Abstract

Self- and Cross-phase modulation of asymmetric femtosecond pulses (~810 nm) propagating through a birefringent single-mode optical fiber (~6.9 cm) is studied both experimentally (using GRENOUILLE) and numerically (by solving a set of coupled nonlinear Schrödinger equations or CNLSEs). An optical spectrogram representation is derived from the electric field of the pulses and is linearly juxtaposed with the corresponding optical spectrum and optical time-trace. The simulations are shown to be in good qualitative agreement with the experiments. Measured input pulse asymmetry, when incorporated into the simulations, is found to be the dominant cause of output spectral asymmetry. The results indicate that it is possible to modulate short pulses both temporally and spectrally by passage through polarization maintaining optical fibers with specified orientation and length.

© 2003 Optical Society of America

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References

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  1. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, 2002).
    [Crossref]
  2. G.P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 2001).
  3. J.M. Dudley, X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, R. Trebino, S. Coen, and R.S. Windeler, “Cross-correlation frequency resolved optical gating analysis of broadband continuum generation in photonic crystal fiber: simulations and experiments,” Opt. Express 10, 1215 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1215.
    [Crossref] [PubMed]
  4. Q. D. Liu, J. T. Chen, Q. Z. Wang, P. P. Ho, and R. R. Alfano, “Single pulse degenerate-cross-phase modulation in a single-mode optical fiber,” Opt. Lett.20, 542–544 (1995).
    [Crossref] [PubMed]
  5. T. Sylvestre, H. Maillotte, E. Lantz, and D. Gindre “Combined spectral effects of pulse walk-off and degenerate cross-phase modulation in birefringent fibers,” Journal of Nonlinear Optical Physics and Materials 6, 313–320 (1997).
  6. Q. D. Liu, L. Shi, P. P. Ho, R. R. Alfano, R.-J. Essiambre, and G.P. Agrawal, “Degenerate cross-phase modulation of femtosecond laser pulses in a birefringent single-mode fiber,” IEEE Photon. Tech. Lett. 9, 1107–1109 (1997).
    [Crossref]
  7. F.G. Omenetto, B.P. Luce, D. Yarotski, and A.J. Taylor, “Observation of chirped soliton dynamics at l= 1.55 mm in a single-mode optical fiber with frequency-resolved optical gating,” Opt. Lett. 24, 1392 (1999).
    [Crossref]
  8. F.G. Omenetto, Y. Chung, D. Yarotski, T. Shaefer, I. Gabitov, and A.J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191 (2002).
    [Crossref]
  9. F.G. Omenetto, J.W. Nicholson, B.P. Luce, D. Yarotski, and A.J. Taylor, “Shaping, propagation and characterization of ultrafast pulses in optical fibers,” Appl. Phys. B 70[Suppl.], S143 (2000).
    [Crossref]
  10. N. Nishizawa and T. Goto, “Experimental analysis of ultrashort pulse propagation in optical fibers around zero-dispersion region using cross-correlation frequency resolved optical gating,” Opt. Express 8, 328 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-6-328.
    [Crossref] [PubMed]
  11. N. Nishizawa and T. Goto, “Trapped pulse generation by femtosecond soliton pulse in birefringent optical fibers,” Opt. Express 10, 256 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-5-256.
    [Crossref] [PubMed]
  12. N. Nishizawa and T. Goto, “Characteristics of pulse trapping by use of ultrashort soliton pulses in optical fibers across the zero-dispersion wavelength,” Opt. Express 10, 1151 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1151.
    [Crossref] [PubMed]
  13. N. Nishizawa and T. Goto, “Ultrafast all optical switching by use of pulse trapping across zero-dispersion wavelength,” Opt. Express 11, 359 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-359.
    [Crossref] [PubMed]
  14. K. Ogawa and M.D. Pelusi, “Characterization of ultrashort optical pulses in a dispersion-managed fiber link using two-photon absorption frequency-resolved optical gating,” Opt. Commun. 198, 83–87 (2001).
    [Crossref]
  15. R.A. Altes, “Detection, estimation, and classification with spectrograms,” J. Acoust. Soc. Am. 674, 1232 (1980).
    [Crossref]
  16. P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Highly simplified device for ultrashort-pulse measurement,” Opt. Lett. 26, 932 (2001).
    [Crossref]
  17. A. Christian Silva, “GRENOUILLE - Practical Issues”, unpublished.
  18. P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Increased-bandwidth in ultrashort-pulse measurement using an angle-dithered nonlinear-optical crystal,” Opt. Express 7, 342 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-10-342.
    [Crossref]
  19. P. O’Shea, M. Kimmel, and R. Trebino, “Increased phase-matching bandwidth in simple ultrashort-laser-pulse measurements,” J. Opt. B 4, 44 (2002).
    [Crossref]
  20. S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express. 11, 491 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-491.
    [Crossref] [PubMed]

