Abstract

In this paper, starting with the Maxwell equations and the corresponding nonlinear wave equation, we derive a pulse evolution equation for a metamaterial with cubic (Kerr) nonlinearity. The given model equation goes beyond the commonly employed slowly evolving wave approximation (SEWA). The dispersive properties of the dielectric susceptibility and magnetic permeability are accounted for in accordance with the Drude model. Using this model equation, propagation properties of a single-cycle pulse and a 4-cycle pulse are studied numerically in a left-handed metamaterial (LHMM) with third-order nonlinearity.

© 2012 Optical Society of America

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  29. A. Kumar, “Ultrashort pulse propagation in a cubic medium including the Raman effect,” Phys. Rev. A 81, 0138071(2010).
  30. I. V. Melnikov, D. Mihalache, F. Moldoveanu, and N.-C. Panoiu, “Quasiadiabatic following of femtosecond optical pulses in a weakly excited semiconductor,” Phys. Rev. A 56, 1569–1576 (1997).
    [CrossRef]
  31. H. Leblond and D. Mihalache, “Few-optical-cycle dissipative solitons,” J. Phys. A 43, 375205 (2010).
    [CrossRef]
  32. H. Leblond, D. Kremer, and D. Mihalache, “Collapse of ultrashort spatiotemporal pulses described by the cubic generalized Kadomtsev-Petviashvili equation,” Phys. Rev. A 81, 0338241 (2010).
  33. H. Leblond and F. Sanchez, “Models for optical solitons in the two-cycle regime,” Phys. Rev. A 67, 0138041 (2003).
    [CrossRef]
  34. J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191–202 (2004).
    [CrossRef]
  35. T. Taniuti and C.-C. Wei, “Reductive perturbation method in nonlinear wave propagation,” J. Phys. Soc. Jpn. 24, 941–946 (1968).
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2010 (4)

J. Zhang, S. Wen, Y. Xiang, Y. Wang, and H. Luo, “Spatiotemporal electromagnetic soliton and spatial ring formation in nonlinear metamaterials,” Phys. Rev. A 81, 0238291(2010).

A. Kumar, “Ultrashort pulse propagation in a cubic medium including the Raman effect,” Phys. Rev. A 81, 0138071(2010).

H. Leblond and D. Mihalache, “Few-optical-cycle dissipative solitons,” J. Phys. A 43, 375205 (2010).
[CrossRef]

H. Leblond, D. Kremer, and D. Mihalache, “Collapse of ultrashort spatiotemporal pulses described by the cubic generalized Kadomtsev-Petviashvili equation,” Phys. Rev. A 81, 0338241 (2010).

2009 (1)

A. Kumar, “A few- and sub-cycle pulse evolution equation in a cubic nonlinear medium,” Theor. Math. Phys. 160, 968–975 (2009).
[CrossRef]

2008 (2)

G. DAguanno, N. Mattiucci, and M. J. Bloemer, “Ultraslow light pulses in a nonlinear metamaterial,” J. Opt. Soc. Am. B 25, 1236–1241 (2008).
[CrossRef]

V. V. Grigoriev and V. V. Kabanov, “Propagation of a localized wavepacket in a metamaterial with a negative index of refraction,” J. Appl. Spectrosc. 75, 192–198 (2008).
[CrossRef]

2007 (1)

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 0338151(2007).

2006 (4)

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 0366171 (2006).
[CrossRef]

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

V. G. Veselago, “Negative refractive index materials,” J. Comp. Theo. Nanoscience 3, 1–30 (2006).

T. Tsurumi, “Propagation of few- to sub-cycle pulse in dispersive media,” J. Phys. Soc. Jpn. 75, 024002 (2006).
[CrossRef]

2005 (5)

G. D’Aguanno, N. Aközbec, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000 (2005).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

M. Scalora, M. S. Syrchin, N. Aközbec, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 0139021(2005).

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 0366141 (2005).

