Abstract

Motivated by recent experimental results, we demonstrate that the ubiquitous pulse propagation equation based on a single generalized nonlinear Schrödinger equation is incomplete and inadequate to explain the formation of the so called negative-frequency resonant radiation emitted by optical solitons. The origin of this deficiency is due to the absence of a peculiar nonlinear coupling between the positive and negative frequency components of the pulse spectrum during propagation, a feature that the slowly-varying envelope approximation is unable to capture. We therefore introduce a conceptually new model, based on the envelope of the analytic signal, that takes into account the full spectral dynamics of all frequency components, is prone to analytical treatment and retains the simulation efficiency of the nonlinear Schrödinger equation. We use our new equation to derive from first principles the phase-matching condition of the negative-frequency resonant radiation observed in previously reported experiments.

© 2013 Optical Society of America

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    [PubMed]
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    [CrossRef]
  31. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A81, 053841 (2010).
    [CrossRef]
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    [CrossRef]
  34. V. Cao Long, P. P. Goldstein, and M. Trippenbach, “On existence of solitons for the 3rd harmonic of a light beam in planar waveguides,” Acta Phys. Pol. A105, 437–444 (2004).

2013 (2)

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, “Negative-frequency dispersive wave generation in quadratic media,” Phys. Rev. A88, 013829 (2013).
[CrossRef]

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Few-cycle optical solitary waves in nonlinear dispersive media,” Phys. Rev. A87, 013805 (2013).
[CrossRef]

2012 (4)

M. Kolesik, P. Townsend Whalen, and J. V. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron.18, 494–506 (2012).
[CrossRef]

F. Biancalana, “Negative frequencies get real,” Physics5, 68 (2012).
[CrossRef]

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

E. Rubino, A. Lotti, F. Belgiorno, S. L. Cacciatori, A. Couairon, U. Leonhardt, and D. Faccio, “Soliton-induced relativistic-scattering and amplification,” Sci. Rep.2, 932 (2012).
[PubMed]

2011 (4)

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett.107, 203901 (2011).
[CrossRef] [PubMed]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P.St.J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett.107, 203902 (2011).
[CrossRef] [PubMed]

S. Amiranashvili and A. Demircan, “Ultrashort optical pulse propagation in terms of analytic signal,” Adv. Opt. Technol.2011, 989515 (2011).

V. Grigoriev and F. Biancalana, “Coupled-mode theory for on-channel nonlinear microcavities,” J. Opt. Soc. B28, 2165–2173 (2011).
[CrossRef]

2010 (3)

M. Conforti, F. Baronio, and C. De Angelis, “Ultrabroadband optical phenomena in quadratic nonlinear media,” IEEE Photonics2, 600–610 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A81, 053841 (2010).
[CrossRef]

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A81, 013819 (2010).
[CrossRef]

2008 (1)

G. Rousseaux, C. Mathis, P. Maïssa, T. G. Philbin, and U. Leonhardt, “Observation of negative-frequency waves in a water tank: a classical analogue to the Hawking effect?,” New J. Phys.10, 053015 (2008).
[CrossRef]

2007 (1)

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006).
[CrossRef]

2005 (3)

D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E72, 016619 (2005).
[CrossRef]

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt.52, 973–986 (2005).
[CrossRef]

R. S. Tasgal, Y. B. Band, and B. Malomed, “Gap solitons in a medium with third-harmonic generation,” Phys. Rev. E72, 016624 (2005).
[CrossRef]

2004 (2)

V. Cao Long, P. P. Goldstein, and M. Trippenbach, “On existence of solitons for the 3rd harmonic of a light beam in planar waveguides,” Acta Phys. Pol. A105, 437–444 (2004).