2003 (2)

N. Nishizawa and T. Goto, “Ultrafast all optical switching by use of pulse trapping across zero-dispersion wavelength,” Opt. Express 11, 359 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-359.
[Crossref] [PubMed]

S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express. 11, 491 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-491.
[Crossref] [PubMed]

2002 (5)

2001 (3)

2000 (2)

P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Increased-bandwidth in ultrashort-pulse measurement using an angle-dithered nonlinear-optical crystal,” Opt. Express 7, 342 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-10-342.
[Crossref]

F.G. Omenetto, J.W. Nicholson, B.P. Luce, D. Yarotski, and A.J. Taylor, “Shaping, propagation and characterization of ultrafast pulses in optical fibers,” Appl. Phys. B 70[Suppl.], S143 (2000).
[Crossref]

1999 (1)

1980 (1)

R.A. Altes, “Detection, estimation, and classification with spectrograms,” J. Acoust. Soc. Am. 674, 1232 (1980).
[Crossref]

Agrawal, G.P.

Q. D. Liu, L. Shi, P. P. Ho, R. R. Alfano, R.-J. Essiambre, and G.P. Agrawal, “Degenerate cross-phase modulation of femtosecond laser pulses in a birefringent single-mode fiber,” IEEE Photon. Tech. Lett. 9, 1107–1109 (1997).
[Crossref]

G.P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 2001).

Akturk, S.

S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express. 11, 491 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-491.
[Crossref] [PubMed]

Alfano, R. R.

Q. D. Liu, L. Shi, P. P. Ho, R. R. Alfano, R.-J. Essiambre, and G.P. Agrawal, “Degenerate cross-phase modulation of femtosecond laser pulses in a birefringent single-mode fiber,” IEEE Photon. Tech. Lett. 9, 1107–1109 (1997).
[Crossref]

Q. D. Liu, J. T. Chen, Q. Z. Wang, P. P. Ho, and R. R. Alfano, “Single pulse degenerate-cross-phase modulation in a single-mode optical fiber,” Opt. Lett.20, 542–544 (1995).
[Crossref] [PubMed]

Altes, R.A.

R.A. Altes, “Detection, estimation, and classification with spectrograms,” J. Acoust. Soc. Am. 674, 1232 (1980).
[Crossref]

Chen, J. T.

Q. D. Liu, J. T. Chen, Q. Z. Wang, P. P. Ho, and R. R. Alfano, “Single pulse degenerate-cross-phase modulation in a single-mode optical fiber,” Opt. Lett.20, 542–544 (1995).
[Crossref] [PubMed]

Chung, Y.

F.G. Omenetto, Y. Chung, D. Yarotski, T. Shaefer, I. Gabitov, and A.J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191 (2002).
[Crossref]

Coen, S.

Dudley, J.M.

Essiambre, R.-J.

Q. D. Liu, L. Shi, P. P. Ho, R. R. Alfano, R.-J. Essiambre, and G.P. Agrawal, “Degenerate cross-phase modulation of femtosecond laser pulses in a birefringent single-mode fiber,” IEEE Photon. Tech. Lett. 9, 1107–1109 (1997).
[Crossref]

Gabitov, I.

F.G. Omenetto, Y. Chung, D. Yarotski, T. Shaefer, I. Gabitov, and A.J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191 (2002).
[Crossref]

Gindre, D.

T. Sylvestre, H. Maillotte, E. Lantz, and D. Gindre “Combined spectral effects of pulse walk-off and degenerate cross-phase modulation in birefringent fibers,” Journal of Nonlinear Optical Physics and Materials 6, 313–320 (1997).

Goto, T.

Gu, X.

Ho, P. P.

Q. D. Liu, J. T. Chen, Q. Z. Wang, P. P. Ho, and R. R. Alfano, “Single pulse degenerate-cross-phase modulation in a single-mode optical fiber,” Opt. Lett.20, 542–544 (1995).
[Crossref] [PubMed]

Q. D. Liu, L. Shi, P. P. Ho, R. R. Alfano, R.-J. Essiambre, and G.P. Agrawal, “Degenerate cross-phase modulation of femtosecond laser pulses in a birefringent single-mode fiber,” IEEE Photon. Tech. Lett. 9, 1107–1109 (1997).
[Crossref]

Kimmel, M.

