Abstract

We report generation of broadband supercontinuum (SC) by noise-like pulses (NLPs) with a central wavelength of 1070 nm propagating through a long piece of standard single-mode fibers (~100 meters) in normal dispersion region far from the zero-dispersion point. Theoretical simulations indicate that the physical mechanism of SC generation is due to nonlinear effects in fibers. The cascaded Raman scattering is responsible for significant spectral broadening in the longer wavelength regions whereas the Kerr effect results in smoothing of SC generated spectrum. The SC exhibits low threshold (43 nJ) and a flat spectrum over 1050-1250 nm.

© 2013 OSA

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  1. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006).
    [CrossRef]
  2. C. Lin, V. T. Nguyen, and W. G. French, “Wideband near-I.R. continuum (0.7 – 2.1 um) generated in low-loss optical fibers,” Electron. Lett.14(25), 822–823 (1978).
    [CrossRef]
  3. S. Martin-Lopez, P. Corredera, and M. Gonzalez-Herraez, “Cavity dispersion management in continuous-wave supercontinuum generation,” Opt. Express17(15), 12785–12793 (2009).
    [CrossRef] [PubMed]
  4. A. Kudlinski, G. Bouwmans, M. Douay, M. Taki, and A. Mussot, “Dispersion-engineered photonic crystal fibers for CW-pumped supercontinuum sources,” J. Lightwave Technol.27(11), 1556–1564 (2009).
    [CrossRef]
  5. M. Horowitz, Y. Barad, and Y. Silberberg, “Noise-like pulses with a broadband spectrum generated from an erbium-doped fiber laser,” Opt. Lett.22(11), 799–801 (1997).
    [CrossRef] [PubMed]
  6. O. Pottiez, R. Grajales-Coutiño, B. Ibarra-Escamilla, E. A. Kuzin, and J. C. Hernández-García, “Adjustable noiselike pulses from a figure-eight fiber laser,” Appl. Opt.50(25), E24–E31 (2011).
    [CrossRef]
  7. L. M. Zhao, D. Y. Tang, J. Wu, X. Q. Fu, and S. C. Wen, “Noise-like pulse in a gain-guided soliton fiber laser,” Opt. Express15(5), 2145–2150 (2007).
    [CrossRef] [PubMed]
  8. S. Kobtsev, S. Kukarin, S. Smirnov, S. Turitsyn, and A. Latkin, “Generation of double-scale femto/pico-second optical lumps in mode-locked fiber lasers,” Opt. Express17(23), 20707–20713 (2009).
    [CrossRef] [PubMed]
  9. J. C. Hernandez-Garcia, O. Pottieza, and J. M. Estudillo-Ayalab, “Supercontinuum generation in a standard fiber pumped by noise-like pulses from a figure-eight fiber laser,” Laser Phys.22(1), 221–226 (2012).
    [CrossRef]
  10. A. K. Zaytsev, C. H. Lin, Y. J. You, F. H. Tsai, C. L. Wang, and C. L. Pan, “Controllable noise-like operation regime in Yb:doped dispersion-mapped fiber ring laser,” Laser Phys. Lett.10(4), 045104 (2013).
    [CrossRef]
  11. G. P. Agrawal, Nonlinear Fiber Optics. 5th Ed. (Elsevier, Academic, 2013).
  12. J. M. Dudley and J. R. Taylor, eds., Supercontinuum Generation in Optical Fibers, (Cambridge University, New York, 2010).
  13. J. Santhanama and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun.222(1-6), 413–420 (2003).
    [CrossRef]
  14. I. Ilev, H. Kumagai, K. Toyoda, and I. Koprinkov, “Highly efficient wideband continuum generation in a single-mode optical fiber by powerful broadband laser pumping,” Appl. Opt.35(15), 2548–2553 (1996).
    [CrossRef] [PubMed]
  15. B. Kibler, B. Barviau, C. Michel, G. Millot, and A. Picozzi, “Thermodynamic approach of supercontinuum generation,” Opt. Fiber Technol.18(5), 257–267 (2012).
    [CrossRef]
  16. B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in a photonic crystal fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(6), 066605 (2011).
    [CrossRef] [PubMed]
  17. R. Song, J. Hou, S. Chen, W. Yang, and Q. Lu, “High power supercontinuum generation in a nonlinear ytterbium-doped fiber amplifier,” Opt. Lett.37(9), 1529–1531 (2012).
    [CrossRef] [PubMed]
  18. R. S. Watt, C. F. Kaminski, and J. Hult, “Generation of supercontinuum radiation in conventional single-mode fibre and its application to broadband absorption spectroscopy,” Appl. Phys. B90(1), 47–53 (2008).
    [CrossRef]
  19. H. Chen, Y. Lei, S. Chen, J. Hou, and Q. Lu, “Experimentally investigate the nonlinear amplifying process of high power picoseconds fiber amplifier,” Opt. & Las. Tech.47, 278–282 (2013).
    [CrossRef]

