Abstract

Electromagnetic pulse propagation in the slow light regime and near a zero group velocity point is relevant to a plethora of potential applications, and has analogies in numerous other wave systems. Unfortunately, the standard frequency-based formulation for pulse propagation is unsuitable for describing the dynamics in such regimes, due to the divergence of the dispersion coefficients. Moreover, in the presence of absorption, it is not clear how to interpret the propagation dynamics due to the drastic change induced by absorption upon the dispersion curves. As a remedy, we present an alternative momentum-based formulation, which is rapidly converging in these regimes, and naturally suitable for lossy and nonlinear media. It is specialized to a waveguide geometry which provides a significant simplification with respect to existing momentum-based schemes. Doing so, we provide a somewhat alternative, yet intuitive picture of the seeming enhanced absorption and nonlinear response in these regimes, and show that light-matter interactions are not enhanced in the slow/stopped light regimes.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2016 (8)

S. Lavdas and N. Panoiu, “Theory of pulsed four-wave mixing in one-dimensional silicon photonic crystal slab waveguides,” Phys. Rev. B 93, 115435 (2016).
[Crossref]

T. Kamalakis, “Derivation of coupled envelope propagation equations in silicon-based photonic crystal waveguides,” J. Opt. Soc. Am. B 33, 2339–2349 (2016).
[Crossref]

F. Blatt, L. S. Simeonov, T. Halfmann, and T. Peters, “Stationary light pulses and narrowband light storage in a laser-cooled ensemble loaded into a hollow-core fiber,” Phys. Rev. A 94, 043833 (2016).
[Crossref]

P. C. Kuan, C. Huang, W. S. Chan, S. Kosen, and S. Y. Lan, “Large fizeau’s light-dragging effect in a moving electromagnetically induced transparent medium,” Nat. Commun. 7, 13030 (2016).
[Crossref]

H. Cui, W. Lin, H. Zhang, and X. Wang, “Backward waves with double zero-group-velocity points in a liquid-filled pipe,” The J. Acoust. Soc. Am. 139, 1179–1194 (2016).
[Crossref] [PubMed]

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Revisiting time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

S. P. Shipman and A. T. Welters, “Pathological scattering by a defect in a slow-light periodic layered medium,” J. Math. Phys. 57, 022902 (2016).
[Crossref]

Y. Sivan, S. Rosenberg, and A. Halstuch, “Coupled-mode theory for electromagnetic pulse propagation in dispersive media undergoing a spatiotemporal perturbation: Exact derivation, numerical validation and peculiar wave mixing,” Phys. Rev. B 93, 144303 (2016).
[Crossref]

2015 (6)

M. Afzelius, N. Gisin, and H. de Riedmatten, “Quantum memory for photons,” Phys. Today 68, 42–47 (2015).
[Crossref]

S. Wuestner, T. W. Pickering, J. M. Hamm, A. F. Page, A. Pusch, and O. Hess, “Ultrafast dynamics of nanoplasmonic stopped-light lasing,” Faraday Discuss. 178, 307 (2015).
[Crossref] [PubMed]

N. Sun, J. Chen, and D. Tang, “Stopping light in an active medium,” The Eur. Phys. J. D 69, 219 (2015).
[Crossref]

K. V. Rajitha, T. N. Dey, J. Evers, and M. Kiffner, “Pulse splitting in light propagation through N-type atomic media due to an interplay of Kerr nonlinearity and group-velocity dispersion,” Phys. Rev. A 92, 023840 (2015).
[Crossref]

G. Grigoryan, V. Chaltykyan, E. Gazazyan, O. Tikhova, and V. Paturyan, “Pulse propagation, population transfer, and light storage in five-level media,” Phys. Rev. A 91, 023802 (2015).
[Crossref]

Y. Chen, Z. Chen, and G. Huang, “Storage and retrieval of vector optical solitons via double electromagnetically induced transparency,” Phys. Rev. A 91, 023820 (2015).
[Crossref]

2014 (3)

K. L. Tsakmakidis, T. W. Pickering, J. M. Hamm, A. F. Page, and O. Hess, “Completely stopped and dispersionless light in plasmonic waveguides,” Phys. Rev. Lett. 112, 167401 (2014).
[Crossref] [PubMed]

K. L. Tsakmakidis, T. W. Pickering, J. M. Hamm, A. F. Page, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nat. Commun. 5, 4972–4979 (2014).
[Crossref]

P. Siddons, “Light propagation through atomic vapours,” J. Phys. B: At. Mol. Opt. Phys 47, 093001 (2014).
[Crossref]

2013 (5)

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
[Crossref] [PubMed]

R. Won, “Quantum memory: Extended storage times,” Nat. Photonics 7, 677 (2013).
[Crossref]

H. Takesue, N. Matsuda, E. Kuramochi, W. J. Munro, and M. Notomi, “An on-chip coupled resonator optical waveguide single-photon buffer,” Nat. Commun. 4, 2725 (2013).
[Crossref] [PubMed]

H. de Riedmatten, “A long-term memory for light,” Physics 80, 6 (2013).

A. Hayrapetyan, K. Grigoryan, R. Petrosyan, and S. Fritzsche, “Propagation of sound waves through a spatially homogeneous but smoothly timedependent medium,” Ann. Phys. 333, 47–65 (2013).
[Crossref]

