Abstract

Pulse propagation in nonlinear waveguides is most frequently modeled by resorting to the generalized nonlinear Schrödinger equation (GNLSE). In recent times, exciting new materials with peculiar nonlinear properties, such as negative nonlinear coefficients and a zero-nonlinearity wavelength, have been demonstrated. Unfortunately, the GNLSE may lead to unphysical results in these cases since, in general, it does not preserve the number of photons and, in the presence of a negative nonlinearity, predicts a blue shift due to Raman scattering. In this paper, we put forth a modified GNLSE that can be used to model the propagation in media with an arbitrary, even negative, nonlinear coefficient. This novel photon-conserving GNLSE (pcGNLSE) ensures preservation of the photon number and can be solved by the same tried and trusted numerical algorithms used for the standard GNLSE. Finally, we compare results for soliton dynamics in fibers with different nonlinear coefficients obtained with the pcGNLSE and the GNLSE.

© 2020 Optical Society of America

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References

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2019 (3)

2018 (3)

A. D. Sánchez, P. I. Fierens, S. M. Hernandez, J. Bonetti, G. Brambilla, and D. F. Grosz, “Anti-Stokes Raman gain enabled by modulation instability in mid-IR waveguides,” J. Opt. Soc. Am. B 35, 2828–2832 (2018).
[Crossref]

F. R. Arteaga-Sierra, A. Antikainen, and G. P. Agrawal, “Soliton dynamics in photonic-crystal fibers with frequency-dependent Kerr nonlinearity,” Phys. Rev. A 98, 013830 (2018).
[Crossref]

A. M. Zheltikov, “Optical shock wave and photon-number conservation,” Phys. Rev. A 98, 043833 (2018).
[Crossref]

2016 (2)

S. Bose, R. Chattopadhyay, S. Roy, and S. K. Bhadra, “Study of nonlinear dynamics in silver-nanoparticle-doped photonic crystal fiber,” J. Opt. Soc. Am. B 33, 1014–1021 (2016).
[Crossref]

S. Bose, A. Sahoo, R. Chattopadhyay, S. Roy, S. K. Bhadra, and G. P. Agrawal, “Implications of a zero-nonlinearity wavelength in photonic crystal fibers doped with silver nanoparticles,” Phys. Rev. A 94, 043835 (2016).
[Crossref]

2011 (1)

O. Vanvincq, J. C. Travers, and A. Kudlinski, “Conservation of the photon number in the generalized nonlinear Schrödinger equation in axially varying optical fibers,” Phys. Rev. A 84, 063820 (2011).
[Crossref]

2010 (3)

R. Driben and J. Herrmann, “Solitary pulse propagation and soliton-induced supercontinuum generation in silica glasses containing silver nanoparticles,” Opt. Lett. 35, 2529–2531 (2010).
[Crossref]

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

S. Amiranashvili and A. Demircan, “Hamiltonian structure of propagation equations for ultrashort optical pulses,” Phys. Rev. A 82, 013812 (2010).
[Crossref]

2009 (1)

2007 (5)

2006 (2)

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 effect in modulation instability in negative-index material,” Opt. Express 14, 1568–1575 (2006).
[Crossref]

2005 (3)

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. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[Crossref]

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

2004 (2)

V. Agranovich, Y. Shen, R. Baughman, and A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[Crossref]

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37–43 (2004).
[Crossref]

2003 (1)

P. Kinsler and G. New, “Few-cycle pulse propagation,” Phys. Rev. A 67, 023813 (2003).
[Crossref]

2002 (1)

S. Coen, D. A. Wardle, and J. D. Harvey, “Observation of non-phase-matched parametric amplification in resonant nonlinear optics,” Phys. Rev. Lett. 89, 273901 (2002).
[Crossref]

2001 (1)

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, “Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber,” IEEE J. Quantum Electron. 37, 398–404 (2001).
[Crossref]

1997 (2)

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

K. Hakuta, M. Suzuki, M. Katsuragawa, and J. Z. Li, “Self-induced phase matching in parametric anti-Stokes stimulated Raman scattering,” Phys. Rev. Lett. 79, 209–212 (1997).
[Crossref]

1989 (2)

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[Crossref]

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

Agranovich, V.