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E70, 016615 (2004).
[CrossRef]

2003 (2)

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science301, 1705–1708 (2003).
[CrossRef] [PubMed]

P.St.J. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003).
[CrossRef] [PubMed]

2001 (1)

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett.87, 203901 (2001).
[CrossRef] [PubMed]

1995 (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51, 2602–2607 (1995).
[CrossRef] [PubMed]

1991 (1)

V. M. Simulik, “Connection between the symmetry properties of the Dirac and Maxwell equations. Conservation laws,” Theor. Math. Phys.87, 386–393 (1991).
[CrossRef]

1970 (1)

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses,” Phys. Rev. Lett.24, 592–594 (1970).
[CrossRef]

1926 (1)

E. C. Titchmarsh, “The zeros of certain integral functions,” Proc. London Math. Soc.25, 283–302 (1926).
[CrossRef]

Agrawal, G. P.

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

Akhmediev, N.

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Few-cycle optical solitary waves in nonlinear dispersive media,” Phys. Rev. A87, 013805 (2013).
[CrossRef]

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51, 2602–2607 (1995).
[CrossRef] [PubMed]

Alfano, R. R.

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses,” Phys. Rev. Lett.24, 592–594 (1970).
[CrossRef]

Amiranashvili, S.

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Few-cycle optical solitary waves in nonlinear dispersive media,” Phys. Rev. A87, 013805 (2013).
[CrossRef]

S. Amiranashvili and A. Demircan, “Ultrashort optical pulse propagation in terms of analytic signal,” Adv. Opt. Technol.2011, 989515 (2011).

Band, Y. B.

R. S. Tasgal, Y. B. Band, and B. Malomed, “Gap solitons in a medium with third-harmonic generation,” Phys. Rev. E72, 016624 (2005).
[CrossRef]

Bandelow, U.

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Few-cycle optical solitary waves in nonlinear dispersive media,” Phys. Rev. A87, 013805 (2013).
[CrossRef]

Baronio, F.

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, “Negative-frequency dispersive wave generation in quadratic media,” Phys. Rev. A88, 013829 (2013).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Ultrabroadband optical phenomena in quadratic nonlinear media,” IEEE Photonics2, 600–610 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A81, 053841 (2010).
[CrossRef]

Belgiorno, F.

E. Rubino, A. Lotti, F. Belgiorno, S. L. Cacciatori, A. Couairon, U. Leonhardt, and D. Faccio, “Soliton-induced relativistic-scattering and amplification,” Sci. Rep.2, 932 (2012).
[PubMed]

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

Biancalana, F.

F. Biancalana, “Negative frequencies get real,” Physics5, 68 (2012).
[CrossRef]

V. Grigoriev and F. Biancalana, “Coupled-mode theory for on-channel nonlinear microcavities,” J. Opt. Soc. B28, 2165–2173 (2011).
[CrossRef]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P.St.J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett.107, 203902 (2011).
[CrossRef] [PubMed]

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett.107, 203901 (2011).
[CrossRef] [PubMed]

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E70, 016615 (2004).
[CrossRef]

Cacciatori, S. L.

E. Rubino, A. Lotti, F. Belgiorno, S. L. Cacciatori, A. Couairon, U. Leonhardt, and D. Faccio, “Soliton-induced relativistic-scattering and amplification,” Sci. Rep.2, 932 (2012).
[PubMed]

Cao Long, V.

V. Cao Long, P. P. Goldstein, and M. Trippenbach, “On existence of solitons for the 3rd harmonic of a light beam in planar waveguides,” Acta Phys. Pol. A105, 437–444 (2004).

Chang, W.

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P.St.J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett.107, 203902 (2011).
[CrossRef] [PubMed]

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett.107, 203901 (2011).
[CrossRef] [PubMed]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006).
[CrossRef]

Conforti, M.

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, “Negative-frequency dispersive wave generation in quadratic media,” Phys. Rev. A88, 013829 (2013).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Ultrabroadband optical phenomena in quadratic nonlinear media,” IEEE Photonics2, 600–610 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A81, 053841 (2010).
[CrossRef]

Couairon, A.

E. Rubino, A. Lotti, F. Belgiorno, S. L. Cacciatori, A. Couairon, U. Leonhardt, and D. Faccio, “Soliton-induced relativistic-scattering and amplification,” Sci. Rep.2, 932 (2012).
[PubMed]

De Angelis, C.