2013 (2)

A. K. Zaytsev, C. H. Lin, Y. J. You, F. H. Tsai, C. L. Wang, and C. L. Pan, “Controllable noise-like operation regime in Yb:doped dispersion-mapped fiber ring laser,” Laser Phys. Lett.10(4), 045104 (2013).
[CrossRef]

H. Chen, Y. Lei, S. Chen, J. Hou, and Q. Lu, “Experimentally investigate the nonlinear amplifying process of high power picoseconds fiber amplifier,” Opt. & Las. Tech.47, 278–282 (2013).
[CrossRef]

2012 (3)

B. Kibler, B. Barviau, C. Michel, G. Millot, and A. Picozzi, “Thermodynamic approach of supercontinuum generation,” Opt. Fiber Technol.18(5), 257–267 (2012).
[CrossRef]

J. C. Hernandez-Garcia, O. Pottieza, and J. M. Estudillo-Ayalab, “Supercontinuum generation in a standard fiber pumped by noise-like pulses from a figure-eight fiber laser,” Laser Phys.22(1), 221–226 (2012).
[CrossRef]

R. Song, J. Hou, S. Chen, W. Yang, and Q. Lu, “High power supercontinuum generation in a nonlinear ytterbium-doped fiber amplifier,” Opt. Lett.37(9), 1529–1531 (2012).
[CrossRef] [PubMed]

2011 (2)

O. Pottiez, R. Grajales-Coutiño, B. Ibarra-Escamilla, E. A. Kuzin, and J. C. Hernández-García, “Adjustable noiselike pulses from a figure-eight fiber laser,” Appl. Opt.50(25), E24–E31 (2011).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in a photonic crystal fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(6), 066605 (2011).
[CrossRef] [PubMed]

2009 (3)

2008 (1)

R. S. Watt, C. F. Kaminski, and J. Hult, “Generation of supercontinuum radiation in conventional single-mode fibre and its application to broadband absorption spectroscopy,” Appl. Phys. B90(1), 47–53 (2008).
[CrossRef]

2007 (1)

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006).
[CrossRef]

2003 (1)

J. Santhanama and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun.222(1-6), 413–420 (2003).
[CrossRef]

1997 (1)

1996 (1)

1978 (1)

C. Lin, V. T. Nguyen, and W. G. French, “Wideband near-I.R. continuum (0.7 – 2.1 um) generated in low-loss optical fibers,” Electron. Lett.14(25), 822–823 (1978).
[CrossRef]

Agrawal, G. P.

J. Santhanama and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun.222(1-6), 413–420 (2003).
[CrossRef]

Barad, Y.

Barviau, B.

B. Kibler, B. Barviau, C. Michel, G. Millot, and A. Picozzi, “Thermodynamic approach of supercontinuum generation,” Opt. Fiber Technol.18(5), 257–267 (2012).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in a photonic crystal fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(6), 066605 (2011).
[CrossRef] [PubMed]

Bouwmans, G.

Chen, H.

H. Chen, Y. Lei, S. Chen, J. Hou, and Q. Lu, “Experimentally investigate the nonlinear amplifying process of high power picoseconds fiber amplifier,” Opt. & Las. Tech.47, 278–282 (2013).
[CrossRef]

Chen, S.

H. Chen, Y. Lei, S. Chen, J. Hou, and Q. Lu, “Experimentally investigate the nonlinear amplifying process of high power picoseconds fiber amplifier,” Opt. & Las. Tech.47, 278–282 (2013).
[CrossRef]

R. Song, J. Hou, S. Chen, W. Yang, and Q. Lu, “High power supercontinuum generation in a nonlinear ytterbium-doped fiber amplifier,” Opt. Lett.37(9), 1529–1531 (2012).
[CrossRef] [PubMed]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006).
[CrossRef]

Corredera, P.

