Abstract

Pulse propagation and shaping are investigated in photorefractive self-pumped phase conjugators in both transmission- and reflection-grating regimes. The dispersion properties of self-pumped phase conjugators are analyzed by taking into account both the grating dispersion and the angular dispersion. The complex transfer functions are obtained by treating the crystal as a linear dispersive medium. We show that the pulse width as a result of the self-pumped phase conjugation is much wider in the reflection regime than in the transmission regime. The experimental results are consistent with the results calculated for the transmission-grating regime, indicating that this type grating is the dominant mechanism in the case of a femtosecond self-pumped phase conjugator.

© 2000 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  33. Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
    [CrossRef]
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    [CrossRef]

1999 (7)

1998 (6)

1997 (4)

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

N. G. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Optical pulse compression in fiber Bragg gratings,” Phys. Rev. Lett. 79, 4566–4569 (1997).
[CrossRef]

H. F. Yau, P. J. Wang, E. Y. Pan, J. Chen, and J. Y. Chang, “Self-pumped phase conjugation with picosecond and femtosecond pulses using BaTiO3,” Opt. Commun. 135, 331–336 (1997).
[CrossRef]

K. Nakagawa, M. Zgonik, and P. Gunter, “Reflection gratings in self-pumped phase-conjugate mirrors,” J. Opt. Soc. Am. B 14, 839–845 (1997).
[CrossRef]

1996 (5)

S. H. Lin, Y. W. Lian, P. Yeh, K. Hsu, and Y. Zhu, “2k-grating-assisted self-pumped phase conjugation: theoretical and experimental studies,” J. Opt. Soc. Am. B 13, 1772–1779 (1996).
[CrossRef]

H. F. Yau, P. J. Wang, E. Y. Pan, and J. Chen, “Self-pumped phase conjugation with femtosecond pulses by use of BaTiO3,” Opt. Lett. 21, 1168–1170 (1996).
[CrossRef] [PubMed]

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

C. M. de Sterke, N. G. R. Broderick, B. J. Eggleton, and M. J. Steel, “Nonlinear optics in fiber gratings,” Opt. Fiber Technol. Mater. Devices Syst. 2, 253–268 (1996).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

1993 (1)

K. Buse, S. Riehemann, S. Loheide, H. Hesse, F. Mersch, and E. Kratzig, “Refractive-indexes of single domain BaTiO3 for different wavelengths and temperatures,” Phys. Status Solidi A 135, K87–K89 (1993).
[CrossRef]

1991 (2)

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

L. H. Aciolo, M. Ulman, E. P. Ippen, J. G. Fujimoto, H. Kong, B. S. Chen, and M. Cronin-Golomb, “Femtosecond temporal encoding in barium titanate,” Opt. Lett. 16, 1984–1986 (1991).
[CrossRef]

1990 (1)

1987 (1)

1986 (1)

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection grating in electro-optic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

1985 (1)

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

1982 (1)

1979 (1)

Aciolo, L. H.

Agrawal, G. P.

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15–22 (1999).
[CrossRef]

Apai, P.

A. Pecchia, M. Laurito, P. Apai, and M. B. Danailov, “Studies of two-wave mixing of very broad-spectrum laser light in BaTiO3,” J. Opt. Soc. Am. B 16, 917–923 (1999).
[CrossRef]

M. B. Danailov, K. Diomande, P. Apai, and R. Szipocs, “Phase conjugation of broadband laser pulses in BaTiO3,” J. Mod. Opt. 45, 5–9 (1998).
[CrossRef]

Broderick, N. G.

N. G. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Optical pulse compression in fiber Bragg gratings,” Phys. Rev. Lett. 79, 4566–4569 (1997).
[CrossRef]

Broderick, N. G. R.

C. M. de Sterke, N. G. R. Broderick, B. J. Eggleton, and M. J. Steel, “Nonlinear optics in fiber gratings,” Opt. Fiber Technol. Mater. Devices Syst. 2, 253–268 (1996).
[CrossRef]

Brown, T. G.

Buse, K.

K. Buse, S. Riehemann, S. Loheide, H. Hesse, F. Mersch, and E. Kratzig, “Refractive-indexes of single domain BaTiO3 for different wavelengths and temperatures,” Phys. Status Solidi A 135, K87–K89 (1993).
[CrossRef]

Chang, J. Y.

H. F. Yau, P. J. Wang, E. Y. Pan, J. Chen, and J. Y. Chang, “Self-pumped phase conjugation with picosecond and femtosecond pulses using BaTiO3,” Opt. Commun. 135, 331–336 (1997).
[CrossRef]

Chen, B. S.

