Abstract

This numerical study demonstrates that Doppler redshift exists in the reflected spectrum of a few-cycle pulse, propagating through a dense medium. It manifests itself in two different forms, a sharp low-frequency spike (LFS) located at the red edge of the reflected spectrum and a relatively broader redshift near the carrier frequency. With the variation of the laser and medium parameters, the dominant reflection mechanism changes between bulk generation of backwards propagation waves and nonlinear reflection near the front face. This leads to the manifestation of Doppler effect changing accordingly between the two different forms. This study unifies the physical mechanism behind the LFS and dynamic nonlinear optical skin effect, which enriches the theoretical explanation of the spectral redshift of few-cycle pulse propagation beyond the intrapulse four-wave mixing.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  23. C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92(4), 047402 (2004).
    [Crossref] [PubMed]
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  25. R. K. Bullough, P. J. Caudrey, J. C. Eilbeck, and J. D. Gibbon, “A general theory of self-induced transparency,” Opt. Quantum Electron. 6(1), 121–140 (1974).

2014 (2)

Z. D. Wang, T. T. Liu, J. L. Ma, and J. Xiao, “Interference resonant propagation and spectral properties of two-colour femtosecond Gaussian pulses in three-level Λ-type atomic medium,” Opt. Commun. 330, 155–159 (2014).
[Crossref]

R. Marskar and U. L. Österberg, “Backpropagation and decay of self-induced-transparency pulses,” Phys. Rev. A 89(2), 023828 (2014).
[Crossref]

2013 (1)

L. Guo, X. T. Xie, and Z. M. Zhan, “Shaping of few-cycle laser pulses via a subwavelength structure,” Chin. Phys. B 22(9), 094212 (2013).
[Crossref]

2012 (1)

Q. Q. Xu, D. Z. Yao, X. N. Liu, Q. Zhou, and G. G. Xiong, “Solitary propagation effect of a well-defined chirped femtosecond laser pulse in a resonance-absorbing medium,” Phys. Rev. A 86(2), 023853 (2012).
[Crossref]

2011 (1)

2010 (1)

X. T. Xie and M. A. Macovei, “Single-cycle gap soliton in a subwavelength structure,” Phys. Rev. Lett 104(7), 073902 (2010).
[Crossref] [PubMed]

2004 (1)

C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92(4), 047402 (2004).
[Crossref] [PubMed]

2002 (1)

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Area evolution of a few-cycle pulse laser in a two-level-atom medium,” Phys. Rev. A 65(3) R031402 (2002).
[Crossref]

2001 (2)

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Spectrum of a few-cycle laser pulse propagation in a two-level atom medium,” Chin. Phys. 10(10), 0941–0945 (2001).
[Crossref]

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Propagation of few-cycle pulse laser in two-level atom medium,” Chin. Phy. Lett. 18(7), 912 (2001).

1999 (1)

V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell–Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83(3), 544–547 (1999).
[Crossref]

1998 (1)

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Coherent spectroscopic effects in the propagation of ultrashort pulses through a two-level system,” Phys. Rev. A 57(1), R36–R39 (1998).
[Crossref]

1997 (2)

1996 (2)

W. Forysiak, R. G. Flesch, and J. V. Moloney, “Doppler shift of self-reflected optical pulses at an interface: dynamic nonlinear optical skin effect,” Phys. Rev. Lett. 76(20), 3695–3698 (1996).
[Crossref] [PubMed]

K. S. Yee, “Numerical solution of initial boundary value problem involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. 14(3), 302–307(1996).

1995 (1)

R. W. Ziolkowski, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. Lett. 52(4), 3082–3094 (1995).

1987 (1)

1985 (1)

L. R. Franco, “Self-reflected wave inside a very dense saturable absorber,” Phys. Rev. Lett. 55(20), 2149–2151 (1985).
[Crossref]

1979 (1)

R. K. Bullough, P. M. Jack, P. W. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20(3), 364–381 (1979).
[Crossref]

1974 (1)

R. K. Bullough, P. J. Caudrey, J. C. Eilbeck, and J. D. Gibbon, “A general theory of self-induced transparency,” Opt. Quantum Electron. 6(1), 121–140 (1974).