2012 (2)

I. Dicaire, A. D. Rossi, S. Combrié, and L. Thévenaz, “Probing molecular absorption under slow-light propagation using a photonic crystal waveguide,” Opt. Lett. 37, 4934–4936 (2012).
[Crossref] [PubMed]

H. Zhang, M. Sabooni, L. Rippe, C. Kim, S. Kröll, L. V. Wang, and P. R. Hemmer, “Slow light for deep tissue imaging with ultrasound modulation,” Appl. Phys. Lett. 100, 131102 (2012).
[Crossref] [PubMed]

2011 (5)

Y. Sivan and J. B. Pendry, “Time-reversal in dynamically-tuned zero-gap periodic systems,” Phys. Rev. Lett. 106, 193902 (2011).
[Crossref]

Y. Sivan and J. B. Pendry, “Theory of wave-front reversal of short pulses in dynamically-tuned zero-gap periodic systems,” Phys. Rev. A 84, 033822 (2011).
[Crossref]

Z. I. Samson, P. Horak, K. F. MacDonald, and N. I. Zheludev, “Femtosecond surface plasmon pulse propagation,” Opt. Lett. 36, 250–252 (2011).
[Crossref] [PubMed]

A. Zadok, A. Eyal, and M. Tur, “Stimulated brillouin scattering slow light in optical fibers,” Appl. Opt. 50, E38–E49 (2011).
[Crossref]

G. A. Wurtz, R. J. Pollard, W. R. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6, 107–111 (2011).
[Crossref] [PubMed]

2010 (5)

C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express 18, 21427–21448 (2010).
[Crossref] [PubMed]

C. Monat, B. Corcoran, D. Pudo, M. E. Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 344–356 (2010).
[Crossref]

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B 82, 041308 (2010).
[Crossref]

J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photonics 2, 287–318 (2010).
[Crossref]

R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B 28, A38–A44 (2010).
[Crossref]

2009 (5)

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17, 2944–2953 (2009).
[Crossref] [PubMed]

Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal waveguide,” Opt. Lett. 34, 1072–1074 (2009).
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A. Archambault, T. V. Teperik, F. Marquier, and J. J. Greffet, “Surface plasmon fourier optics,” Phys. Rev. B 79, 195414 (2009).
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L. Thévenaz, S. Chin, I. Dicaire, J.-C. Beugnot, S. F. Mafang, and M. G. Herraez, “Experimental verification of the effect of slow light on molecular absorption,” Proceeding SPIE 7503, 75034W (2009).
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2008 (6)

M. Reza, M. M. Dignam, and S. Hughes, “Can light be stopped in realistic metamaterials?” Nature 455, E10–E11 (2008).
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J. I. L. Chen, G. von Freymann, S. Y. Choi, V. Kitaev, and G. A. Ozin, “Slow photons in the fast lane in chemistry,” J. Mater. Chem. 18, 369–373 (2008).
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F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multimode optical fibers,” J. Opt. Soc. Am. B 25, 1645–1654 (2008).
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T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
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J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, “Observation of spontaneous raman scattering in silicon slow-light photonic crystal waveguides,” Appl. Phys. Lett. 93, 251105 (2008).
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M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-q coupled nanocavities,” Nat. Photonics 2, 741–747 (2008).
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2007 (5)

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666–2670 (2007).
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G. Brennen, E. Giacobino, and C. Simon, “Focus on quantum memory,” New J. Phys. 17, 050201 (2007).
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T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1, 49–52 (2007).
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M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75, 053807 (2007).
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Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).
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2006 (4)

R. Atkinson, W. R. Hendren, G. A. Wurtz, W. Dickson, A. V. Zayats, P. Evans, and R. J. Pollard, “Anisotropic optical properties of arrays of gold nanorods embedded in alumina,” Phys. Rev. B 73, 235402 (2006).
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J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
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A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media 16, 293–382 (2006).
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J. I. L. Chen, G. von Freymann, S. Y. Choi, V. Kitaev, and G. A. Ozin, “Amplified photochemistry with slow photons,” Adv. Mater. 18, 1915–1919 (2006).
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2005 (4)

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
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M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
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B. Maes, P. Bienstman, and R. Baets, “Bloch modes and self-localized waveguides in nonlinear photonic crystals,” J. Opt. Soc. Am. B 22, 613–619 (2005).
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J. Mendonca and A. Guerreiro, “Time refraction and the quantum properties of vacuum,” Phys. Rev. A 72, 063805 (2005).
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2002 (3)

A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, J. Musser, B. S. Ham, and P. R. Hemmer, “Observation of ultraslow and stored light pulses in a solid,” Phys. Rev. Lett. 88, 023602 (2002).
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U. Leonhardt, “A laboratory analogue of the event horizon using slow light in an atomic medium,” Nature. 415, 406–409 (2002).
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M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Top. Appl. Phys. 84, 17–43 (2002).
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2001 (2)

V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E 63, 027602 (2001).
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N. Bhat and J. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E 64, 056604 (2001).
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2000 (1)

J. E. Sipe, “Vector k · p approach for photonic band structures,” Phys. Rev. E 62, 5672–5677 (2000).
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1999 (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
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M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
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1998 (1)

N. Aközbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
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1996 (1)

C. de Sterke, D. Salinas, and J. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
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1995 (1)

P. Tran, “Photonic-band-structure calculation of material possessing kerr nonlinearity,” Phys. Rev. B 52, 10673–10676 (1995).
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1994 (1)

J. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through non-uniform grating structures,” J. Opt. Soc. Am. B 11, 1307–1320 (1994).
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1991 (1)

1990 (1)

C. de Sterke and J. Sipe, “Switching dynamics of finite periodic nonlinear media: A numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
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1988 (1)

C. de Sterke and J. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
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M. Afzelius, N. Gisin, and H. de Riedmatten, “Quantum memory for photons,” Phys. Today 68, 42–47 (2015).
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Agrawal, G. P.

