Abstract

Controllable and dispersive magnetic permeability in negative-index materials (NIMs) offers greatly enhanced design freedom to alter the linear and nonlinear properties. This makes it possible for us to control the propagation of ultrashort pulses at will. In this paper, we will investigate the self-focusing of ultrashort pulses associated with anomalous nonlinear self-steepening (SS) effect and anomalous space–time focus (linear SS) effect in bulk NIMs with a nonlinear electric polarization, trying to disclose some unusual behaviors different from those in naturally positive-index materials. It is found that negative SS acts to push the peak intensity toward the trailing edge of a pulse and the resulting spectrum exhibits a broad redshifted trail; however, positive SS leads to the occurrence of the opposite situation in both temporal and spectral domains, that is, the peak intensity will move toward the leading edge of the pulse by shifting the spectrum toward the blue side. In addition, compared with the case in positive-index materials (PIMs), another most notable property of NIMs is that, whether self-focusing of ultrashort pulses will occur at the leading or the trailing edge can be manipulated freely by engineering the SS effect by choosing the size of split-ring resonator circuit elements. Our analysis is performed by directly numerically solving nonlinear Schrödinger equations as well as by using the moment method, both showing consistent results. These findings demonstrate that NIMs can provide us unique opportunities unattainable in PIMs to manipulate ultrashort pulse propagation.

© 2013 Optical Society of America

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2012

2011

Y. Xiang, X. Dai, S. Wen, J. Guo, and D. Fang, “Controllable Raman soliton self-frequency shift in nonlinear metamaterials,” Phys. Rev. A 84, 033815 (2011).
[CrossRef]

A. K. Sarma and M. Saha, “Modulational instability of coupled nonlinear field equations for pulse propagation in a negative index material embedded into a Kerr medium,” J. Opt. Soc. Am. B 28, 944–948 (2011).
[CrossRef]

2010

A. Joseph and K. Porsezian, “Stability criterion for Gaussian pulse propagation through negative index materials,” Phys. Rev. A 81, 023805 (2010).
[CrossRef]

P. Li, R. Yang, and Z. Xu, “Gray solitary-wave solutions in nonlinear negative-index materials,” Phys. Rev. A 82, 046603 (2010).

J. Zhang, S. Wen, Y. Xiang, Y. Wang, and H. Luo, “Spatiotemporal electromagnetic soliton and spatial formation in nonlinear metamaterials,” Phys. Rev. A 81, 023829 (2010).
[CrossRef]

P. Kinsler, “Unidirectional optical pulse propagation equation for materials with both electric and magnetic responses,” Phys. Rev. A 81, 023808 (2010).
[CrossRef]

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

A. D. Boardman, O. Hess, R. C. Mitchell-Thomas, Y. G. Rapoport, and L. Velasco, “Temporal solitons in magnetooptic and metamaterial waveguides,” Photon. Nanostr. Fundam. Appl. 8, 228–243 (2010).
[CrossRef]

2009

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Y. Hu and H. Zhuo, “Investigation of quasi-steady-state self-focusing in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 26, B68–B73 (2009).
[CrossRef]

2008

Y. Hu, S. Wen, H. Zhuo, K. You, and D. Fan, “Focusing properties of Gaussian beams by a slab of Kerr-type left-handed metamaterial,” Opt. Express 16, 4774–4784 (2008).
[CrossRef]

Y. Hu, S. Wen, Y. Wang, and D. Fan, “Propagation of Gaussian beams in negative-index metamaterials with cubic nonlinearity,” Opt. Commun. 281, 2663–2669 (2008).
[CrossRef]

A. D. Boardman, N. King, R. C. Mitchell-Thomas, V. Malnev, and V. G. Rapoport, “Gain control and diffraction-managed solitons in metamaterials,” Metamaterials 2, 145–154 (2008).
[CrossRef]

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

S. Feng and K. Halterman, “Parametrically shielding electromagnetic fields by nonlinear metamaterials,” Phys. Rev. Lett. 100, 063901 (2008).
[CrossRef]

2007

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic beam propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[CrossRef]

A. Couairon and A. Mysrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[CrossRef]

2006

A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index metamaterials by optical parametric amplification,” Opt. Lett. 31, 2169–2171 (2006).
[CrossRef]