Lantz, E.

T. Sylvestre, H. Maillotte, E. Lantz, and D. Gindre “Combined spectral effects of pulse walk-off and degenerate cross-phase modulation in birefringent fibers,” Journal of Nonlinear Optical Physics and Materials 6, 313–320 (1997).

Liu, Q. D.

Q. D. Liu, J. T. Chen, Q. Z. Wang, P. P. Ho, and R. R. Alfano, “Single pulse degenerate-cross-phase modulation in a single-mode optical fiber,” Opt. Lett.20, 542–544 (1995).
[Crossref] [PubMed]

Q. D. Liu, L. Shi, P. P. Ho, R. R. Alfano, R.-J. Essiambre, and G.P. Agrawal, “Degenerate cross-phase modulation of femtosecond laser pulses in a birefringent single-mode fiber,” IEEE Photon. Tech. Lett. 9, 1107–1109 (1997).
[Crossref]

Luce, B.P.

F.G. Omenetto, J.W. Nicholson, B.P. Luce, D. Yarotski, and A.J. Taylor, “Shaping, propagation and characterization of ultrafast pulses in optical fibers,” Appl. Phys. B 70[Suppl.], S143 (2000).
[Crossref]

F.G. Omenetto, B.P. Luce, D. Yarotski, and A.J. Taylor, “Observation of chirped soliton dynamics at l= 1.55 mm in a single-mode optical fiber with frequency-resolved optical gating,” Opt. Lett. 24, 1392 (1999).
[Crossref]

Maillotte, H.

T. Sylvestre, H. Maillotte, E. Lantz, and D. Gindre “Combined spectral effects of pulse walk-off and degenerate cross-phase modulation in birefringent fibers,” Journal of Nonlinear Optical Physics and Materials 6, 313–320 (1997).

Nicholson, J.W.

F.G. Omenetto, J.W. Nicholson, B.P. Luce, D. Yarotski, and A.J. Taylor, “Shaping, propagation and characterization of ultrafast pulses in optical fibers,” Appl. Phys. B 70[Suppl.], S143 (2000).
[Crossref]

Nishizawa, N.

O’Shea, P.

Ogawa, K.

K. Ogawa and M.D. Pelusi, “Characterization of ultrashort optical pulses in a dispersion-managed fiber link using two-photon absorption frequency-resolved optical gating,” Opt. Commun. 198, 83–87 (2001).
[Crossref]

Omenetto, F.G.

F.G. Omenetto, Y. Chung, D. Yarotski, T. Shaefer, I. Gabitov, and A.J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191 (2002).
[Crossref]

F.G. Omenetto, J.W. Nicholson, B.P. Luce, D. Yarotski, and A.J. Taylor, “Shaping, propagation and characterization of ultrafast pulses in optical fibers,” Appl. Phys. B 70[Suppl.], S143 (2000).
[Crossref]

F.G. Omenetto, B.P. Luce, D. Yarotski, and A.J. Taylor, “Observation of chirped soliton dynamics at l= 1.55 mm in a single-mode optical fiber with frequency-resolved optical gating,” Opt. Lett. 24, 1392 (1999).
[Crossref]

Pelusi, M.D.

K. Ogawa and M.D. Pelusi, “Characterization of ultrashort optical pulses in a dispersion-managed fiber link using two-photon absorption frequency-resolved optical gating,” Opt. Commun. 198, 83–87 (2001).
[Crossref]

Shaefer, T.

F.G. Omenetto, Y. Chung, D. Yarotski, T. Shaefer, I. Gabitov, and A.J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191 (2002).
[Crossref]

Shi, L.

Q. D. Liu, L. Shi, P. P. Ho, R. R. Alfano, R.-J. Essiambre, and G.P. Agrawal, “Degenerate cross-phase modulation of femtosecond laser pulses in a birefringent single-mode fiber,” IEEE Photon. Tech. Lett. 9, 1107–1109 (1997).
[Crossref]

Silva, A. Christian

A. Christian Silva, “GRENOUILLE - Practical Issues”, unpublished.

Sylvestre, T.

T. Sylvestre, H. Maillotte, E. Lantz, and D. Gindre “Combined spectral effects of pulse walk-off and degenerate cross-phase modulation in birefringent fibers,” Journal of Nonlinear Optical Physics and Materials 6, 313–320 (1997).

Taylor, A.J.