Aközbek, N.

N. Aközbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998).
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Archambault, A.

A. Archambault, T. V. Teperik, F. Marquier, and J. J. Greffet, “Surface plasmon fourier optics,” Phys. Rev. B 79, 195414 (2009).
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Atkinson, R.

R. Atkinson, W. R. Hendren, G. A. Wurtz, W. Dickson, A. V. Zayats, P. Evans, and R. J. Pollard, “Anisotropic optical properties of arrays of gold nanorods embedded in alumina,” Phys. Rev. B 73, 235402 (2006).
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Baba, T.

Bacot, V.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Revisiting time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
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Baets, R.

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
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Beugnot, J.-C.

L. Thévenaz, S. Chin, I. Dicaire, J.-C. Beugnot, S. F. Mafang, and M. G. Herraez, “Experimental verification of the effect of slow light on molecular absorption,” Proceeding SPIE 7503, 75034W (2009).
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Bhat, N.

N. Bhat and J. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E 64, 056604 (2001).
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Bienstman, P.

Bigelow, M. S.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
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Blatt, F.

F. Blatt, L. S. Simeonov, T. Halfmann, and T. Peters, “Stationary light pulses and narrowband light storage in a laser-cooled ensemble loaded into a hollow-core fiber,” Phys. Rev. A 94, 043833 (2016).
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X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B 82, 041308 (2010).
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R. W. Boyd, “Material slow light and structural slow light: similarities and differences for nonlinear optics,” J. Opt. Soc. Am. B 28, A38–A44 (2010).
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Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
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G. Brennen, E. Giacobino, and C. Simon, “Focus on quantum memory,” New J. Phys. 17, 050201 (2007).
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G. Grigoryan, V. Chaltykyan, E. Gazazyan, O. Tikhova, and V. Paturyan, “Pulse propagation, population transfer, and light storage in five-level media,” Phys. Rev. A 91, 023802 (2015).
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P. C. Kuan, C. Huang, W. S. Chan, S. Kosen, and S. Y. Lan, “Large fizeau’s light-dragging effect in a moving electromagnetically induced transparent medium,” Nat. Commun. 7, 13030 (2016).
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X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B 82, 041308 (2010).
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N. Sun, J. Chen, and D. Tang, “Stopping light in an active medium,” The Eur. Phys. J. D 69, 219 (2015).
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Chen, J. I. L.

J. I. L. Chen, G. von Freymann, S. Y. Choi, V. Kitaev, and G. A. Ozin, “Slow photons in the fast lane in chemistry,” J. Mater. Chem. 18, 369–373 (2008).
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J. I. L. Chen, G. von Freymann, S. Y. Choi, V. Kitaev, and G. A. Ozin, “Amplified photochemistry with slow photons,” Adv. Mater. 18, 1915–1919 (2006).
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Y. Chen, Z. Chen, and G. Huang, “Storage and retrieval of vector optical solitons via double electromagnetically induced transparency,” Phys. Rev. A 91, 023820 (2015).
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Y. Chen, Z. Chen, and G. Huang, “Storage and retrieval of vector optical solitons via double electromagnetically induced transparency,” Phys. Rev. A 91, 023820 (2015).
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Chin, S.

L. Thévenaz, S. Chin, I. Dicaire, J.-C. Beugnot, S. F. Mafang, and M. G. Herraez, “Experimental verification of the effect of slow light on molecular absorption,” Proceeding SPIE 7503, 75034W (2009).
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Choi, S. Y.

J. I. L. Chen, G. von Freymann, S. Y. Choi, V. Kitaev, and G. A. Ozin, “Slow photons in the fast lane in chemistry,” J. Mater. Chem. 18, 369–373 (2008).
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J. I. L. Chen, G. von Freymann, S. Y. Choi, V. Kitaev, and G. A. Ozin, “Amplified photochemistry with slow photons,” Adv. Mater. 18, 1915–1919 (2006).
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Corcoran, B.

C. Monat, B. Corcoran, D. Pudo, M. E. Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 344–356 (2010).
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C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17, 2944–2953 (2009).
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B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
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Cui, H.

H. Cui, W. Lin, H. Zhang, and X. Wang, “Backward waves with double zero-group-velocity points in a liquid-filled pipe,” The J. Acoust. Soc. Am. 139, 1179–1194 (2016).
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M. Afzelius, N. Gisin, and H. de Riedmatten, “Quantum memory for photons,” Phys. Today 68, 42–47 (2015).
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H. de Riedmatten, “A long-term memory for light,” Physics 80, 6 (2013).

de Sterke, C.