V. Agranovich, Y. Shen, R. Baughman, and A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[Crossref]

Agrawal, G. P.

F. R. Arteaga-Sierra, A. Antikainen, and G. P. Agrawal, “Soliton dynamics in photonic-crystal fibers with frequency-dependent Kerr nonlinearity,” Phys. Rev. A 98, 013830 (2018).
[Crossref]

S. Bose, A. Sahoo, R. Chattopadhyay, S. Roy, S. K. Bhadra, and G. P. Agrawal, “Implications of a zero-nonlinearity wavelength in photonic crystal fibers doped with silver nanoparticles,” Phys. Rev. A 94, 043835 (2016).
[Crossref]

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

Akozbek, N.

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]

Amiranashvili, S.

S. Amiranashvili, M. Radziunas, U. Bandelow, and R. Čiegis, “Numerical methods for accurate description of ultrashort pulses in optical fibers,” Commun. Nonlinear Sci. Numer. Simul. 67, 391–402 (2019).
[Crossref]

S. Amiranashvili and A. Demircan, “Hamiltonian structure of propagation equations for ultrashort optical pulses,” Phys. Rev. A 82, 013812 (2010).
[Crossref]

S. Amiranashvili, “Hamiltonian framework for short optical pulses,” in New Approaches to Nonlinear Waves (Springer, 2016), pp. 153–196.

Antikainen, A.

F. R. Arteaga-Sierra, A. Antikainen, and G. P. Agrawal, “Soliton dynamics in photonic-crystal fibers with frequency-dependent Kerr nonlinearity,” Phys. Rev. A 98, 013830 (2018).
[Crossref]

Arteaga-Sierra, F. R.

F. R. Arteaga-Sierra, A. Antikainen, and G. P. Agrawal, “Soliton dynamics in photonic-crystal fibers with frequency-dependent Kerr nonlinearity,” Phys. Rev. A 98, 013830 (2018).
[Crossref]

Bandelow, U.

S. Amiranashvili, M. Radziunas, U. Bandelow, and R. Čiegis, “Numerical methods for accurate description of ultrashort pulses in optical fibers,” Commun. Nonlinear Sci. Numer. Simul. 67, 391–402 (2019).
[Crossref]

Baughman, R.

V. Agranovich, Y. Shen, R. Baughman, and A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[Crossref]

Bhadra, S. K.

S. Bose, R. Chattopadhyay, S. Roy, and S. K. Bhadra, “Study of nonlinear dynamics in silver-nanoparticle-doped photonic crystal fiber,” J. Opt. Soc. Am. B 33, 1014–1021 (2016).
[Crossref]

S. Bose, A. Sahoo, R. Chattopadhyay, S. Roy, S. K. Bhadra, and G. P. Agrawal, “Implications of a zero-nonlinearity wavelength in photonic crystal fibers doped with silver nanoparticles,” Phys. Rev. A 94, 043835 (2016).
[Crossref]

Bloemer, M. J.

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]

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

Bonetti, J.

Bose, S.

S. Bose, A. Sahoo, R. Chattopadhyay, S. Roy, S. K. Bhadra, and G. P. Agrawal, “Implications of a zero-nonlinearity wavelength in photonic crystal fibers doped with silver nanoparticles,” Phys. Rev. A 94, 043835 (2016).
[Crossref]

S. Bose, R. Chattopadhyay, S. Roy, and S. K. Bhadra, “Study of nonlinear dynamics in silver-nanoparticle-doped photonic crystal fiber,” J. Opt. Soc. Am. B 33, 1014–1021 (2016).
[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]

Brambilla, G.

Chattopadhyay, R.