F.G. Omenetto, Y. Chung, D. Yarotski, T. Shaefer, I. Gabitov, and A.J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191 (2002).
[Crossref]

F.G. Omenetto, J.W. Nicholson, B.P. Luce, D. Yarotski, and A.J. Taylor, “Shaping, propagation and characterization of ultrafast pulses in optical fibers,” Appl. Phys. B 70[Suppl.], S143 (2000).
[Crossref]

F.G. Omenetto, B.P. Luce, D. Yarotski, and A.J. Taylor, “Observation of chirped soliton dynamics at l= 1.55 mm in a single-mode optical fiber with frequency-resolved optical gating,” Opt. Lett. 24, 1392 (1999).
[Crossref]

Trebino, R.

Wang, Q. Z.

Q. D. Liu, J. T. Chen, Q. Z. Wang, P. P. Ho, and R. R. Alfano, “Single pulse degenerate-cross-phase modulation in a single-mode optical fiber,” Opt. Lett.20, 542–544 (1995).
[Crossref] [PubMed]

Windeler, R.S.

Xu, L.

Yarotski, D.

F.G. Omenetto, Y. Chung, D. Yarotski, T. Shaefer, I. Gabitov, and A.J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191 (2002).
[Crossref]

F.G. Omenetto, J.W. Nicholson, B.P. Luce, D. Yarotski, and A.J. Taylor, “Shaping, propagation and characterization of ultrafast pulses in optical fibers,” Appl. Phys. B 70[Suppl.], S143 (2000).
[Crossref]

F.G. Omenetto, B.P. Luce, D. Yarotski, and A.J. Taylor, “Observation of chirped soliton dynamics at l= 1.55 mm in a single-mode optical fiber with frequency-resolved optical gating,” Opt. Lett. 24, 1392 (1999).
[Crossref]

Zeek, E.

Appl. Phys. B (1)

F.G. Omenetto, J.W. Nicholson, B.P. Luce, D. Yarotski, and A.J. Taylor, “Shaping, propagation and characterization of ultrafast pulses in optical fibers,” Appl. Phys. B 70[Suppl.], S143 (2000).
[Crossref]

J. Acoust. Soc. Am. (1)

R.A. Altes, “Detection, estimation, and classification with spectrograms,” J. Acoust. Soc. Am. 674, 1232 (1980).
[Crossref]

J. Opt. B (1)

P. O’Shea, M. Kimmel, and R. Trebino, “Increased phase-matching bandwidth in simple ultrashort-laser-pulse measurements,” J. Opt. B 4, 44 (2002).
[Crossref]

Opt. Commun. (2)

K. Ogawa and M.D. Pelusi, “Characterization of ultrashort optical pulses in a dispersion-managed fiber link using two-photon absorption frequency-resolved optical gating,” Opt. Commun. 198, 83–87 (2001).
[Crossref]

F.G. Omenetto, Y. Chung, D. Yarotski, T. Shaefer, I. Gabitov, and A.J. Taylor, “Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers,” Opt. Commun. 208, 191 (2002).
[Crossref]

Opt. Express (6)

J.M. Dudley, X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, R. Trebino, S. Coen, and R.S. Windeler, “Cross-correlation frequency resolved optical gating analysis of broadband continuum generation in photonic crystal fiber: simulations and experiments,” Opt. Express 10, 1215 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1215.
[Crossref] [PubMed]

P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, “Increased-bandwidth in ultrashort-pulse measurement using an angle-dithered nonlinear-optical crystal,” Opt. Express 7, 342 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-10-342.
[Crossref]

N. Nishizawa and T. Goto, “Experimental analysis of ultrashort pulse propagation in optical fibers around zero-dispersion region using cross-correlation frequency resolved optical gating,” Opt. Express 8, 328 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-6-328.
[Crossref] [PubMed]

N. Nishizawa and T. Goto, “Trapped pulse generation by femtosecond soliton pulse in birefringent optical fibers,” Opt. Express 10, 256 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-5-256.
[Crossref] [PubMed]

N. Nishizawa and T. Goto, “Characteristics of pulse trapping by use of ultrashort soliton pulses in optical fibers across the zero-dispersion wavelength,” Opt. Express 10, 1151 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1151.
[Crossref] [PubMed]

N. Nishizawa and T. Goto, “Ultrafast all optical switching by use of pulse trapping across zero-dispersion wavelength,” Opt. Express 11, 359 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-359.
[Crossref] [PubMed]

Opt. Express. (1)

S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express. 11, 491 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-491.
[Crossref] [PubMed]

Opt. Lett. (2)

Other (6)

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, 2002).
[Crossref]

G.P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 2001).

