Abstract

The plane-wave scattering properties of a sphere of material having an ideal, homogeneous, and causal permittivity ε(f), and permeability μ(f) were investigated through detailed three-dimensional finite-difference time-domain, method-of-moments, and series-solution simulations. A Lorentzian functional form was chosen for ε(f) and μ(f), as it yields causal responses and allows us to study the physics of the left-handed-medium (LHM) regime. Our interest lies mainly in the frequency range where negative refraction [Re(n)<0] is observed. We found that when operating in the LHM regime, an impedance-matched sphere responds with scattering features strikingly different from those found in ordinary materials. In particular, we found zero backscattering and forward scattering that exceeds that of a metal sphere of similar size. The equality of E- and H-plane patterns was proved analytically and numerically, and the possibility of internal subwavelength focusing with a zero index sphere is also reported.

© 2004 Optical Society of America

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  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  2. J. B. Pendry, S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys. Condens. Matter 14, 8463–8479 (2002).
    [CrossRef]
  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  4. F. J. Rachford, D. L. Smith, P. F. Loschialpo, D. W. Forester, “Calculations and measurements of wire and/or split-ring negative index media,” Phys. Rev. E 66, 036613 (2002).
    [CrossRef]
  5. P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
    [CrossRef]
  6. P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left-handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
    [CrossRef]
  7. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
    [CrossRef] [PubMed]
  8. R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
    [CrossRef]
  9. R. A. Shelby, D. R. Smith, S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77 (2001).
    [CrossRef] [PubMed]
  10. J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and Enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
    [CrossRef]
  11. R. W. Ziolkowski, A. D. Kipple, “Causality and double-negative metamaterials,” Phys. Rev. E 68, 026615 (2003).
    [CrossRef]
  12. J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
    [CrossRef] [PubMed]
  13. D. R. Fredkin, A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753–1755 (2002).
    [CrossRef]
  14. P. R. Berman, “Goos–Hanchen shift in negatively refractive media,” Phys. Rev. E 66, 067603 (2002).
    [CrossRef]
  15. I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
    [CrossRef]
  16. M. K. Karkkainen, S. I. Maslovski, “Wave propagation, refraction and focusing phenomena in Lorentzian double-negative materials: a theoretical and numerical study,” Microwave Opt. Technol. Lett. 37, 4–7 (2003).
    [CrossRef]
  17. S. Foteinopoulou, E. N. Economou, C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
    [CrossRef] [PubMed]
  18. A. A. Houck, J. B. Brock, I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401 (2003).
    [CrossRef]
  19. D. R. Smith, D. C. Vier, N. Kroll, S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246–2248 (2000).
    [CrossRef]
  20. M. Kerker, D.-S. Wang, G. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. 73, 765–767 (1983).
    [CrossRef]
  21. R. Ruppin, “Extinction properties of a sphere with negative permittivity and permeability,” Solid State Commun. 116, 411–415 (2000).
    [CrossRef]
  22. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
    [CrossRef]
  23. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  24. The Finite Difference Time Domain simulations were performed with MAXTDA. MAXTDA, written at GTRI, was recently modified by Georgia Tech Research Institute (GTRI) to include causal Lorentzian functional forms for permittivity and permeability.
  25. L. N. Medgyesi-Mitschang, J. M. Putnam, M. B. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 11, 1383–1398 (1994).
    [CrossRef]
  26. J. M. Putnam, M. B. Gedera, “CARLOS-3D: a general-purpose 3-D method of moments scattering code,” IEEE Antennas Propag. Mag., April1993, pp. 69–71.
  27. V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963).
    [CrossRef]
  28. D. S. Jones, The Theory of Electromagnetism (MacMillan, New York, 1964).
  29. R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  30. A. Taflove, Computational Electrodynamics, The Finite-Difference Time-Domain Method (Artech House, Norwood, Mass., 1995).
  31. G. T. Ruck, Editor, Radar Cross Section Handbook (Plenum, New York, 1970).
  32. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere—Part I: direct reflection and transmission; Part II: theory of the rainbow and the glory,” J. Math. Phys. 10, 82–177 (1969).
    [CrossRef]

2004 (1)

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left-handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

2003 (6)

R. W. Ziolkowski, A. D. Kipple, “Causality and double-negative metamaterials,” Phys. Rev. E 68, 026615 (2003).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

M. K. Karkkainen, S. I. Maslovski, “Wave propagation, refraction and focusing phenomena in Lorentzian double-negative materials: a theoretical and numerical study,” Microwave Opt. Technol. Lett. 37, 4–7 (2003).
[CrossRef]

S. Foteinopoulou, E. N. Economou, C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[CrossRef] [PubMed]

A. A. Houck, J. B. Brock, I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

2002 (5)

