Abstract

A scheme to generate individually modulated femtosecond pulse string by multilayer volume holographic grating (MVHG) is proposed. Based on Kogelnik’s coupled-wave theory and matrix optics, temporal and spectral expressions of diffracted field are given when a femtosecond pulse is diffracted by a MVHG. It is shown that the number of diffracted sub-pulses in the pulse string equals to the number of grating layers of the MVHG, peak intensity and duration of each diffracted sub-pulse depend on thickness of the corresponding grating layer, whereas pulse interval between adjacent sub-pulses is related to thickness of the corresponding buffer layer. Thus by modulating parameters of the MVHG, individually modulated femtosecond pulse string can be acquired. Based on Bragg selectivity of the volume grating and phase shift provided by the buffer layers, we give an explanation on these phenomena. The result is useful to design MVHG-based devices employed in optical communications, pulse shaping and processing.

© 2014 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]

2014 (1)

2013 (7)

X. Yan, Y. Dai, Z. Gao, Y. Chen, X. Yang, and G. Ma, “Femtosecond pulse shaping by modulating the refractive index modulation of volume holographic grating,” Opt. Express 21(6), 7560–7569 (2013).
[Crossref] [PubMed]

N. Zhang, X. Li, L. Jiang, X. Shi, C. Li, and Y. Lu, “Femtosecond double-pulse fabrication of hierarchical nanostructures based on electron dynamics control for high surface-enhanced Raman scattering,” Opt. Lett. 38(18), 3558–3561 (2013).
[Crossref] [PubMed]

P. Liu, L. Jiang, J. Hu, X. Yan, B. Xia, and Y. Lu, “Etching rate enhancement by shaped femtosecond pulse train electron dynamics control for microchannels fabrication in fused silica glass,” Opt. Lett. 38(22), 4613–4616 (2013).
[Crossref] [PubMed]

T. J.-Y. Derrien, J. Krüger, T. E. Itina, S. Höhm, A. Rosenfeld, and J. Bonse, “Rippled area formed by surface plasmon polaritons upon femtosecond laser double-pulse irradiation of silicon,” Opt. Express 21(24), 29643–29655 (2013).
[Crossref] [PubMed]

X. Yan, M. Qian, L. Gao, X. Yang, Y. Dai, X. Yan, and G. Ma, “Pulse splitting by modulating the thickness of buffer layer of two-layer volume holographic grating,” Opt. Express 21(26), 31852–31861 (2013).
[Crossref] [PubMed]

M. Barberoglou, G. D. Tsibidis, D. Gray, E. Magoulakis, C. Fotakis, E. Stratakis, and P. A. Loukakos, “The influence of ultra-fast temporal energy regulation on the morphology of Si surfaces through femtosecond double pulse laser irradiation,” Appl. Phys., A Mater. Sci. Process. 113(2), 273–283 (2013).
[Crossref]

S. Höhm, A. Rosenfeld, J. Krüger, and J. Bonse, “Area dependence of femtosecond laser-induced periodic surface structures for varying band gap materials after double pulse excitation,” Appl. Surf. Sci. 278, 7–12 (2013).
[Crossref]

2012 (1)

2011 (1)

L. Guo, A. Yan, and S. Fu, “Diffraction properties of multi-layer volume holographic gratings under an ultrashort pulsed beam with arbitrary temporal profiles,” Optik (Stuttg.) 122(19), 1692–1696 (2011).
[Crossref]

2010 (3)

E. F. Pen and M. Yu. Rodionov, “Properties of multilayer nonuniform holographic structures,” Quantum Electron. 40(10), 919–924 (2010).
[Crossref]

C. Yang, X. Yan, R. Zhu, H. Zou, and F. Han, “Diffraction study of volume holographic gratings in dispersive photorefractive material for femtosecond pulse readout,” Optik (Stuttg.) 121(12), 1138–1143 (2010).
[Crossref]