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A81, 053841 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Ultrabroadband optical phenomena in quadratic nonlinear media,” IEEE Photonics2, 600–610 (2010).
[CrossRef]

Demircan, A.

S. Amiranashvili and A. Demircan, “Ultrashort optical pulse propagation in terms of analytic signal,” Adv. Opt. Technol.2011, 989515 (2011).

Dudley, J. M.

Faccio, D.

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, “Negative-frequency dispersive wave generation in quadratic media,” Phys. Rev. A88, 013829 (2013).
[CrossRef]

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

E. Rubino, A. Lotti, F. Belgiorno, S. L. Cacciatori, A. Couairon, U. Leonhardt, and D. Faccio, “Soliton-induced relativistic-scattering and amplification,” Sci. Rep.2, 932 (2012).
[PubMed]

Genty, G.

Goldstein, P. P.

V. Cao Long, P. P. Goldstein, and M. Trippenbach, “On existence of solitons for the 3rd harmonic of a light beam in planar waveguides,” Acta Phys. Pol. A105, 437–444 (2004).

Grigoriev, V.

V. Grigoriev and F. Biancalana, “Coupled-mode theory for on-channel nonlinear microcavities,” J. Opt. Soc. B28, 2165–2173 (2011).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3. (Artech House, 2005).

Herrmann, J.

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett.87, 203901 (2001).
[CrossRef] [PubMed]

Hölzer, P.

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett.107, 203901 (2011).
[CrossRef] [PubMed]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P.St.J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett.107, 203902 (2011).
[CrossRef] [PubMed]

Husakou, A. V.

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett.87, 203901 (2001).
[CrossRef] [PubMed]

Joly, N. Y.

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P.St.J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett.107, 203902 (2011).
[CrossRef] [PubMed]

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett.107, 203901 (2011).
[CrossRef] [PubMed]

Karlsson, M.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51, 2602–2607 (1995).
[CrossRef] [PubMed]

Kehr, S. C.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

Kibler, B.

Kinsler, P.

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A81, 013819 (2010).
[CrossRef]

G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, “Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides,” Opt. Express15, 5382–5387 (2007).
[CrossRef] [PubMed]

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt.52, 973–986 (2005).
[CrossRef]

Knight, J. C.

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science301, 1705–1708 (2003).
[CrossRef] [PubMed]

Kolesik, M.

M. Kolesik, P. Townsend Whalen, and J. V. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron.18, 494–506 (2012).
[CrossRef]

König, F.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

Kuklewicz, C. E.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

Leonhardt, U.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

E. Rubino, A. Lotti, F. Belgiorno, S. L. Cacciatori, A. Couairon, U. Leonhardt, and D. Faccio, “Soliton-induced relativistic-scattering and amplification,” Sci. Rep.2, 932 (2012).
[PubMed]

G. Rousseaux, C. Mathis, P. Maïssa, T. G. Philbin, and U. Leonhardt, “Observation of negative-frequency waves in a water tank: a classical analogue to the Hawking effect?,” New J. Phys.10, 053015 (2008).
[CrossRef]

Lotti, A.

E. Rubino, A. Lotti, F. Belgiorno, S. L. Cacciatori, A. Couairon, U. Leonhardt, and D. Faccio, “Soliton-induced relativistic-scattering and amplification,” Sci. Rep.2, 932 (2012).
[PubMed]

Luan, F.

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science301, 1705–1708 (2003).
[CrossRef] [PubMed]

Maïssa, P.

G. Rousseaux, C. Mathis, P. Maïssa, T. G. Philbin, and U. Leonhardt, “Observation of negative-frequency waves in a water tank: a classical analogue to the Hawking effect?,” New J. Phys.10, 053015 (2008).
[CrossRef]

Malomed, B.

R. S. Tasgal, Y. B. Band, and B. Malomed, “Gap solitons in a medium with third-harmonic generation,” Phys. Rev. E72, 016624 (2005).
[CrossRef]

Mandl, F.

