Portí, B-I Wu, J. A. Morente, A. Salinas, and J. A. Kong, "Time domain simulation of electromagnetic cloaking structure with TLM method," Opt. Express 16, 6461-6470 (2008).
[CrossRef] [PubMed]

C. Blanchard, J. A. Portí, J. A. Morente, A. Salinas, and E. A. Navarro, "Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method," J. Appl. Phys. 102, 064101 (2007).
[CrossRef]

J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, and C. Blanchard, "On the circuit description of TLM nodes," Int. J. Electron. 93, 479-491 (2006).
[CrossRef]

J. A. Portí, J. A. Morente, and M. C. Carrión, "Simple derivation of scattering matrix for TLM nodes," Electron. Lett. 34, 1763-1764 (1998).
[CrossRef]

Navarro, E. A.

C. Blanchard, J. A. Portí, J. A. Morente, A. Salinas, and E. A. Navarro, "Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method," J. Appl. Phys. 102, 064101 (2007).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

Portí, J.

Portí, J. A.

C. Blanchard, J. A. Portí, J. A. Morente, A. Salinas, and E. A. Navarro, "Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method," J. Appl. Phys. 102, 064101 (2007).
[CrossRef]

J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, and C. Blanchard, "On the circuit description of TLM nodes," Int. J. Electron. 93, 479-491 (2006).
[CrossRef]

J. A. Portí, J. A. Morente, and M. C. Carrión, "Simple derivation of scattering matrix for TLM nodes," Electron. Lett. 34, 1763-1764 (1998).
[CrossRef]

Rodríguez-Sola, M.

J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, and C. Blanchard, "On the circuit description of TLM nodes," Int. J. Electron. 93, 479-491 (2006).
[CrossRef]

Salinas, A.

C. Blanchard, J. Portí, B-I Wu, J. A. Morente, A. Salinas, and J. A. Kong, "Time domain simulation of electromagnetic cloaking structure with TLM method," Opt. Express 16, 6461-6470 (2008).
[CrossRef] [PubMed]

C. Blanchard, J. A. Portí, J. A. Morente, A. Salinas, and E. A. Navarro, "Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method," J. Appl. Phys. 102, 064101 (2007).
[CrossRef]

J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, and C. Blanchard, "On the circuit description of TLM nodes," Int. J. Electron. 93, 479-491 (2006).
[CrossRef]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

So, P. P. M.

P. P. M. So, H. Du, and W. J. R. Hoefer, "Modeling of metamaterials with negative refractive index using 2-D shunt and 3-D SCN TLM networks," IEEE Trans. Microwave Theory Tech. 53, 1496-1505 (2005).
[CrossRef]

Sun, X.

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Wu, B-I

Yao, P.

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Appl. Phys. Lett.

Z. Liang, P. Yao, X. Sun, and X. Jiang, "The physical picture and the essential elements of the dynamical process for dispersive cloaking structures," Appl. Phys. Lett. 92, 131118 (2008).
[CrossRef]

Electron. Lett.

J. A. Portí, J. A. Morente, and M. C. Carrión, "Simple derivation of scattering matrix for TLM nodes," Electron. Lett. 34, 1763-1764 (1998).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

P. B. Johns, "A symmetrical condensed node for the TLM method," IEEE Trans. Microwave Theory Tech. 35, 370-377 (1987).
[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]

P. P. M. So, H. Du, and W. J. R. Hoefer, "Modeling of metamaterials with negative refractive index using 2-D shunt and 3-D SCN TLM networks," IEEE Trans. Microwave Theory Tech. 53, 1496-1505 (2005).
[CrossRef]

Int. J. Electron.

J. A. Portí, J. A. Morente, A. Salinas, M. Rodríguez-Sola, and C. Blanchard, "On the circuit description of TLM nodes," Int. J. Electron. 93, 479-491 (2006).
[CrossRef]

J. Appl. Phys.

C. Blanchard, J. A. Portí, J. A. Morente, A. Salinas, and E. A. Navarro, "Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method," J. Appl. Phys. 102, 064101 (2007).
[CrossRef]

Opt. Express

Proc. Inst. Elec. Eng.

P. B. Johns and R. L. Beurle, "Numerical solution of 2-dimensional scattering problems using a transmission-line matrix," Proc. Inst. Elec. Eng. 118, 1203-1208 (1971).
[CrossRef]

Science

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

Other

C. Christopoulos, The Transmission-Line Modeling Method, The Institute of Electrical and Electronic Engineers (New York and Oxford University Press, Oxford, 1995).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Splitting of the TLM series node into 3 sub-circuits: (a) series sub-circuit for Hz , (b) parallel sub-circuits for Ey , and (c) parallel sub-circuits for Ex .

Fig. 2.
Fig. 2.

Splitting of the TLM parallel node into 3 sub-circuits: (a) parallel sub-circuit for Ez , (b) series sub-circuits for Hy , and (c) series sub-circuits for Hx .