J. Bonse and J. Kruger, “Pulse number dependence of laser-induced periodic surface structures for femtosecond laser irradiation of silicon,” J. Appl. Phys. 108(3), 034903 (2010).
[Crossref]

2009 (2)

A. Yan, L. Liu, Y. Zhi, D. Liu, and J. Sun, “Bragg diffraction of multilayer volume holographic gratings under ultrashort laser pulse readout,” J. Opt. Soc. Am. A 26(1), 135–141 (2009).
[Crossref] [PubMed]

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. B 96(1), 71–77 (2009).
[Crossref]

2007 (3)

2005 (1)

2003 (2)

2001 (1)

R. K. Shelton, L. S. Ma, H. C. Kapteyn, M. M. Murnane, J. L. Hall, and J. Ye, “Phase-coherent optical pulse synthesis from separate femtosecond lasers,” Science 293(5533), 1286–1289 (2001).
[Crossref] [PubMed]

2000 (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[Crossref]

1995 (1)

1994 (1)

R. De Vré and L. Hesselink, “Analysis of photorefractive stratified volume holographic optical elements,” J. Soc. Opt. Am. B 11(9), 1800–1808 (1994).
[Crossref]

1993 (1)

1992 (1)

1988 (1)

1980 (1)

A. P. Yakimovich, “Multilayer three-dimensional holographic gratings,” Opt. Spectrosc. 49, 85–88 (1980).

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Barberoglou, M.

M. Barberoglou, G. D. Tsibidis, D. Gray, E. Magoulakis, C. Fotakis, E. Stratakis, and P. A. Loukakos, “The influence of ultra-fast temporal energy regulation on the morphology of Si surfaces through femtosecond double pulse laser irradiation,” Appl. Phys., A Mater. Sci. Process. 113(2), 273–283 (2013).
[Crossref]

Bartels, A.

A. Ehlers, I. Riemann, S. Martin, R. Le Harzic, A. Bartels, C. Janke, and K. Konig, “High (1GHz) repetition rate compact femtosecond laser: A powerful multiphoton tool for nanomedicine and nanobiotechnology,” J. Appl. Phys. 102(1), 014701 (2007).
[Crossref]

Bergquist, J. C.

Bize, S.

Bonse, J.

T. J.-Y. Derrien, J. Krüger, T. E. Itina, S. Höhm, A. Rosenfeld, and J. Bonse, “Rippled area formed by surface plasmon polaritons upon femtosecond laser double-pulse irradiation of silicon,” Opt. Express 21(24), 29643–29655 (2013).
[Crossref] [PubMed]

S. Höhm, A. Rosenfeld, J. Krüger, and J. Bonse, “Area dependence of femtosecond laser-induced periodic surface structures for varying band gap materials after double pulse excitation,” Appl. Surf. Sci. 278, 7–12 (2013).
[Crossref]

J. Bonse and J. Kruger, “Pulse number dependence of laser-induced periodic surface structures for femtosecond laser irradiation of silicon,” J. Appl. Phys. 108(3), 034903 (2010).
[Crossref]

Chambers, D. M.

Che, W.

Chen, Y.

Dai, Y.

De Vré, R.

R. De Vré and L. Hesselink, “Analysis of photorefractive stratified volume holographic optical elements,” J. Soc. Opt. Am. B 11(9), 1800–1808 (1994).
[Crossref]

Derrien, T. J.-Y.

Diddams, S. A.

Ehlers, A.

A. Ehlers, I. Riemann, S. Martin, R. Le Harzic, A. Bartels, C. Janke, and K. Konig, “High (1GHz) repetition rate compact femtosecond laser: A powerful multiphoton tool for nanomedicine and nanobiotechnology,” J. Appl. Phys. 102(1), 014701 (2007).
[Crossref]

Fotakis, C.