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 0166261 (2005).
[CrossRef]

2004 (5)

W. T. Lu, J. B. Sokoloff, and S. Sridhar, “Refraction of electromagnetic energy for wavepackets incident on a negative-index medium is always negative,” Phys. Rev. E 69, 0266041 (2004).

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 1651121 (2004).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (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]

J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191–202 (2004).
[CrossRef]

2003 (1)

H. Leblond and F. Sanchez, “Models for optical solitons in the two-cycle regime,” Phys. Rev. A 67, 0138041 (2003).
[CrossRef]

2002 (1)

M. Bayindir, K. Aydin, E. Ozbay, P. Marko, and C. M. Soukoulis, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120–122 (2002).
[CrossRef]

2000 (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

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]

1999 (3)

D. Z. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[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]

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal dynamics of electromagnetic pulses in saturating nonlinear optical media with normal group velocity dispersion,” Phys. Rev. E 60, 7622–7625 (1999).
[CrossRef]

1998 (1)

1997 (2)

I. V. Melnikov, D. Mihalache, F. Moldoveanu, and N.-C. Panoiu, “Quasiadiabatic following of femtosecond optical pulses in a weakly excited semiconductor,” Phys. Rev. A 56, 1569–1576 (1997).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

1996 (1)

R. G. Flesch, A. Pushkarev, and J. V. Moloney, “Carrier wave shocking of femtosecond optical pulses,” Phys. Rev. Lett. 76, 2488–2491 (1996).
[CrossRef]

1995 (1)

1968 (2)

T. Taniuti and C.-C. Wei, “Reductive perturbation method in nonlinear wave propagation,” J. Phys. Soc. Jpn. 24, 941–946 (1968).
[CrossRef]

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

Agranovich, V. M.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 1651121 (2004).
[CrossRef]

Aközbec, N.

G. D’Aguanno, N. Aközbec, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000 (2005).
[CrossRef]

M. Scalora, M. S. Syrchin, N. Aközbec, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 0139021(2005).

Alexopoulos, N. G.

D. Z. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[CrossRef]

Aydin, K.

M. Bayindir, K. Aydin, E. Ozbay, P. Marko, and C. M. Soukoulis, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120–122 (2002).
[CrossRef]

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]

Baughman, R. H.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 1651121 (2004).
[CrossRef]

Bayindir, M.

M. Bayindir, K. Aydin, E. Ozbay, P. Marko, and C. M. Soukoulis, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120–122 (2002).
[CrossRef]

Berezhiani, V. I.

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal dynamics of electromagnetic pulses in saturating nonlinear optical media with normal group velocity dispersion,” Phys. Rev. E 60, 7622–7625 (1999).
[CrossRef]

Bloemer, M. J.

G. DAguanno, N. Mattiucci, and M. J. Bloemer, “Ultraslow light pulses in a nonlinear metamaterial,” J. Opt. Soc. Am. B 25, 1236–1241 (2008).
[CrossRef]

G. D’Aguanno, N. Aközbec, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000 (2005).
[CrossRef]

M. Scalora, M. S. Syrchin, N. Aközbec, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 0139021(2005).

Brabec, T.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Broas, R. F. J.

D. Z. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[CrossRef]

Burger, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

D’Aguanno, G.

M. Scalora, M. S. Syrchin, N. Aközbec, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 0139021(2005).

G. D’Aguanno, N. Aközbec, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000 (2005).
[CrossRef]

DAguanno, G.

Dai, X.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 0338151(2007).

Enkrich, C.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Fan, D.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 0338151(2007).

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 0366171 (2006).
[CrossRef]

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]

Flesch, R. G.

R. G. Flesch, A. Pushkarev, and J. V. Moloney, “Carrier wave shocking of femtosecond optical pulses,” Phys. Rev. Lett. 76, 2488–2491 (1996).
[CrossRef]

Fu, X.