J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. 97, 073903 (2006).
[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, 036617 (2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self steepening in modulation instability in negative-index material,” Opt. Express 14, 1568–1575 (2006).
[CrossRef]

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

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef]

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006).
[CrossRef]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef]

2005

V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2005).
[CrossRef]

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
[CrossRef]

M. Scalora, M. S. Syrchin, N. Akozbek, 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, 013902 (2005).
[CrossRef]

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 13, 4922–4930 (2005).
[CrossRef]

G. D’Aguanno, N. Akozbek, 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]

2004

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Tunable transmission and bistability in left-handed band-gap structures,” Appl. Phys. Lett. 85, 1451–1453 (2004).
[CrossRef]

2003

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]

A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

S. Champeaux and L. Bergé, “Femtosecond pulse compression in pressure-gas cells filled with argon,” Phys. Rev. E 68, 066603 (2003).
[CrossRef]

H. Ward and L. Bergé, “Temporal shaping of femtosecond solitary pulses in photoionized media,” Phys. Rev. Lett. 90, 053901 (2003).
[CrossRef]

2000

A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (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]

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

1999

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

1998

M. Trippenbach and Y. B. Band, “Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A 57, 4791–4803 (1998).
[CrossRef]

J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
[CrossRef]

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

1997

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

M. Trippenbach and Y. B. Band, “Dynamics of short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A 56, 4242–4253 (1997).
[CrossRef]

1996

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996).
[CrossRef]

1992

1990

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Akozbek, N.

G. D’Aguanno, N. Akozbek, 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. Akozbek, 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, 013902 (2005).
[CrossRef]

Band, Y. B.

M. Trippenbach and Y. B. Band, “Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A 57, 4791–4803 (1998).
[CrossRef]

M. Trippenbach and Y. B. Band, “Dynamics of short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A 56, 4242–4253 (1997).
[CrossRef]

Bergé, L.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[CrossRef]

S. Champeaux and L. Bergé, “Femtosecond pulse compression in pressure-gas cells filled with argon,” Phys. Rev. E 68, 066603 (2003).
[CrossRef]

H. Ward and L. Bergé, “Temporal shaping of femtosecond solitary pulses in photoionized media,” Phys. Rev. Lett. 90, 053901 (2003).
[CrossRef]

Bloemer, M. J.

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

M. Scalora, M. S. Syrchin, N. Akozbek, 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, 013902 (2005).
[CrossRef]

G. D’Aguanno, N. Akozbek, 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]

Boardman, A. D.

A. D. Boardman, O. Hess, R. C. Mitchell-Thomas, Y. G. Rapoport, and L. Velasco, “Temporal solitons in magnetooptic and metamaterial waveguides,” Photon. Nanostr. Fundam. Appl. 8, 228–243 (2010).
[CrossRef]

A. D. Boardman, N. King, R. C. Mitchell-Thomas, V. Malnev, and V. G. Rapoport, “Gain control and diffraction-managed solitons in metamaterials,” Metamaterials 2, 145–154 (2008).
[CrossRef]

Brabec, T.

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

Brueck, S. R. J.

Cai, W.

Champeaux, S.

S. Champeaux and L. Bergé, “Femtosecond pulse compression in pressure-gas cells filled with argon,” Phys. Rev. E 68, 066603 (2003).
[CrossRef]

Chettiar, U. K.

Clement, T. S.

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

Couairon, A.

A. Couairon and A. Mysrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
[CrossRef]

D’Aguanno, G.

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[CrossRef]

M. Scalora, M. S. Syrchin, N. Akozbek, 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, 013902 (2005).
[CrossRef]

G. D’Aguanno, N. Akozbek, 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]

Dai, X.

Y. Xiang, J. Wu, X. Dai, S. Wen, J. Guo, and Q. Wang, “Manipulating dispersive wave generation by anomalous self-steepening effect in metamaterials,” Opt. Express 20, 26828–26836 (2012).
[CrossRef]

Y. Xiang, X. Dai, S. Wen, J. Guo, and D. Fang, “Controllable Raman soliton self-frequency shift in nonlinear metamaterials,” Phys. Rev. A 84, 033815 (2011).
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Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
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S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic beam propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Diddams, S. A.

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

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Drachev, V. P.

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef]

Fan, D.