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

D. R. Fredkin, A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753–1755 (2002).
[CrossRef]

P. R. Berman, “Goos–Hanchen shift in negatively refractive media,” Phys. Rev. E 66, 067603 (2002).
[CrossRef]

J. B. Pendry, S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys. Condens. Matter 14, 8463–8479 (2002).
[CrossRef]

F. J. Rachford, D. L. Smith, P. F. Loschialpo, D. W. Forester, “Calculations and measurements of wire and/or split-ring negative index media,” Phys. Rev. E 66, 036613 (2002).
[CrossRef]

2001 (3)

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

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

R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

2000 (4)

R. Ruppin, “Extinction properties of a sphere with negative permittivity and permeability,” Solid State Commun. 116, 411–415 (2000).
[CrossRef]

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

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

D. R. Smith, D. C. Vier, N. Kroll, S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246–2248 (2000).
[CrossRef]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and Enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

1994 (1)

1993 (1)

J. M. Putnam, M. B. Gedera, “CARLOS-3D: a general-purpose 3-D method of moments scattering code,” IEEE Antennas Propag. Mag., April1993, pp. 69–71.

1983 (1)

1969 (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere—Part I: direct reflection and transmission; Part II: theory of the rainbow and the glory,” J. Math. Phys. 10, 82–177 (1969).
[CrossRef]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

1963 (1)

V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Berman, P. R.

P. R. Berman, “Goos–Hanchen shift in negatively refractive media,” Phys. Rev. E 66, 067603 (2002).
[CrossRef]

Brock, J. B.

A. A. Houck, J. B. Brock, I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef]

Chuang, I. L.

A. A. Houck, J. B. Brock, I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef]

Economou, E. N.

S. Foteinopoulou, E. N. Economou, C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[CrossRef] [PubMed]

Forester, D. W.

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left-handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

F. J. Rachford, D. L. Smith, P. F. Loschialpo, D. W. Forester, “Calculations and measurements of wire and/or split-ring negative index media,” Phys. Rev. E 66, 036613 (2002).
[CrossRef]

Foteinopoulou, S.

S. Foteinopoulou, E. N. Economou, C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[CrossRef] [PubMed]

Fredkin, D. R.

D. R. Fredkin, A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753–1755 (2002).
[CrossRef]

Gedera, M. B.

L. N. Medgyesi-Mitschang, J. M. Putnam, M. B. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 11, 1383–1398 (1994).
[CrossRef]

J. M. Putnam, M. B. Gedera, “CARLOS-3D: a general-purpose 3-D method of moments scattering code,” IEEE Antennas Propag. Mag., April1993, pp. 69–71.

Giles, G. L.

Grzegorczyk, T. M.

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

Heyman, E.

R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and Enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Houck, A. A.

A. A. Houck, J. B. Brock, I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef]

Jones, D. S.

D. S. Jones, The Theory of Electromagnetism (MacMillan, New York, 1964).

Karkkainen, M. K.

M. K. Karkkainen, S. I. Maslovski, “Wave propagation, refraction and focusing phenomena in Lorentzian double-negative materials: a theoretical and numerical study,” Microwave Opt. Technol. Lett. 37, 4–7 (2003).
[CrossRef]

Kerker, M.

Kipple, A. D.

R. W. Ziolkowski, A. D. Kipple, “Causality and double-negative metamaterials,” Phys. Rev. E 68, 026615 (2003).
[CrossRef]

Kivshar, Y. S.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

Kong, J. A.

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

Kroll, N.

D. R. Smith, D. C. Vier, N. Kroll, S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246–2248 (2000).
[CrossRef]

Loschialpo, P. F.

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left-handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

F. J. Rachford, D. L. Smith, P. F. Loschialpo, D. W. Forester, “Calculations and measurements of wire and/or split-ring negative index media,” Phys. Rev. E 66, 036613 (2002).
[CrossRef]

Maslovski, S. I.

M. K. Karkkainen, S. I. Maslovski, “Wave propagation, refraction and focusing phenomena in Lorentzian double-negative materials: a theoretical and numerical study,” Microwave Opt. Technol. Lett. 37, 4–7 (2003).
[CrossRef]

Medgyesi-Mitschang, L. N.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Monzon, C.

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left-handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

Nemat-Nasser, S. C.

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

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

Nussenzveig, H. M.

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere—Part I: direct reflection and transmission; Part II: theory of the rainbow and the glory,” J. Math. Phys. 10, 82–177 (1969).
[CrossRef]

Pacheco, J.

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, 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, S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys. Condens. Matter 14, 8463–8479 (2002).
[CrossRef]

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

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and Enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Putnam, J. M.

L. N. Medgyesi-Mitschang, J. M. Putnam, M. B. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 11, 1383–1398 (1994).
[CrossRef]

J. M. Putnam, M. B. Gedera, “CARLOS-3D: a general-purpose 3-D method of moments scattering code,” IEEE Antennas Propag. Mag., April1993, pp. 69–71.