Douay, M.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006).
[CrossRef]

Estudillo-Ayalab, J. M.

J. C. Hernandez-Garcia, O. Pottieza, and J. M. Estudillo-Ayalab, “Supercontinuum generation in a standard fiber pumped by noise-like pulses from a figure-eight fiber laser,” Laser Phys.22(1), 221–226 (2012).
[CrossRef]

French, W. G.

C. Lin, V. T. Nguyen, and W. G. French, “Wideband near-I.R. continuum (0.7 – 2.1 um) generated in low-loss optical fibers,” Electron. Lett.14(25), 822–823 (1978).
[CrossRef]

Fu, X. Q.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006).
[CrossRef]

Gonzalez-Herraez, M.

Grajales-Coutiño, R.

Hernandez-Garcia, J. C.

J. C. Hernandez-Garcia, O. Pottieza, and J. M. Estudillo-Ayalab, “Supercontinuum generation in a standard fiber pumped by noise-like pulses from a figure-eight fiber laser,” Laser Phys.22(1), 221–226 (2012).
[CrossRef]

Hernández-García, J. C.

Horowitz, M.

Hou, J.

H. Chen, Y. Lei, S. Chen, J. Hou, and Q. Lu, “Experimentally investigate the nonlinear amplifying process of high power picoseconds fiber amplifier,” Opt. & Las. Tech.47, 278–282 (2013).
[CrossRef]

R. Song, J. Hou, S. Chen, W. Yang, and Q. Lu, “High power supercontinuum generation in a nonlinear ytterbium-doped fiber amplifier,” Opt. Lett.37(9), 1529–1531 (2012).
[CrossRef] [PubMed]

Hult, J.

R. S. Watt, C. F. Kaminski, and J. Hult, “Generation of supercontinuum radiation in conventional single-mode fibre and its application to broadband absorption spectroscopy,” Appl. Phys. B90(1), 47–53 (2008).
[CrossRef]

Ibarra-Escamilla, B.

Ilev, I.

Kaminski, C. F.

R. S. Watt, C. F. Kaminski, and J. Hult, “Generation of supercontinuum radiation in conventional single-mode fibre and its application to broadband absorption spectroscopy,” Appl. Phys. B90(1), 47–53 (2008).
[CrossRef]

Kibler, B.

B. Kibler, B. Barviau, C. Michel, G. Millot, and A. Picozzi, “Thermodynamic approach of supercontinuum generation,” Opt. Fiber Technol.18(5), 257–267 (2012).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in a photonic crystal fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(6), 066605 (2011).
[CrossRef] [PubMed]

Kobtsev, S.

Koprinkov, I.

Kudlinski, A.

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in a photonic crystal fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(6), 066605 (2011).
[CrossRef] [PubMed]

A. Kudlinski, G. Bouwmans, M. Douay, M. Taki, and A. Mussot, “Dispersion-engineered photonic crystal fibers for CW-pumped supercontinuum sources,” J. Lightwave Technol.27(11), 1556–1564 (2009).
[CrossRef]

Kukarin, S.

Kumagai, H.

Kuzin, E. A.

Latkin, A.

Lei, Y.

H. Chen, Y. Lei, S. Chen, J. Hou, and Q. Lu, “Experimentally investigate the nonlinear amplifying process of high power picoseconds fiber amplifier,” Opt. & Las. Tech.47, 278–282 (2013).
[CrossRef]

Lin, C.

C. Lin, V. T. Nguyen, and W. G. French, “Wideband near-I.R. continuum (0.7 – 2.1 um) generated in low-loss optical fibers,” Electron. Lett.14(25), 822–823 (1978).
[CrossRef]

Lin, C. H.

A. K. Zaytsev, C. H. Lin, Y. J. You, F. H. Tsai, C. L. Wang, and C. L. Pan, “Controllable noise-like operation regime in Yb:doped dispersion-mapped fiber ring laser,” Laser Phys. Lett.10(4), 045104 (2013).
[CrossRef]

Lu, Q.

H. Chen, Y. Lei, S. Chen, J. Hou, and Q. Lu, “Experimentally investigate the nonlinear amplifying process of high power picoseconds fiber amplifier,” Opt. & Las. Tech.47, 278–282 (2013).
[CrossRef]

R. Song, J. Hou, S. Chen, W. Yang, and Q. Lu, “High power supercontinuum generation in a nonlinear ytterbium-doped fiber amplifier,” Opt. Lett.37(9), 1529–1531 (2012).
[CrossRef] [PubMed]

Martin-Lopez, S.