Fig. 3.
Fig. 3.

(Online color) Far field pattern of a cloaking structure at 2GHz, and its Scattering Width versus frequency for 0°, 45°, 90°, 135°, 180°.

Equations (20)

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4 Z 0 Δ t 2 + Z z Z 0 Δ t 2 = μ z μ 0 Δ x Δ y Δ z , 2 Y 0 Δ t 2 + Y y Y 0 Δ t 2 = ε y ε 0 Δ x Δ z Δ y , 2 Y 0 Δ t 2 + Y x Y 0 Δ t 2 = ε x ε 0 Δ y Δ z Δ x .
[ a x c b x c g i x 0 c a y c b y g 0 i y b x c a x c g i x 0 c b y c a y g 0 i y d d d d f 0 0 e x 0 e x 0 0 h x 0 0 e y 0 e y 0 0 h y ] ,
a j = 2 2 + Y j 2 4 + Z z , c = 2 4 + Z z , e j = 2 2 + Y j , g = 2 4 + Z z , i j = 2 Y j Y j + 2 .
b j = Y j 2 + Y j + 2 4 + Z z , d = 2 Z z 4 + Z z , f = 4 Z z 4 + Z z , h j = Y j 2 Y j + 2 , ( with j = { x , y } )
H z = 2 ( V 1 i + V 2 i + V 3 i V 4 i + V 5 i ) Δ z Z 0 ( 4 + Z z ) , E y = 2 ( V 2 i + V 4 i + Y y V 7 i ) Δ y ( 2 + Y y ) , E x = 2 ( V 1 i + V 3 i + Y x V 6 i ) Δ x ( 2 + Y x ) .
V 1 i = 0.5 ( Δ x E x Z 0 Δ z H z ) V 5 i = 0.5 Z z Z 0 Δ z H z
V 2 i = 0.5 ( Δ y E y + Z 0 Δ z H z ) V 6 i = 0.5 Δ x E x
V 3 i = 0.5 ( Δ x E x + Z 0 Δ z H z ) V 7 i = 0.5 Δ y E y .
V 4 i = 0.5 ( Δ y E y Z 0 Δ z H z )
Z z = Δ t 2 ω 2 4 [ 2 μ z μ 0 Δ t Z 0 Δ x Δ y Δ z 4 ] , Y y = Δ t 2 ω 2 4 [ ε y ε 0 Δ t Y 0 Δ x Δ z Δ y 4 ] , Y x = Δ t 2 ω 2 4 [ ε x ε 0 Δ t Y 0 Δ y Δ z Δ x 4 ] .
4 Y 0 Δ t 2 + Y z Y 0 Δ t 2 = ε z ε 0 Δ x Δ y Δ z , 2 Z 0 Δ t 2 + Z y Z 0 Δ t 2 = μ y μ 0 Δ x Δ z Δ y , 2 Z 0 Δ t 2 + Z x Z 0 Δ t 2 = μ x μ 0 Δ y Δ z Δ x .
[ a x c b x c g i x 0 c a y c b y g 0 i y b x c a x c g i x 0 c b y c a y g 0 i y c c c c c f 0 e x 0 e x 0 0 h x 0 0 e y 0 e y 0 0 h y ] ,
a j = 2 4 + Y z 2 2 + Z j c = 2 4 + Y z f = Y z 4 Y z + 4 h j = 2 Z j 2 + Z j
b j = Y z 2 + Y z Z j 4 + Z j e j = 2 Z j 2 + Z j g = 2 Y z Y z + 4 i j = 2 2 + Z j . ( with j = { x , y } )
E z = 2 ( V 1 i + V 2 i + V 3 i + V 4 i + Y z V 5 i ) Δ z ( 4 + Y z ) , H y = 2 ( V 2 i + V 4 i + V 7 i ) Δ y Z 0 ( 2 + Z y ) , H x = 2 ( V 1 i V 3 i + V 6 i ) Δ x Z 0 ( 2 + Z x ) ,
V 1 i = 0.5 ( Δ z E z Z 0 Δ x H x ) V 5 i = 0.5 Δ z E z
V 2 i = 0.5 ( Δ z E z + Z 0 Δ y H y ) V 6 i = 0.5 Z x Z 0 Δ x H x
V 3 i = 0.5 ( Δ z E z + Z 0 Δ x H x ) V 7 i = 0.5 Z y Z 0 Δ y H y .
V 4 i = 0.5 ( Δ z E z + Z 0 Δ y H y )
Y z = Δ t 2 ω 2 4 [ 2 ε z ε 0 Δ t Y 0 Δ x Δ y Δ z 4 ] , Z y = Δ t 2 ω 2 4 [ μ y μ 0 Δ t Z 0 Δ x Δ z Δ y 4 ] , Z x = Δ t 2 ω 2 4 [ μ x μ 0 Δ t Z 0 Δ y Δ z Δ x 4 ] .

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