Abstract

The question of losses in metamaterials that are based upon magnetic resonances deriving from split-ring arrays is addressed through the use of active inclusions designed from diode arrays. A full discussion of the way in which the current-voltage characteristics of the inclusions are deployed to produce gain is given, and the question of the overall stability of the new material is investigated in a quantitative way that is linked to absolute and convective instability. It is shown that instability associated with the inclusion of negative resistance devices can be avoided through a scheme that is approximately scalable from gigahertz to at least the low terahertz regimes. Furthermore, absolute instabilities that potentially complicate any active gain media can be controlled through a suitable choice of the system parameters. The latter step reduces the working frequency window over which spatial gain is available, leading to the need for compromise. Full numerical details are given with the conclusion that construction of such arrays of diode inclusions is possible and that a practical gain window is accessible.

© 2007 Optical Society of America

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  6. A. D. Boardman, P. Egan, L. Velasco, and N. King, "Control of planar nonlinear guided waves and spatial solitons with a left-handed medium," J. Opt. A, Pure Appl. Opt. 7, S57-S67 (2005).
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  33. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
  35. R. J. Briggs, Electron-Stream Interactions with Plasmas (MIT Press, 1964).
  36. A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
    [CrossRef]

2006 (3)

2005 (6)

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, "Saturation of the magnetic response of split-ring resonators at optical frequencies," Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef] [PubMed]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

A. D. Boardman, L. Velasco, N. King, and Y. Rapoport, "Ultra-narrow bright spatial solitons interacting with left-handed surfaces," J. Opt. Soc. Am. B 22, 1443-1452 (2005).
[CrossRef]

A. D. Boardman, N. King, and L. Velasco, "Negative refraction in perspective," Electromagnetics 25, 365-389 (2005).
[CrossRef]

A. D. Boardman, P. Egan, L. Velasco, and N. King, "Control of planar nonlinear guided waves and spatial solitons with a left-handed medium," J. Opt. A, Pure Appl. Opt. 7, S57-S67 (2005).
[CrossRef]

A. Boardman, N. King, Y. Rapoport, and L. Velasco, "Gyrotropic impact upon negatively refracting surfaces," New J. Phys. 7, 1-24 (2005).
[CrossRef]

2004 (2)

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

2003 (4)

J. O. Dimmock, "Losses in left-handed materials," Opt. Express 11, 2397-2402 (2003).
[CrossRef] [PubMed]

M. Lapine, M. Gorkunov, and K. H. Ringhofer, "Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements," Phys. Rev. E 67, 065601 (2003).
[CrossRef]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

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

2002 (3)

A. Grbic and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas Propag. Mag. 1, 10-13 (2002).

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

2001 (3)

J. Pendry, "Electromagnetic materials enter the negative age," Phys. World 14, 47-51 (2001).

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 055625 (2001).
[CrossRef]

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 (1)

D. R. Smith and N. Kroll, "Negative refractive index in left-handed materials," Phys. Rev. Lett. 85, 2933-2936 (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 (1)

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

1995 (1)

M. Reddy, R. Y. Yu, H. Kroemer, M. J. W. Rodwell, S. C. Martin, R. E. Muller, and R. P. Smith, "Bias stabilization for resonant tunnel-diode oscillators," IEEE Microw. Guid. Wave Lett. 5, 219-221 (1995).
[CrossRef]

1990 (1)

C. Kidner, I. Mehdi, J. R. East, and G. I. Haddad, "Power and stability limitations of resonant tunneling diodes," IEEE Trans. Microwave Theory Tech. 38, 864-872 (1990).
[CrossRef]

1988 (1)

W. W. Lam, C. F. Jou, H. Z. Chen, K. S. Stolt, N. C. Luhmann, and D. B. Rutledge, "Millimeter-wave diode-grid phase shifters," IEEE Trans. Microwave Theory Tech. 36, 902-907 (1988).
[CrossRef]

1958 (1)

P. A. Sturrock, "Kinematics of growing waves," Phys. Rev. 112, 1488-1503 (1958).
[CrossRef]

1906 (1)

J. C. M. Garnett, "Colours in metal glasses, in metallic films, and in metallic solutions II," Philos. Trans. R. Soc. London, Ser. A 393, 237-288 (1906).