Rachford, F. J.

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left-handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

F. J. Rachford, D. L. Smith, P. F. Loschialpo, D. W. Forester, “Calculations and measurements of wire and/or split-ring negative index media,” Phys. Rev. E 66, 036613 (2002).
[CrossRef]

Ramakrishna, S. A.

J. B. Pendry, S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys. Condens. Matter 14, 8463–8479 (2002).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and Enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Ron, A.

D. R. Fredkin, A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753–1755 (2002).
[CrossRef]

Ruppin, R.

R. Ruppin, “Extinction properties of a sphere with negative permittivity and permeability,” Solid State Commun. 116, 411–415 (2000).
[CrossRef]

Schelleng, J.

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left-handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

Schultz, S.

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

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

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

D. R. Smith, D. C. Vier, N. Kroll, S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246–2248 (2000).
[CrossRef]

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

Shelby, R. A.

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

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

Smith, D. L.

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left-handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

F. J. Rachford, D. L. Smith, P. F. Loschialpo, D. W. Forester, “Calculations and measurements of wire and/or split-ring negative index media,” Phys. Rev. E 66, 036613 (2002).
[CrossRef]

Smith, D. R.

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

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

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

D. R. Smith, D. C. Vier, N. Kroll, S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246–2248 (2000).
[CrossRef]

Soukoulis, C. M.

S. Foteinopoulou, E. N. Economou, C. M. Soukoulis, “Refraction in media with a negative refractive index,” Phys. Rev. Lett. 90, 107402 (2003).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and Enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

Taflove, A.

A. Taflove, Computational Electrodynamics, The Finite-Difference Time-Domain Method (Artech House, Norwood, Mass., 1995).

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

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[CrossRef]

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[CrossRef]

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J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

Zhang, Y.

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

Ziolkowski, R. W.

R. W. Ziolkowski, A. D. Kipple, “Causality and double-negative metamaterials,” Phys. Rev. E 68, 026615 (2003).
[CrossRef]

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[CrossRef]

Ann. Phys. (Leipzig) (1)

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[CrossRef]

Appl. Phys. Lett. (4)

D. R. Smith, D. C. Vier, N. Kroll, S. Schultz, “Direct calculation of permeability and permittivity for a left-handed metamaterial,” Appl. Phys. Lett. 77, 2246–2248 (2000).
[CrossRef]

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

D. R. Fredkin, A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753–1755 (2002).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

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IEEE Trans. Antennas Propag. (1)

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[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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[CrossRef]

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[CrossRef]

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Microwave Opt. Technol. Lett. (1)

M. K. Karkkainen, S. I. Maslovski, “Wave propagation, refraction and focusing phenomena in Lorentzian double-negative materials: a theoretical and numerical study,” Microwave Opt. Technol. Lett. 37, 4–7 (2003).
[CrossRef]

Phys. Rev. E (6)

R. W. Ziolkowski, A. D. Kipple, “Causality and double-negative metamaterials,” Phys. Rev. E 68, 026615 (2003).
[CrossRef]

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[CrossRef]

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P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left-handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

R. W. Ziolkowski, E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Phys. Rev. Lett. (5)

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

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

J. Pacheco, T. M. Grzegorczyk, B. I. Wu, Y. Zhang, J. A. Kong, “Power propagation in homogeneous isotropic frequency-dispersive left-handed media,” Phys. Rev. Lett. 89, 257401 (2002).
[CrossRef] [PubMed]

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Science (1)

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[CrossRef]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Other (5)

D. S. Jones, The Theory of Electromagnetism (MacMillan, New York, 1964).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

The Finite Difference Time Domain simulations were performed with MAXTDA. MAXTDA, written at GTRI, was recently modified by Georgia Tech Research Institute (GTRI) to include causal Lorentzian functional forms for permittivity and permeability.

A. Taflove, Computational Electrodynamics, The Finite-Difference Time-Domain Method (Artech House, Norwood, Mass., 1995).

G. T. Ruck, Editor, Radar Cross Section Handbook (Plenum, New York, 1970).

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Figures (13)

Fig. 1
Fig. 1

Identity of the E- and H-plane scattering patterns. (a) Original geometry showing incident fields and scattering planes, (b) Effect of duality operation on the original problem. On comparison of both figures and a π/2 rotation, the identity of the E- and H-plane patterns is established.

Fig. 2
Fig. 2

Near backscattering (small sector bistatic) results for illumination at 0 deg. The figure shows bistatic BOR-CARLOS calculations, with data points every degree. The calculations involved a 10-cm-radius sphere with ε=μ=-1-j0.001. The deep backscatter null (-112 dB) demonstrates compliance with Weston’s theorem and is a proof of the adequacy of the BOR solution.