C. de Sterke, D. Salinas, and J. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
[Crossref]

C. de Sterke and J. Sipe, “Switching dynamics of finite periodic nonlinear media: A numerical study,” Phys. Rev. A 42, 2858–2869 (1990).
[Crossref] [PubMed]

C. de Sterke and J. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149–5165 (1988).
[Crossref]

de Sterke, C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[Crossref]

J. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through non-uniform grating structures,” J. Opt. Soc. Am. B 11, 1307–1320 (1994).
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C. M. de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991).
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Dey, T. N.

K. V. Rajitha, T. N. Dey, J. Evers, and M. Kiffner, “Pulse splitting in light propagation through N-type atomic media due to an interplay of Kerr nonlinearity and group-velocity dispersion,” Phys. Rev. A 92, 023840 (2015).
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I. Dicaire, A. D. Rossi, S. Combrié, and L. Thévenaz, “Probing molecular absorption under slow-light propagation using a photonic crystal waveguide,” Opt. Lett. 37, 4934–4936 (2012).
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L. Thévenaz, S. Chin, I. Dicaire, J.-C. Beugnot, S. F. Mafang, and M. G. Herraez, “Experimental verification of the effect of slow light on molecular absorption,” Proceeding SPIE 7503, 75034W (2009).
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Dickson, W.

R. Atkinson, W. R. Hendren, G. A. Wurtz, W. Dickson, A. V. Zayats, P. Evans, and R. J. Pollard, “Anisotropic optical properties of arrays of gold nanorods embedded in alumina,” Phys. Rev. B 73, 235402 (2006).
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Dignam, M. M.

M. Reza, M. M. Dignam, and S. Hughes, “Can light be stopped in realistic metamaterials?” Nature 455, E10–E11 (2008).
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Dissanayake, C. M.

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Ebnali-Heidari, M.

Eddi, A.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Revisiting time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
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Eggleton, B. J.

C. Monat, B. Corcoran, D. Pudo, M. E. Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 344–356 (2010).
[Crossref]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17, 2944–2953 (2009).
[Crossref] [PubMed]

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
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Evans, P.

R. Atkinson, W. R. Hendren, G. A. Wurtz, W. Dickson, A. V. Zayats, P. Evans, and R. J. Pollard, “Anisotropic optical properties of arrays of gold nanorods embedded in alumina,” Phys. Rev. B 73, 235402 (2006).
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Evers, J.

K. V. Rajitha, T. N. Dey, J. Evers, and M. Kiffner, “Pulse splitting in light propagation through N-type atomic media due to an interplay of Kerr nonlinearity and group-velocity dispersion,” Phys. Rev. A 92, 023840 (2015).
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A. Figotin and I. Vitebskiy, “Slow light in photonic crystals,” Waves Random Complex Media 16, 293–382 (2006).
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Fink, M.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Revisiting time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
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M. Fink, “Acoustic time-reversal mirrors: Imaging of complex media with acoustic and seismic waves,” Top. Appl. Phys. 84, 17–43 (2002).
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Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
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Fort, E.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Revisiting time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
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A. Hayrapetyan, K. Grigoryan, R. Petrosyan, and S. Fritzsche, “Propagation of sound waves through a spatially homogeneous but smoothly timedependent medium,” Ann. Phys. 333, 47–65 (2013).
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M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
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Gaeta, A. L.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

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Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005).
[Crossref] [PubMed]

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G. Grigoryan, V. Chaltykyan, E. Gazazyan, O. Tikhova, and V. Paturyan, “Pulse propagation, population transfer, and light storage in five-level media,” Phys. Rev. A 91, 023802 (2015).
[Crossref]

Giacobino, E.

G. Brennen, E. Giacobino, and C. Simon, “Focus on quantum memory,” New J. Phys. 17, 050201 (2007).
[Crossref]

Gisin, N.

M. Afzelius, N. Gisin, and H. de Riedmatten, “Quantum memory for photons,” Phys. Today 68, 42–47 (2015).
[Crossref]

Gopal, V.

M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75, 053807 (2007).
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A. Archambault, T. V. Teperik, F. Marquier, and J. J. Greffet, “Surface plasmon fourier optics,” Phys. Rev. B 79, 195414 (2009).
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Grigoryan, G.

G. Grigoryan, V. Chaltykyan, E. Gazazyan, O. Tikhova, and V. Paturyan, “Pulse propagation, population transfer, and light storage in five-level media,” Phys. Rev. A 91, 023802 (2015).
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Grigoryan, K.

A. Hayrapetyan, K. Grigoryan, R. Petrosyan, and S. Fritzsche, “Propagation of sound waves through a spatially homogeneous but smoothly timedependent medium,” Ann. Phys. 333, 47–65 (2013).
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Grillet, C.

C. Monat, B. Corcoran, D. Pudo, M. E. Heidari, C. Grillet, M. D. Pelusi, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron. 16, 344–356 (2010).
[Crossref]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009).
[Crossref]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express 17, 2944–2953 (2009).
[Crossref] [PubMed]

Guerreiro, A.

J. Mendonca and A. Guerreiro, “Time refraction and the quantum properties of vacuum,” Phys. Rev. A 72, 063805 (2005).
[Crossref]

Halfmann, T.