1904 (1)

J. C. Garnett, "Colours in metal glasses and in metallic films," Philos. Trans. R. Soc. London, Ser. A 370, 385-420 (1904).

Basov, D. N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Boardman, A.

A. Boardman, N. King, Y. Rapoport, and L. Velasco, "Gyrotropic impact upon negatively refracting surfaces," New J. Phys. 7, 1-24 (2005).
[CrossRef]

Boardman, A. D.

A. D. Boardman, N. King, and L. Velasco, "Negative refraction in perspective," Electromagnetics 25, 365-389 (2005).
[CrossRef]

A. D. Boardman, P. Egan, L. Velasco, and N. King, "Control of planar nonlinear guided waves and spatial solitons with a left-handed medium," J. Opt. A, Pure Appl. Opt. 7, S57-S67 (2005).
[CrossRef]

A. D. Boardman, L. Velasco, N. King, and Y. Rapoport, "Ultra-narrow bright spatial solitons interacting with left-handed surfaces," J. Opt. Soc. Am. B 22, 1443-1452 (2005).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Briggs, R. J.

R. J. Briggs, Electron-Stream Interactions with Plasmas (MIT Press, 1964).

Brock, J. B.

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

Chen, H. Z.

W. W. Lam, C. F. Jou, H. Z. Chen, K. S. Stolt, N. C. Luhmann, and D. B. Rutledge, "Millimeter-wave diode-grid phase shifters," IEEE Trans. Microwave Theory Tech. 36, 902-907 (1988).
[CrossRef]

Chuang, I. L.

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

Dimmock, J. O.

East, J. R.

C. Kidner, I. Mehdi, J. R. East, and G. I. Haddad, "Power and stability limitations of resonant tunneling diodes," IEEE Trans. Microwave Theory Tech. 38, 864-872 (1990).
[CrossRef]

Economou, E. N.

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, "Saturation of the magnetic response of split-ring resonators at optical frequencies," Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef] [PubMed]

Efros, A. L.

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

Egan, P.

A. D. Boardman, P. Egan, L. Velasco, and N. King, "Control of planar nonlinear guided waves and spatial solitons with a left-handed medium," J. Opt. A, Pure Appl. Opt. 7, S57-S67 (2005).
[CrossRef]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Eleftheriades, G. V.

A. Grbic and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

Engheta, N.

N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas Propag. Mag. 1, 10-13 (2002).

Fang, N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Garnett, J. C.

J. C. Garnett, "Colours in metal glasses and in metallic films," Philos. Trans. R. Soc. London, Ser. A 370, 385-420 (1904).

Garnett, J. C. M.

J. C. M. Garnett, "Colours in metal glasses, in metallic films, and in metallic solutions II," Philos. Trans. R. Soc. London, Ser. A 393, 237-288 (1906).

Gorkunov, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, "Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements," Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Grbic, A.

A. Grbic and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Haddad, G. I.

C. Kidner, I. Mehdi, J. R. East, and G. I. Haddad, "Power and stability limitations of resonant tunneling diodes," IEEE Trans. Microwave Theory Tech. 38, 864-872 (1990).
[CrossRef]

Helszajn, J.

J. Helszajn, Passive and Active Microwave Circuits (Wiley, 1978).

Heyman, E.

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 055625 (2001).
[CrossRef]

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]

Houck, A. A.

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

Hurwitz, A.

A. Hurwitz, "On the conditions under which an equation has only roots with negative real parts," in Selected Papers on Mathematical Trends in Control Theory, R.Bellman and R.Kalaba, eds. (Dover, 1964).

Jou, C. F.

W. W. Lam, C. F. Jou, H. Z. Chen, K. S. Stolt, N. C. Luhmann, and D. B. Rutledge, "Millimeter-wave diode-grid phase shifters," IEEE Trans. Microwave Theory Tech. 36, 902-907 (1988).
[CrossRef]

Kafesaki, M.

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, "Saturation of the magnetic response of split-ring resonators at optical frequencies," Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef] [PubMed]

Kidner, C.

C. Kidner, I. Mehdi, J. R. East, and G. I. Haddad, "Power and stability limitations of resonant tunneling diodes," IEEE Trans. Microwave Theory Tech. 38, 864-872 (1990).
[CrossRef]

King, N.