Michel, C.

B. Kibler, B. Barviau, C. Michel, G. Millot, and A. Picozzi, “Thermodynamic approach of supercontinuum generation,” Opt. Fiber Technol.18(5), 257–267 (2012).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in a photonic crystal fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(6), 066605 (2011).
[CrossRef] [PubMed]

Millot, G.

B. Kibler, B. Barviau, C. Michel, G. Millot, and A. Picozzi, “Thermodynamic approach of supercontinuum generation,” Opt. Fiber Technol.18(5), 257–267 (2012).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in a photonic crystal fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(6), 066605 (2011).
[CrossRef] [PubMed]

Mussot, A.

Nguyen, V. T.

C. Lin, V. T. Nguyen, and W. G. French, “Wideband near-I.R. continuum (0.7 – 2.1 um) generated in low-loss optical fibers,” Electron. Lett.14(25), 822–823 (1978).
[CrossRef]

Pan, C. L.

A. K. Zaytsev, C. H. Lin, Y. J. You, F. H. Tsai, C. L. Wang, and C. L. Pan, “Controllable noise-like operation regime in Yb:doped dispersion-mapped fiber ring laser,” Laser Phys. Lett.10(4), 045104 (2013).
[CrossRef]

Picozzi, A.

B. Kibler, B. Barviau, C. Michel, G. Millot, and A. Picozzi, “Thermodynamic approach of supercontinuum generation,” Opt. Fiber Technol.18(5), 257–267 (2012).
[CrossRef]

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in a photonic crystal fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(6), 066605 (2011).
[CrossRef] [PubMed]

Pottiez, O.

Pottieza, O.

J. C. Hernandez-Garcia, O. Pottieza, and J. M. Estudillo-Ayalab, “Supercontinuum generation in a standard fiber pumped by noise-like pulses from a figure-eight fiber laser,” Laser Phys.22(1), 221–226 (2012).
[CrossRef]

Santhanama, J.

J. Santhanama and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun.222(1-6), 413–420 (2003).
[CrossRef]

Silberberg, Y.

Smirnov, S.

Song, R.

Taki, M.

Tang, D. Y.

Toyoda, K.

Tsai, F. H.

A. K. Zaytsev, C. H. Lin, Y. J. You, F. H. Tsai, C. L. Wang, and C. L. Pan, “Controllable noise-like operation regime in Yb:doped dispersion-mapped fiber ring laser,” Laser Phys. Lett.10(4), 045104 (2013).
[CrossRef]

Turitsyn, S.

Wang, C. L.

A. K. Zaytsev, C. H. Lin, Y. J. You, F. H. Tsai, C. L. Wang, and C. L. Pan, “Controllable noise-like operation regime in Yb:doped dispersion-mapped fiber ring laser,” Laser Phys. Lett.10(4), 045104 (2013).
[CrossRef]

Watt, R. S.

R. S. Watt, C. F. Kaminski, and J. Hult, “Generation of supercontinuum radiation in conventional single-mode fibre and its application to broadband absorption spectroscopy,” Appl. Phys. B90(1), 47–53 (2008).
[CrossRef]

Wen, S. C.

Wu, J.

Yang, W.

You, Y. J.

A. K. Zaytsev, C. H. Lin, Y. J. You, F. H. Tsai, C. L. Wang, and C. L. Pan, “Controllable noise-like operation regime in Yb:doped dispersion-mapped fiber ring laser,” Laser Phys. Lett.10(4), 045104 (2013).
[CrossRef]

Zaytsev, A. K.

A. K. Zaytsev, C. H. Lin, Y. J. You, F. H. Tsai, C. L. Wang, and C. L. Pan, “Controllable noise-like operation regime in Yb:doped dispersion-mapped fiber ring laser,” Laser Phys. Lett.10(4), 045104 (2013).
[CrossRef]

Zhao, L. M.

