Abstract

Based on the modified Kogelnik’s coupled-wave theory, time- and frequency-domain diffractions of a femtosecond pulse from transmitted volume holographic gratings (VHGs) are theoretically studied. Results show that when the refractive index modulation of the VHG changes in a certain range, the number of temporal diffracted pulse will evolve from one to two, then to three, and this pulse number evolution is periodic. This particular phenomenon can be explained by diffraction intensity spectrum and the overmodulation effect of refractive index modulation of transmitted VHG. Moreover, we find centers of all temporal diffracted pulses translate along the negative time axis, and the translation is irrelevant to the refractive index modulations. We will use time delay of volume grating to give a reasonable explanation.

© 2013 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. B. Yang, X. Yan, Y. Yang, and H. Zhang, “Study on the instantaneous characteristics of diffracted and transmitted light of static photorefractive grating illuminated by ultra-short pulse laser,” Opt. Laser Technol.40(7), 906–911 (2008).
    [CrossRef]
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    [CrossRef]
  10. C. Wang, L. Liu, A. Yan, D. Liu, D. Li, and W. Qu, “Pulse shaping properties of volume holographic gratings in anisotropic media,” J. Opt. Soc. Am. A23(12), 3191–3196 (2006).
    [CrossRef] [PubMed]
  11. A. Yan, L. Liu, Y. Zhi, D. Liu, and J. Sun, “Bragg diffraction of multiplayer volume holographic gratings under ultrashort laser pulse readout,” J. Opt. Soc. Am. A26(1), 135–141 (2009).
    [CrossRef]
  12. A. Yan, L. Liu, L. Wang, D. Liu, J. Sun, and L. Wan, “Pulse shaping and diffraction properties of multi-layers reflection volume holographic gratings,” Appl. Phys. B96(1), 71–77 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  25. S. Gallego, M. Ortuño, C. Neipp, C. Garcia, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun.215(4-6), 263–269 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2011 (2)

Y. Yi, D. Liu, and H. Liu, “Instantaneous characteristics study of the diffracted and transmitted light of a static photorefractive reflection volume holographic grating read by an ultrashort pulse laser,” J. Opt.13(3), 035701 (2011).
[CrossRef]

M. P. Hernández-Garay, O. Martínez-Matos, J. G. Izquierdo, M. L. Calvo, P. Vaveliuk, P. Cheben, and L. Bañares, “Femtosecond spectral pulse shaping with holographic gratings recorded in photopolymerizable glasses,” Opt. Express19(2), 1516–1527 (2011).
[CrossRef] [PubMed]

2009 (3)

2008 (1)

B. Yang, X. Yan, Y. Yang, and H. Zhang, “Study on the instantaneous characteristics of diffracted and transmitted light of static photorefractive grating illuminated by ultra-short pulse laser,” Opt. Laser Technol.40(7), 906–911 (2008).
[CrossRef]

2006 (2)

C. Wang, L. Liu, A. Yan, D. Liu, D. Li, and W. Qu, “Pulse shaping properties of volume holographic gratings in anisotropic media,” J. Opt. Soc. Am. A23(12), 3191–3196 (2006).
[CrossRef] [PubMed]

S. Iwai, Y. Ishige, S. Tanaka, Y. Okimoto, Y. Tokura, and H. Okamoto, “Coherent control of charge and lattice dynamics in a photoinduced neutral-to-ionic transition of a charge-transfer compound,” Phys. Rev. Lett.96(5), 057403 (2006).
[CrossRef] [PubMed]

2005 (1)

T. Nagata, M. Kamata, and M. Obara, “Optical waveguide fabrication with double pulse femtosecond lasers,” Appl. Phys. Lett.86(25), 251103 (2005).
[CrossRef]

2004 (1)

X. Yan, B. Yang, and B. Yu, “Diffraction study of photorefractive hologram under ultrashort pulse readout,” Optik (Stuttg.)115(11-12), 512–516 (2004).
[CrossRef]

2003 (1)

S. Gallego, M. Ortuño, C. Neipp, C. Garcia, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun.215(4-6), 263–269 (2003).
[CrossRef]

2001 (3)