A. D. Boardman, N. King, and L. Velasco, "Negative refraction in perspective," Electromagnetics 25, 365-389 (2005).
[CrossRef]

A. Boardman, N. King, Y. Rapoport, and L. Velasco, "Gyrotropic impact upon negatively refracting surfaces," New J. Phys. 7, 1-24 (2005).
[CrossRef]

A. D. Boardman, P. Egan, L. Velasco, and N. King, "Control of planar nonlinear guided waves and spatial solitons with a left-handed medium," J. Opt. A, Pure Appl. Opt. 7, S57-S67 (2005).
[CrossRef]

A. D. Boardman, L. Velasco, N. King, and Y. Rapoport, "Ultra-narrow bright spatial solitons interacting with left-handed surfaces," J. Opt. Soc. Am. B 22, 1443-1452 (2005).
[CrossRef]

Kishimoto, S.

Y. Ookawa, S. Kishimoto, K. Maezawa, and T. Mizutani, "Novel resonant tunneling diode oscillator capable of large output power operation," IEICE Trans. Electron. E89C, 999-1004 (2006).
[CrossRef]

Kivshar, Y. S.

I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, "Tunable split-ring resonators for nonlinear negative-index metamaterials," Opt. Express 14, 9344-9349 (2006).
[CrossRef] [PubMed]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Koltenbah, B. E. C.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Koschny, T.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, "Saturation of the magnetic response of split-ring resonators at optical frequencies," Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef] [PubMed]

Kroemer, H.

M. Reddy, R. Y. Yu, H. Kroemer, M. J. W. Rodwell, S. C. Martin, R. E. Muller, and R. P. Smith, "Bias stabilization for resonant tunnel-diode oscillators," IEEE Microw. Guid. Wave Lett. 5, 219-221 (1995).
[CrossRef]

Kroll, N.

D. R. Smith and N. Kroll, "Negative refractive index in left-handed materials," Phys. Rev. Lett. 85, 2933-2936 (2000).
[CrossRef] [PubMed]

Lam, W. W.

W. W. Lam, C. F. Jou, H. Z. Chen, K. S. Stolt, N. C. Luhmann, and D. B. Rutledge, "Millimeter-wave diode-grid phase shifters," IEEE Trans. Microwave Theory Tech. 36, 902-907 (1988).
[CrossRef]

Lapine, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, "Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements," Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Li, K.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Luhmann, N. C.

W. W. Lam, C. F. Jou, H. Z. Chen, K. S. Stolt, N. C. Luhmann, and D. B. Rutledge, "Millimeter-wave diode-grid phase shifters," IEEE Trans. Microwave Theory Tech. 36, 902-907 (1988).
[CrossRef]

Maezawa, K.

Y. Ookawa, S. Kishimoto, K. Maezawa, and T. Mizutani, "Novel resonant tunneling diode oscillator capable of large output power operation," IEICE Trans. Electron. E89C, 999-1004 (2006).
[CrossRef]

Martin, S. C.

M. Reddy, R. Y. Yu, H. Kroemer, M. J. W. Rodwell, S. C. Martin, R. E. Muller, and R. P. Smith, "Bias stabilization for resonant tunnel-diode oscillators," IEEE Microw. Guid. Wave Lett. 5, 219-221 (1995).
[CrossRef]

Mehdi, I.

C. Kidner, I. Mehdi, J. R. East, and G. I. Haddad, "Power and stability limitations of resonant tunneling diodes," IEEE Trans. Microwave Theory Tech. 38, 864-872 (1990).
[CrossRef]

Mizutani, T.

Y. Ookawa, S. Kishimoto, K. Maezawa, and T. Mizutani, "Novel resonant tunneling diode oscillator capable of large output power operation," IEICE Trans. Electron. E89C, 999-1004 (2006).
[CrossRef]

Morrison, S. K.

Muller, R. E.

M. Reddy, R. Y. Yu, H. Kroemer, M. J. W. Rodwell, S. C. Martin, R. E. Muller, and R. P. Smith, "Bias stabilization for resonant tunnel-diode oscillators," IEEE Microw. Guid. Wave Lett. 5, 219-221 (1995).
[CrossRef]

Ookawa, Y.