S. Bose, R. Chattopadhyay, S. Roy, and S. K. Bhadra, “Study of nonlinear dynamics in silver-nanoparticle-doped photonic crystal fiber,” J. Opt. Soc. Am. B 33, 1014–1021 (2016).
[Crossref]

S. Bose, A. Sahoo, R. Chattopadhyay, S. Roy, S. K. Bhadra, and G. P. Agrawal, “Implications of a zero-nonlinearity wavelength in photonic crystal fibers doped with silver nanoparticles,” Phys. Rev. A 94, 043835 (2016).
[Crossref]

Ciegis, R.

S. Amiranashvili, M. Radziunas, U. Bandelow, and R. Čiegis, “Numerical methods for accurate description of ultrashort pulses in optical fibers,” Commun. Nonlinear Sci. Numer. Simul. 67, 391–402 (2019).
[Crossref]

Coen, S.

S. Coen, D. A. Wardle, and J. D. Harvey, “Observation of non-phase-matched parametric amplification in resonant nonlinear optics,” Phys. Rev. Lett. 89, 273901 (2002).
[Crossref]

D’Aguanno, G.

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]

Dai, X.

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 pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[Crossref]

Demircan, A.

S. Amiranashvili and A. Demircan, “Hamiltonian structure of propagation equations for ultrashort optical pulses,” Phys. Rev. A 82, 013812 (2010).
[Crossref]

Driben, R.

Dudley, J.

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, 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).
[Crossref]

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

Fierens, P. I.

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, 036617 (2006).
[Crossref]

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

Genty, G.

Grosz, D. F.

Hakuta, K.

K. Hakuta, M. Suzuki, M. Katsuragawa, and J. Z. Li, “Self-induced phase matching in parametric anti-Stokes stimulated Raman scattering,” Phys. Rev. Lett. 79, 209–212 (1997).
[Crossref]

Harvey, J. D.

S. Coen, D. A. Wardle, and J. D. Harvey, “Observation of non-phase-matched parametric amplification in resonant nonlinear optics,” Phys. Rev. Lett. 89, 273901 (2002).
[Crossref]

Haus, H. A.

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[Crossref]

Hernandez, S. M.

Herrmann, J.

Hu, Y.

Hult, J.

Husakou, A.

Karasawa, N.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, “Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber,” IEEE J. Quantum Electron. 37, 398–404 (2001).
[Crossref]

Katsuragawa, M.

K. Hakuta, M. Suzuki, M. Katsuragawa, and J. Z. Li, “Self-induced phase matching in parametric anti-Stokes stimulated Raman scattering,” Phys. Rev. Lett. 79, 209–212 (1997).
[Crossref]

Kibler, B.

Kinsler, P.

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

G. Genty, P. Kinsler, B. Kibler, and J. Dudley, “Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides,” Opt. Express 15, 5382–5387 (2007).
[Crossref]

P. Kinsler and G. New, “Few-cycle pulse propagation,” Phys. Rev. A 67, 023813 (2003).
[Crossref]

Kourakis, I.

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

Kudlinski, A.

O. Vanvincq, J. C. Travers, and A. Kudlinski, “Conservation of the photon number in the generalized nonlinear Schrödinger equation in axially varying optical fibers,” Phys. Rev. A 84, 063820 (2011).
[Crossref]

Lægsgaard, J.

Lai, Y.

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[Crossref]

Lazarides, N.

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

Li, J. Z.

K. Hakuta, M. Suzuki, M. Katsuragawa, and J. Z. Li, “Self-induced phase matching in parametric anti-Stokes stimulated Raman scattering,” Phys. Rev. Lett. 79, 209–212 (1997).
[Crossref]

Linale, N.

Mattiucci, N.