C. Neipp, I. Pascual, and A. Beléndez, “Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions,” J. Opt. A, Pure Appl. Opt.3(6), 504–513 (2001).
[CrossRef]

D. Felinto, C. A. C. Bosco, L. H. Acioli, and S. S. Vianna, “Accumulative effects in temporal coherent control,” Phys. Rev. A64(6), 063413 (2001).
[CrossRef]

T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett.26(8), 557–559 (2001).
[CrossRef] [PubMed]

2000 (2)

1999 (1)

C. Iaconis and I. A. Walmsley, “Self-Referencing spectral interferometry for measuring ultrashort optical pulse,” IEEE J. Quantum Electron.35(4), 501–509 (1999).
[CrossRef]

1998 (1)

1995 (1)

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron.19(3), 161–237 (1995).
[CrossRef]

1993 (1)

M. S. Luo, S. L. Chuang, P. C. Planken, I. Brener, and M. C. Nuss, “Coherent double-pulse control of quantum beats in a coupled quantum well,” Phys. Rev. B Condens. Matter48(15), 11043–11050 (1993).
[CrossRef] [PubMed]

1983 (1)

G. C. Valley, “Short-pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron.19(11), 1637–1645 (1983).
[CrossRef]

1979 (1)

C. T. Chen, D. M. Kim, and D. von der Linde, “Efficient hologram recording in LiNbO3:Fe using optical pulses,” Appl. Phys. Lett.34(5), 321–324 (1979).
[CrossRef]

1975 (1)

D. von der Linde and A. M. Glass, “Photorefractive effects for reversible holographic storage of information,” Appl. Phys. (Berl.)8(2), 85–100 (1975).
[CrossRef]

1974 (1)

P. Shah, T. A. Rabson, F. K. Tittel, and T. K. Gaylord, “Volume holographic recording and storage in Fe-doped LiNbO3 using optical pulses,” Appl. Phys. Lett.24(3), 130–131 (1974).
[CrossRef]

1973 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J.48, 2909–2947 (1969).

Acioli, L. H.

D. Felinto, C. A. C. Bosco, L. H. Acioli, and S. S. Vianna, “Accumulative effects in temporal coherent control,” Phys. Rev. A64(6), 063413 (2001).
[CrossRef]

Bañares, L.

Beléndez, A.

S. Gallego, M. Ortuño, C. Neipp, C. Garcia, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun.215(4-6), 263–269 (2003).
[CrossRef]

C. Neipp, I. Pascual, and A. Beléndez, “Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions,” J. Opt. A, Pure Appl. Opt.3(6), 504–513 (2001).
[CrossRef]

Bosco, C. A. C.

D. Felinto, C. A. C. Bosco, L. H. Acioli, and S. S. Vianna, “Accumulative effects in temporal coherent control,” Phys. Rev. A64(6), 063413 (2001).
[CrossRef]

Brener, I.

M. S. Luo, S. L. Chuang, P. C. Planken, I. Brener, and M. C. Nuss, “Coherent double-pulse control of quantum beats in a coupled quantum well,” Phys. Rev. B Condens. Matter48(15), 11043–11050 (1993).
[CrossRef] [PubMed]

Brixner, T.

Brost, G. A.

Calvo, M. L.

Canioni, L.

Cheben, P.

Chen, C. T.

C. T. Chen, D. M. Kim, and D. von der Linde, “Efficient hologram recording in LiNbO3:Fe using optical pulses,” Appl. Phys. Lett.34(5), 321–324 (1979).
[CrossRef]

Chuang, S. L.

M. S. Luo, S. L. Chuang, P. C. Planken, I. Brener, and M. C. Nuss, “Coherent double-pulse control of quantum beats in a coupled quantum well,” Phys. Rev. B Condens. Matter48(15), 11043–11050 (1993).
[CrossRef] [PubMed]

Ding, Y.

Efimov, O. M.

Felinto, D.

D. Felinto, C. A. C. Bosco, L. H. Acioli, and S. S. Vianna, “Accumulative effects in temporal coherent control,” Phys. Rev. A64(6), 063413 (2001).
[CrossRef]

Gallego, S.