Y. Ookawa, S. Kishimoto, K. Maezawa, and T. Mizutani, "Novel resonant tunneling diode oscillator capable of large output power operation," IEICE Trans. Electron. E89C, 999-1004 (2006).
[CrossRef]

Padilla, W. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Parazzoli, C. G.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Pendry, J.

J. Pendry, "Electromagnetic materials enter the negative age," Phys. World 14, 47-51 (2001).

Pendry, J. B.

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, "Saturation of the magnetic response of split-ring resonators at optical frequencies," Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[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]

Pokrovsky, A. L.

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

Popov, A. K.

Rapoport, Y.

A. D. Boardman, L. Velasco, N. King, and Y. Rapoport, "Ultra-narrow bright spatial solitons interacting with left-handed surfaces," J. Opt. Soc. Am. B 22, 1443-1452 (2005).
[CrossRef]

A. Boardman, N. King, Y. Rapoport, and L. Velasco, "Gyrotropic impact upon negatively refracting surfaces," New J. Phys. 7, 1-24 (2005).
[CrossRef]

Reddy, M.

M. Reddy, R. Y. Yu, H. Kroemer, M. J. W. Rodwell, S. C. Martin, R. E. Muller, and R. P. Smith, "Bias stabilization for resonant tunnel-diode oscillators," IEEE Microw. Guid. Wave Lett. 5, 219-221 (1995).
[CrossRef]

Ringhofer, K. H.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, "Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements," Phys. Rev. E 67, 065601 (2003).
[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]

Rodwell, M. J. W.

M. Reddy, R. Y. Yu, H. Kroemer, M. J. W. Rodwell, S. C. Martin, R. E. Muller, and R. P. Smith, "Bias stabilization for resonant tunnel-diode oscillators," IEEE Microw. Guid. Wave Lett. 5, 219-221 (1995).
[CrossRef]

Routh, E. J.

E. J. Routh, A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion, Reprinted ed. (Taylor and Francis, 1975).

Rutledge, D. B.

W. W. Lam, C. F. Jou, H. Z. Chen, K. S. Stolt, N. C. Luhmann, and D. B. Rutledge, "Millimeter-wave diode-grid phase shifters," IEEE Trans. Microwave Theory Tech. 36, 902-907 (1988).
[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]

Scott, A.

A. Scott, Active and Nonlinear Wave Propagation in Electromagnetics (Wiley, 1970).

Shadrivov, I. V.

I. V. Shadrivov, S. K. Morrison, and Y. S. Kivshar, "Tunable split-ring resonators for nonlinear negative-index metamaterials," Opt. Express 14, 9344-9349 (2006).
[CrossRef] [PubMed]

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Shalaev, V. M.

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]

Smith, D. R.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

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 and N. Kroll, "Negative refractive index in left-handed materials," Phys. Rev. Lett. 85, 2933-2936 (2000).
[CrossRef] [PubMed]

Smith, R. P.

M. Reddy, R. Y. Yu, H. Kroemer, M. J. W. Rodwell, S. C. Martin, R. E. Muller, and R. P. Smith, "Bias stabilization for resonant tunnel-diode oscillators," IEEE Microw. Guid. Wave Lett. 5, 219-221 (1995).
[CrossRef]

Soukoulis, C. M.

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, "Saturation of the magnetic response of split-ring resonators at optical frequencies," Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef] [PubMed]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (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]

Stolt, K. S.

W. W. Lam, C. F. Jou, H. Z. Chen, K. S. Stolt, N. C. Luhmann, and D. B. Rutledge, "Millimeter-wave diode-grid phase shifters," IEEE Trans. Microwave Theory Tech. 36, 902-907 (1988).
[CrossRef]

Sturrock, P. A.

P. A. Sturrock, "Kinematics of growing waves," Phys. Rev. 112, 1488-1503 (1958).
[CrossRef]

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Tanielian, M.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Velasco, L.