M. Barberoglou, G. D. Tsibidis, D. Gray, E. Magoulakis, C. Fotakis, E. Stratakis, and P. A. Loukakos, “The influence of ultra-fast temporal energy regulation on the morphology of Si surfaces through femtosecond double pulse laser irradiation,” Appl. Phys., A Mater. Sci. Process. 113(2), 273–283 (2013).
[Crossref]

Fu, S.

L. Guo, A. Yan, and S. Fu, “Diffraction properties of multi-layer volume holographic gratings under an ultrashort pulsed beam with arbitrary temporal profiles,” Optik (Stuttg.) 122(19), 1692–1696 (2011).
[Crossref]

Gao, L.

Gao, Z.

Gaylord, T. K.

Glytsis, E. N.

Granger, A.

Gray, D.

M. Barberoglou, G. D. Tsibidis, D. Gray, E. Magoulakis, C. Fotakis, E. Stratakis, and P. A. Loukakos, “The influence of ultra-fast temporal energy regulation on the morphology of Si surfaces through femtosecond double pulse laser irradiation,” Appl. Phys., A Mater. Sci. Process. 113(2), 273–283 (2013).
[Crossref]

Guo, L.

L. Guo, A. Yan, and S. Fu, “Diffraction properties of multi-layer volume holographic gratings under an ultrashort pulsed beam with arbitrary temporal profiles,” Optik (Stuttg.) 122(19), 1692–1696 (2011).
[Crossref]

Guo, X.

Hall, J. L.

Han, B.

Han, F.

C. Yang, X. Yan, R. Zhu, H. Zou, and F. Han, “Diffraction study of volume holographic gratings in dispersive photorefractive material for femtosecond pulse readout,” Optik (Stuttg.) 121(12), 1138–1143 (2010).
[Crossref]

Hesselink, L.

R. De Vré and L. Hesselink, “Analysis of photorefractive stratified volume holographic optical elements,” J. Soc. Opt. Am. B 11(9), 1800–1808 (1994).
[Crossref]

Höhm, S.

S. Höhm, A. Rosenfeld, J. Krüger, and J. Bonse, “Area dependence of femtosecond laser-induced periodic surface structures for varying band gap materials after double pulse excitation,” Appl. Surf. Sci. 278, 7–12 (2013).
[Crossref]

T. J.-Y. Derrien, J. Krüger, T. E. Itina, S. Höhm, A. Rosenfeld, and J. Bonse, “Rippled area formed by surface plasmon polaritons upon femtosecond laser double-pulse irradiation of silicon,” Opt. Express 21(24), 29643–29655 (2013).
[Crossref] [PubMed]

Hollberg, L. W.

Holman, K. W.

Hu, J.

Itina, T. E.

Janke, C.

A. Ehlers, I. Riemann, S. Martin, R. Le Harzic, A. Bartels, C. Janke, and K. Konig, “High (1GHz) repetition rate compact femtosecond laser: A powerful multiphoton tool for nanomedicine and nanobiotechnology,” J. Appl. Phys. 102(1), 014701 (2007).
[Crossref]

Jiang, L.

Johnson, R. V.

Jones, D. J.

Jones, R. J.

Kapteyn, H. C.

R. K. Shelton, L. S. Ma, H. C. Kapteyn, M. M. Murnane, J. L. Hall, and J. Ye, “Phase-coherent optical pulse synthesis from separate femtosecond lasers,” Science 293(5533), 1286–1289 (2001).
[Crossref] [PubMed]

Kim, S.

Kitching, J.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Konig, K.

A. Ehlers, I. Riemann, S. Martin, R. Le Harzic, A. Bartels, C. Janke, and K. Konig, “High (1GHz) repetition rate compact femtosecond laser: A powerful multiphoton tool for nanomedicine and nanobiotechnology,” J. Appl. Phys. 102(1), 014701 (2007).
[Crossref]

Kruger, J.