Chen, J.

H. F. Yau, P. J. Wang, E. Y. Pan, J. Chen, and J. Y. Chang, “Self-pumped phase conjugation with picosecond and femtosecond pulses using BaTiO3,” Opt. Commun. 135, 331–336 (1997).
[CrossRef]

H. F. Yau, P. J. Wang, E. Y. Pan, and J. Chen, “Self-pumped phase conjugation with femtosecond pulses by use of BaTiO3,” Opt. Lett. 21, 1168–1170 (1996).
[CrossRef] [PubMed]

Chi, M.

Cronin-Golomb, M.

Danailov, M. B.

A. Pecchia, M. Laurito, P. Apai, and M. B. Danailov, “Studies of two-wave mixing of very broad-spectrum laser light in BaTiO3,” J. Opt. Soc. Am. B 16, 917–923 (1999).
[CrossRef]

M. B. Danailov, K. Diomande, P. Apai, and R. Szipocs, “Phase conjugation of broadband laser pulses in BaTiO3,” J. Mod. Opt. 45, 5–9 (1998).
[CrossRef]

de Sterke, C. M.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

C. M. de Sterke, N. G. R. Broderick, B. J. Eggleton, and M. J. Steel, “Nonlinear optics in fiber gratings,” Opt. Fiber Technol. Mater. Devices Syst. 2, 253–268 (1996).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Ding, Y.

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

Diomande, K.

M. B. Danailov, K. Diomande, P. Apai, and R. Szipocs, “Phase conjugation of broadband laser pulses in BaTiO3,” J. Mod. Opt. 45, 5–9 (1998).
[CrossRef]

Dou, S. X.

Eggleton, B. J.

Fainman, Y.

Feinberg, J.

Fekete, D.

Fujimoto, J. G.

Gunter, P.

Haus, H. A.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

Hesse, H.

K. Buse, S. Riehemann, S. Loheide, H. Hesse, F. Mersch, and E. Kratzig, “Refractive-indexes of single domain BaTiO3 for different wavelengths and temperatures,” Phys. Status Solidi A 135, K87–K89 (1993).
[CrossRef]

Hsu, K.

Ibsen, M.

N. G. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Optical pulse compression in fiber Bragg gratings,” Phys. Rev. Lett. 79, 4566–4569 (1997).
[CrossRef]

Ippen, E. P.

Kong, H.

Kratzig, E.

K. Buse, S. Riehemann, S. Loheide, H. Hesse, F. Mersch, and E. Kratzig, “Refractive-indexes of single domain BaTiO3 for different wavelengths and temperatures,” Phys. Status Solidi A 135, K87–K89 (1993).
[CrossRef]

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Kukhtarev, N. V.

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection grating in electro-optic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

Laming, R. I.

N. G. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Optical pulse compression in fiber Bragg gratings,” Phys. Rev. Lett. 79, 4566–4569 (1997).
[CrossRef]

Laurito, M.

Lenz, G.

Lian, Y. W.

Lin, S. H.

Litchinitser, L.

Litchinitser, N. M.

Loheide, S.

K. Buse, S. Riehemann, S. Loheide, H. Hesse, F. Mersch, and E. Kratzig, “Refractive-indexes of single domain BaTiO3 for different wavelengths and temperatures,” Phys. Status Solidi A 135, K87–K89 (1993).
[CrossRef]

Matsumoto, H.

C. Yang, K. Minoshima, K. Seta, H. Matsumoto, and Y. Zhu, “Generation of self-pumped phase conjugation from the −c-face of BaTiO3 with femtosecond pulses,” Appl. Opt. 38, 1704–1708 (1999).
[CrossRef]

C. Yang, K. Minoshima, K. Seta, and H. Matsumoto, “Characterization of femtosecond self-pumped phase conjugation in BaTiO3,” Appl. Phys. Lett. 74, 2062–2064 (1999).
[CrossRef]

Melloch, M. R.

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

Mersch, F.

K. Buse, S. Riehemann, S. Loheide, H. Hesse, F. Mersch, and E. Kratzig, “Refractive-indexes of single domain BaTiO3 for different wavelengths and temperatures,” Phys. Status Solidi A 135, K87–K89 (1993).
[CrossRef]

Minoshima, K.

C. Yang, K. Minoshima, K. Seta, and H. Matsumoto, “Characterization of femtosecond self-pumped phase conjugation in BaTiO3,” Appl. Phys. Lett. 74, 2062–2064 (1999).
[CrossRef]

C. Yang, K. Minoshima, K. Seta, H. Matsumoto, and Y. Zhu, “Generation of self-pumped phase conjugation from the −c-face of BaTiO3 with femtosecond pulses,” Appl. Opt. 38, 1704–1708 (1999).
[CrossRef]

Nakagawa, K.