F. Mandl and G. Shaw, Quantum Field Theory (John Wiley, 1996).

Mathis, C.

G. Rousseaux, C. Mathis, P. Maïssa, T. G. Philbin, and U. Leonhardt, “Observation of negative-frequency waves in a water tank: a classical analogue to the Hawking effect?,” New J. Phys.10, 053015 (2008).
[CrossRef]

McLenaghan, J.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2010).

Moloney, J. V.

M. Kolesik, P. Townsend Whalen, and J. V. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron.18, 494–506 (2012).
[CrossRef]

Nazarkin, A.

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett.107, 203901 (2011).
[CrossRef] [PubMed]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P.St.J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett.107, 203902 (2011).
[CrossRef] [PubMed]

New, G. H. C.

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt.52, 973–986 (2005).
[CrossRef]

Nold, J.

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett.107, 203901 (2011).
[CrossRef] [PubMed]

Philbin, T. G.

G. Rousseaux, C. Mathis, P. Maïssa, T. G. Philbin, and U. Leonhardt, “Observation of negative-frequency waves in a water tank: a classical analogue to the Hawking effect?,” New J. Phys.10, 053015 (2008).
[CrossRef]

Rohr, S.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

Rousseaux, G.

G. Rousseaux, C. Mathis, P. Maïssa, T. G. Philbin, and U. Leonhardt, “Observation of negative-frequency waves in a water tank: a classical analogue to the Hawking effect?,” New J. Phys.10, 053015 (2008).
[CrossRef]

Rubino, E.

E. Rubino, A. Lotti, F. Belgiorno, S. L. Cacciatori, A. Couairon, U. Leonhardt, and D. Faccio, “Soliton-induced relativistic-scattering and amplification,” Sci. Rep.2, 932 (2012).
[PubMed]

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

Russell, P. St.J.

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett.107, 203901 (2011).
[CrossRef] [PubMed]

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science301, 1705–1708 (2003).
[CrossRef] [PubMed]

Russell, P.St.J.

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P.St.J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett.107, 203902 (2011).
[CrossRef] [PubMed]

P.St.J. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003).
[CrossRef] [PubMed]

Sakurai, J. J.

J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, 1967), p. 169.

Saleh, M. F.

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P.St.J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett.107, 203902 (2011).
[CrossRef] [PubMed]

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett.107, 203901 (2011).
[CrossRef] [PubMed]

Shapiro, S. L.

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses,” Phys. Rev. Lett.24, 592–594 (1970).
[CrossRef]

Shaw, G.

F. Mandl and G. Shaw, Quantum Field Theory (John Wiley, 1996).

Simulik, V. M.

V. M. Simulik, “Connection between the symmetry properties of the Dirac and Maxwell equations. Conservation laws,” Theor. Math. Phys.87, 386–393 (1991).
[CrossRef]

Skryabin, D. V.

D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E72, 016619 (2005).
[CrossRef]

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E70, 016615 (2004).
[CrossRef]

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science301, 1705–1708 (2003).
[CrossRef] [PubMed]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3. (Artech House, 2005).

Tasgal, R. S.

R. S. Tasgal, Y. B. Band, and B. Malomed, “Gap solitons in a medium with third-harmonic generation,” Phys. Rev. E72, 016624 (2005).
[CrossRef]

Titchmarsh, E. C.

E. C. Titchmarsh, “The zeros of certain integral functions,” Proc. London Math. Soc.25, 283–302 (1926).
[CrossRef]

Townsend, D.

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

Townsend Whalen, P.

M. Kolesik, P. Townsend Whalen, and J. V. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron.18, 494–506 (2012).
[CrossRef]

Travers, J. C.

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett.107, 203901 (2011).
[CrossRef] [PubMed]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P.St.J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett.107, 203902 (2011).
[CrossRef] [PubMed]

Trillo, S.

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, “Negative-frequency dispersive wave generation in quadratic media,” Phys. Rev. A88, 013829 (2013).
[CrossRef]

Trippenbach, M.