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 0366171 (2006).
[CrossRef]

Gaeta, A.

Grigoriev, V. V.

V. V. Grigoriev and V. V. Kabanov, “Propagation of a localized wavepacket in a metamaterial with a negative index of refraction,” J. Appl. Spectrosc. 75, 192–198 (2008).
[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]

Kabanov, V. V.

V. V. Grigoriev and V. V. Kabanov, “Propagation of a localized wavepacket in a metamaterial with a negative index of refraction,” J. Appl. Spectrosc. 75, 192–198 (2008).
[CrossRef]

Koschny, T.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Kourakis, I.

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 0166261 (2005).
[CrossRef]

Krausz, F.

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Kremer, D.

H. Leblond, D. Kremer, and D. Mihalache, “Collapse of ultrashort spatiotemporal pulses described by the cubic generalized Kadomtsev-Petviashvili equation,” Phys. Rev. A 81, 0338241 (2010).

Kumar, A.

A. Kumar, “Ultrashort pulse propagation in a cubic medium including the Raman effect,” Phys. Rev. A 81, 0138071(2010).

A. Kumar, “A few- and sub-cycle pulse evolution equation in a cubic nonlinear medium,” Theor. Math. Phys. 160, 968–975 (2009).
[CrossRef]

Lawrence, B. L.

Lazarides, N.

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 0366141 (2005).

Leblond, H.

H. Leblond, D. Kremer, and D. Mihalache, “Collapse of ultrashort spatiotemporal pulses described by the cubic generalized Kadomtsev-Petviashvili equation,” Phys. Rev. A 81, 0338241 (2010).

H. Leblond and D. Mihalache, “Few-optical-cycle dissipative solitons,” J. Phys. A 43, 375205 (2010).
[CrossRef]

H. Leblond and F. Sanchez, “Models for optical solitons in the two-cycle regime,” Phys. Rev. A 67, 0138041 (2003).
[CrossRef]

Linden, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Lu, W. T.

W. T. Lu, J. B. Sokoloff, and S. Sridhar, “Refraction of electromagnetic energy for wavepackets incident on a negative-index medium is always negative,” Phys. Rev. E 69, 0266041 (2004).

Luo, H.

J. Zhang, S. Wen, Y. Xiang, Y. Wang, and H. Luo, “Spatiotemporal electromagnetic soliton and spatial ring formation in nonlinear metamaterials,” Phys. Rev. A 81, 0238291(2010).

Marko, P.

M. Bayindir, K. Aydin, E. Ozbay, P. Marko, and C. M. Soukoulis, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120–122 (2002).
[CrossRef]

Mattiucci, N.

G. DAguanno, N. Mattiucci, and M. J. Bloemer, “Ultraslow light pulses in a nonlinear metamaterial,” J. Opt. Soc. Am. B 25, 1236–1241 (2008).
[CrossRef]

G. D’Aguanno, N. Aközbec, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000 (2005).
[CrossRef]

M. Scalora, M. S. Syrchin, N. Aközbec, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 0139021(2005).

Melnikov, I. V.

I. V. Melnikov, D. Mihalache, F. Moldoveanu, and N.-C. Panoiu, “Quasiadiabatic following of femtosecond optical pulses in a weakly excited semiconductor,” Phys. Rev. A 56, 1569–1576 (1997).
[CrossRef]

Mihalache, D.

H. Leblond and D. Mihalache, “Few-optical-cycle dissipative solitons,” J. Phys. A 43, 375205 (2010).
[CrossRef]

H. Leblond, D. Kremer, and D. Mihalache, “Collapse of ultrashort spatiotemporal pulses described by the cubic generalized Kadomtsev-Petviashvili equation,” Phys. Rev. A 81, 0338241 (2010).

