Abstract

The process of generating periodic optical waveforms includes the generation and phasing of several harmonics of a fundamental frequency. In this work, we show that simultaneous generation and phasing of the harmonics can be performed in a monolithic aperiodic optical superlattice (AOS). Stable periodic waveforms can thus be delivered to a predetermined location by simply sending a laser beam through a properly designed and fabricated AOS crystal. A detailed mathematical description for generating the domain pattern in such an AOS crystal is given and the process is numerically demonstrated. The waveform that is generated from a monolithic AOS is highly reproducible and phase stable. We also use propagation in air as an example to show how any predictable phase and amplitude modifications such as air dispersion that will alter the desired waveform can be pre-compensated in the design phase of the AOS crystal.

© 2009 OSA

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  1. Z. Jiang, D. S. Seo, S.-D. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Four-User, 2.5-Gb/s, Spectrally Coded OCDMA System Demonstration Using Low-Power Nonlinear Processing,” J. Lightwave Technol. 23(1), 143–158 (2005).
    [CrossRef]
  2. M. C. Stowe, F. C. Cruz, A. Marian, and J. Ye, “High resolution atomic coherent control via spectral phase manipulation of an optical frequency comb,” Phys. Rev. Lett. 96(15), 153001 (2006).
    [CrossRef] [PubMed]
  3. A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
    [CrossRef] [PubMed]
  4. A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5(8), 1563–1572 (1988).
    [CrossRef]
  5. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25(1), 25–27 (2000).
    [CrossRef]
  6. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
    [CrossRef] [PubMed]
  7. S. E. Harris and A. V. Sokolov, “Subfemtosecond Pulse Generation by Molecular Modulation,” Phys. Rev. Lett. 81(14), 2894–2897 (1998).
    [CrossRef]
  8. T. W. Hänsch, “A proposed sub-femtosecond pulse synthesizer using separate phase-locked laser oscillators,” Opt. Commun. 80(1), 71–75 (1990).
    [CrossRef]
  9. I. V. Shutov and A. S. Chirkin, “Consecutive high-order harmonic generation and formation of subfemtosecond light pulses in aperiodical nonlinear photonic crystals,” Phys. Rev. A 78(1), 013827 (2008).
    [CrossRef]
  10. M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
    [CrossRef] [PubMed]
  11. W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  15. D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical Properties of Lithium-Rich Lithium Niobate Fabricated by Vapor Transport Equilibration,” IEEE J. Quantum Electron. 26(1), 135–138 (1990).
    [CrossRef]
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    [CrossRef]

2008 (2)

I. V. Shutov and A. S. Chirkin, “Consecutive high-order harmonic generation and formation of subfemtosecond light pulses in aperiodical nonlinear photonic crystals,” Phys. Rev. A 78(1), 013827 (2008).
[CrossRef]

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

2007 (1)

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[CrossRef]

2006 (1)

M. C. Stowe, F. C. Cruz, A. Marian, and J. Ye, “High resolution atomic coherent control via spectral phase manipulation of an optical frequency comb,” Phys. Rev. Lett. 96(15), 153001 (2006).
[CrossRef] [PubMed]

2005 (2)

2003 (1)

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

2000 (1)

1998 (1)

S. E. Harris and A. V. Sokolov, “Subfemtosecond Pulse Generation by Molecular Modulation,” Phys. Rev. Lett. 81(14), 2894–2897 (1998).
[CrossRef]

1997 (1)

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi–Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science 278(5339), 843–846 (1997).
[CrossRef]

1993 (1)

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[CrossRef] [PubMed]

1990 (2)

T. W. Hänsch, “A proposed sub-femtosecond pulse synthesizer using separate phase-locked laser oscillators,” Opt. Commun. 80(1), 71–75 (1990).
[CrossRef]

D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical Properties of Lithium-Rich Lithium Niobate Fabricated by Vapor Transport Equilibration,” IEEE J. Quantum Electron. 26(1), 135–138 (1990).
[CrossRef]

1988 (1)

1972 (1)

Baltuska, A.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

Byer, R. L.

D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical Properties of Lithium-Rich Lithium Niobate Fabricated by Vapor Transport Equilibration,” IEEE J. Quantum Electron. 26(1), 135–138 (1990).
[CrossRef]

Chen, W. J.

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

Chirkin, A. S.

I. V. Shutov and A. S. Chirkin, “Consecutive high-order harmonic generation and formation of subfemtosecond light pulses in aperiodical nonlinear photonic crystals,” Phys. Rev. A 78(1), 013827 (2008).
[CrossRef]

Corkum, P. B.

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[CrossRef] [PubMed]

Cruz, F. C.