Nolte, D. D.

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

Oba, K.

Ouellete, F.

Ouellette, F.

Pan, E. Y.

H. F. Yau, P. J. Wang, E. Y. Pan, J. Chen, and J. Y. Chang, “Self-pumped phase conjugation with picosecond and femtosecond pulses using BaTiO3,” Opt. Commun. 135, 331–336 (1997).
[CrossRef]

H. F. Yau, P. J. Wang, E. Y. Pan, and J. Chen, “Self-pumped phase conjugation with femtosecond pulses by use of BaTiO3,” Opt. Lett. 21, 1168–1170 (1996).
[CrossRef] [PubMed]

Pecchia, A.

Pepper, D. M.

Richardson, D. J.

N. G. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Optical pulse compression in fiber Bragg gratings,” Phys. Rev. Lett. 79, 4566–4569 (1997).
[CrossRef]

Riehemann, S.

K. Buse, S. Riehemann, S. Loheide, H. Hesse, F. Mersch, and E. Kratzig, “Refractive-indexes of single domain BaTiO3 for different wavelengths and temperatures,” Phys. Status Solidi A 135, K87–K89 (1993).
[CrossRef]

Ringhofer, K. H.

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection grating in electro-optic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

Semenets, T. I.

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection grating in electro-optic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

Seta, K.

C. Yang, K. Minoshima, K. Seta, H. Matsumoto, and Y. Zhu, “Generation of self-pumped phase conjugation from the −c-face of BaTiO3 with femtosecond pulses,” Appl. Opt. 38, 1704–1708 (1999).
[CrossRef]

C. Yang, K. Minoshima, K. Seta, and H. Matsumoto, “Characterization of femtosecond self-pumped phase conjugation in BaTiO3,” Appl. Phys. Lett. 74, 2062–2064 (1999).
[CrossRef]

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Slusher, R. E.

Song, H.

St. J. Russell, P.

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

Steel, M. J.

C. M. de Sterke, N. G. R. Broderick, B. J. Eggleton, and M. J. Steel, “Nonlinear optics in fiber gratings,” Opt. Fiber Technol. Mater. Devices Syst. 2, 253–268 (1996).
[CrossRef]

Sun, P. C.

Szipocs, R.

M. B. Danailov, K. Diomande, P. Apai, and R. Szipocs, “Phase conjugation of broadband laser pulses in BaTiO3,” J. Mod. Opt. 45, 5–9 (1998).
[CrossRef]

Taverner, D.

N. G. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Optical pulse compression in fiber Bragg gratings,” Phys. Rev. Lett. 79, 4566–4569 (1997).
[CrossRef]

Tomberger, G.

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection grating in electro-optic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

Ulman, M.

Wang, P. J.

H. F. Yau, P. J. Wang, E. Y. Pan, J. Chen, and J. Y. Chang, “Self-pumped phase conjugation with picosecond and femtosecond pulses using BaTiO3,” Opt. Commun. 135, 331–336 (1997).
[CrossRef]

H. F. Yau, P. J. Wang, E. Y. Pan, and J. Chen, “Self-pumped phase conjugation with femtosecond pulses by use of BaTiO3,” Opt. Lett. 21, 1168–1170 (1996).
[CrossRef] [PubMed]

Weiner, A. M.

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

Winful, H. G.

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

Wong, W. S.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

Yang, C.

Yariv, A.

Yau, H. F.

H. F. Yau, P. J. Wang, E. Y. Pan, J. Chen, and J. Y. Chang, “Self-pumped phase conjugation with picosecond and femtosecond pulses using BaTiO3,” Opt. Commun. 135, 331–336 (1997).
[CrossRef]

H. F. Yau, P. J. Wang, E. Y. Pan, and J. Chen, “Self-pumped phase conjugation with femtosecond pulses by use of BaTiO3,” Opt. Lett. 21, 1168–1170 (1996).
[CrossRef] [PubMed]

Ye, P.

Yeh, P.

Zgonik, M.

Zhang, X.

Zhu, Y.