1972 (2)

J. C. Eilbeck, “Reflection of short pulses in linear optics,” Gen. Phys. 5(9), 1355–1363 (1972).
[Crossref]

J. C. Eilbeck and R. K. Bullough, “The method of characteristics in the theory of resonant or nonresonant nonlinear optics,” Gen. Phys. 5(6), 820–829 (1972).
[Crossref]

Allen, L.

L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).

Baltuska, A.

Bullough, R. K.

R. K. Bullough, P. M. Jack, P. W. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20(3), 364–381 (1979).
[Crossref]

R. K. Bullough, P. J. Caudrey, J. C. Eilbeck, and J. D. Gibbon, “A general theory of self-induced transparency,” Opt. Quantum Electron. 6(1), 121–140 (1974).

J. C. Eilbeck and R. K. Bullough, “The method of characteristics in the theory of resonant or nonresonant nonlinear optics,” Gen. Phys. 5(6), 820–829 (1972).
[Crossref]

Caudrey, P. J.

R. K. Bullough, P. J. Caudrey, J. C. Eilbeck, and J. D. Gibbon, “A general theory of self-induced transparency,” Opt. Quantum Electron. 6(1), 121–140 (1974).

De Silvestri, S.

Eberly, J.H.

L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).

Eilbeck, J. C.

R. K. Bullough, P. J. Caudrey, J. C. Eilbeck, and J. D. Gibbon, “A general theory of self-induced transparency,” Opt. Quantum Electron. 6(1), 121–140 (1974).

J. C. Eilbeck, “Reflection of short pulses in linear optics,” Gen. Phys. 5(9), 1355–1363 (1972).
[Crossref]

J. C. Eilbeck and R. K. Bullough, “The method of characteristics in the theory of resonant or nonresonant nonlinear optics,” Gen. Phys. 5(6), 820–829 (1972).
[Crossref]

Elsaesser, T.

C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92(4), 047402 (2004).
[Crossref] [PubMed]

Ferencz, K.

Flesch, R. G.

W. Forysiak, R. G. Flesch, and J. V. Moloney, “Doppler shift of self-reflected optical pulses at an interface: dynamic nonlinear optical skin effect,” Phys. Rev. Lett. 76(20), 3695–3698 (1996).
[Crossref] [PubMed]

Forysiak, W.

W. Forysiak, R. G. Flesch, and J. V. Moloney, “Doppler shift of self-reflected optical pulses at an interface: dynamic nonlinear optical skin effect,” Phys. Rev. Lett. 76(20), 3695–3698 (1996).
[Crossref] [PubMed]

Franco, L. R.

L. R. Franco, “Propagation of light in a nonlinear absorber,” J. Opt. Soc. Am. B 4(11), 1878–1884 (1987).
[Crossref]

L. R. Franco, “Self-reflected wave inside a very dense saturable absorber,” Phys. Rev. Lett. 55(20), 2149–2151 (1985).
[Crossref]

Gaeta, A. L.

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Coherent spectroscopic effects in the propagation of ultrashort pulses through a two-level system,” Phys. Rev. A 57(1), R36–R39 (1998).
[Crossref]

Gibbon, J. D.

R. K. Bullough, P. J. Caudrey, J. C. Eilbeck, and J. D. Gibbon, “A general theory of self-induced transparency,” Opt. Quantum Electron. 6(1), 121–140 (1974).

Guo, L.

L. Guo, X. T. Xie, and Z. M. Zhan, “Shaping of few-cycle laser pulses via a subwavelength structure,” Chin. Phys. B 22(9), 094212 (2013).
[Crossref]

Herrmann, J.

V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell–Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83(3), 544–547 (1999).
[Crossref]

Hey, R.

C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92(4), 047402 (2004).
[Crossref] [PubMed]

Jack, P. M.

R. K. Bullough, P. M. Jack, P. W. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20(3), 364–381 (1979).
[Crossref]

Kalosha, V. P.

V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell–Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83(3), 544–547 (1999).
[Crossref]

Kitchenside, P. W.