F. Blatt, L. S. Simeonov, T. Halfmann, and T. Peters, “Stationary light pulses and narrowband light storage in a laser-cooled ensemble loaded into a hollow-core fiber,” Phys. Rev. A 94, 043833 (2016).
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G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
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Figures (6)

Fig. 1
Fig. 1 (a) Cross section of a one dimensional plasmonic MDM waveguide of width d. The solid line features a transverse profile of the magnetic field propagating in z direction, corresponding to the antisymmetric branch. (b) Dispersion curve of antisymmetric plasmonic modes in a MDM slab waveguide where the core width is 35 nm, the permittivity of the dielectric is D = 2.5 and the parameters of the permittivity of the silver (see Eq. (45)) are = 5, ωP = 1.4 × 1016s−1 and γ = 3.2 × 1013s−1 [73]. In cases where we ignore absorption, we set γ = 0. The red segment on the left is a stopped light regime where forward and backward modes coexist. Here, β0 = 1.51 × 108m−1 (λ0 ≈ 363.6nm) and Δβ = 0.15 × 108m−1, where β0 lies exactly at the ZGVP. The second red segment lies in a slow light regime where β0 = 3 × 108m−1(λ0 ≈ 362.9nm) and Δβ = 0.3 × 108m−1. Here β0 lies at a point where the group velocity is 5.51 × 10−4c. The green line on the left represents the light line, corresponding to c / D. In the presence of absorption, the dispersion curve (and the ZGVP) is shifted slightly (not shown). (c) The values of Tabs = 1/ℑ(ω0) corresponding to the mode shown in (b).
Fig. 2
Fig. 2 Pulse propagation at (a) ZGVP and (b) slow light regime, where the spectral width of the pulse, Δβ and the center value of the propagation constant, β0 are the corresponding values shown in Fig. 1(b). The green dashed line is A(z, t = Tdisp) where M = 2, the blue dotted line corresponds to M = 3 and the red solid line represents the exact solution M =∞.
Fig. 3
Fig. 3 Spatio-temporal pulse dynamics of the amplitude of the electric field (color represents the value of |A(z, t)|e−ℑ(ω0)t) in the linear case. Plots (a) and (b) represent the ZGVP regime case in the absence and presence of absorption, respectively. Here, β 0 = 1.51 × 10 8 1 m, Δ β = 0.015 × 10 8 1 m, Tdisp = 7.7 × 10−10s; in (b), Tabs = 6.3 × 10−14s, such that TabsTdisp. Plots (c) and (d) represent the slow light regime case in the absence and presence of absorption, respectively. Here, β 0 = 3 × 10 8 1 m, Δ β = 0.03 × 10 8 1 m, Tdisp = 1.39 × 10−9s , Tabs = 6.26 × 10−14s in (d) such that TabsTdisp. In both cases, the total time of propagation is Tdisp and 10Tabs, respectively.
Fig. 4
Fig. 4 The spatial spectrum associated with the results shown in Figs. 3(a)–(d).
Fig. 5
Fig. 5 Spatio-temporal dynamics of the amplitude of the electric field (color represents the value |A(z, t)|e−ℑ(ω0)t) for a pulse in the non-linear case. Plots (a) and (b) represent pulse propagation in the ZGVP regime in the absence and presence of absorption, respectively. Here, the time scales of the dispersion time and the absorption time are exactly as in Figs 3(a)–(b). The nonlinearity time scale is TNL = 0.2Tdisp in the absence of absorption (a) and TNL = 0.1Tabs when absorption is present (b). Plots (c) and (d) represent pulse propagation in the slow light regime in the absence and presence of absorption, respectively. Here, the dispersion and absorption time scales are exactly as in Figs 3(c)–(d). The nonlinearity time scale is TNL = 0.2Tdisp in the absence of absorption (c) and TNL = 0.1Tabs when absorption is present (d). In both cases, the total time of propagation is TNL and 10Tabs, respectively.
Fig. 6
Fig. 6 The spatial spectrum associated with the results shown in Figs. 5(a)–(d).

Equations (74)