Appl. Opt. (3)

Appl. Phys. B (1)

N. V. Kukhtarev, T. I. Semenets, K. H. Ringhofer, and G. Tomberger, “Phase conjugation by reflection grating in electro-optic crystals,” Appl. Phys. B 41, 259–263 (1986).
[CrossRef]

Appl. Phys. Lett. (2)

C. Yang, K. Minoshima, K. Seta, and H. Matsumoto, “Characterization of femtosecond self-pumped phase conjugation in BaTiO3,” Appl. Phys. Lett. 74, 2062–2064 (1999).
[CrossRef]

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

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

Y. Ding, D. D. Nolte, M. R. Melloch, and A. M. Weiner, “Time-domain image processing using dynamic holography,” IEEE J. Sel. Top. Quantum Electron. 4, 332–341 (1998).
[CrossRef]

J. Mod. Opt. (2)

M. B. Danailov, K. Diomande, P. Apai, and R. Szipocs, “Phase conjugation of broadband laser pulses in BaTiO3,” J. Mod. Opt. 45, 5–9 (1998).
[CrossRef]

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

J. Opt. Soc. Am. B (7)

Opt. Commun. (2)

H. F. Yau, P. J. Wang, E. Y. Pan, J. Chen, and J. Y. Chang, “Self-pumped phase conjugation with picosecond and femtosecond pulses using BaTiO3,” Opt. Commun. 135, 331–336 (1997).
[CrossRef]

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15–22 (1999).
[CrossRef]

Opt. Express (1)

Opt. Fiber Technol. Mater. Devices Syst. (1)

C. M. de Sterke, N. G. R. Broderick, B. J. Eggleton, and M. J. Steel, “Nonlinear optics in fiber gratings,” Opt. Fiber Technol. Mater. Devices Syst. 2, 253–268 (1996).
[CrossRef]

Opt. Lett. (7)

Phys. Rev. Lett. (2)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

N. G. Broderick, D. Taverner, D. J. Richardson, M. Ibsen, and R. I. Laming, “Optical pulse compression in fiber Bragg gratings,” Phys. Rev. Lett. 79, 4566–4569 (1997).
[CrossRef]

Phys. Status Solidi A (1)

K. Buse, S. Riehemann, S. Loheide, H. Hesse, F. Mersch, and E. Kratzig, “Refractive-indexes of single domain BaTiO3 for different wavelengths and temperatures,” Phys. Status Solidi A 135, K87–K89 (1993).
[CrossRef]

Rev. Mod. Phys. (1)

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

Other (3)

See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995), and references therein.

See, for example, J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, and Applications on Femtosecond Time Scale (Academic, San Diego, Calif., 1995).

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993), Chap. 2.

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Figures (9)

Fig. 1
Fig. 1

(a) Transmission-grating-based self-pumped phase conjugator; (b) reflection-grating-based self-pumped phase conjugator; (c) transmission grating; (d) reflection grating. (K1, Ω1) indicate the incident beam; (K2, Ω2) indicate the diffracted beam; θ1,2 are the incident and the diffracted angles. l1 and l are the thickness of grating and the distance between the crystal surface and the grating, respectively. θ is the incident angle in air.

Fig. 2
Fig. 2

(a) First-order dispersion k, α1, and β1 versus the wavelength. Solid, dashed, and dotted curves represent k, α1, and β1, respectively. The Bragg angles are 22.5° and 60° for the transmission and the reflection gratings, respectively. (b) α1 and β1 versus the Bragg angle at 450 and 800 nm.

Fig. 3
Fig. 3

(a) Second-order dispersion k, α2, and β2 versus the wavelength for four grating periods. (b) The second-order dispersion α2 and β2 versus Bragg angle for two wavelengths. The second-order dispersion of the grating changes sign at the zero-dispersion angle θD.

Fig. 4
Fig. 4

Broadening factor of the SPPC from the TG-SPPCM at 450 nm versus the grating period. Solid, dashed, and dotted curves represent the chirp parameter C=0,-1, 1, respectively. The SPPC’s broaden for the three cases.

Fig. 5
Fig. 5

Broadening factor of the SPPC from the TG-SPPCM at 800 nm versus the grating period. Solid, dashed, and dotted curves represent the chirp parameter C=0,-1, 1, respectively. The SPPC is compressed in the regimes of 485 nm<Λ<800 nm for C=-1 and Λ>825 nm for C=1.

Fig. 6
Fig. 6

Broadening factor of the SPPC from the 2k-grating versus the wavelength for three chirp parameters. The period of the 2k grating relates to the wavelength as Λ=λ/2n0.

Fig. 7
Fig. 7

Pulse shapes of SPPC’s from the TG-SPPCM at 450 and 800 nm for two chirp parameters. The solid curve represents a pump pulse of unchirped Gaussian shape. Dashed (R) and dashed-dotted (R) curves are the SPPC pulse shapes at 800 nm for C=0 and C=-1, respectively. Dotted (B) and dashed-dotted-dotted (B) curves are the SPPC pulse shapes at 450 nm for C=0 and C=-1, respectively. The SPPC pulse at 800 nm with C=-1 is compressed.