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]

Morita, R.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, “Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber,” IEEE J. Quantum Electron. 37, 398–404 (2001).
[Crossref]

Nakagawa, N.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, “Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber,” IEEE J. Quantum Electron. 37, 398–404 (2001).
[Crossref]

Nakamura, S.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, “Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber,” IEEE J. Quantum Electron. 37, 398–404 (2001).
[Crossref]

New, G.

P. Kinsler and G. New, “Few-cycle pulse propagation,” Phys. Rev. A 67, 023813 (2003).
[Crossref]

Pendry, J. B.

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37–43 (2004).
[Crossref]

Poliakov, E. Y.

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]

Radziunas, M.

S. Amiranashvili, M. Radziunas, U. Bandelow, and R. Čiegis, “Numerical methods for accurate description of ultrashort pulses in optical fibers,” Commun. Nonlinear Sci. Numer. Simul. 67, 391–402 (2019).
[Crossref]

Roy, S.

S. Bose, A. Sahoo, R. Chattopadhyay, S. Roy, S. K. Bhadra, and G. P. Agrawal, “Implications of a zero-nonlinearity wavelength in photonic crystal fibers doped with silver nanoparticles,” Phys. Rev. A 94, 043835 (2016).
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S. Bose, R. Chattopadhyay, S. Roy, and S. K. Bhadra, “Study of nonlinear dynamics in silver-nanoparticle-doped photonic crystal fiber,” J. Opt. Soc. Am. B 33, 1014–1021 (2016).
[Crossref]

Sahoo, A.

S. Bose, A. Sahoo, R. Chattopadhyay, S. Roy, S. K. Bhadra, and G. P. Agrawal, “Implications of a zero-nonlinearity wavelength in photonic crystal fibers doped with silver nanoparticles,” Phys. Rev. A 94, 043835 (2016).
[Crossref]

Sánchez, A. D.

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]

Shen, Y.

V. Agranovich, Y. Shen, R. Baughman, and A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[Crossref]

Y. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

Shibata, M.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, “Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber,” IEEE J. Quantum Electron. 37, 398–404 (2001).
[Crossref]

Shigekawa, H.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, “Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber,” IEEE J. Quantum Electron. 37, 398–404 (2001).
[Crossref]

Shukla, P.

I. Kourakis and P. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
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J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37–43 (2004).
[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, 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).
[Crossref]

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

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K. Hakuta, M. Suzuki, M. Katsuragawa, and J. Z. Li, “Self-induced phase matching in parametric anti-Stokes stimulated Raman scattering,” Phys. Rev. Lett. 79, 209–212 (1997).
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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]

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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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[Crossref]

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O. Vanvincq, J. C. Travers, and A. Kudlinski, “Conservation of the photon number in the generalized nonlinear Schrödinger equation in axially varying optical fibers,” Phys. Rev. A 84, 063820 (2011).
[Crossref]

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

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O. Vanvincq, J. C. Travers, and A. Kudlinski, “Conservation of the photon number in the generalized nonlinear Schrödinger equation in axially varying optical fibers,” Phys. Rev. A 84, 063820 (2011).
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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. Coen, D. A. Wardle, and J. D. Harvey, “Observation of non-phase-matched parametric amplification in resonant nonlinear optics,” Phys. Rev. Lett. 89, 273901 (2002).
[Crossref]

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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 pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
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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 effect in modulation instability in negative-index material,” Opt. Express 14, 1568–1575 (2006).
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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, 033815 (2007).
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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).
[Crossref]

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

Yamashita, M.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, “Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber,” IEEE J. Quantum Electron. 37, 398–404 (2001).
[Crossref]

Zakhidov, A.

V. Agranovich, Y. Shen, R. Baughman, and A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[Crossref]

Zheltikov, A. M.