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2 ( r , t ) μ 0 2 t 2 ( R ( r , t ) * ( r , t ) ) 1 c 2 2 t 2 ( R NL ( r , t ) ( r , t ) ) = 0 .
( r , t ) = e i β 0 z ω t [ ( d t A ( z , t ) e i ( ω ω 0 ) t ) ( d t ω t [ e ^ ( r , ω ) ] e i ω t ) ] ,
2 e ^ ( r , ω ) β 2 ( ω ) e ^ ( r , ω ) + μ 0 ( r , ω ) ω 2 e ^ ( r , ω ) = 0 .
a ^ ( z , ω ω 0 ) 2 e ^ ( r , ω ) + 2 i β 0 μ 0 ( r , ω ) e ^ ( r , ω ) a ^ ( z , ω ω 0 ) z β 0 2 μ 0 ( r , ω ) e ^ ( r , ω ) a ^ ( z , ω ω 0 ) + ω 2 μ 0 ( r , ω ) e ^ ( r , ω ) a ^ ( z , ω ω 0 ) 1 c ( ω ω 0 ) 2 t ω ( R NL ( r , t ) ( r , t ) ) = 0 ,
R ( r , t ) * ( r , t ) = ω t [ ( r , ω ) e ^ ( r , ω ) a ^ ( z , ω ω 0 ) e i β 0 z ] .
t ω [ A ( z , t ) e i ω 0 t ] = a ^ ( z , ω ω 0 ) .
β 2 ( ω ) = m = 0 1 m ! d m ( β 2 ) d ω m | ω 0 ( ω ω 0 ) m .
i 2 β 0 2 A ( z , t ) z 2 + A ( z , t ) z + m = 1 D ^ m n A ( z , t ) t m + i L NL | A ( z , t ) | 2 A ( z , t ) = 0 ,
D ^ m = 1 2 β 0 ( i ) m 1 m ! d m ( β 2 ) d ω m | ω 0 ,
L NL = 2 P 0 NL ω 0 | e ^ ( r , ω 0 ) | 4 d r ,
P 0 = β 0 μ 0 ω 0 | e ^ ( r , ω 0 ) | 2 d r ,
z * = z , t * = t z / v g .
i 2 β 0 2 A ( z * , t * ) z * 2 + A ( z * , t * ) z * i 2 β 0 v g 2 A ( Z * , t * ) z * t * i 2 β 0 2 A ( z * , t * ) t * 2 + m = 0 D ^ m m A ( z * , t * ) t * m A ( z * , t * ) + i L NL | A ( z * , t * ) | 2 A ( z * , t * ) = 0 ,
v g ( d β d ω ) 1 | ω 0 = β 0 1 ,
z ^ = z L disp ,
t ^ = t * T P ,
i 2 β 0 L disp 2 A ( z ^ , t ^ ) z ^ 2 + A ( z ^ , t ^ ) z ^ i 2 β 0 v g T P 2 A ( z ^ , t ^ ) z ^ t ^ i 2 A ( z ^ , t ^ ) t ^ 2 + L disp m = 3 D ^ m m A ( z ^ , t ^ ) t * m + i L disp L NL | A ( z ^ , t ^ ) | 2 A ( z ^ , t ^ ) = 0 .
D ^ m T P D ^ m 1 .
A ( z * , t * ) z * i 2 β 0 2 A ( z * , t * ) t * 2 + i L NL | A ( z * , t * ) | 2 A ( Z * , t * ) = 0 ,
× × ( r , t ) μ 0 2 t 2 ( R ( r , t ) * ( r , t ) ) 1 c 2 2 t 2 ( R NL ( r , t ) ( r , t ) ) = 0 .
R ( r , t ) * ( r , t ) = n = 0 i n n ! d n ( r , ω ) d ω n | ω 0 n ( r , z , t ) t n .
( r , z , t ) = β z [ ̃ ( r , β , t ) ] = β z [ a ̃ ( β β 0 , t ) e ̃ ( r , β ) e ̃ i ω ( β ) t ] ,
( r , z , t ) = e i ω 0 t β z [ ( d z A ( z , t ) e i ( β β 0 ) z ) ( d z β z [ e ̃ ( r , β ) ] e i β z ) ] = e i ω 0 t d ζ A ( z ζ , t ) e i β 0 ζ β ζ [ e ̃ ( r , β ) ] ,
A ( z , t ) = 2 π e β 0 z β z [ a ( β β 0 , t ) ] ,
a ( β β 0 , t ) = a ̃ ( β β 0 , t ) e i [ ω ( β ) ω 0 ] t ,
[ ( 2 e ̃ y x y 2 e ̃ x y 2 + β 2 e ̃ x + i β 2 e ̃ z x ) x ^ + ( 2 e ̃ x x y 2 e ̃ y x 2 + β 2 e ̃ y + i β 2 e ̃ z y ) y ^ + ( i β ( e ̃ x x + e ̃ y y ) 2 e ̃ x x 2 2 e ̃ y y 2 ) z ^ ] + μ 0 ( r , β ) ω 2 e ̃ ( r , β ) = 0 .