R. K. Bullough, P. M. Jack, P. W. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20(3), 364–381 (1979).
[Crossref]

Krausz, F.

Liu, T. T.

Z. D. Wang, T. T. Liu, J. L. Ma, and J. Xiao, “Interference resonant propagation and spectral properties of two-colour femtosecond Gaussian pulses in three-level Λ-type atomic medium,” Opt. Commun. 330, 155–159 (2014).
[Crossref]

Liu, X. N.

Q. Q. Xu, D. Z. Yao, X. N. Liu, Q. Zhou, and G. G. Xiong, “Solitary propagation effect of a well-defined chirped femtosecond laser pulse in a resonance-absorbing medium,” Phys. Rev. A 86(2), 023853 (2012).
[Crossref]

Luo, C. W.

C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92(4), 047402 (2004).
[Crossref] [PubMed]

Ma, J. L.

Z. D. Wang, T. T. Liu, J. L. Ma, and J. Xiao, “Interference resonant propagation and spectral properties of two-colour femtosecond Gaussian pulses in three-level Λ-type atomic medium,” Opt. Commun. 330, 155–159 (2014).
[Crossref]

Macovei, M. A.

X. T. Xie and M. A. Macovei, “Single-cycle gap soliton in a subwavelength structure,” Phys. Rev. Lett 104(7), 073902 (2010).
[Crossref] [PubMed]

Marskar, R.

R. Marskar and U. L. Österberg, “Backpropagation and decay of self-induced-transparency pulses,” Phys. Rev. A 89(2), 023828 (2014).
[Crossref]

R. Marskar and U. L. Österberg, “Multilevel Maxwell–Bloch simulations in inhomogeneously broadened media,” Opt. Express 19(18), 16784–16796 (2011).
[Crossref] [PubMed]

Moloney, J. V.

W. Forysiak, R. G. Flesch, and J. V. Moloney, “Doppler shift of self-reflected optical pulses at an interface: dynamic nonlinear optical skin effect,” Phys. Rev. Lett. 76(20), 3695–3698 (1996).
[Crossref] [PubMed]

Nisoli, M.

Österberg, U. L.

R. Marskar and U. L. Österberg, “Backpropagation and decay of self-induced-transparency pulses,” Phys. Rev. A 89(2), 023828 (2014).
[Crossref]

R. Marskar and U. L. Österberg, “Multilevel Maxwell–Bloch simulations in inhomogeneously broadened media,” Opt. Express 19(18), 16784–16796 (2011).
[Crossref] [PubMed]

Ploog, K. H.

C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92(4), 047402 (2004).
[Crossref] [PubMed]

Pshenichnikov, M. S.

Ranka, J. K.

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Coherent spectroscopic effects in the propagation of ultrashort pulses through a two-level system,” Phys. Rev. A 57(1), R36–R39 (1998).
[Crossref]

Reimann, K.

C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92(4), 047402 (2004).
[Crossref] [PubMed]

Sartania, S.

Saunders, R.

R. K. Bullough, P. M. Jack, P. W. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20(3), 364–381 (1979).
[Crossref]

Schirmer, R. W.

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Coherent spectroscopic effects in the propagation of ultrashort pulses through a two-level system,” Phys. Rev. A 57(1), R36–R39 (1998).
[Crossref]

Spielmann, Ch.

Svelto, O.

Szipöcs, R.

Wang, Z. D.

Z. D. Wang, T. T. Liu, J. L. Ma, and J. Xiao, “Interference resonant propagation and spectral properties of two-colour femtosecond Gaussian pulses in three-level Λ-type atomic medium,” Opt. Commun. 330, 155–159 (2014).
[Crossref]

Wang, Z. Y.

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Area evolution of a few-cycle pulse laser in a two-level-atom medium,” Phys. Rev. A 65(3) R031402 (2002).
[Crossref]

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Propagation of few-cycle pulse laser in two-level atom medium,” Chin. Phy. Lett. 18(7), 912 (2001).

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Spectrum of a few-cycle laser pulse propagation in a two-level atom medium,” Chin. Phys. 10(10), 0941–0945 (2001).
[Crossref]

Wegener, M.