M. C. Stowe, F. C. Cruz, A. Marian, and J. Ye, “High resolution atomic coherent control via spectral phase manipulation of an optical frequency comb,” Phys. Rev. Lett. 96(15), 153001 (2006).
[CrossRef] [PubMed]

Fejer, M. M.

Z. Jiang, D. S. Seo, S.-D. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Four-User, 2.5-Gb/s, Spectrally Coded OCDMA System Demonstration Using Low-Power Nonlinear Processing,” J. Lightwave Technol. 23(1), 143–158 (2005).
[CrossRef]

D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical Properties of Lithium-Rich Lithium Niobate Fabricated by Vapor Transport Equilibration,” IEEE J. Quantum Electron. 26(1), 135–138 (1990).
[CrossRef]

Gohle, Ch.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

Goulielmakis, E.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

Hänsch, T. W.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

T. W. Hänsch, “A proposed sub-femtosecond pulse synthesizer using separate phase-locked laser oscillators,” Opt. Commun. 80(1), 71–75 (1990).
[CrossRef]

Harris, S. E.

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[CrossRef] [PubMed]

S. E. Harris and A. V. Sokolov, “Subfemtosecond Pulse Generation by Molecular Modulation,” Phys. Rev. Lett. 81(14), 2894–2897 (1998).
[CrossRef]

Hentschel, M.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

Heritage, J. P.

Holzwarth, R.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

Hsieh, Z. M.

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

Huang, C.-B.

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[CrossRef]

Huang, S. W.

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

Jiang, Z.

Jundt, D. H.

D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical Properties of Lithium-Rich Lithium Niobate Fabricated by Vapor Transport Equilibration,” IEEE J. Quantum Electron. 26(1), 135–138 (1990).
[CrossRef]

Kirschner, E. M.

Krausz, F.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

Kung, A. H.

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

Lai, C. J.

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

Langrock, C.

Leaird, D. E.

Lee, C. K.

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

Lin, C. H.

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

Marian, A.

M. C. Stowe, F. C. Cruz, A. Marian, and J. Ye, “High resolution atomic coherent control via spectral phase manipulation of an optical frequency comb,” Phys. Rev. Lett. 96(15), 153001 (2006).
[CrossRef] [PubMed]

Ming, N. B.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi–Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science 278(5339), 843–846 (1997).
[CrossRef]

Pan, C. L.

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

Pan, R. P.

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

Peck, E.

Ranka, J. K.

Reeder, K.

Roussev, R. V.

Scrinzi, A.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

Seo, D. S.

Shutov, I. V.

I. V. Shutov and A. S. Chirkin, “Consecutive high-order harmonic generation and formation of subfemtosecond light pulses in aperiodical nonlinear photonic crystals,” Phys. Rev. A 78(1), 013827 (2008).
[CrossRef]

Shverdin, M. Y.

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[CrossRef] [PubMed]

Sokolov, A. V.

S. E. Harris and A. V. Sokolov, “Subfemtosecond Pulse Generation by Molecular Modulation,” Phys. Rev. Lett. 81(14), 2894–2897 (1998).
[CrossRef]

Stentz, A. J.

Stowe, M. C.

M. C. Stowe, F. C. Cruz, A. Marian, and J. Ye, “High resolution atomic coherent control via spectral phase manipulation of an optical frequency comb,” Phys. Rev. Lett. 96(15), 153001 (2006).
[CrossRef] [PubMed]

Su, H. Y.

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

Tang, T. T.

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

Udem, Th.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

Uiberacker, M.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

Walker, D. R.

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[CrossRef] [PubMed]

Weiner, A. M.

Windeler, R. S.

Yakovlev, V. S.

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

Yang, S.-D.

Yavuz, D. D.

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[CrossRef] [PubMed]

Ye, J.

M. C. Stowe, F. C. Cruz, A. Marian, and J. Ye, “High resolution atomic coherent control via spectral phase manipulation of an optical frequency comb,” Phys. Rev. Lett. 96(15), 153001 (2006).
[CrossRef] [PubMed]

Yin, G. Y.