Y. Hu, S. Wen, H. Zhuo, K. You, and D. Fan, “Focusing properties of Gaussian beams by a slab of Kerr-type left-handed metamaterial,” Opt. Express 16, 4774–4784 (2008).
[CrossRef]

Y. Hu, S. Wen, Y. Wang, and D. Fan, “Propagation of Gaussian beams in negative-index metamaterials with cubic nonlinearity,” Opt. Commun. 281, 2663–2669 (2008).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic beam propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
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S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self steepening in modulation instability in negative-index material,” Opt. Express 14, 1568–1575 (2006).
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S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
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Fan, W.

Fang, D.

Y. Xiang, X. Dai, S. Wen, J. Guo, and D. Fang, “Controllable Raman soliton self-frequency shift in nonlinear metamaterials,” Phys. Rev. A 84, 033815 (2011).
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M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Tunable transmission and bistability in left-handed band-gap structures,” Appl. Phys. Lett. 85, 1451–1453 (2004).
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N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
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S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
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S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self steepening in modulation instability in negative-index material,” Opt. Express 14, 1568–1575 (2006).
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A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000).
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J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
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J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996).
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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).
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Y. Xiang, J. Wu, X. Dai, S. Wen, J. Guo, and Q. Wang, “Manipulating dispersive wave generation by anomalous self-steepening effect in metamaterials,” Opt. Express 20, 26828–26836 (2012).
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Y. Xiang, X. Dai, S. Wen, J. Guo, and D. Fang, “Controllable Raman soliton self-frequency shift in nonlinear metamaterials,” Phys. Rev. A 84, 033815 (2011).
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S. Feng and K. Halterman, “Parametrically shielding electromagnetic fields by nonlinear metamaterials,” Phys. Rev. Lett. 100, 063901 (2008).
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A. D. Boardman, O. Hess, R. C. Mitchell-Thomas, Y. G. Rapoport, and L. Velasco, “Temporal solitons in magnetooptic and metamaterial waveguides,” Photon. Nanostr. Fundam. Appl. 8, 228–243 (2010).
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A. Joseph and K. Porsezian, “Stability criterion for Gaussian pulse propagation through negative index materials,” Phys. Rev. A 81, 023805 (2010).
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L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
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N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
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King, N.

A. D. Boardman, N. King, R. C. Mitchell-Thomas, V. Malnev, and V. G. Rapoport, “Gain control and diffraction-managed solitons in metamaterials,” Metamaterials 2, 145–154 (2008).
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[CrossRef]

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Tunable transmission and bistability in left-handed band-gap structures,” Appl. Phys. Lett. 85, 1451–1453 (2004).
[CrossRef]

A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
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M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
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N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
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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).
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N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
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P. Li, R. Yang, and Z. Xu, “Gray solitary-wave solutions in nonlinear negative-index materials,” Phys. Rev. A 82, 046603 (2010).

Linden, S.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef]

Luo, H.

J. Zhang, S. Wen, Y. Xiang, Y. Wang, and H. Luo, “Spatiotemporal electromagnetic soliton and spatial formation in nonlinear metamaterials,” Phys. Rev. A 81, 023829 (2010).
[CrossRef]

Malloy, K. J.

Malnev, V.

A. D. Boardman, N. King, R. C. Mitchell-Thomas, V. Malnev, and V. G. Rapoport, “Gain control and diffraction-managed solitons in metamaterials,” Metamaterials 2, 145–154 (2008).
[CrossRef]

Mattiucci, N.

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

M. Scalora, M. S. Syrchin, N. Akozbek, 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, 013902 (2005).
[CrossRef]

G. D’Aguanno, N. Akozbek, 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).
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S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

Mishra, A. K.

Mitchell-Thomas, R. C.

A. D. Boardman, O. Hess, R. C. Mitchell-Thomas, Y. G. Rapoport, and L. Velasco, “Temporal solitons in magnetooptic and metamaterial waveguides,” Photon. Nanostr. Fundam. Appl. 8, 228–243 (2010).
[CrossRef]

A. D. Boardman, N. King, R. C. Mitchell-Thomas, V. Malnev, and V. G. Rapoport, “Gain control and diffraction-managed solitons in metamaterials,” Metamaterials 2, 145–154 (2008).
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J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. 97, 073903 (2006).
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A. Couairon and A. Mysrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
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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]

Nuter, R.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[CrossRef]

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S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

Osgood, R. M.