V. Cao Long, P. P. Goldstein, and M. Trippenbach, “On existence of solitons for the 3rd harmonic of a light beam in planar waveguides,” Acta Phys. Pol. A105, 437–444 (2004).

Tyrrell, J. C. A.

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt.52, 973–986 (2005).
[CrossRef]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2010).

Westerberg, N.

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, “Negative-frequency dispersive wave generation in quadratic media,” Phys. Rev. A88, 013829 (2013).
[CrossRef]

Yulin, A. V.

D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E72, 016619 (2005).
[CrossRef]

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E70, 016615 (2004).
[CrossRef]

Acta Phys. Pol. A (1)

V. Cao Long, P. P. Goldstein, and M. Trippenbach, “On existence of solitons for the 3rd harmonic of a light beam in planar waveguides,” Acta Phys. Pol. A105, 437–444 (2004).

Adv. Opt. Technol. (1)

S. Amiranashvili and A. Demircan, “Ultrashort optical pulse propagation in terms of analytic signal,” Adv. Opt. Technol.2011, 989515 (2011).

IEEE J. Sel. Top. Quantum Electron. (1)

M. Kolesik, P. Townsend Whalen, and J. V. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron.18, 494–506 (2012).
[CrossRef]

IEEE Photonics (1)

M. Conforti, F. Baronio, and C. De Angelis, “Ultrabroadband optical phenomena in quadratic nonlinear media,” IEEE Photonics2, 600–610 (2010).
[CrossRef]

J. Mod. Opt. (1)

J. C. A. Tyrrell, P. Kinsler, and G. H. C. New, “Pseudospectral spatial-domain: a new method for nonlinear pulse propagation in the few-cycle regime with arbitrary dispersion,” J. Mod. Opt.52, 973–986 (2005).
[CrossRef]

J. Opt. Soc. B (1)

V. Grigoriev and F. Biancalana, “Coupled-mode theory for on-channel nonlinear microcavities,” J. Opt. Soc. B28, 2165–2173 (2011).
[CrossRef]

New J. Phys. (1)

G. Rousseaux, C. Mathis, P. Maïssa, T. G. Philbin, and U. Leonhardt, “Observation of negative-frequency waves in a water tank: a classical analogue to the Hawking effect?,” New J. Phys.10, 053015 (2008).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (5)

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A81, 013819 (2010).
[CrossRef]

M. Conforti, N. Westerberg, F. Baronio, S. Trillo, and D. Faccio, “Negative-frequency dispersive wave generation in quadratic media,” Phys. Rev. A88, 013829 (2013).
[CrossRef]

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51, 2602–2607 (1995).
[CrossRef] [PubMed]

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Few-cycle optical solitary waves in nonlinear dispersive media,” Phys. Rev. A87, 013805 (2013).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A81, 053841 (2010).
[CrossRef]

Phys. Rev. E (3)

R. S. Tasgal, Y. B. Band, and B. Malomed, “Gap solitons in a medium with third-harmonic generation,” Phys. Rev. E72, 016624 (2005).
[CrossRef]

D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E72, 016619 (2005).
[CrossRef]

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E70, 016615 (2004).
[CrossRef]

Phys. Rev. Lett. (5)

P. Hölzer, W. Chang, J. C. Travers, A. Nazarkin, J. Nold, N. Y. Joly, M. F. Saleh, F. Biancalana, and P. St.J. Russell, “Femtosecond nonlinear fiber optics in the ionization regime,” Phys. Rev. Lett.107, 203901 (2011).
[CrossRef] [PubMed]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P.St.J. Russell, and F. Biancalana, “Theory of photoionization-induced blueshift of ultrashort solitons in gas-filled hollow-core photonic crystal fibers,” Phys. Rev. Lett.107, 203902 (2011).
[CrossRef] [PubMed]

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett.87, 203901 (2001).
[CrossRef] [PubMed]

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses,” Phys. Rev. Lett.24, 592–594 (1970).
[CrossRef]