A. M. Zheltikov, “Optical shock wave and photon-number conservation,” Phys. Rev. A 98, 043833 (2018).
[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]

Commun. Nonlinear Sci. Numer. Simul. (1)

S. Amiranashvili, M. Radziunas, U. Bandelow, and R. Čiegis, “Numerical methods for accurate description of ultrashort pulses in optical fibers,” Commun. Nonlinear Sci. Numer. Simul. 67, 391–402 (2019).
[Crossref]

IEEE J. Quantum Electron. (2)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[Crossref]

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, “Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband optical pulses in a fused-silica fiber,” IEEE J. Quantum Electron. 37, 398–404 (2001).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (4)

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. A (9)

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989).
[Crossref]

P. Kinsler and G. New, “Few-cycle pulse propagation,” Phys. Rev. A 67, 023813 (2003).
[Crossref]

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

S. Amiranashvili and A. Demircan, “Hamiltonian structure of propagation equations for ultrashort optical pulses,” Phys. Rev. A 82, 013812 (2010).
[Crossref]

A. M. Zheltikov, “Optical shock wave and photon-number conservation,” Phys. Rev. A 98, 043833 (2018).
[Crossref]

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, 033815 (2007).
[Crossref]

S. Bose, A. Sahoo, R. Chattopadhyay, S. Roy, S. K. Bhadra, and G. P. Agrawal, “Implications of a zero-nonlinearity wavelength in photonic crystal fibers doped with silver nanoparticles,” Phys. Rev. A 94, 043835 (2016).
[Crossref]

F. R. Arteaga-Sierra, A. Antikainen, and G. P. Agrawal, “Soliton dynamics in photonic-crystal fibers with frequency-dependent Kerr nonlinearity,” Phys. Rev. A 98, 013830 (2018).
[Crossref]

O. Vanvincq, J. C. Travers, and A. Kudlinski, “Conservation of the photon number in the generalized nonlinear Schrödinger equation in axially varying optical fibers,” Phys. Rev. A 84, 063820 (2011).
[Crossref]

Phys. Rev. B (1)

V. Agranovich, Y. Shen, R. Baughman, and A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[Crossref]

Phys. Rev. E (3)

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

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

Phys. Rev. Lett. (4)

K. Hakuta, M. Suzuki, M. Katsuragawa, and J. Z. Li, “Self-induced phase matching in parametric anti-Stokes stimulated Raman scattering,” Phys. Rev. Lett. 79, 209–212 (1997).
[Crossref]

S. Coen, D. A. Wardle, and J. D. Harvey, “Observation of non-phase-matched parametric amplification in resonant nonlinear optics,” Phys. Rev. Lett. 89, 273901 (2002).
[Crossref]

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

Phys. Today (1)

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today 57(6), 37–43 (2004).
[Crossref]

Other (4)

S. Amiranashvili, “Hamiltonian framework for short optical pulses,” in New Approaches to Nonlinear Waves (Springer, 2016), pp. 153–196.

J. Bonetti, A. D. Sánchez, S. M. Hernandez, and D. F. Grosz, “A simple approach to the quantum theory of nonlinear fiber optics,” arXiv 1902.00561 (2019).

Y. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

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

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Figures (4)

Fig. 1.
Fig. 1. Propagation of a fundamental soliton in the anomalous dispersion regime of a medium with a positive nonlinear coefficient and $ \gamma (\omega ) = {\gamma _0}(1 + \omega /{\omega _{\textit 0}}) $. Top panel: input spectrum (dashed-dotted line) and output spectra (GNLSE, solid line; pcGNLSE, dotted line). Spectral evolution: (a) GNLSE and (b) pcGNLSE. Bottom panel: energy evolution along propagation. In this scenario, the GNLSE and pcGNLSE equations yield the exact same spectral shifts and energy evolution.
Fig. 2.
Fig. 2. Propagation of a fundamental soliton in the normal dispersion regime of a medium with a negative nonlinear coefficient. Top panel: input spectrum (dashed-dotted line) and output spectra (GNLSE, solid line; pcGNLSE, dotted line). Spectral evolution: (a) GNLSE and (b) pcGNLSE. Bottom panel: energy evolution along propagation. In this scenario, the pcGNLSE predicts a physically sound soliton red shift, while the GNLSE produces an unphysical blue shift.
Fig. 3.
Fig. 3. Propagation of a fundamental soliton in the anomalous dispersion regime of a medium with a positive nonlinear coefficient and a negative self-steepening parameter. Top panel: input spectrum (dashed-dotted line) and output spectra (GNLSE, solid line; pcGNLSE, dotted line). Spectral evolution: (a) GNLSE and (b) pcGNLSE. In this scenario, the GNLSE and the pcGNLSE predict different soliton red shifts.
Fig. 4.
Fig. 4. Energy and photon number evolution corresponding to Fig. 3: GNLSE, solid line; pcGNLSE, dotted line. Note that, while the pcGNLSE consistently conserves the photon count and reflects energy losses due to Raman scattering, the GNLSE predicts the increase in both photon number and energy.