J. Bonse and J. Kruger, “Pulse number dependence of laser-induced periodic surface structures for femtosecond laser irradiation of silicon,” J. Appl. Phys. 108(3), 034903 (2010).
[Crossref]

Krüger, J.

S. Höhm, A. Rosenfeld, J. Krüger, and J. Bonse, “Area dependence of femtosecond laser-induced periodic surface structures for varying band gap materials after double pulse excitation,” Appl. Surf. Sci. 278, 7–12 (2013).
[Crossref]

T. J.-Y. Derrien, J. Krüger, T. E. Itina, S. Höhm, A. Rosenfeld, and J. Bonse, “Rippled area formed by surface plasmon polaritons upon femtosecond laser double-pulse irradiation of silicon,” Opt. Express 21(24), 29643–29655 (2013).
[Crossref] [PubMed]

Le Harzic, R.

A. Ehlers, I. Riemann, S. Martin, R. Le Harzic, A. Bartels, C. Janke, and K. Konig, “High (1GHz) repetition rate compact femtosecond laser: A powerful multiphoton tool for nanomedicine and nanobiotechnology,” J. Appl. Phys. 102(1), 014701 (2007).
[Crossref]

Leng, N.

Lessard, R. A.

Li, C.

Li, X.

Liu, D.

A. Yan, L. Liu, Y. Zhi, D. Liu, and J. Sun, “Bragg diffraction of multilayer volume holographic gratings under ultrashort laser pulse readout,” J. Opt. Soc. Am. A 26(1), 135–141 (2009).
[Crossref] [PubMed]

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. B 96(1), 71–77 (2009).
[Crossref]

Liu, 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. B 96(1), 71–77 (2009).
[Crossref]

A. Yan, L. Liu, Y. Zhi, D. Liu, and J. Sun, “Bragg diffraction of multilayer volume holographic gratings under ultrashort laser pulse readout,” J. Opt. Soc. Am. A 26(1), 135–141 (2009).
[Crossref] [PubMed]

Liu, P.

Loukakos, P. A.

M. Barberoglou, G. D. Tsibidis, D. Gray, E. Magoulakis, C. Fotakis, E. Stratakis, and P. A. Loukakos, “The influence of ultra-fast temporal energy regulation on the morphology of Si surfaces through femtosecond double pulse laser irradiation,” Appl. Phys., A Mater. Sci. Process. 113(2), 273–283 (2013).
[Crossref]

Lu, Y.

Ma, G.

Ma, L.

Ma, L. S.

R. K. Shelton, L. S. Ma, H. C. Kapteyn, M. M. Murnane, J. L. Hall, and J. Ye, “Phase-coherent optical pulse synthesis from separate femtosecond lasers,” Science 293(5533), 1286–1289 (2001).
[Crossref] [PubMed]

Magnusson, R.

Magoulakis, E.

M. Barberoglou, G. D. Tsibidis, D. Gray, E. Magoulakis, C. Fotakis, E. Stratakis, and P. A. Loukakos, “The influence of ultra-fast temporal energy regulation on the morphology of Si surfaces through femtosecond double pulse laser irradiation,” Appl. Phys., A Mater. Sci. Process. 113(2), 273–283 (2013).
[Crossref]

Martin, S.

A. Ehlers, I. Riemann, S. Martin, R. Le Harzic, A. Bartels, C. Janke, and K. Konig, “High (1GHz) repetition rate compact femtosecond laser: A powerful multiphoton tool for nanomedicine and nanobiotechnology,” J. Appl. Phys. 102(1), 014701 (2007).
[Crossref]

Murnane, M. M.

R. K. Shelton, L. S. Ma, H. C. Kapteyn, M. M. Murnane, J. L. Hall, and J. Ye, “Phase-coherent optical pulse synthesis from separate femtosecond lasers,” Science 293(5533), 1286–1289 (2001).
[Crossref] [PubMed]

Nordin, G. P.

Pen, E. F.