S. Gallego, M. Ortuño, C. Neipp, C. Garcia, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun.215(4-6), 263–269 (2003).
[CrossRef]

Garcia, C.

S. Gallego, M. Ortuño, C. Neipp, C. Garcia, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun.215(4-6), 263–269 (2003).
[CrossRef]

Gaylord, T. K.

P. Shah, T. A. Rabson, F. K. Tittel, and T. K. Gaylord, “Volume holographic recording and storage in Fe-doped LiNbO3 using optical pulses,” Appl. Phys. Lett.24(3), 130–131 (1974).
[CrossRef]

T. K. Gaylord, T. A. Rabson, F. K. Tittel, and C. R. Quick, “Pulsed writing of solid state holograms,” Appl. Opt.12(2), 414–415 (1973).
[CrossRef] [PubMed]

Gerber, G.

Glass, A. M.

D. von der Linde and A. M. Glass, “Photorefractive effects for reversible holographic storage of information,” Appl. Phys. (Berl.)8(2), 85–100 (1975).
[CrossRef]

Glebov, L. B.

Hernández-Garay, M. P.

Hsu, K. Y.

Iaconis, C.

C. Iaconis and I. A. Walmsley, “Self-Referencing spectral interferometry for measuring ultrashort optical pulse,” IEEE J. Quantum Electron.35(4), 501–509 (1999).
[CrossRef]

Ishige, Y.

S. Iwai, Y. Ishige, S. Tanaka, Y. Okimoto, Y. Tokura, and H. Okamoto, “Coherent control of charge and lattice dynamics in a photoinduced neutral-to-ionic transition of a charge-transfer compound,” Phys. Rev. Lett.96(5), 057403 (2006).
[CrossRef] [PubMed]

Iwai, S.

S. Iwai, Y. Ishige, S. Tanaka, Y. Okimoto, Y. Tokura, and H. Okamoto, “Coherent control of charge and lattice dynamics in a photoinduced neutral-to-ionic transition of a charge-transfer compound,” Phys. Rev. Lett.96(5), 057403 (2006).
[CrossRef] [PubMed]

Izquierdo, J. G.

Kamata, M.

T. Nagata, M. Kamata, and M. Obara, “Optical waveguide fabrication with double pulse femtosecond lasers,” Appl. Phys. Lett.86(25), 251103 (2005).
[CrossRef]

Kanan, A.

Kim, D. M.

C. T. Chen, D. M. Kim, and D. von der Linde, “Efficient hologram recording in LiNbO3:Fe using optical pulses,” Appl. Phys. Lett.34(5), 321–324 (1979).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J.48, 2909–2947 (1969).

Li, D.

Lin, S. H.

Liu, D.

Y. Yi, D. Liu, and H. Liu, “Instantaneous characteristics study of the diffracted and transmitted light of a static photorefractive reflection volume holographic grating read by an ultrashort pulse laser,” J. Opt.13(3), 035701 (2011).
[CrossRef]

A. Yan, L. Liu, L. Wang, D. Liu, J. Sun, and L. Wan, “Pulse shaping and diffraction properties of multi-layers reflection volume holographic gratings,” Appl. Phys. B96(1), 71–77 (2009).
[CrossRef]

A. Yan, L. Liu, Y. Zhi, D. Liu, and J. Sun, “Bragg diffraction of multiplayer volume holographic gratings under ultrashort laser pulse readout,” J. Opt. Soc. Am. A26(1), 135–141 (2009).
[CrossRef]

C. Wang, L. Liu, A. Yan, D. Liu, D. Li, and W. Qu, “Pulse shaping properties of volume holographic gratings in anisotropic media,” J. Opt. Soc. Am. A23(12), 3191–3196 (2006).
[CrossRef] [PubMed]

Liu, H.

Y. Yi, D. Liu, and H. Liu, “Instantaneous characteristics study of the diffracted and transmitted light of a static photorefractive reflection volume holographic grating read by an ultrashort pulse laser,” J. Opt.13(3), 035701 (2011).
[CrossRef]

Liu, L.

Lumeau, J.

Luo, M. S.