Appl. Opt. (2)

Appl. Phys. B (1)

R. S. Watt, C. F. Kaminski, and J. Hult, “Generation of supercontinuum radiation in conventional single-mode fibre and its application to broadband absorption spectroscopy,” Appl. Phys. B90(1), 47–53 (2008).
[CrossRef]

Electron. Lett. (1)

C. Lin, V. T. Nguyen, and W. G. French, “Wideband near-I.R. continuum (0.7 – 2.1 um) generated in low-loss optical fibers,” Electron. Lett.14(25), 822–823 (1978).
[CrossRef]

J. Lightwave Technol. (1)

Laser Phys. (1)

J. C. Hernandez-Garcia, O. Pottieza, and J. M. Estudillo-Ayalab, “Supercontinuum generation in a standard fiber pumped by noise-like pulses from a figure-eight fiber laser,” Laser Phys.22(1), 221–226 (2012).
[CrossRef]

Laser Phys. Lett. (1)

A. K. Zaytsev, C. H. Lin, Y. J. You, F. H. Tsai, C. L. Wang, and C. L. Pan, “Controllable noise-like operation regime in Yb:doped dispersion-mapped fiber ring laser,” Laser Phys. Lett.10(4), 045104 (2013).
[CrossRef]

Opt. & Las. Tech. (1)

H. Chen, Y. Lei, S. Chen, J. Hou, and Q. Lu, “Experimentally investigate the nonlinear amplifying process of high power picoseconds fiber amplifier,” Opt. & Las. Tech.47, 278–282 (2013).
[CrossRef]

Opt. Commun. (1)

J. Santhanama and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun.222(1-6), 413–420 (2003).
[CrossRef]

Opt. Express (3)

Opt. Fiber Technol. (1)

B. Kibler, B. Barviau, C. Michel, G. Millot, and A. Picozzi, “Thermodynamic approach of supercontinuum generation,” Opt. Fiber Technol.18(5), 257–267 (2012).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot, and A. Picozzi, “Emergence of spectral incoherent solitons through supercontinuum generation in a photonic crystal fiber,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.84(6), 066605 (2011).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006).
[CrossRef]

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics. 5th Ed. (Elsevier, Academic, 2013).

J. M. Dudley and J. R. Taylor, eds., Supercontinuum Generation in Optical Fibers, (Cambridge University, New York, 2010).

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Figures (6)

Fig. 1
Fig. 1

Schematic of the experimental setup: FC, fiber coupler; HWP, half-wave plate; QWP, quarter-wave plate, GP, grating pair; PI-ISO, polarization-insensitive isolator; ISO, Faraday isolator; M1,M2, mirrors; MM LD, multi-mode laser diodes.

Fig. 2
Fig. 2

The experimentally measured spectrum (a) and intensity autocorrelation trace (b) of NLPs irradiated by the oscillator in Fig. 1.

Fig. 3
Fig. 3

Building-up dynamics (a) and steady-state waveform (b) in time domain of a fiber ring oscillator generating NLPs.

Fig. 4
Fig. 4

Calculated SC evolution in spectral (a, c, e) and time (b, d, f) domains for different pump pulses: noise-like pulses (a, b), 40 ps-wide mode-locked Gaussian pulses (c, d), and 200 fs-wide mode-locked Gaussian pulses (e, f).

Fig. 5
Fig. 5

(a) Experimental (dotted) and simulated (solid) SC spectra generated in 100 m of SMF by pumping with 3W average NLP input power and simulation conditions; (b) Experimental (dotted) and simulated (solid) SC spectra generated in 100 m of SMF by pumping with different average NLP input powers (1 W, 2 W, and 3 W);

Fig. 6
Fig. 6

Simulated spectral (a) and temporal (b) evolutions of 3W average power NLP propagating through 100 m of SMF (the longitudinal step is 10 m).

Equations (3)

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{ A x z =iγ{ | A x | 2 A x + 2 3 | A y | 2 A x + 1 3 A y 2 A x }+g( E pulse ) A x i 2 β 2 2 A x t 2 A y z =iγ{ | A y | 2 A y + 2 3 | A x | 2 A y + 1 3 A x 2 A y }+g( E pulse ) A y i 2 β 2 2 A y t 2 ,
A ( z,ω ) z =i γω ω 0 exp{ L( ω )z }F{ A( z,T ) R( T ) | A( z,T T ) | 2 d T },
R( t )=( 1 f R )δ( t )+ f R τ 1 2 + τ 2 2 τ 1 τ 2 2 exp( t τ 2 )sin( t τ 1 )Θ( t ),

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