E. F. Pen and M. Yu. Rodionov, “Properties of multilayer nonuniform holographic structures,” Quantum Electron. 40(10), 919–924 (2010).
[Crossref]

Peng, J.

Qi, Y.

Qian, M.

Riemann, I.

A. Ehlers, I. Riemann, S. Martin, R. Le Harzic, A. Bartels, C. Janke, and K. Konig, “High (1GHz) repetition rate compact femtosecond laser: A powerful multiphoton tool for nanomedicine and nanobiotechnology,” J. Appl. Phys. 102(1), 014701 (2007).
[Crossref]

Robertsson, L.

Rodionov, M. Yu.

E. F. Pen and M. Yu. Rodionov, “Properties of multilayer nonuniform holographic structures,” Quantum Electron. 40(10), 919–924 (2010).
[Crossref]

Rosenfeld, A.

S. Höhm, A. Rosenfeld, J. Krüger, and J. Bonse, “Area dependence of femtosecond laser-induced periodic surface structures for varying band gap materials after double pulse excitation,” Appl. Surf. Sci. 278, 7–12 (2013).
[Crossref]

T. J.-Y. Derrien, J. Krüger, T. E. Itina, S. Höhm, A. Rosenfeld, and J. Bonse, “Rippled area formed by surface plasmon polaritons upon femtosecond laser double-pulse irradiation of silicon,” Opt. Express 21(24), 29643–29655 (2013).
[Crossref] [PubMed]

Shelton, R. K.

R. K. Shelton, L. S. Ma, H. C. Kapteyn, M. M. Murnane, J. L. Hall, and J. Ye, “Phase-coherent optical pulse synthesis from separate femtosecond lasers,” Science 293(5533), 1286–1289 (2001).
[Crossref] [PubMed]

Shi, X.

Song, L.

Stratakis, E.

M. Barberoglou, G. D. Tsibidis, D. Gray, E. Magoulakis, C. Fotakis, E. Stratakis, and P. A. Loukakos, “The influence of ultra-fast temporal energy regulation on the morphology of Si surfaces through femtosecond double pulse laser irradiation,” Appl. Phys., A Mater. Sci. Process. 113(2), 273–283 (2013).
[Crossref]

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. B 96(1), 71–77 (2009).
[Crossref]

A. Yan, L. Liu, Y. Zhi, D. Liu, and J. Sun, “Bragg diffraction of multilayer volume holographic gratings under ultrashort laser pulse readout,” J. Opt. Soc. Am. A 26(1), 135–141 (2009).
[Crossref] [PubMed]

Tanguay, A. R.

Tsibidis, G. D.

M. Barberoglou, G. D. Tsibidis, D. Gray, E. Magoulakis, C. Fotakis, E. Stratakis, and P. A. Loukakos, “The influence of ultra-fast temporal energy regulation on the morphology of Si surfaces through femtosecond double pulse laser irradiation,” Appl. Phys., A Mater. Sci. Process. 113(2), 273–283 (2013).
[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. B 96(1), 71–77 (2009).
[Crossref]

Wang, H.

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. B 96(1), 71–77 (2009).
[Crossref]

Wang, S. S.

Weiner, A. M.

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[Crossref]

Wu, S. D.

Wu, Y. M.

Xia, B.

Xiang, H.

Xiao, H.

Xu, C.

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Yan, A.

L. Guo, A. Yan, and S. Fu, “Diffraction properties of multi-layer volume holographic gratings under an ultrashort pulsed beam with arbitrary temporal profiles,” Optik (Stuttg.) 122(19), 1692–1696 (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. B 96(1), 71–77 (2009).
[Crossref]

A. Yan, L. Liu, Y. Zhi, D. Liu, and J. Sun, “Bragg diffraction of multilayer volume holographic gratings under ultrashort laser pulse readout,” J. Opt. Soc. Am. A 26(1), 135–141 (2009).
[Crossref] [PubMed]

Yan, X.