I. V. Melnikov, D. Mihalache, F. Moldoveanu, and N.-C. Panoiu, “Quasiadiabatic following of femtosecond optical pulses in a weakly excited semiconductor,” Phys. Rev. A 56, 1569–1576 (1997).
[CrossRef]

Miklaszewski, R.

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal dynamics of electromagnetic pulses in saturating nonlinear optical media with normal group velocity dispersion,” Phys. Rev. E 60, 7622–7625 (1999).
[CrossRef]

Moldoveanu, F.

I. V. Melnikov, D. Mihalache, F. Moldoveanu, and N.-C. Panoiu, “Quasiadiabatic following of femtosecond optical pulses in a weakly excited semiconductor,” Phys. Rev. A 56, 1569–1576 (1997).
[CrossRef]

Moloney, J. V.

R. G. Flesch, A. Pushkarev, and J. V. Moloney, “Carrier wave shocking of femtosecond optical pulses,” Phys. Rev. Lett. 76, 2488–2491 (1996).
[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]

Ozbay, E.

M. Bayindir, K. Aydin, E. Ozbay, P. Marko, and C. M. Soukoulis, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120–122 (2002).
[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]

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]

Panoiu, N.-C.

I. V. Melnikov, D. Mihalache, F. Moldoveanu, and N.-C. Panoiu, “Quasiadiabatic following of femtosecond optical pulses in a weakly excited semiconductor,” Phys. Rev. A 56, 1569–1576 (1997).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191–202 (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]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[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]

Poliakov, E. Y.

M. Scalora, M. S. Syrchin, N. Aközbec, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 0139021(2005).

Pushkarev, A.

R. G. Flesch, A. Pushkarev, and J. V. Moloney, “Carrier wave shocking of femtosecond optical pulses,” Phys. Rev. Lett. 76, 2488–2491 (1996).
[CrossRef]

Ranka, J. K.

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]

Sanchez, F.

H. Leblond and F. Sanchez, “Models for optical solitons in the two-cycle regime,” Phys. Rev. A 67, 0138041 (2003).
[CrossRef]

Scalora, M.

M. Scalora, M. S. Syrchin, N. Aközbec, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 0139021(2005).

G. D’Aguanno, N. Aközbec, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000 (2005).
[CrossRef]

Schmidt, F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

Schultz, S.

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]

Shen, Y. R.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 1651121 (2004).
[CrossRef]

Shukla, P. K.

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 0166261 (2005).
[CrossRef]

Shurig, D.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

Sievenpiper, D. Z.

D. Z. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[CrossRef]

Skarka, V.

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal dynamics of electromagnetic pulses in saturating nonlinear optical media with normal group velocity dispersion,” Phys. Rev. E 60, 7622–7625 (1999).
[CrossRef]

Smith, D. R.

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[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]

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]

Sokoloff, J. B.

W. T. Lu, J. B. Sokoloff, and S. Sridhar, “Refraction of electromagnetic energy for wavepackets incident on a negative-index medium is always negative,” Phys. Rev. E 69, 0266041 (2004).

Soukoulis, C. M.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

M. Bayindir, K. Aydin, E. Ozbay, P. Marko, and C. M. Soukoulis, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120–122 (2002).
[CrossRef]

Sridhar, S.

W. T. Lu, J. B. Sokoloff, and S. Sridhar, “Refraction of electromagnetic energy for wavepackets incident on a negative-index medium is always negative,” Phys. Rev. E 69, 0266041 (2004).

Stegeman, G.

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]

Su, W.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 0338151(2007).

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 0366171 (2006).
[CrossRef]

Syrchin, M. S.

M. Scalora, M. S. Syrchin, N. Aközbec, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 0139021(2005).

Tang, Z.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 0338151(2007).

Taniuti, T.

T. Taniuti and C.-C. Wei, “Reductive perturbation method in nonlinear wave propagation,” J. Phys. Soc. Jpn. 24, 941–946 (1968).
[CrossRef]

Torruellas, W.

Tsironis, G. P.