Fig. 3
Fig. 3

Forward-scattering response of a 10-cm-radius LHM sphere as a function of electrical size. All four independent approaches are represented. We imposed the condition ε=μ=-1-j0.001 at all frequencies.

Fig. 4
Fig. 4

Forward scattering from a 10-cm-radius LHM sphere compared with that from a metal sphere of the same size. While ε=μ and LHM in each case, the index assumes three values: -1/2, -1, and -2. The series solution shows that a LHM sphere casts a larger shadow than a metal sphere of similar size.

Fig. 5
Fig. 5

Identity of E- and H-plane scattering patterns. The CARLOS-3D simulation considered a slightly lossy LHM sphere with a 10 cm radius and ε=μ=-1-j0.001 and was performed at 6 GHz. The calculation validates our analytical finding on the identity of E- and H-plane patterns described in Subsection 2.B.

Fig. 6
Fig. 6

BOR-CARLOS bistatic data for matched 10-cm-radius RHM/LHM spheres of index n=±1.5-j0.001 at frequency 8 GHz. Only one curve is shown for each case because the E- and H-plane scattering patterns were found to be identical. The first sidelobes of the LHM sphere are ∼10 dB larger than those of the corresponding RHM sphere. This will result in dramatic differences in the glory.

Fig. 7
Fig. 7

Bistatic E-plane scattering patterns for a slightly lossy LHM sphere with a 10 cm radius and ε=μ=-1-j0.001. The CARLOS-3D calculations span a decade of frequencies from 0.6 GHz to 6 GHz, which result in k0a in the range from 2π/5 to 4π. The H-plane data are identical and are not shown for the sake of brevity in the presentation. Only half of the polar pattern is shown. The arrow indicates the direction of plane-wave incidence.

Fig. 8
Fig. 8

Bistatic H-plane scattering patterns for a slightly lossy LHM sphere with a 10 cm radius and ε=μ=-1-j0.001. The CARLOS-BOR calculations span 5 GHz to 10 GHz, which result in k0a in the range from 20π/6 to 20π/3. The H-plane data are identical and are not shown. Only half of the polar pattern is shown. The arrow indicates the direction of plane wave incidence.

Fig. 9
Fig. 9

Global bistatic H-plane dBsm plot for a slightly lossy LHM sphere with a 10 cm radius and ε=μ=-1-j0.001. In a global plot, the radial variable corresponds to the frequency. The CARLOS-3D calculations shown here span a decade of frequencies from 0.6 GHz to 6 GHz, which result in k0a in the range from 0.4π to 4π. The H-plane plot is identical and is not shown. Only half of the plot is shown because of symmetry. As in the previous figures, the incident plane wave is impinging from the top.

Fig. 10
Fig. 10

Side view of global bistatic H-plane dBsm surface for a slightly lossy LHM sphere with a 10 cm radius and ε=μ=-1-j0.001. The H-plane surface is identical is and not shown. Only half of the surface is presented because of symmetry. The deep monostatic null is evident in this 3D view of the surface.

Fig. 11
Fig. 11

Total near fields of a 3 cm Lorentzian sphere once the harmonic state has been reached (10 GHz). The sphere operates in the LHM regime with an index n=-1. The snapshot corresponds to the Y=0 plane [E plane in Fig. 2(a)], and the incident wave is impinging from the left. The shielding effect is evident. By rotating the figure by 90 degrees counterclockwise, one can see the face of a joker. Further simulations in 2D and 3D indicate that the shape is commonplace with rounded LHM structures; it is and a trademark of LHM regimes.

Fig. 12
Fig. 12

Magnitude of the time-harmonic electric near field corresponding to 3 cm matched spheres at 9.344 GHz. The snapshots are simultaneous and correspond to the E plane. The incident wave is impinging from the left. (a) LHM Lorentzian sphere, G=0.04 GHz, εDC=μDC=4.0, and fo=6.3 GHz, resulting in n=-1.5-j0.02 at 9.344 GHz. (b) Matched RHM sphere with n=1.5-j0.02. A circle is included to delineate the sphere contour.

Fig. 13
Fig. 13

Total time-harmonic near fields of a zero-index 3-cm-radius Lorentzian sphere. For G=0.04 GHz, εDC=μDC=4.0, and fo=6.3 GHz, Re(n)=0 at 12.6 GHz. The snapshots correspond to the E plane (top figures) and H Plane (bottom figures). The incident wave is impinging from the left. Snapshots (a) and (b) are separated by roughly a quarter of a period. A stationary focal point is clearly visible near the center of the sphere, with (a) showing a null at the precise location where (b) shows a peak.

Equations (1)

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F(f)=1+K-11+j(fG/fo2)-(f/fo)2.

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