E. Rubino, J. McLenaghan, S. C. Kehr, F. Belgiorno, D. Townsend, S. Rohr, C. E. Kuklewicz, U. Leonhardt, F. König, and D. Faccio, “Negative-frequency resonant radiation,” Phys. Rev. Lett.108, 253901 (2012).
[CrossRef] [PubMed]

Physics (1)

F. Biancalana, “Negative frequencies get real,” Physics5, 68 (2012).
[CrossRef]

Proc. London Math. Soc. (1)

E. C. Titchmarsh, “The zeros of certain integral functions,” Proc. London Math. Soc.25, 283–302 (1926).
[CrossRef]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006).
[CrossRef]

Sci. Rep. (1)

E. Rubino, A. Lotti, F. Belgiorno, S. L. Cacciatori, A. Couairon, U. Leonhardt, and D. Faccio, “Soliton-induced relativistic-scattering and amplification,” Sci. Rep.2, 932 (2012).
[PubMed]

Science (2)

P.St.J. Russell, “Photonic crystal fibers,” Science299, 358–362 (2003).
[CrossRef] [PubMed]

D. V. Skryabin, F. Luan, J. C. Knight, and P. St.J. Russell, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science301, 1705–1708 (2003).
[CrossRef] [PubMed]

Theor. Math. Phys. (1)

V. M. Simulik, “Connection between the symmetry properties of the Dirac and Maxwell equations. Conservation laws,” Theor. Math. Phys.87, 386–393 (1991).
[CrossRef]

Other (6)

J. H. Eberly, L. Mandel, and E. Wolf, eds., Coherence and Quantum Optics VII (Springer, 1996), p. 313.

J. J. Sakurai, Advanced Quantum Mechanics (Addison-Wesley, 1967), p. 169.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2010).

F. Mandl and G. Shaw, Quantum Field Theory (John Wiley, 1996).

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

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3. (Artech House, 2005).

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Figures (4)

Fig. 1
Fig. 1

Dispersion curves for bulk silica. All possible combinations of forward/backward waves and positive/negative energy states appear in this plot.

Fig. 2
Fig. 2

(a) Phase-matching curve (normalized to β0) derived by using Eqs. (1011) in bulk silica. q/β0, (2Δkq)/β0 and (−2Δk + 3q)/β0 are indicated by the green, red and gray horizontal lines, respectively. RR, NRR and TH-RR frequencies are indicated by circles. (b) Δk for bulk silica vs. pump wavelength.

Fig. 3
Fig. 3

(a) Contour plot of the spectral evolution of a short sech pulse in bulk silica, obtained by direct simulation of Eq. (8), when THG is neglected. The pulse is pumped at λ0 = 2 μm, with a peak intensity of 1.4 TW/cm2 and a duration t0 = 15 fsec. The formation of RR and NRR is clearly visible. Vertical black dashed lines indicate the position of the radiations as predicted by Eqs. (1011), compare with Fig. 2(a). (b) Same as (a) when also switching off the second nonlinear term inside the square brackets of Eq. (8), i.e. the conjugated Kerr term. The NRR line has completely disappeared. (c) Same as (a) but when switching off the shock operator, and for a peak intensity 2.6 TW/cm2. (d) Results obtained with the UPPE of Eq. (4), using the same parameters as in (a). All plots are in logarithmic scale.

Fig. 4
Fig. 4

Comparison between the amplitudes of the generated NRR when the conjugated Kerr term (CKT) is present but THG is absent (red line), when THG is present but conjugated Kerr term is absent (black line), and in the case when both terms are present (blue line). Equation (8) has been used in the simulation, and parameters are the same as in Fig. 3. Spectra are recorded after z = 5 mm of propagation, and the vertical scale is logarithmic.