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 0366141 (2005).

Tsurumi, T.

T. Tsurumi, “Propagation of few- to sub-cycle pulse in dispersive media,” J. Phys. Soc. Jpn. 75, 024002 (2006).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “Negative refractive index materials,” J. Comp. Theo. Nanoscience 3, 1–30 (2006).

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

Vier, D. C.

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]

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]

Wang, Y.

J. Zhang, S. Wen, Y. Xiang, Y. Wang, and H. Luo, “Spatiotemporal electromagnetic soliton and spatial ring formation in nonlinear metamaterials,” Phys. Rev. A 81, 0238291(2010).

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 0366171 (2006).
[CrossRef]

Wegener, M.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Wei, C.-C.

T. Taniuti and C.-C. Wei, “Reductive perturbation method in nonlinear wave propagation,” J. Phys. Soc. Jpn. 24, 941–946 (1968).
[CrossRef]

Wen, S.

J. Zhang, S. Wen, Y. Xiang, Y. Wang, and H. Luo, “Spatiotemporal electromagnetic soliton and spatial ring formation in nonlinear metamaterials,” Phys. Rev. A 81, 0238291(2010).

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 0338151(2007).

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 0366171 (2006).
[CrossRef]

Wright, E. M.

Xiang, Y.

J. Zhang, S. Wen, Y. Xiang, Y. Wang, and H. Luo, “Spatiotemporal electromagnetic soliton and spatial ring formation in nonlinear metamaterials,” Phys. Rev. A 81, 0238291(2010).

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 0338151(2007).

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 0366171 (2006).
[CrossRef]

Yablonovitch, E.

D. Z. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[CrossRef]

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]

Zakhidov, A. A.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 1651121 (2004).
[CrossRef]

Zhang, J.

J. Zhang, S. Wen, Y. Xiang, Y. Wang, and H. Luo, “Spatiotemporal electromagnetic soliton and spatial ring formation in nonlinear metamaterials,” Phys. Rev. A 81, 0238291(2010).

Zhang, L.

D. Z. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[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]

Zheltikov, A. M.

G. D’Aguanno, N. Aközbec, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Dispersion-free pulse propagation in a negative-index material,” Opt. Lett. 30, 1998–2000 (2005).
[CrossRef]

M. Scalora, M. S. Syrchin, N. Aközbec, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 0139021(2005).

Zhou, J.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Zhou, J. F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

Zschiedrich, L.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

M. Bayindir, K. Aydin, E. Ozbay, P. Marko, and C. M. Soukoulis, “Transmission properties of composite metamaterials in free space,” Appl. Phys. Lett. 81, 120–122 (2002).
[CrossRef]

Contemp. Phys. (1)

J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191–202 (2004).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

D. Z. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopoulos, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. 47, 2059–2074 (1999).
[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]

J. Appl. Spectrosc. (1)

V. V. Grigoriev and V. V. Kabanov, “Propagation of a localized wavepacket in a metamaterial with a negative index of refraction,” J. Appl. Spectrosc. 75, 192–198 (2008).
[CrossRef]

J. Comp. Theo. Nanoscience (1)

V. G. Veselago, “Negative refractive index materials,” J. Comp. Theo. Nanoscience 3, 1–30 (2006).

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

J. Phys. A (1)

H. Leblond and D. Mihalache, “Few-optical-cycle dissipative solitons,” J. Phys. A 43, 375205 (2010).
[CrossRef]

J. Phys. Soc. Jpn. (2)

T. Taniuti and C.-C. Wei, “Reductive perturbation method in nonlinear wave propagation,” J. Phys. Soc. Jpn. 24, 941–946 (1968).
[CrossRef]

T. Tsurumi, “Propagation of few- to sub-cycle pulse in dispersive media,” J. Phys. Soc. Jpn. 75, 024002 (2006).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (6)

H. Leblond, D. Kremer, and D. Mihalache, “Collapse of ultrashort spatiotemporal pulses described by the cubic generalized Kadomtsev-Petviashvili equation,” Phys. Rev. A 81, 0338241 (2010).