A. Boardman, N. King, Y. Rapoport, and L. Velasco, "Gyrotropic impact upon negatively refracting surfaces," New J. Phys. 7, 1-24 (2005).
[CrossRef]

A. D. Boardman, N. King, and L. Velasco, "Negative refraction in perspective," Electromagnetics 25, 365-389 (2005).
[CrossRef]

A. D. Boardman, P. Egan, L. Velasco, and N. King, "Control of planar nonlinear guided waves and spatial solitons with a left-handed medium," J. Opt. A, Pure Appl. Opt. 7, S57-S67 (2005).
[CrossRef]

A. D. Boardman, L. Velasco, N. King, and Y. Rapoport, "Ultra-narrow bright spatial solitons interacting with left-handed surfaces," J. Opt. Soc. Am. B 22, 1443-1452 (2005).
[CrossRef]

Vier, D. C.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Yen, T. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[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]

Yu, R. Y.

M. Reddy, R. Y. Yu, H. Kroemer, M. J. W. Rodwell, S. C. Martin, R. E. Muller, and R. P. Smith, "Bias stabilization for resonant tunnel-diode oscillators," IEEE Microw. Guid. Wave Lett. 5, 219-221 (1995).
[CrossRef]

Zhang, X.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Zharov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Zhou, J.

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, "Saturation of the magnetic response of split-ring resonators at optical frequencies," Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef] [PubMed]

Ziolkowski, R. W.

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 055625 (2001).
[CrossRef]

Electromagnetics (1)

A. D. Boardman, N. King, and L. Velasco, "Negative refraction in perspective," Electromagnetics 25, 365-389 (2005).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

N. Engheta, "An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability," IEEE Antennas Propag. Mag. 1, 10-13 (2002).

IEEE Microw. Guid. Wave Lett. (1)

M. Reddy, R. Y. Yu, H. Kroemer, M. J. W. Rodwell, S. C. Martin, R. E. Muller, and R. P. Smith, "Bias stabilization for resonant tunnel-diode oscillators," IEEE Microw. Guid. Wave Lett. 5, 219-221 (1995).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

W. W. Lam, C. F. Jou, H. Z. Chen, K. S. Stolt, N. C. Luhmann, and D. B. Rutledge, "Millimeter-wave diode-grid phase shifters," IEEE Trans. Microwave Theory Tech. 36, 902-907 (1988).
[CrossRef]

C. Kidner, I. Mehdi, J. R. East, and G. I. Haddad, "Power and stability limitations of resonant tunneling diodes," IEEE Trans. Microwave Theory Tech. 38, 864-872 (1990).
[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]

IEICE Trans. Electron. (1)

Y. Ookawa, S. Kishimoto, K. Maezawa, and T. Mizutani, "Novel resonant tunneling diode oscillator capable of large output power operation," IEICE Trans. Electron. E89C, 999-1004 (2006).
[CrossRef]

J. Appl. Phys. (1)

A. Grbic and G. V. Eleftheriades, "Experimental verification of backward-wave radiation from a negative refractive index metamaterial," J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

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

A. D. Boardman, P. Egan, L. Velasco, and N. King, "Control of planar nonlinear guided waves and spatial solitons with a left-handed medium," J. Opt. A, Pure Appl. Opt. 7, S57-S67 (2005).
[CrossRef]

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

New J. Phys. (1)

A. Boardman, N. King, Y. Rapoport, and L. Velasco, "Gyrotropic impact upon negatively refracting surfaces," New J. Phys. 7, 1-24 (2005).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Philos. Trans. R. Soc. London, Ser. A (2)

J. C. M. Garnett, "Colours in metal glasses, in metallic films, and in metallic solutions II," Philos. Trans. R. Soc. London, Ser. A 393, 237-288 (1906).

J. C. Garnett, "Colours in metal glasses and in metallic films," Philos. Trans. R. Soc. London, Ser. A 370, 385-420 (1904).