Equations (28)

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

× E = 1 c t B ,
× B = 1 c t E ,
γ μ μ ψ = 0 ,
i E ω z + β ( ω ) E ω + ω 2 c n ( ω ) P NL , ω = 0 ,
A ( z , t ) ( z , t ) e i β 0 z + i ω 0 t ,
P NL ( z , t ) = χ ( 3 ) 8 [ A 3 e 3 i ω 0 t + 3 i β 0 z + A * 3 e 3 i ω 0 t 3 i β 0 z + 3 | A | 2 e i ω 0 t + i β 0 z + 3 | A | 2 A * e i ω 0 t i β 0 z ] .
A p ( z , t ) = 3 χ ( 3 ) 4 [ | A | 2 A + | A | 2 A * e 2 i ω 0 t 2 i β 0 z + 1 3 A 3 e 2 i ω 0 t + 2 i β 0 z ] +
i ξ A + D ^ ( i τ ) A + γ S ^ ( i τ ) [ | A | 2 A + | A | 2 A * e 2 i ω 0 τ + 2 i Δ k ξ + 1 3 A 3 e 2 i ω 0 τ 2 i Δ k ξ ] + = 0 ,
( i ξ + D ^ ) g + γ F 2 g * e 2 i q ξ + 2 γ F 2 g = ( D ^ + 1 2 β 2 τ 2 ) F e i q ξ + γ F 3 e 2 i ω 0 τ + 2 i Δ k ξ i q ξ 1 3 γ F 3 e 2 i ω 0 τ 2 i Δ k ξ + 3 i q ξ .
D ( Δ ω ) = q ,
D ( Δ ω ) = 2 Δ k q ,
D ( Δ ω ) = 2 Δ k + 3 q .
d N d ξ = d d ξ + | A ( ξ , τ ) | 2 d τ = + [ A ξ A * + c . c . ] d τ = 0 .
d N d ξ = d τ { i γ | A | 4 γ ω 0 ( | A | 2 A ) τ A * i γ | A | 4 γ ω 0 ( | A | 2 A * ) τ } A = = γ ω 0 d τ { 2 ( | A | 2 ) τ | A | 2 + A τ A * | A | 2 + A τ * A | A | 2 } = = γ ω 0 d τ 3 ( | A | 2 ) τ | A | 2 = 0 .
d N d ξ = i γ | A | 2 A 2 * e 2 i ϕ + i γ 3 | A | 2 A 2 e 2 i ϕ γ ω 0 [ ( | A | 2 A * ) τ + 2 i ω 0 | A | 2 A * ] A * e 2 i ϕ + γ ω 0 [ A 2 A τ 2 i ω 0 3 A 3 ] A * e 2 i ϕ + c . c . = = i γ | A | 2 A 2 * e 2 i ϕ + i γ | A | 2 A 2 e 2 i ϕ γ ω 0 [ A τ A 3 * + 2 A τ * A * | A | 2 ] e 2 i ϕ + γ ω 0 [ A τ A | A | 2 ] e 2 i ϕ + c . c . = = γ 2 ω 0 ( A τ A 3 * + 3 | A | 2 A * A τ * ) e 2 i ϕ + ( A 3 A τ * + 3 | A | 2 A A τ ) e 2 i ϕ + 2 ( A τ A 3 * + 2 | A | 2 A * A τ * ) e 2 i ϕ 2 | A | 2 A A τ e 2 i ϕ + c . c . = = γ 2 ω 0 ( 2 A τ A 3 * + 6 | A | 2 A * A τ * 2 A τ A 3 * 6 | A | 2 A * A τ * ) e 2 i ϕ + c . c . = 0 .
[ E ( t ) ] = 1 π t E ( t ) = + E ( t t ) π t d t
[ E ( t ) e ± i ω 0 t ] = + E ( t t ) e ± i ω 0 ( t t ) π t d t = e ± i ω 0 t [ e i ω 0 t π t E ( t ) ]
{ i π t } = sgn ( ω ) , { i e ± i ω 0 t π t } = sgn ( ω ± ω 0 )