H. Leblond and F. Sanchez, “Models for optical solitons in the two-cycle regime,” Phys. Rev. A 67, 0138041 (2003).
[CrossRef]

A. Kumar, “Ultrashort pulse propagation in a cubic medium including the Raman effect,” Phys. Rev. A 81, 0138071(2010).

I. V. Melnikov, D. Mihalache, F. Moldoveanu, and N.-C. Panoiu, “Quasiadiabatic following of femtosecond optical pulses in a weakly excited semiconductor,” Phys. Rev. A 56, 1569–1576 (1997).
[CrossRef]

J. Zhang, S. Wen, Y. Xiang, Y. Wang, and H. Luo, “Spatiotemporal electromagnetic soliton and spatial ring formation in nonlinear metamaterials,” Phys. Rev. A 81, 0238291(2010).

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 0338151(2007).

Phys. Rev. B (1)

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 1651121 (2004).
[CrossRef]

Phys. Rev. E (5)

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 0366141 (2005).

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 0166261 (2005).
[CrossRef]

W. T. Lu, J. B. Sokoloff, and S. Sridhar, “Refraction of electromagnetic energy for wavepackets incident on a negative-index medium is always negative,” Phys. Rev. E 69, 0266041 (2004).

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal dynamics of electromagnetic pulses in saturating nonlinear optical media with normal group velocity dispersion,” Phys. Rev. E 60, 7622–7625 (1999).
[CrossRef]

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 0366171 (2006).
[CrossRef]

Phys. Rev. Lett. (6)

R. G. Flesch, A. Pushkarev, and J. V. Moloney, “Carrier wave shocking of femtosecond optical pulses,” Phys. Rev. Lett. 76, 2488–2491 (1996).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef]

M. Scalora, M. S. Syrchin, N. Aközbec, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 0139021(2005).

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

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]

Science (3)

J. B. Pendry, D. Shurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (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]

Sov. Phys. Usp. (1)

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

Theor. Math. Phys. (1)

A. Kumar, “A few- and sub-cycle pulse evolution equation in a cubic nonlinear medium,” Theor. Math. Phys. 160, 968–975 (2009).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Frequency dependence of the real parts of the dielectric permittivity, Re(ε), and the magnetic susceptibility, Re(μ), and the real and imaginary parts, Re(n) and Im(n), of the refractive index for γ¯=5×10-4.

Fig. 2.
Fig. 2.

The dimensionless electric field, E¯, of the input (dotted line) and the output (solid curve) single-cycle pulse as a function of the dimensionless time, t¯, after a propagation distance of ξ=9λp for ωm/ωp=0.8, χ(3)=10-10(esu), and E0=3×104Statvolts/cm.

Fig. 3.
Fig. 3.

The dimensionless electric field, E¯, of the input (solid line) and the output (dotted line) single-cycle pulse as a function of the dimensionless time, t¯, after a propagation distance of 7λp for χ(3)=10-10(esu) and E0=3×104Statvolts/cm. The solid line corresponds to the input pulse, while the dashed and dotted lines represent the output pulse with ωm/ωp=0.8 and ωm/ωp=1.2, respectively.

Fig. 4.
Fig. 4.

Input (dashed line) and the output (solid line) dimensionless electric field of 4-cycle pulse after a propagation distance of 0.9λp for ωm/ωp=0.8, χ(3)=10-10(esu), and E0=3×104Statvolts/cm.

Fig. 5.
Fig. 5.