Equations (16)

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

A ~ ω z = i β ( ω ) A ~ ω + i γ ( ω ) F ( A | A | 2 ) + i f R γ ( ω ) F × ( A 0 h ( τ ) | A ( t τ ) | 2 d τ A | A | 2 ) ,
Φ p z = 2 f R γ ( ω p ) h ~ μ I ( ω 0 + ω p ) P p P s ,
Φ s z = 2 f R γ ( ω s ) h ~ μ I ( ω 0 + ω s ) P p P s ,
d ρ d z = i [ H ^ K e r r + H ^ R , ρ ] + 0 [ L ^ μ ρ L ^ μ 1 2 { ρ , L ^ μ L ^ μ } ] d μ ,
H ^ Kerr = γ 0 4 π ω 0 A ^ ω 1 A ^ ω 2 A ^ ω 1 μ A ^ ω 2 + μ d ω 1 d ω 2 d μ ,
H ^ R = γ 0 f R ( h ~ μ R 1 ) 4 π ω 0 A ^ ω 1 A ^ ω 2 A ^ ω 1 μ A ^ ω 2 + μ d ω 1 d ω 2 d μ ,
L ^ μ = γ 0 f R h ~ μ I π ω 0 A ^ ω μ A ^ ω d ω .
d A ^ ω d z = i [ A ^ ω , H ^ K e r r + H ^ R ] + 1 2 0 [ L ^ μ , A ^ ω ] L ^ μ L ^ μ [ L ^ μ , A ^ ω ] d μ .
H ^ Kerr = 1 8 π ( B ^ ω 1 B ^ ω 2 C ^ ω 1 μ C ^ ω 2 + μ + C ^ ω 1 C ^ ω 2 B ^ ω 1 μ B ^ ω 2 + μ ) d ω 1 d ω 2 d μ ,
H ^ R = f R ( h ~ μ R 1 ) 4 π B ^ ω 1 B ^ ω 2 B ^ ω 1 μ B ^ ω 2 + μ d ω 1 d ω 2 d μ ,
L ^ μ = f R h ~ μ I π B ^ ω μ B ^ ω d ω .
A ~ ω z = i β ( ω ) A ~ ω + i γ ¯ ( ω ) 2 F ( C B 2 ) + i γ ¯ ( ω ) 2 F ( B C 2 ) + i f R γ ¯ ( ω ) F ( B 0 h R ( τ ) | B ( t τ ) | 2 d τ B | B | 2 ) ,
B ~ ω = γ ( ω ) ω 0 + ω 4 A ~ ω , C ~ ω = ( γ ( ω ) ω 0 + ω 4 ) A ~ ω ,
γ ¯ ( ω ) = γ ( ω ) × ( ω 0 + ω ) 3 4 .
Φ p z = 2 f R | γ ( ω s ) γ ( ω p ) | h ~ μ I ( ω 0 + ω s ) ( ω 0 + ω p ) P p P s ,
Φ s z = 2 f R | γ ( ω s ) γ ( ω p ) | h ~ μ I ( ω 0 + ω s ) ( ω 0 + ω p ) P p P s .