Padilla, W. J.

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

Panoiu, N. C.

Pendry, J. B.

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

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
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J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

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M. Scalora, M. S. Syrchin, N. Akozbek, 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, 013902 (2005).
[CrossRef]

Popov, A. K.

Porsezian, K.

A. Joseph and K. Porsezian, “Stability criterion for Gaussian pulse propagation through negative index materials,” Phys. Rev. A 81, 023805 (2010).
[CrossRef]

Ramakrishna, S. A.

S. O’Brien, D. McPeake, S. A. Ramakrishna, and J. B. Pendry, “Near-infrared photonic band gaps and nonlinear effects in negative magnetic metamaterials,” Phys. Rev. B 69, 241101 (2004).
[CrossRef]

Ranka, J. K.

J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
[CrossRef]

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996).
[CrossRef]

Rapoport, V. G.

A. D. Boardman, N. King, R. C. Mitchell-Thomas, V. Malnev, and V. G. Rapoport, “Gain control and diffraction-managed solitons in metamaterials,” Metamaterials 2, 145–154 (2008).
[CrossRef]

Rapoport, Y. G.

A. D. Boardman, O. Hess, R. C. Mitchell-Thomas, Y. G. Rapoport, and L. Velasco, “Temporal solitons in magnetooptic and metamaterial waveguides,” Photon. Nanostr. Fundam. Appl. 8, 228–243 (2010).
[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]

Rompotis, N.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Rothenberg, J. E.

Saha, M.

Sarma, A. K.

Sarychev, A. K.

Scalora, M.

M. Scalora, M. S. Syrchin, N. Akozbek, 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, 013902 (2005).
[CrossRef]

G. D’Aguanno, N. Akozbek, 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]

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J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996).
[CrossRef]

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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]

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I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529–534 (2006).
[CrossRef]

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Tunable transmission and bistability in left-handed band-gap structures,” Appl. Phys. Lett. 85, 1451–1453 (2004).
[CrossRef]

A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

Shalaev, V. M.

Shukla, P. K.

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

Shurig, D.

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

Silberberg, Y.

Skupin, S.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J. P. Wolf, “Ultrashort filaments of light in weakly ionized optically transparent media,” Rep. Prog. Phys. 70, 1633–1713 (2007).
[CrossRef]

Smith, D. R.

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

Soukoulis, C. M.

Su, W.

Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic beam propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[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, 036617 (2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self steepening in modulation instability in negative-index material,” Opt. Express 14, 1568–1575 (2006).
[CrossRef]

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M. Scalora, M. S. Syrchin, N. Akozbek, 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, 013902 (2005).
[CrossRef]

Tang, Z.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic beam propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
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M. Trippenbach and Y. B. Band, “Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A 57, 4791–4803 (1998).
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N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
[CrossRef]

Tsitsas, N. L.

N. L. Tsitsas, N. Rompotis, I. Kourakis, P. G. Kevrekidis, and D. J. Frantzeskakis, “Higher-order effects and ultrashort solitons in left-handed metamaterials,” Phys. Rev. E 79, 037601 (2009).
[CrossRef]

Van Engen, A. G.

A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, “Propagation dynamics of intense femtosecond pulses: multiple splittings, coalescence, and continuum generation,” Phys. Rev. Lett. 82, 1430–1433 (1999).
[CrossRef]

Velasco, L.

A. D. Boardman, O. Hess, R. C. Mitchell-Thomas, Y. G. Rapoport, and L. Velasco, “Temporal solitons in magnetooptic and metamaterial waveguides,” Photon. Nanostr. Fundam. Appl. 8, 228–243 (2010).
[CrossRef]

Vier, D. C.

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

Wang, Q.

Wang, Y.

J. Zhang, S. Wen, Y. Xiang, Y. Wang, and H. Luo, “Spatiotemporal electromagnetic soliton and spatial formation in nonlinear metamaterials,” Phys. Rev. A 81, 023829 (2010).
[CrossRef]

Y. Hu, S. Wen, Y. Wang, and D. Fan, “Propagation of Gaussian beams in negative-index metamaterials with cubic nonlinearity,” Opt. Commun. 281, 2663–2669 (2008).
[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, 036617 (2006).
[CrossRef]

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H. Ward and L. Bergé, “Temporal shaping of femtosecond solitary pulses in photoionized media,” Phys. Rev. Lett. 90, 053901 (2003).
[CrossRef]

Wegener, M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef]

Wen, S.