Fig. 8
Fig. 8

Pulse shapes of SPPC from 2k-grating-based SPPCM. Solid, dashed, and dotted curves represent the pump pulse, SPPC at 800 nm, and SPPC at 450 nm, respectively.

Fig. 9
Fig. 9

Zero-dispersion angle versus wavelength. Solid and dashed curves represent the transmission grating and the reflection grating, respectively.

Equations (30)

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E=A1 exp[i(ωt-k1xx-k1zz)]+A2 exp[i(ωt-k2xx-k2zz)]+c.c.,
kjx=kj cos θj,kjz=kj sin θj(j=1, 2),
k2(Ω)sin θ2(Ω)=k1 sin θ1±K,
dθ2dΩ=-tan θ2k2 dk2dΩ.
Δα(Ω)=k2(Ω)cos θ2(Ω)-k1 cos θ1.
α1=1cos θ2 dk2dΩωl,θB,
α2=-tan2 θ22k2 cos θ2 dk2dΩ2+12 cos θ2 d2k2dΩ2ωl,θB,
k2(Ω)cos θ2(Ω)-k1 cos θ1=0,
Δβ(Ω)=k2(Ω)sin θ2(Ω)-k1 sin θ1±K.
dθ2dΩ=1k2 tan θ2 dk2dΩ,
β1=1sin θ2 dk2dΩωl,θB,
β2=-12k2 sin θ2 tan2 θ2 dk2dΩ2+12 sin θ2 d2k2dΩ2ωl,θB.
b2=-4λ32πc2 l1 sin2 θn02(n02-sin2 θ) dn0dλ2+L(2n0Λ cos θB)2 1-λn0 dn0dλ2+l4 sin2 θ(n0 cos θ)2 dn0dλ2,
R(Ω)=R0T exp[ilΔα(Ω)/2]sin c[lΔα(Ω)/2],
sin c[lΔα(Ω)/2]exp{-(1/6)[lΔα(Ω)/2]2}.
HT(Ω)=R0T exp-ib0+ilα12-b1Ω+il2α1224+ilα22-b2Ω2.
HR(Ω)=R0R exp-ib0+ilβ12-b1Ω+il2β1224+ilβ22-b2Ω2.
Epc(z, T)=12π -H(Ω)E(0, Ω)×exp(-iΩT)exp-i kz2Ω2dΩ,
Epc(z, T)=|Epc|exp[-(T+Tc)2/2T12+ϕ(z, T)],
T1T0=1+l2α1212T021/21+2b2+kz-lα2T02(1+l2α12/12T02)21/2.
ϕ(z, T)=(2b2+kz-lα2)(T+b1-lα1/2)22T041+l2αl212T02+2b2+kz-lα2T022-12 tan-1 (2b2+kz-lα2)/T021+l2α1212T02-b0.
δω=ϕT=(2b2+kz-lα2)(T+b1-lα1/2)T041+l2α1212T02+2b2+kz-lα2T022.
T1T0=11+C21+l2α12(1+C2)12T021/21+[2b2+kz-lα2-CT02/(1+C2)]2[T02/(1+C2)]2[1+l2α12(1+C2)/12T02]21/2.
ϕ(z, T)=12 [2b2+kz-lα2-CT02/(1+C2)](T+b1-lα1/2)2[T02/(1+C2)+l2α12/12]2+[2b2+kz-lα2-CT02/(1+C2)]2-12 tan-1 2b2+kz-lα2-CT02/(1+C2)T02/(1+C2)+l2α12/12+const,
δω=[2b2+kz-lα2-CT02/(1+C2)](T+b1-lα1/2)[T02/(1+C2)+l2α12/12]2+[2b2+kz-lα2-CT02/(1+C2)]2.
Tpcmin=(1+l2α12/12T02)1/2T0.
Tpcmin=[1+l2α12(1+C2)/12T02]1/2T0/1+C2.
θDT=tan-1(d2k/dΩ2)/[k(dk/dΩ)2]
θDR=cot-1(d2k/dΩ2)/[k(dk/dΩ)2]
RP=α3(Ω-ωl)3α2(Ω-ωl)2sin2 θk cos2 θ dkdΩ×1+(dk/dΩ)2/ksin2 θ(dk/dΩ)2/k cos2 θ-(d2k/dΩ2)×|Δωl|,

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