M. Wegener, Extreme Nonlinear Optics (Springer, 2005).

Wei, Z.

Wiersma, D. A.

Woerner, M.

C. W. Luo, K. Reimann, M. Woerner, T. Elsaesser, R. Hey, and K. H. Ploog, “Phase-resolved nonlinear response of a two-dimensional electron gas under femtosecond intersubband excitation,” Phys. Rev. Lett. 92(4), 047402 (2004).
[Crossref] [PubMed]

Xiao, J.

Z. D. Wang, T. T. Liu, J. L. Ma, and J. Xiao, “Interference resonant propagation and spectral properties of two-colour femtosecond Gaussian pulses in three-level Λ-type atomic medium,” Opt. Commun. 330, 155–159 (2014).
[Crossref]

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Area evolution of a few-cycle pulse laser in a two-level-atom medium,” Phys. Rev. A 65(3) R031402 (2002).
[Crossref]

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Spectrum of a few-cycle laser pulse propagation in a two-level atom medium,” Chin. Phys. 10(10), 0941–0945 (2001).
[Crossref]

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Propagation of few-cycle pulse laser in two-level atom medium,” Chin. Phy. Lett. 18(7), 912 (2001).

Xie, X. T.

L. Guo, X. T. Xie, and Z. M. Zhan, “Shaping of few-cycle laser pulses via a subwavelength structure,” Chin. Phys. B 22(9), 094212 (2013).
[Crossref]

X. T. Xie and M. A. Macovei, “Single-cycle gap soliton in a subwavelength structure,” Phys. Rev. Lett 104(7), 073902 (2010).
[Crossref] [PubMed]

Xiong, G. G.

Q. Q. Xu, D. Z. Yao, X. N. Liu, Q. Zhou, and G. G. Xiong, “Solitary propagation effect of a well-defined chirped femtosecond laser pulse in a resonance-absorbing medium,” Phys. Rev. A 86(2), 023853 (2012).
[Crossref]

Xu, Q. Q.

Q. Q. Xu, D. Z. Yao, X. N. Liu, Q. Zhou, and G. G. Xiong, “Solitary propagation effect of a well-defined chirped femtosecond laser pulse in a resonance-absorbing medium,” Phys. Rev. A 86(2), 023853 (2012).
[Crossref]

Xu, Z. Z.

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Area evolution of a few-cycle pulse laser in a two-level-atom medium,” Phys. Rev. A 65(3) R031402 (2002).
[Crossref]

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Propagation of few-cycle pulse laser in two-level atom medium,” Chin. Phy. Lett. 18(7), 912 (2001).

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Spectrum of a few-cycle laser pulse propagation in a two-level atom medium,” Chin. Phys. 10(10), 0941–0945 (2001).
[Crossref]

Yao, D. Z.

Q. Q. Xu, D. Z. Yao, X. N. Liu, Q. Zhou, and G. G. Xiong, “Solitary propagation effect of a well-defined chirped femtosecond laser pulse in a resonance-absorbing medium,” Phys. Rev. A 86(2), 023853 (2012).
[Crossref]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problem involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. 14(3), 302–307(1996).

Zhan, Z. M.

L. Guo, X. T. Xie, and Z. M. Zhan, “Shaping of few-cycle laser pulses via a subwavelength structure,” Chin. Phys. B 22(9), 094212 (2013).
[Crossref]

Zhou, Q.

Q. Q. Xu, D. Z. Yao, X. N. Liu, Q. Zhou, and G. G. Xiong, “Solitary propagation effect of a well-defined chirped femtosecond laser pulse in a resonance-absorbing medium,” Phys. Rev. A 86(2), 023853 (2012).
[Crossref]

Ziolkowski, R. W.

R. W. Ziolkowski, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. Lett. 52(4), 3082–3094 (1995).

Chin. Phy. Lett. (1)

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Propagation of few-cycle pulse laser in two-level atom medium,” Chin. Phy. Lett. 18(7), 912 (2001).