(a) The dimensionless intensity, |E¯|2, as a function of the dimensionless transverse radial distance, r¯, and the dimensionless propagation distance, ξ, of a single-cycle pulse after a propagation distance of 4×10-4λp for ωm/ωp=0.8, χ(3)=10-10(esu), and E0=3×104Statvolts/cm. (b) The dimensionless intensity, |E¯|2, as a function of the dimensionless time, t¯, and the dimensionless propagation distance, ξ, of a single-cycle pulse after a propagation distance of 4×10-4λp for ωm/ωp=0.8, χ(3)=10-10(esu), and E0=3×104Statvolts/cm.

Fig. 6.
Fig. 6.

(a) The dimensionless intensity, |E¯|2, as a function of the dimensionless time, t¯, and the dimensionless propagation distance, ξ, of a single-cycle pulse after a propagation distance of 8×10-4λp for ωm/ωp=0.8, χ(3)=10-10(esu), and E0=3×104Statvolts/cm. (b) The dimensionless intensity, |E¯|2, as a function of the dimensionless transverse coordinates, x¯ and y¯, of a single-cycle pulse after a propagation distance of 8×10-4λp for ωm/ωp=0.8, χ(3)=10-10(esu), and E0=3×104Statvolts/cm.

Fig. 7.
Fig. 7.

(a) The dimensionless intensity, |E¯|2, as a function of the dimensionless time, t¯, and the dimensionless propagation distance, ξ, of a 4-cycle pulse after a propagation distance of 8×10-4λp for ωm/ωp=0.8, χ(3)=10-10(esu), and E0=3×104Statvolts/cm. (b) The dimensionless intensity, |E¯|2, as a function of the dimensionless transverse coordinates, x¯ and y¯, of a 4-cycle pulse after a propagation distance of 8×10-4λp for ωm/ωp=0.8, χ(3)=10-10(esu), and E0=3×104Statvolts/cm.

Equations (35)