{ ( t ) } F { Q ^ [ E ( t ) ] } = { E ( t ) i [ E ( t ) ] } = E ω + sgn ( ω ) E ω = 2 E ω > 0
{ Q ^ [ E ( t ) e i ω 0 t ] } = E ω ω 0 + sgn ( ω ) E ω ω 0 = 2 E Δ ω > ω 0 .
Q ^ [ P N L ( z , t ) ] = 3 χ ( 3 ) 8 [ | A | 2 A e i ω 0 t + i β 0 z i [ e i ω 0 t π t | A | 2 A ] e i ω 0 t + i β 0 z + | A | 2 A * e i ω 0 t i β 0 z + i [ e i ω 0 t π t | A | 2 A * ] e i ω 0 t i β 0 z + 2 3 A 3 e 3 i ω 0 t + 3 i β 0 z ]
A p ( z , t ) = 3 χ ( 3 ) 8 { | A | 2 A i [ e i ω 0 t π t | A | 2 A ] + + | A | 2 A * e 2 i ω 0 t 2 i β 0 z i [ e i ω 0 t π t | A | 2 A * ] e 2 i ω 0 t 2 i β 0 z + 2 3 A 3 e 2 i ω 0 t + 2 i β 0 z }
i A ξ + D ^ ( i τ ) A + γ 2 ( 1 + i ω 0 τ ) { | A | 2 A i [ e i ω 0 τ π τ | A | 2 A ] + + | A | 2 A * e 2 i ϕ i [ e i ω 0 t π τ | A | 2 A * ] e 2 i ϕ + 2 3 A 3 e 2 i ϕ } = 0
d N d ξ = γ 2 d τ i [ ( 1 + i ω 0 τ ) ( i e i ω 0 τ π τ | A | 2 A ) ] A * + i [ ( 1 i ω 0 τ ) ( i e i ω 0 τ π τ | A | 2 A * ) ] A = = γ 2 d ω i [ ( 1 + ω ω 0 ) sgn ( ω + ω 0 ) { | A | 2 A } ] [ A ] * + + i [ ( 1 ω ω 0 ) sgn ( ω ω 0 ) { | A | 2 A * } ] [ A * ] * = = γ 2 d ω i [ ( 1 + ω ω 0 ) { | A | 2 A } ] [ A ] * i [ ( 1 ω ω 0 ) { | A | 2 A * } ] [ A * ] * = = γ 2 d τ i [ ( 1 + i ω 0 τ ) | A | 2 A ] A * i [ ( 1 i ω 0 τ ) | A | 2 A * ] A = = γ 2 d τ [ i | A | 4 1 ω 0 ( | A | 2 A ) τ A * i | A | 4 1 ω 0 ( | A | 2 A * ) τ A ] = = γ 2 ω 0 d τ [ 2 ( | A | 2 ) τ | A | 2 + A τ A * | A | 2 + A τ * A | A | 2 ] = γ 2 ω 0 d τ 3 ( | A | 2 ) τ | A | 2 = 0 ,
d N d ξ = γ 2 d τ i { ( 1 + i ω 0 τ ) [ ( i e i ω 0 t π τ | A | 2 A * ) e 2 i ϕ ] } A * + i { ( 1 i ω 0 τ ) [ ( i e i ω 0 t π τ | A | 2 A ) e 2 i ϕ ] } A = = γ 2 d ω i { ( 1 + ω ω 0 ) [ sgn ( ω + ω 0 ) { | A | 2 A * } ω + 2 ω 0 ] } [ A ] * e 2 i Δ k ξ + + i { ( 1 ω ω 0 ) [ sgn ( ω ω 0 ) { | A | 2 A } ω 2 ω 0 ] } [ A * ] * e 2 i Δ k ξ = = γ 2 d τ i { ( 1 + i ω 0 τ ) [ | A | 2 A * e 2 i ϕ ] } A * i { ( 1 i ω 0 τ ) [ | A | 2 A e 2 i ϕ ] } A .
n ( ω ) = 1 + j = 1 m B j ω j 2 ω j 2 ω 2 ,
n ( ω ) 1 + B R ω R 2 ω R 2 ω 2 1 μ 2 ( ω ω R ) ,
n ( ω ) 1 μ 2 ( ω ω R + i γ ) ,

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