M. S. Luo, S. L. Chuang, P. C. Planken, I. Brener, and M. C. Nuss, “Coherent double-pulse control of quantum beats in a coupled quantum well,” Phys. Rev. B Condens. Matter48(15), 11043–11050 (1993).
[CrossRef] [PubMed]

Martínez-Matos, O.

Nagata, T.

T. Nagata, M. Kamata, and M. Obara, “Optical waveguide fabrication with double pulse femtosecond lasers,” Appl. Phys. Lett.86(25), 251103 (2005).
[CrossRef]

Neipp, C.

S. Gallego, M. Ortuño, C. Neipp, C. Garcia, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun.215(4-6), 263–269 (2003).
[CrossRef]

C. Neipp, I. Pascual, and A. Beléndez, “Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions,” J. Opt. A, Pure Appl. Opt.3(6), 504–513 (2001).
[CrossRef]

Nolte, D. D.

Nuss, M. C.

M. S. Luo, S. L. Chuang, P. C. Planken, I. Brener, and M. C. Nuss, “Coherent double-pulse control of quantum beats in a coupled quantum well,” Phys. Rev. B Condens. Matter48(15), 11043–11050 (1993).
[CrossRef] [PubMed]

Obara, M.

T. Nagata, M. Kamata, and M. Obara, “Optical waveguide fabrication with double pulse femtosecond lasers,” Appl. Phys. Lett.86(25), 251103 (2005).
[CrossRef]

Okamoto, H.

S. Iwai, Y. Ishige, S. Tanaka, Y. Okimoto, Y. Tokura, and H. Okamoto, “Coherent control of charge and lattice dynamics in a photoinduced neutral-to-ionic transition of a charge-transfer compound,” Phys. Rev. Lett.96(5), 057403 (2006).
[CrossRef] [PubMed]

Okimoto, Y.

S. Iwai, Y. Ishige, S. Tanaka, Y. Okimoto, Y. Tokura, and H. Okamoto, “Coherent control of charge and lattice dynamics in a photoinduced neutral-to-ionic transition of a charge-transfer compound,” Phys. Rev. Lett.96(5), 057403 (2006).
[CrossRef] [PubMed]

Ortuño, M.

S. Gallego, M. Ortuño, C. Neipp, C. Garcia, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun.215(4-6), 263–269 (2003).
[CrossRef]

Pascual, I.

S. Gallego, M. Ortuño, C. Neipp, C. Garcia, A. Beléndez, and I. Pascual, “Overmodulation effects in volume holograms recorded on photopolymers,” Opt. Commun.215(4-6), 263–269 (2003).
[CrossRef]

C. Neipp, I. Pascual, and A. Beléndez, “Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions,” J. Opt. A, Pure Appl. Opt.3(6), 504–513 (2001).
[CrossRef]

Planken, P. C.

M. S. Luo, S. L. Chuang, P. C. Planken, I. Brener, and M. C. Nuss, “Coherent double-pulse control of quantum beats in a coupled quantum well,” Phys. Rev. B Condens. Matter48(15), 11043–11050 (1993).
[CrossRef] [PubMed]

Qu, W.

Quick, C. R.

Rabson, T. A.

P. Shah, T. A. Rabson, F. K. Tittel, and T. K. Gaylord, “Volume holographic recording and storage in Fe-doped LiNbO3 using optical pulses,” Appl. Phys. Lett.24(3), 130–131 (1974).
[CrossRef]

T. K. Gaylord, T. A. Rabson, F. K. Tittel, and C. R. Quick, “Pulsed writing of solid state holograms,” Appl. Opt.12(2), 414–415 (1973).
[CrossRef] [PubMed]

Shah, P.

P. Shah, T. A. Rabson, F. K. Tittel, and T. K. Gaylord, “Volume holographic recording and storage in Fe-doped LiNbO3 using optical pulses,” Appl. Phys. Lett.24(3), 130–131 (1974).
[CrossRef]

Siiman, L. A.

Smirnov, V. I.

Sun, J.