Phys. Rev. (1)

P. A. Sturrock, "Kinematics of growing waves," Phys. Rev. 112, 1488-1503 (1958).
[CrossRef]

Phys. Rev. E (4)

I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004).
[CrossRef]

R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 055625 (2001).
[CrossRef]

M. Lapine, M. Gorkunov, and K. H. Ringhofer, "Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements," Phys. Rev. E 67, 065601 (2003).
[CrossRef]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Phys. Rev. Lett. (5)

J. Zhou, T. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, "Saturation of the magnetic response of split-ring resonators at optical frequencies," Phys. Rev. Lett. 95, 223902 (2005).
[CrossRef] [PubMed]

D. R. Smith and N. Kroll, "Negative refractive index in left-handed materials," Phys. Rev. Lett. 85, 2933-2936 (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]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

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

Phys. World (1)

J. Pendry, "Electromagnetic materials enter the negative age," Phys. World 14, 47-51 (2001).

Science (2)

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

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

Solid State Commun. (1)

A. L. Pokrovsky and A. L. Efros, "Sign of refractive index and group velocity in left-handed media," Solid State Commun. 124, 283-287 (2002).
[CrossRef]

Other (6)

J. Helszajn, Passive and Active Microwave Circuits (Wiley, 1978).

R. J. Briggs, Electron-Stream Interactions with Plasmas (MIT Press, 1964).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

A. Scott, Active and Nonlinear Wave Propagation in Electromagnetics (Wiley, 1970).

A. Hurwitz, "On the conditions under which an equation has only roots with negative real parts," in Selected Papers on Mathematical Trends in Control Theory, R.Bellman and R.Kalaba, eds. (Dover, 1964).

E. J. Routh, A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion, Reprinted ed. (Taylor and Francis, 1975).

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Figures (10)

Fig. 1
Fig. 1

Simple idea of a diode pair oscillator circuit biased by voltage V b [27]. The dynamic operation depends upon the variation of the voltage at the central node with the current flowing into it.

Fig. 2
Fig. 2

Typical (I,V) characteristic curves at the central node of the representative diode pair shown in Fig. 1 and previously studied by Ookawa et al. [27]. Only a few bias voltage steps are displayed, but the interesting feature is the small negative slopes that are possible around the origin.

Fig. 3
Fig. 3

Equivalent circuit of a split-ring resonator with an attached diode. The dotted double-headed arrow simply signifies the presence of a separate stable dc biasing circuit.

Fig. 4
Fig. 4

(a) and (c) Real and (b) and (d) imaginary parts of the permeability for (a) and (b) g 0 = 0.1 ω 0 C g and (c) and (d) g 0 = 0.15 ω 0 C g . The solid curves are for the passive case, whereas the dashed curve is with the effect of active inclusions.

Fig. 5
Fig. 5

Sketches to illustrate (a) absolute and (b) convective instability.

Fig. 6
Fig. 6

(a) Active ( g 0 = 0.105 ω 0 C g ) and (b) passive complex permeabilities for an illustrative example of what happens when F is small ( F = 0.03 ) because of the lower density of metaparticles. The unshaded part of the figure indicates the available operational window implied by avoiding regions of absolute instability. The solid curves are for the real parts case, whereas the dashed curve is with the imaginary parts case.

Fig. 7
Fig. 7

Passive metamaterial: g 0 = 0 . (a) ( K , K ) phase plane for real Ω. The gray scale intensity denotes frequency. (b) ( K , K ) dependence upon real Ω.

Fig. 8
Fig. 8

g 0 = 0.15 ω 0 C g . (a) ( K , K ) phase plane for real Ω. The gray scale intensity denotes frequency. (b) ( K , K ) dependence upon real Ω. The shading indicates the excluded absolute instability regions. (c) and (d) ( Ω , Ω ) phase plane for real K. The gray scale intensity denotes wave number. Note that in (c) there is a branch that is approximately coincident with the vertical axis. The important branch is where Ω 1 where the resonance of μ is located.

Fig. 9
Fig. 9

( Ω , Ω ) phase plane for real K in the vicinity of the critical branch about the μ resonance for (a) g 0 = 0.10 ω 0 C g , (b) g 0 = 0.15 ω 0 C g , and (c) g 0 = 0.20 ω 0 C g . The gray scale intensity denotes wave number.

Fig. 10
Fig. 10

(a) Variation of the frequency range over which gain occurs for real frequency, complex wave number propagation, as a function of conductance. (b) The outcome of a real wave number complex frequency behavior as a function of conductance.