Q. D. Liu, J. T. Chen, Q. Z. Wang, P. P. Ho, and R. R. Alfano, “Single pulse degenerate-cross-phase modulation in a single-mode optical fiber,” Opt. Lett.20, 542–544 (1995).
[Crossref] [PubMed]

T. Sylvestre, H. Maillotte, E. Lantz, and D. Gindre “Combined spectral effects of pulse walk-off and degenerate cross-phase modulation in birefringent fibers,” Journal of Nonlinear Optical Physics and Materials 6, 313–320 (1997).

Q. D. Liu, L. Shi, P. P. Ho, R. R. Alfano, R.-J. Essiambre, and G.P. Agrawal, “Degenerate cross-phase modulation of femtosecond laser pulses in a birefringent single-mode fiber,” IEEE Photon. Tech. Lett. 9, 1107–1109 (1997).
[Crossref]

A. Christian Silva, “GRENOUILLE - Practical Issues”, unpublished.

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Figures (5)

Fig. 1.
Fig. 1.

Block diagram of the experimental setup (not drawn to scale). The optical isolator prevents feedback into the mode-locked Ti:Sapphire laser from the input end of the fiber. The mirrors M1, M2, M3, M4 are placed only when measuring the FROG traces of the pulses input to the fiber. The input half-wave plate, polarizer 1 and polarizer 2 are used such that three possible configurations are studied - θin =θout =±45°,0°, where θ is the angle between the polarization of the input(output) light and the slow axis of the optical fiber. The output half-wave plate is used to rotate the axis of polarization of the output light to match with the axis of the nonlinear crystal in the GRENOUILLE setup. The optical spectrum analyzer is present as a cross-check for the FROG recovered pulses.

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2.

Coupled into the fiber FROG traces: measured (c) and assumed in the simulation (g). Measured, from the FROG algorithm recovered spectrum (a) and time trace (d). Used in the simulation spectrum (e) and time trace (h). Spectrograms: measured (b) and used for the simulation (f).

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Fig. 3.
Fig. 3.

Experimental (a to d) and Simulated (e to h) figures for θ=-45°. t-λ spectrograms (b & f) for θ=-45° juxtaposed with corresponding time-trace (d & h), optical spectrum (a & e) and SHG-FROG trace (c & g).

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4.

Experimental (a to d) and Simulated (e to h) figures for θ=+45°. t-λ spectrograms (b & f) for θ=+45° juxtaposed with corresponding time-trace (d & h), optical spectrum (a & e) and SHG-FROG trace (c & g).

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5.

Experimental (a to d) and Simulated (e to h) figures for θ=0°. t-λ spectrograms (b & f) for θ=0° juxtaposed with corresponding time-trace (d & h), optical spectrum (a & e) and SHG-FROG trace (c & g).

Download Full Size | PPT Slide | PDF

Equations (13)

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

∂ A ∂ Z = i γ P 0 ( ∣ A ∣ 2 A + 2 3 ∣ B ∣ 2 A − T R T 0 A ∂ ∣ A ∣ 2 ∂ τ ) − i β ( 2 ) 2 T 0 2 ∂ 2 A ∂ τ 2 + β ( 3 ) 6 T 0 3 ∂ 3 A ∂ τ 3 − α A 2
∂ B ∂ Z = i γ P 0 ( ∣ B ∣ 2 B + 2 3 ∣ A ∣ 2 B − T R T 0 B ∂ ∣ B ∣ 2 ∂ τ ) − d T 0 ∂ B ∂ τ − i β ( 2 ) 2 T 0 2 ∂ 2 B ∂ τ 2 + β ( 3 ) 6 T 0 3 ∂ 3 B ∂ τ 3 − α B 2
L NL = 1 γ P 0 ∼ 1.3 mm
L w = T 0 d ∼ 6.3 cm
L D 2 = T 0 2 β ( 2 ) ∼ 54 cm
L D 3 = T 0 3 β ( 3 ) ∼ 26 m
L IRS = T 0 T R L NL ∼ 6 cm
L α = 1 α ∼ 700 m
I FROG ( ω , τ ) = ∣ ∫ − ∞ + ∞ dt E ( t ) E ( t − τ ) e i ω τ ∣ 2
S ( ω , τ ) = ∣ ∫ − ∞ + ∞ dt E ( t ) g ( t − τ ) e − iωτ ∣ 2 ,
g ( t − τ ) = exp − ( t − τ − t win ) 2 2 .
U x ( t , 0 ) = exp ⁡ [ 1 + τ − exp ⁡ ( − τ ) + ic τ 2 ]
U x ( t , 0 ) = exp ⁡ [ − τ 2 2 ( 1 − ic − τ 3 + τ 2 12 − τ 3 60 + … ) ] ,

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