[ ( 2 e ̃ y x y 2 e ̃ x y 2 + β 2 e ̃ x + i β 2 e ̃ z x ) x ^ + ( 2 e ̃ x x y 2 e ̃ y x 2 + β 2 e ̃ y + i β 2 e ̃ z y ) y ^ + ( i β ( e ̃ x x + e ̃ y y ) 2 e ̃ x x 2 2 e ̃ y y 2 ) z ^ ] a ( β β 0 , t ) + μ 0 ω 0 2 ( r , ω 0 ) e ̃ ( r ̃ , β ) a ( β β 0 , t ) + i μ 0 e ̃ ( r , β ) ( 2 ω 0 ( r ) + ω 0 2 d ( r , ω ) d ω ) a ( β β 0 , t ) t + μ 0 e ̃ ( r , β ) n = 2 i n ( 1 ( n 2 ) ! d n 2 ( r , ω ) d ω n 2 | ω 0 + 2 ( n 1 ) ! ω 0 d n 1 ( r , ω ) d ω n 1 | ω 0 + 1 n ! ω 0 2 d n ( r , ω ) d ω n | ω 0 ) n a ( β β 0 , t ) t n + e i ω 0 t 2 t 2 z β [ R NL ( r , z , t ) ( r , z , t ) ] = 0 .
( ( r , ω ) ω 2 ( r , ω 0 ) ω 0 2 ) a ( β β 0 , t ) e ̃ ( r , β ) i e ̃ ( r , β ) ( 2 ω 0 ( r , ω ) + ω 0 2 d ( r , ω ) d ω | ω 0 ) a ( β β 0 , t ) t e ̃ ( r , β ) n = 2 i n ( 1 ( n 2 ) ! d n 2 ( r , ω ) d ω n 2 | ω 0 + 2 ( n 1 ) ! ω 0 d n 1 ( r , ω ) d ω n 1 | ω 0 + 1 n ! ω 0 2 d n ( r , ω ) d ω n | ω 0 ) n a ( β β 0 , t ) t n e i ω o t 2 t 2 z β [ R NL ( r , z , t ) ( r , z , t ) ] = 0 .
a ( β β 0 , t ) t + i ω 0 U ( β , β 0 ) [ ( ( r , ω ) ω 2 ( r , ω 0 ) ω 0 2 ) | e ̃ ( r , β ) | 2 d r ] ( β β 0 , t ) + i ω 0 U ( β , β 0 ) n = 2 i n [ ( 1 ( n 2 ) ! d n 2 ( r , ω ) d ω n 2 | ω 0 + 2 ( n 1 ) ! ω 0 d n 1 ( r , ω ) d ω n 1 | ω 0 + 1 n ! ω 0 2 d n ( r , ω ) d ω n | ω 0 ) | e ̃ ( r , β ) | 2 d r ] n a ( β β 0 , t ) t n + i e i ω 0 t 2 ω 0 U ( β , β 0 ) 2 t 2 z β [ R NL ( r , z , t ) ( r , z , t ) ] e ̃ * ( r , β ) d r = 0 ,
U ( β , β 0 ) = ( r , ω ) + 1 2 ω ( r , ω ) d ω | ω ( β 0 ) | e ̃ ( r , β ) | 2 d r .
a ( β , β , t ) t + i m = 0 D ̃ m ( β β 0 ) m a ( β β 0 , t ) + n = 2 m = 0 i n + 1 D ̃ n , m ( β β 0 ) m n a ( β β 0 , t ) t n i e i ω 0 t 2 ω 0 U ( β , β 0 ) 2 t 2 z β [ R NL ( r , z , t ) ( r , z , t ) ] e ̃ * ( r , β ) d r = 0 ,
D ̃ m = 1 m ! 1 2 ω 0 d m d β m [ [ ω 2 ( β ) ( r , ω ( β ) ) ω 0 2 ( r , ω ) ] | e ̃ ( r , β ) | 2 d r U ( β , β 0 ) ] | β 0 ,
D ̃ n , m = 1 m ! 1 2 ω 0 d m d β m [ 1 U ( β , β 0 ) ( 1 ( n 2 ) ! d n 2 ( r , ω ) d ω n 2 + 2 ( n 1 ) ! ω 0 d n 1 ( r , ω ) d ω n 1 + 1 n ! ω 0 2 d n ( r , ω ) d ω n ) | e ̃ ( r , β ) | 2 d r ] | ω 0 .
i D ̃ 2 , 0 A ( z , t ) t 2 + A ( z , t ) t + ω 0 A ( z , t ) z i D ̃ 2 2 A ( z , t ) z 2 + m = 3 ( i ) m 1 D ̃ m m A ( z , t ) z m + n = 2 m = 0 m 0 if n = 2 ( 1 ) m i n + m + 1 D ̃ n , m n + m A ( z , t ) t n z m + i e i ω 0 t 2 ω 0 β z [ 1 U ( β , β 0 ) 2 t 2 z β [ R NL ( r , z , t ) ( r , z , t ) ] e ̃ * ( r , β ) d r ] = 0 ,
D ̃ 2 = ω 0 2 + D ̃ 2 , 0 ω 0 2 ,
D ̃ 2 , 0 = 2 2 ω 0 U ( β , β 0 ) ( 2 ( r , ω 0 ) + 4 ω 0 d ( r , ω ) d ω | ω 0 + ω 0 2 d 2 ( r , ω ) d ω 2 | ω 0 ) | e ̃ ( r , β ) | 2 d r .
v g ( ω 0 ) .
z * = z v g t ,
t * = t .
z = z * ,
t = t * v g z * .
i D ̃ 2 , 0 2 A ( z * , t * ) t * 2 + A ( z * , t * ) t * + i ( ω 0 ) A ( z * , t * ) z * [ i 2 ω 0 ( 2 ( ω 0 ) v g + i ( ω 0 ) 2 ) D ̃ 2 , 0 ] 2 A ( z * , t * ) z * 2 2 i v g D ̃ 2 , 0 2 A ( z * , t * ) z * t * + m = 3 ( i ) m 1 D ̃ m m A ( z * , t * ) z * m + n = 2 m = 0 m 0 if n = 2 k = 0 n ( n k ) i n + 1 ( i ) m ( v g ) k D ̃ n , m n + m A ( z * , t * ) t * n k z * m + k + i T NL | A ( z * , t * ) | 2 A ( z * , t * ) = 0 ,