Y. Xiang, J. Wu, X. Dai, S. Wen, J. Guo, and Q. Wang, “Manipulating dispersive wave generation by anomalous self-steepening effect in metamaterials,” Opt. Express 20, 26828–26836 (2012).
[CrossRef]

Y. Xiang, X. Dai, S. Wen, J. Guo, and D. Fang, “Controllable Raman soliton self-frequency shift in nonlinear metamaterials,” Phys. Rev. A 84, 033815 (2011).
[CrossRef]

J. Zhang, S. Wen, Y. Xiang, Y. Wang, and H. Luo, “Spatiotemporal electromagnetic soliton and spatial formation in nonlinear metamaterials,” Phys. Rev. A 81, 023829 (2010).
[CrossRef]

Y. Hu, S. Wen, H. Zhuo, K. You, and D. Fan, “Focusing properties of Gaussian beams by a slab of Kerr-type left-handed metamaterial,” Opt. Express 16, 4774–4784 (2008).
[CrossRef]

Y. Hu, S. Wen, Y. Wang, and D. Fan, “Propagation of Gaussian beams in negative-index metamaterials with cubic nonlinearity,” Opt. Commun. 281, 2663–2669 (2008).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058–3063 (2007).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic beam propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[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, 036617 (2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self steepening in modulation instability in negative-index material,” Opt. Express 14, 1568–1575 (2006).
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Figures (5)

Fig. 1.
Fig. 1.

Linear SS coefficient ε, nonlinear SS coefficient s1, and GVD parameter β2 versus normalized frequency ω/ωpe for (a) ωpm/ωpe=0.8 and (b) ωpm/ωpe=0.95. Here, β2 is calculated in units of 1/cωpe and s=0.02.

Fig. 2.
Fig. 2.

Numerical prediction for (a) on-axis normalized intensity profile and (b) corresponding spectra. The black curve corresponds to the case without negative linear SS at z=1.69, whereas the other curves correspond to the case with the inclusion of negative linear SS at z=1.69, 1.71, and 1.86. ω/ωpe=0.63.

Fig. 3.
Fig. 3.

Numerical prediction for (a) on-axis normalized intensity profile and (b) corresponding spectra. The black curve corresponds to the case with the inclusion of just the negative nonlinear SS effect at z=1.69, whereas the other curves correspond to the case with the inclusion of both negative linear and nonlinear SS at z=1.69, 1.71, and 1.86. ω/ωpe=0.69.

Fig. 4.
Fig. 4.

Numerical prediction for (a) on-axis normalized intensity profile and (b) corresponding spectra. The black curve corresponds to the case with the inclusion of just the positive nonlinear SS effect at z=1.69, whereas the other curves correspond to the case with the inclusion of both negative linear SS and positive nonlinear SS at z=1.69, 1.71, and 1.86. ω/ωpe=0.15.

Fig. 5.
Fig. 5.

Normalized peak intensity as a function of normalized propagation distance z for ω/ωpe=0.15, 0.63, and 0.69.

Equations (12)

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

Eξ=i2β0(1+iετ)1Δ2Eiβ222Eτ2+iγ(1+isτ)|E|2E,
ε=sω0β1β0,
s1=s(2ω0β1β0+ω0μrμrω),
Ez=isgn(n)4(1+iετ)1Δ2Eisgn(β2)2α2Eτ2+isgn(γ)p(1+is1τ)|E|2E,
εr(ω)=1ωpe2ω2+iωγe,μr(ω)=1ωpm2ω2+iωγm,
E(τ,z)=aexp[iΦiΩ(τT)(1+iC)(τT)2/2w2],
dTdz=3sgn(γ)pIs18πw,
dΩdz=sgn(γ)pCs1I2πw3,
dCdz=sgn(γ)pI2πw+4sgn(γ)ps1ΩI2πw,
T=3sgn(γ)ps1Iz8πw,
Ωp2s1I2z24πw4,
Csgn(γ)zpI2πw.

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