Yang, C.

C. Yang, X. Yan, R. Zhu, H. Zou, and F. Han, “Diffraction study of volume holographic gratings in dispersive photorefractive material for femtosecond pulse readout,” Optik (Stuttg.) 121(12), 1138–1143 (2010).
[Crossref]

Yang, D.

Yang, X.

Ye, J.

Zhang, G.

Zhang, N.

Zhao, J.

Zhi, Y.

Zhu, R.

C. Yang, X. Yan, R. Zhu, H. Zou, and F. Han, “Diffraction study of volume holographic gratings in dispersive photorefractive material for femtosecond pulse readout,” Optik (Stuttg.) 121(12), 1138–1143 (2010).
[Crossref]

Zou, H.

C. Yang, X. Yan, R. Zhu, H. Zou, and F. Han, “Diffraction study of volume holographic gratings in dispersive photorefractive material for femtosecond pulse readout,” Optik (Stuttg.) 121(12), 1138–1143 (2010).
[Crossref]

Appl. Opt. (4)

Appl. Phys. B (1)

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. B 96(1), 71–77 (2009).
[Crossref]

Appl. Phys., A Mater. Sci. Process. (1)

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[Crossref]

Appl. Surf. Sci. (1)

S. Höhm, A. Rosenfeld, J. Krüger, and J. Bonse, “Area dependence of femtosecond laser-induced periodic surface structures for varying band gap materials after double pulse excitation,” Appl. Surf. Sci. 278, 7–12 (2013).
[Crossref]

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[Crossref]

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[Crossref]

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J. Soc. Opt. Am. B (1)

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[Crossref]

Opt. Express (5)

Opt. Lett. (5)

Opt. Spectrosc. (1)

A. P. Yakimovich, “Multilayer three-dimensional holographic gratings,” Opt. Spectrosc. 49, 85–88 (1980).

Optik (Stuttg.) (2)

L. Guo, A. Yan, and S. Fu, “Diffraction properties of multi-layer volume holographic gratings under an ultrashort pulsed beam with arbitrary temporal profiles,” Optik (Stuttg.) 122(19), 1692–1696 (2011).
[Crossref]

C. Yang, X. Yan, R. Zhu, H. Zou, and F. Han, “Diffraction study of volume holographic gratings in dispersive photorefractive material for femtosecond pulse readout,” Optik (Stuttg.) 121(12), 1138–1143 (2010).
[Crossref]

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E. F. Pen and M. Yu. Rodionov, “Properties of multilayer nonuniform holographic structures,” Quantum Electron. 40(10), 919–924 (2010).
[Crossref]

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[Crossref]

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R. K. Shelton, L. S. Ma, H. C. Kapteyn, M. M. Murnane, J. L. Hall, and J. Ye, “Phase-coherent optical pulse synthesis from separate femtosecond lasers,” Science 293(5533), 1286–1289 (2001).
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J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale (Elsevier Inc. 2006).

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Figures (10)