A. Yan, L. Liu, L. Wang, D. Liu, J. Sun, and L. Wan, “Pulse shaping and diffraction properties of multi-layers reflection volume holographic gratings,” Appl. Phys. B96(1), 71–77 (2009).
[CrossRef]

A. Yan, L. Liu, Y. Zhi, D. Liu, and J. Sun, “Bragg diffraction of multiplayer volume holographic gratings under ultrashort laser pulse readout,” J. Opt. Soc. Am. A26(1), 135–141 (2009).
[CrossRef]

Tanaka, S.

S. Iwai, Y. Ishige, S. Tanaka, Y. Okimoto, Y. Tokura, and H. Okamoto, “Coherent control of charge and lattice dynamics in a photoinduced neutral-to-ionic transition of a charge-transfer compound,” Phys. Rev. Lett.96(5), 057403 (2006).
[CrossRef] [PubMed]

Tittel, F. K.

P. Shah, T. A. Rabson, F. K. Tittel, and T. K. Gaylord, “Volume holographic recording and storage in Fe-doped LiNbO3 using optical pulses,” Appl. Phys. Lett.24(3), 130–131 (1974).
[CrossRef]

T. K. Gaylord, T. A. Rabson, F. K. Tittel, and C. R. Quick, “Pulsed writing of solid state holograms,” Appl. Opt.12(2), 414–415 (1973).
[CrossRef] [PubMed]

Tokura, Y.

S. Iwai, Y. Ishige, S. Tanaka, Y. Okimoto, Y. Tokura, and H. Okamoto, “Coherent control of charge and lattice dynamics in a photoinduced neutral-to-ionic transition of a charge-transfer compound,” Phys. Rev. Lett.96(5), 057403 (2006).
[CrossRef] [PubMed]

Valley, G. C.

G. C. Valley, “Short-pulse grating formation in photorefractive materials,” IEEE J. Quantum Electron.19(11), 1637–1645 (1983).
[CrossRef]

Vaveliuk, P.

Vianna, S. S.

D. Felinto, C. A. C. Bosco, L. H. Acioli, and S. S. Vianna, “Accumulative effects in temporal coherent control,” Phys. Rev. A64(6), 063413 (2001).
[CrossRef]

von der Linde, D.

C. T. Chen, D. M. Kim, and D. von der Linde, “Efficient hologram recording in LiNbO3:Fe using optical pulses,” Appl. Phys. Lett.34(5), 321–324 (1979).
[CrossRef]

D. von der Linde and A. M. Glass, “Photorefractive effects for reversible holographic storage of information,” Appl. Phys. (Berl.)8(2), 85–100 (1975).
[CrossRef]

Walmsley, I. A.

C. Iaconis and I. A. Walmsley, “Self-Referencing spectral interferometry for measuring ultrashort optical pulse,” IEEE J. Quantum Electron.35(4), 501–509 (1999).
[CrossRef]

Wan, L.

A. Yan, L. Liu, L. Wang, D. Liu, J. Sun, and L. Wan, “Pulse shaping and diffraction properties of multi-layers reflection volume holographic gratings,” Appl. Phys. B96(1), 71–77 (2009).
[CrossRef]

Wang, C.

Wang, L.

A. Yan, L. Liu, L. Wang, D. Liu, J. Sun, and L. Wan, “Pulse shaping and diffraction properties of multi-layers reflection volume holographic gratings,” Appl. Phys. B96(1), 71–77 (2009).
[CrossRef]

Weiner, A. M.

Yan, A.

Yan, X.

B. Yang, X. Yan, Y. Yang, and H. Zhang, “Study on the instantaneous characteristics of diffracted and transmitted light of static photorefractive grating illuminated by ultra-short pulse laser,” Opt. Laser Technol.40(7), 906–911 (2008).
[CrossRef]

X. Yan, B. Yang, and B. Yu, “Diffraction study of photorefractive hologram under ultrashort pulse readout,” Optik (Stuttg.)115(11-12), 512–516 (2004).
[CrossRef]

Yang, B.

B. Yang, X. Yan, Y. Yang, and H. Zhang, “Study on the instantaneous characteristics of diffracted and transmitted light of static photorefractive grating illuminated by ultra-short pulse laser,” Opt. Laser Technol.40(7), 906–911 (2008).
[CrossRef]

X. Yan, B. Yang, and B. Yu, “Diffraction study of photorefractive hologram under ultrashort pulse readout,” Optik (Stuttg.)115(11-12), 512–516 (2004).
[CrossRef]

Yang, Y.