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∇⃗×E⃗+1cB⃗t=0,
∇⃗×H⃗-1cD⃗Lt=4πcP⃗NLt,
∇⃗·D⃗=0,
∇⃗·B⃗=0,
∇⃗2E⃗-εμc22E⃗t2=4πμc22P⃗NLt2,
P⃗NL=χ(3)|E⃗|2E⃗,
z¯=zλp,r¯=r¯λp(xλp,yλp),t¯=ctλp
E¯=E⃗E0,P¯NL=χ¯(3)|E¯|2E¯,
(2z¯2+¯2)E¯-εμ2E¯t¯2=μ2P¯NLt¯2,
(2z¯2+¯2)E¯=-(ω¯ε(ω¯))(ω¯μ(ω¯))E¯~-(ω¯μ(ω¯))ω¯P¯~NL,
(2z¯2+¯2)E¯=-[n=01n!n(ω¯μ)ω¯n|ω¯=0(it¯)n][m=01m!m(ω¯ε)ω¯m|ω¯=0(it¯)m]E¯-[n=01n!n(ω¯μ)ω¯n|ω¯=0(it¯)n](it¯)P¯NL.
z¯=ϵz¯,R¯=ϵar¯,T¯=ϵt¯,
(ϵ22z¯2+ϵ2a¯2)E¯(R¯,z¯,T¯)=[n=01n!n(ω¯μ)ω¯n|ω¯=0(iϵT¯)n]×[m=01m!m(ω¯ε)ω¯m|ω¯=0(iϵT¯)m]E¯(R¯,z¯,T¯)[n=01n!n(ω¯μ)ω¯n|ω¯=0(iϵT¯)n]×(iϵT¯)χ¯(3)[(E¯·E¯)E¯(R¯,z¯,T¯)].
ω¯ϵ(ω¯)=β1(ω¯)+i2α1(ω¯),β1(ω¯)=Re[ω¯|ϵ|eiθ1],α1(ω¯)=2Im[ω¯|ϵ|eiθ1],
ω¯μ(ω¯)=β2(ω¯)+i2α2(ω¯),β2(ω¯)=Re[ω¯|μ|eiθ2],α2(ω¯)=2Im[ω¯|μ|eiθ2],
β10(2m)=(d2mβ1dω¯2m)ω¯=0=0,α10(2m+1)=(d2m+1a1dω¯2m+1)ω¯=0=0,
β20(2m)=(d2mβ2dω¯2m)ω¯=0=0,α20(2m+1)=(d2m+1a2dω¯2m+1)ω¯=0=0,
[m=01m!m(ω¯ε)ω¯m|ω¯=0(iϵT¯)m]=[β10(1)(iϵT¯)+D^1(T¯)],
[n=01n!n(ω¯μ)ω¯n|ω¯=0(iϵT¯)n]=[β20(1)(iϵT¯)+D^2(T¯)],
β10(1)β1ω¯,|ω¯=0,β20(1)β2ω¯,|ω¯=0,
D^1(T¯)=m=0i2m+3[ϵ2m+3β10(2m+3)(2m+3)!(T¯)2m+3+ϵ2m+2α10(2m+2)2×(2m+2)!(T¯)2m+2],
D^2(T¯)=n=0i2n+3[ϵ2n+3β20(2n+3)(2n+3)!(T¯)2n+3+ϵ2n+2α20(2n+2)2×(2n+2)!(T¯)2n+2].
τ=T¯+z¯(β10(1)β20(1))-12T¯+z¯V,ξ=ϵσz¯,
(ϵ2+2σ2ξ2+2ϵ2+σV2τξ+ϵ2V22τ2+ϵ2a¯2)E¯=[ϵ2β10(1)β20(1)2τ2-ϵ2β20(1)D^1(τ)(iϵτ)-ϵ2β10(1)D^2(τ)(iϵτ)-ϵ4D^1(τ)D^2(τ)]E¯-χ¯(3)(iϵτ)[β20(1)(iϵτ)+ϵ2D^2(τ)]|E¯|2E¯,
D^1(τ)=m=0i2m+3[ϵ2m+1β10(2m+3)(2m+3)!(τ)2m+3+ϵ2mα10(2m+2)2×(2m+2)!(τ)2m+2],
D^2(τ)=n=0i2n+3[ϵ2n+1β20(2n+3)(2n+3)!(τ)2n+3+ϵ2nα20(2n+2)2×(2n+2)!(τ)2n+2].
α10(2)=ϵα10(2),α20(2)=ϵα20(2),χ¯(3)=ϵ2χ¯(3).
2τξE¯=12β10(1)β20(1)¯2E¯112β10(1)β20(1)(β10(1)β20(3)+β20(1)β10(3))4E¯τ418β10(1)β20(1)(β10(1)α20(2)+β20(1)α10(2))3E¯τ3+β20(1)12β10(1)β20(1)χ¯(3)2(|E¯|2E¯)τ2.
ε(ω¯)=1-1ω¯2+iω¯γ¯e,
μ(ω¯)=1-ωm2/ωp2ω¯2+iω¯γ¯m,
2τξE¯=112β10(1)β20(1)(β10(1)β20(3)+β20(1)β10(3))4E¯τ418β10(1)β20(1)(β10(1)α20(2)+β20(1)α10(2))3E¯τ3+β20(1)12β10(1)β20(1)χ¯(3)2(|E¯|2E¯)τ2.
E=e12(1.67t¯8.8871)2cos(ω¯t¯).
2E¯τξ=12β10(1)β20(1)χ¯(3)β20(1)2(|E¯|2E¯)τ2,
2E¯τξ=-12β10(1)β20(1)¯2E¯-112β10(1)β20(1)(β10(1)β20(3)+β20(1)β10(3))4E¯τ4-18β10(1)β20(1)(β10(1)α20(2)+β20(1)α10(2))3E¯τ3+12β10(1)β20(1)χ¯(3)β20(1)2(|E¯|2E¯)τ2.
E¯=exp(-12((1.67t¯)4×8.8871)2-r¯2)cos(ω¯t¯).

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