T NL = 2 U ( β 0 , β 0 ) NL ω 0 | e ̃ ( r , β 0 ) | 4 d r ,
z ̃ = z * Z ,
t ̃ = t * T disp ,
if 2 A ( z ̃ , t ̃ ) t ̃ 2 + A ( z ̃ , t ̃ ) t ̃ 2 i g 2 A ( z ̃ , t ̃ ) z ̃ t ̃ + i q A ( z ̃ , t ̃ ) z ̃ i 2 2 A ( z ̃ , t ̃ ) z ̃ 2 + s 2 A ( z ̃ , t ̃ ) z ̃ 2 + m = 3 ( i ) m 1 T disp D ̃ m Z m m A ( z ̃ , t ̃ ) z ̃ m + n = 2 m = 0 m 0 if n = 2 k = 0 n ( n k ) i n + 1 ( i ) m ( ( v g ) Z ) k D ̃ n , m T disp n k 1 1 Z m n + m A ( z ̃ , t ̃ ) t ̃ n k z ̃ m + k + T disp T NL | A ( z ̃ , t ̃ ) | 2 A ( z ̃ , t ̃ ) = 0 ,
f D ̃ 2 , 0 T disp ,
g v g D ̃ 2 , 0 Z ,
q ( ω 0 ) T disp Z = ( ω 0 ) ω 0 Z ,
s ( ω 0 ) ω 0 D ̃ 2 , 0 ( 2 v g + i ( ω 0 ) ) .
A ( z , t ) t i ω 0 2 2 A ( z , t ) z 2 + m = 3 ( i ) m 1 D ̃ m m A ( z , t ) z m + i T NL | A ( z , t ) | 2 A ( z , t ) = 0 ,
M ( ω ) = ω P 2 ω 2 + i γ ω ,
A ( z , t ) = 2 π e i β z 0 β z [ a ( β β 0 , t = 0 ) e i Ω M ( β ) t ] .
R M , M max ( t ) = | A ( M ) ( z , t ) A ( M max ) ( z , t ) | 2 d z | A ( M max ) ( z , t ) | 2 d z ,
U ( β , β 0 ) = s ( ( r , ω 0 ) + 1 2 ω 0 d ( r , ω ) d ω | ω 0 ) | e ̃ ( r , β 0 ) | 2 d r , = 1 2 ( ( r , ω 0 ) + d [ ( r , ω ) ] d ω | ω 0 ) | e ̃ ( r , β 0 ) | 2 d r .
( r , ω 0 ) | e ̃ ( r , β 0 ) | 2 d x d y = ( r , ω 0 ) ( | e ̃ x | 2 + | e ̃ y | 2 + | e ̃ z | 2 ) d x d y = ( r , ω 0 ) ( 1 ω 0 ( r , ω 0 ) ( h z y + i β 0 h y ) e ̃ x * + 1 ω 0 ( r , ω 0 ) ( h z x i β 0 h x ) e ̃ y * + 1 ω 0 ( r , ω 0 ) ( h y x h x y ) e ̃ z * ) d x d y = 1 ω 0 ( h z y e ̃ x * + i β 0 h y e ̃ x * + h z x e ̃ y * i β 0 h x e ̃ y * + h y x e ̃ z * h x y e ̃ z * ) d x d y ,
e ̃ x = i ω 0 ( r , ω 0 ) ( i β 0 h y h z y ) ,
e ̃ y = i ω 0 ( r , ω 0 ) ( h x y i β 0 h y ) ,
e ̃ z = i ω 0 ( r , ω 0 ) ( h x y h y x ) .
( r , ω 0 ) | e ̃ ( r , β 0 ) | 2 d x d y = 1 ω 0 ( h z e ̃ x * y + i β 0 h y e ̃ x * h z e ̃ y * x + i β 0 h y e ̃ x * h z e ̃ z * x + h x e ̃ z * y ) d x d y ,
h r i e ̃ r j d r k | r i = r i = = 0 ,
e ̃ z x = i β 0 e ̃ x + i ω 0 μ 0 h y ,
e ̃ z x = i β 0 e ̃ x + i ω 0 μ 0 h y ,
( r , ω 0 ) | e ̃ ( r , β 0 ) | 2 d x d y = 1 ω 0 ( h z ( e ̃ x * y e ̃ y * x ) + i ω 0 μ 0 ( | h x | 2 + | h y | 2 ) ) d x d y .
e ̃ x * y e ̃ y * x = i ω 0 μ 0 h z * ,
( r , ω 0 ) | e ̃ ( r , β 0 ) | 2 d x d y = μ 0 ( | h x | 2 + | h y | 2 + | h z | 2 ) d x d y = μ 0 | h ( r , β 0 ) | 2 d x d y .
U ( β , β 0 ) = 1 2 ( d ( ( r , ω 0 ) ω ) d ω | ω 0 | e ̃ ( r , β 0 ) | 2 + μ 0 | h ( r , β 0 ) | 2 ) d x d y .
R n ~ d ( r , ω 0 ) ( r , ω ) d ω | ω 0 d r ( r , ω ) | ω 0 d r .
D ̃ n , m ~ 1 ω 0 n 1 D ̃ 1 , m .
R m ~ d | e ̃ ( r , β ) | 2 d β | β 0 d r | e ̃ ( r , β ) | 2 d r ,
| e ̃ ( r , β ) | ~ e | k r | ,
| k | = β 2 ( r , β ) ( ω / c ) 2 .
R m ~ r | e ̃ ( r , β ) | 2 d r | e ̃ ( r , β ) | 2 d r .
D ̃ n , m ~ D ̃ n , 0 β 0 m .

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