Chin. Phys. (1)

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Spectrum of a few-cycle laser pulse propagation in a two-level atom medium,” Chin. Phys. 10(10), 0941–0945 (2001).
[Crossref]

Chin. Phys. B (1)

L. Guo, X. T. Xie, and Z. M. Zhan, “Shaping of few-cycle laser pulses via a subwavelength structure,” Chin. Phys. B 22(9), 094212 (2013).
[Crossref]

Gen. Phys. (2)

J. C. Eilbeck, “Reflection of short pulses in linear optics,” Gen. Phys. 5(9), 1355–1363 (1972).
[Crossref]

J. C. Eilbeck and R. K. Bullough, “The method of characteristics in the theory of resonant or nonresonant nonlinear optics,” Gen. Phys. 5(6), 820–829 (1972).
[Crossref]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problem involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. 14(3), 302–307(1996).

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

Opt. Commun. (1)

Z. D. Wang, T. T. Liu, J. L. Ma, and J. Xiao, “Interference resonant propagation and spectral properties of two-colour femtosecond Gaussian pulses in three-level Λ-type atomic medium,” Opt. Commun. 330, 155–159 (2014).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

R. K. Bullough, P. J. Caudrey, J. C. Eilbeck, and J. D. Gibbon, “A general theory of self-induced transparency,” Opt. Quantum Electron. 6(1), 121–140 (1974).

Phys. Rev. A (4)

J. Xiao, Z. Y. Wang, and Z. Z. Xu, “Area evolution of a few-cycle pulse laser in a two-level-atom medium,” Phys. Rev. A 65(3) R031402 (2002).
[Crossref]

Q. Q. Xu, D. Z. Yao, X. N. Liu, Q. Zhou, and G. G. Xiong, “Solitary propagation effect of a well-defined chirped femtosecond laser pulse in a resonance-absorbing medium,” Phys. Rev. A 86(2), 023853 (2012).
[Crossref]

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Coherent spectroscopic effects in the propagation of ultrashort pulses through a two-level system,” Phys. Rev. A 57(1), R36–R39 (1998).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a) The reflected field of a few-cycle pulse propagation through a dense TLA medium, which consists of a leading (thick line) and tail parts (thin line). (b) The corresponding reflected (solid line) and reemitted field (dashed line) spectra. (c) The velocity variations of the moving absorption front versus time and the corresponding frequencies obtained with ω / ω 0 = c υ c + υ. The horizontal dashed line is the central frequency of the LFS, ω/ω0 = 0.215, and the two horizontal solid linea represent its FWMH τ = 0.063ω0.
Fig. 2
Fig. 2 (a)(d) The reflected spectrum. (b)(e) The spatial evolution of the absorption front. (c)(f) The time evolution of the absorption front’s velocity (circles) and the backpropagation wave’s frequency (squares). The top and bottom correspond to τp = 10fs and τp = 40fs, respectively. The dashed line in (d) is the spectrum of the incident pulse.
Fig. 3
Fig. 3 The reflected (solid line) and reemitted spectra (dashed line) (a)(e). Time evolution of velocities (b)(f) and corresponding frequencies (b)(g). The top and bottom correspond to 3π and 4π pulses, respectively. Subscripts 1 and 2 in (f)(g) stand for the first and second moving fronts respectively. (c) The instantaneous fields of 2π (dotted line), 3π (dashed line), and 4π (solid line) incident pulses. The insert in (a) is the enlarged view to the LFS for a 3π incident pulse.
Fig. 4
Fig. 4 Dependence of amplitude (blue squares) and frequency (red circles) of the LFS on medium density for 2π (a) and 4π (b) pulses, respectively. (c)The variation of the reflected spectra with increasing medium density for a 2π pulse.
Fig. 5
Fig. 5 The left and right are the reflected and transmitted spectra, respectively. The top and bottom are that for ωc = 0.1fs−1 and ωc = 1.0fs−1, respectively.

Equations (1)

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t H y = 1 μ 0 z E x , t E x = 1 ε 0 z H y 1 ε 0 t P x . t u = 1 T 2 u ω 0 v , t v = 1 T 2 v + ω 0 u + 2 Ω w , t w = 1 T 1 ( w w 0 ) 2 Ω v .

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