Equations (32)

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

I D = ( g 0 + i ω g 1 ) U D + b U D 2 U D .
u = I ( R i ω L ) + U D .
I = I D + I C = [ g 0 + i ω ( g 1 + C g ) b U D 2 ] U D .
( ω r 2 g 0 L ω r 2 + i ω Γ + i ω ) I = u L + b ω r 2 g 0 L ω r 2 + i ω u I ( R i ω L ) 2 [ u I ( R i ω L ) ] ,
ω r 2 = 1 L [ C g + g 1 ] 1 L C g , Γ = R L .
I = i ω u L ω 2 ω r 2 ω 2 + g 0 2 C g 2 ω 2 i ω [ Γ ( ω r 2 ω 2 + g 0 2 C g 2 ) g 0 C g ] .
μ = 1 + F Ω 2 Ω 2 Ω 2 + ( g 0 C ) 2 Ω 2 i Ω [ γ g 0 C Ω 2 + ( g 0 C ) 2 ] ,
F = n A 2 μ 0 L , F = F 1 + F 3 , ω 0 2 = ω r 2 1 + F 3 , Γ = Γ 1 + F 3 ,
Ω = ω ω 0 , γ = Γ ω 0 , C = ω 0 C g ,
μ = 1 + F Ω 2 [ 1 Ω 2 + ( g 0 C ) 2 1 ] Ω 2 [ 1 Ω 2 + ( g 0 C ) 2 1 ] 2 + [ γ g 0 C Ω 2 + ( g 0 C ) 2 ] 2 ,
μ = F Ω [ γ g 0 C Ω 2 + ( g 0 C ) 2 ] Ω 2 [ 1 Ω 2 + ( g 0 C ) 2 1 ] 2 + [ γ g 0 C Ω 2 + ( g 0 C ) 2 ] 2 .
( 1 F ) Ω 6 [ 2 F γ 2 2 g 0 2 C 2 ( 1 F ) ] Ω 4 + [ 1 + 2 g 0 C γ ( g 0 C γ 1 ) + g 0 2 C 2 ( F 2 ) + g 0 4 C 4 ( 1 F ) ] Ω 2 + g 0 2 C 2 ( γ g 0 C 1 ) 2 < 0 .
g 0 C = g 0 L C g .
μ = 1 F Ω 2 Ω 2 + i Ω [ γ C g 0 ] .
g 0 C = 1 2 γ [ 1 ± 1 ( 2 γ Ω cross ) 2 ] ,
K 2 = k 2 c 0 2 ω 0 2 = Ω 2 ε ( Ω ) μ ( Ω ) .
Ω ZERO = ± 4 ( 1 F ) [ 1 γ ( g 0 C ) ] [ ( F 1 ) ( g 0 C ) + γ ] 2 2 ( 1 F ) ,
Ω ZERO = ( g 0 C ) ( 1 F ) γ 2 ( 1 F ) .
g 0 Γ C g ( 1 F ) = g 0 MAX .
Ω POLE = ± 4 ( 1 γ g 0 C ) ( g 0 C γ ) 2 2 ,
Ω POLE = ( g 0 C γ ) 2 .
γ g 0 C ( 1 + g 0 2 C 2 ) .
K = ± Ω ε μ = ± Ω 1 2 [ ε μ ε μ + ε μ + i ( ε μ + ε μ ) 1 ε μ ε μ + ε μ ] ,
a 5 Ω 5 + i a 4 Ω 4 + a 3 Ω 3 + i a 2 Ω 2 + a 1 Ω + i a 0 = 0 ,
a 5 = F 1 ,
a 4 = ( 1 F ) ( g 0 C ν ) γ ,
a 3 = K 2 ν g 0 C [ 1 + F ] + γ ν Ω p 2 [ F 1 ] + 1 γ g 0 C ,
a 4 = ν [ 1 γ g 0 C ] Ω p 2 g 0 C [ 1 F ] + Ω p 2 γ K 2 [ g 0 C γ ν ] ,
a 1 = Ω p 2 [ 1 γ g 0 C ] K 2 ( 1 + ν γ [ γ + ν ] g 0 C ) ,
a 0 = K 2 ν [ 1 γ g 0 C ] .
K < 0 ± ( ε μ + ε μ ) < 0 .
K ± ( μ + ν Ω μ ) < 0 μ < ν Ω μ ( μ < 0 ) .

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