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[CrossRef] [PubMed]

Zhu, S. N.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi–Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science 278(5339), 843–846 (1997).
[CrossRef]

Zhu, Y. Y.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi–Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science 278(5339), 843–846 (1997).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical Properties of Lithium-Rich Lithium Niobate Fabricated by Vapor Transport Equilibration,” IEEE J. Quantum Electron. 26(1), 135–138 (1990).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

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

Nat. Photonics (1)

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[CrossRef]

Nature (1)

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

T. W. Hänsch, “A proposed sub-femtosecond pulse synthesizer using separate phase-locked laser oscillators,” Opt. Commun. 80(1), 71–75 (1990).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

I. V. Shutov and A. S. Chirkin, “Consecutive high-order harmonic generation and formation of subfemtosecond light pulses in aperiodical nonlinear photonic crystals,” Phys. Rev. A 78(1), 013827 (2008).
[CrossRef]

Phys. Rev. Lett. (5)

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[CrossRef] [PubMed]

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[CrossRef] [PubMed]

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[CrossRef] [PubMed]

S. E. Harris and A. V. Sokolov, “Subfemtosecond Pulse Generation by Molecular Modulation,” Phys. Rev. Lett. 81(14), 2894–2897 (1998).
[CrossRef]

M. C. Stowe, F. C. Cruz, A. Marian, and J. Ye, “High resolution atomic coherent control via spectral phase manipulation of an optical frequency comb,” Phys. Rev. Lett. 96(15), 153001 (2006).
[CrossRef] [PubMed]

Science (1)

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi–Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science 278(5339), 843–846 (1997).
[CrossRef]

Other (1)

N. Bloembergen, Nonlinear Optics (World Scientific, 1996).

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Figures (4)

Fig. 1
Fig. 1

(left frame) Absolute values of the normalized effective second order nonlinear coefficients |d˜(Δk)| calculated from Eq. (4) of the designed aperiodic optical superlattice and (right frame) simulation results of the electric field evolution in the designed AOS.

Fig. 2
Fig. 2

(large frames) Simulation results of the phase evolution inside the designed AOS and (small frames) the corresponding waveforms (solid curve) and envelopes (dotted curve). (a) For AOS generation parameters of phase ζ(211)=ζ(321)=ζ(431)=ζ(541)=0 , simulation results agree with expected values φCE=0.5π and φm=0.5π .For ζ(211)=ζ(321)=ζ(431)=ζ(541)=0.5π , 1.0π and 1.5 π from (b) to (d), the expected values are (b) φCE=0 , φm=0 , (c) φCE=0.5π , φm=0.5π , (d) φCE=1.0π , φm=1.0π , respectively. In a stable periodic waveform, φCE affects the shape of the waveform, and φm produces a timeshift. This is confirmed with the results shown in the small frame figures on the right where the envelopes shift sequentially by 1/4 period from (a) to (d), which is due to the 0.5π phase shift of φm from one figure to the next.

Fig. 3
Fig. 3

Generated electromagnetic waveform (solid curve) and the corresponding envelope (dotted curve) which are (a) at the crystal exit and (b) at 1 m away from the crystal exit. The generation amplitude parameters are set to a(211)=1.000 , a(321)=0.996 , a(431)=1.174 , a(541)=1.294 , which are the same as those for calculating Fig. 2. The generation phase parameters are set to ζ(211)=1.332π , ζ(321)=0.030π , ζ(431)=0.186π , ζ(541)=1.942π to compensate the phase distortion due to dispersion in air.

Fig. 4
Fig. 4

Plots of simulated waveforms (solid curve) and envelopes (dotted curve). The generation amplitude parameters are set to a(211)=1.000 , a(321)=2.188 , a(532)=12.119 , a(752)=21.416 and the generation phase parameters are set to ζ(211)=ζ(321)=ζ(431)=ζ(541)=0 , 0.25π, 0.5π, 0.75π from (a) to (d) respectively. From Eq. (8) and with φ1=0 , it can be predicted that φCE will change from 0.5π to −0.25π. The expected values of the CEP are (a) φCE=0.5π , φm=0.5π . (b) φCE=0.25π , φm=0.25π . (c) φCE=0 , φm=0 . (d) φCE=0.25π , φm=0.25π , which agree with above waveforms.

Tables (1)

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Table 1 Phase mismatch of selected processesa

Equations (8)

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φn=φCE+nφm,
Erz=iωp+ωq=ωrωrd(2)(z)cnrEpEqexp(iΔk(rpq)z),
Erz=i(rpq)ωrcnrd˜(rpq)EpEq,
d˜(rpq)d˜(Δk(rpq))=1Ld(2)(z)exp(iΔk(rpq)z)dz.
φn(final)=φn+ϕn=π2+φd˜(n(n1)1)+φn1+φ1+ϕn,φ1(final)=φ1+ϕ1,
π2φd˜(211)+2ϕ1ϕ2=π2φd˜(321)+ϕ1+ϕ2ϕ3=,
d(2)(z)sgn((rpq)Na(rpq)sin(Δk(rpq)z+ζ(rpq))),
d(2)(z)sgn(z/l1lz/ll(rpq)Na(rpq)cos(Δk(rpq)z'+ζ(rpq))dz'),

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