B. Yang, X. Yan, Y. Yang, and H. Zhang, “Study on the instantaneous characteristics of diffracted and transmitted light of static photorefractive grating illuminated by ultra-short pulse laser,” Opt. Laser Technol.40(7), 906–911 (2008).
[CrossRef]

Yeh, P.

Yi, Y.

Y. Yi, D. Liu, and H. Liu, “Instantaneous characteristics study of the diffracted and transmitted light of a static photorefractive reflection volume holographic grating read by an ultrashort pulse laser,” J. Opt.13(3), 035701 (2011).
[CrossRef]

Yu, B.

X. Yan, B. Yang, and B. Yu, “Diffraction study of photorefractive hologram under ultrashort pulse readout,” Optik (Stuttg.)115(11-12), 512–516 (2004).
[CrossRef]

Zhang, H.

B. Yang, X. Yan, Y. Yang, and H. Zhang, “Study on the instantaneous characteristics of diffracted and transmitted light of static photorefractive grating illuminated by ultra-short pulse laser,” Opt. Laser Technol.40(7), 906–911 (2008).
[CrossRef]

Zheng, Z.

Zhi, Y.

Appl. Opt. (1)

Appl. Phys. (Berl.) (1)

D. von der Linde and A. M. Glass, “Photorefractive effects for reversible holographic storage of information,” Appl. Phys. (Berl.)8(2), 85–100 (1975).
[CrossRef]

Appl. Phys. B (1)

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Figures (6)

Fig. 1
Fig. 1

Recording and readout of a volume holographic grating. Two continuous waves symmetrically incident on the material to write an unslanted photorefractive VHG with the grating vector along coordinate axis x, then a femtosecond pulse incidents, a diffracted pulse and a transmitted pulse are emerged.

Fig. 2
Fig. 2

Intensity distributions of input Gaussian pulses with different pulse durations.

Fig. 3
Fig. 3

Distributions of temporal diffraction intensity when refractive index modulation of the VHG changes in the range of (a): Δn = 1.0 × 10−4~2.0 × 10−4, (b): Δn = 2.0 × 10−4~3.0 × 10−4 .

Fig. 4
Fig. 4

Distributions of temporal diffraction intensity when refractive index modulation of the VHG changes from 1.0 × 10−3 to 9.0 × 10−3 with the index-step of 2.0 × 10−3.

Fig. 5
Fig. 5

Normalized diffraction intensity spectrum distributions when refractive index modulation changes in the range of (a): Δn = 1.0 × 10−4~1.8 × 10−4, (b): Δn = 2.0 × 10−4~2.8 × 10−4.

Fig. 6
Fig. 6

Distributions of temporal diffraction intensity when duration of the input pulse is Δτ = 200fs, other parameters are the same as those of Fig. 3.

Equations (13)

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E r (t)=exp(i ω 0 t t 2 / T 2 ),
E r (ω)= 1 2π E r (t) exp(iωt)dt= T 2 π exp[ T 2 (ω ω 0 ) 2 4 ].
cos θ r ' d E t ( ω,z ) dz = E d ( ω,z ),
cos θ r ' d E d ( ω,z ) dz i 2πc K 2 n 0 ( 1 ω 1 ω 0 ) E d ( ω,z )= E t ( ω,z ),
E d (ω,d)=iνexp(iξ) sin ν 2 +ξ 2 ν 2 + ξ 2 E r (ω),
I d (ω,d)= ν 2 sin 2 ν 2 +ξ 2 ν 2 + ξ 2 E r 2 (ω).
E d (t,d)= E d ( ω,d)exp(-iωt)dω.
I d (t,d)= | - E d (ω,d)exp(-iωt)dω | 2 .
ν=π.
(Δn) p =π2ccos θ r '/(ωd).
V g =l ( φ ω ) 1 ,
τ g = l V g = φ ω .
τ g = π 2 cd Λ 2 n 0 cos θ r ' ω 0 2 .

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