Fig. 1
Fig. 1 Readout structure of a transmitted multilayer volume holographic grating (MVHG).
Fig. 2
Fig. 2 Temporal intensity distribution of the incident Gaussian pulse with duration of 100fs.
Fig. 3
Fig. 3 Temporal intensity distributions of the diffracted pulse when the number of grating layers is: (a) N = 1, 2, 3; (b) N = 4.
Fig. 4
Fig. 4 Temporal intensity distributions of the diffracted pulses in a 3-layer MVHG when thickness of (a) the first buffer layer changes in the range of 2mm to 10mm, and (b) the second buffer layer changes in the range of 2mm to 8mm, the other thickness keeps constant.
Fig. 5
Fig. 5 Temporal intensity distributions of the diffracted pulses in a 4-layer MVHG when the second buffer layer thickness changes in the range of 2mm to 8mm, whereas the thickness of the first and the third buffer layers holds constant at 8mm.
Fig. 6
Fig. 6 Pulse interval between adjacent two diffracted sub-pulses is in linearly proportional to the buffer layer thickness.
Fig. 7
Fig. 7 Temporal intensity distributions of the diffracted pulse in a 3-layer MVHG when thickness of (a): the first grating layer, (b): the second grating layer, (c): the third grating layer changes in the range of 1mm to 5mm, while thicknesses of other two grating layers keep constant, (d): the first and the third grating layers change simultaneously.
Fig. 8
Fig. 8 (a): Temporal intensity distributions of the diffracted pulse in a 4-layer MVHG when thickness of the fourth grating layer changes from 2mm to 4mm. (b): Pulse duration as a function of thickness of the fourth grating layer.
Fig. 9
Fig. 9 Summary of the influences of parameters of the MVHG on the diffracted pulse string.
Fig. 10
Fig. 10 Diffraction of a spectral component of the femtosecond pulse by m-layer MVHG.

Equations (17)

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R= e iξ [cos ξ 2 + ν 2 + iξ ξ 2 + ν 2 sin ξ 2 + ν 2 ],
S=i e iξ C R C S ν sin ξ 2 + ν 2 ξ 2 + ν 2 .
ν= π T i Δn λ 0 C R C S ,ξ= T i δ 2 C S , C R =cos θ R , C S =cos θ R K β cos θ K ; δ=ΔθKsin( θ K θ 0 ) Δλ K 2 4π n 0 , θ 0 = | θ R θ S | 2 , θ K = π 2 ( θ 0 θ R ); β= 2π n 0 λ 0 ,n= n 0 +Δncos( K r ),2Λsin θ 0 = λ 0 n 0 ,| K |= 2π Λ ; θ= θ 0 +Δθ,λ= λ 0 +Δλ.
M i =[ m i11 m i12 m i21 m i22 ].
m i11 =exp(jξ)[cosα+ jξ α sinα], m i12 =j ν α C S C R sinαexp(jξ), m i21 =j ν α C R C S sinαexp(jξ), m i22 =exp(jξ)[cosα jξ α sinα].
[ D i ]=[ exp( i k br d i ) 0 0 exp( i k bd d i ) ]=exp(i k br d i )[ 1 0 0 exp(2iζ d i ) ].
[ R( T d ,ω) S( T d ,ω) ]=[ M c ]×[ R(0,ω) S(0,ω) ].
u 0 (t)=exp(i ω 0 t t 2 / T 2 ),
U 0 ( ω )= 1 2π u 0 ( t )exp( iωt )dt= T 2 π exp[ T 2 (ω ω 0 ) 2 4 ].
S( T d ,t )= S( T d ,ω )exp( iωt )dω.
I S ( T d ,t )= | S( T d ,t ) | 2 .
S( T d ,ω)=( M 222 M 121 e 2jζd + M 221 M 111 ) U 0 (0,ω),
s( T d ,t)= s 1 (t+d)+ s 2 (t).
S( T d ,ω)=( M 322 M 222 M 221 e 2jζ( d 1 + d 2 ) + M 321 M 212 M 221 e 2jζ d 1 + M 322 M 221 M 111 e 2jζ d 2 + M 321 M 211 M 111 ) U 0 (0,ω).
S( T d ,ω)=( M 322 M 222 M 221 e 2jζ( d 1 + d 2 ) + M 322 M 221 M 111 e 2jζ d 2 + M 321 M 211 M 111 ) U 0 (0,ω).
s( T d ,t)= s 1 (t+( d 2 + d 1 ))+ s 2 (t+ d 2 )+ s 3 (t).
s( T d ,t)= s 1 (t+( d m1 + d m2 +...+ d 2 + d 1 ))...+ s m2 (t+( d m1 + d m2 ))+ s m1 (t+ d m1 )+ s m (t).

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