Abstract

We show that a sawtooth phase-modulation is the optimal profile for grating assisted phase matching (GAPM). Perfect (sharp) sawtooth modulation fully corrects the phase-mismatch, exhibiting conversion equal to conventional phase matching, while smoothened, approximate sawtooth structures are more efficient than sinusoidal or square GAPM modulations that were previously studied. As an example, we demonstrate numerically optically-induced sawtooth GAPM for high harmonic generation. Sawtooth GAPM is the most efficient method for increasing the conversion efficiency of high harmonic generation through quasi-phase-matching, with an ultimate efficiency that closely matches the ideal phase-matching case.

© 2010 OSA

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  1. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
    [CrossRef]
  2. S. Somekh and A. Yariv, “Phase-matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972).
    [CrossRef]
  3. J. P. van der Ziel, M. Ilegems, P. W. Foy, and R. M. Mikulyak, “Phase-Matched Second-Harmonic Generation in a Periodic GaAs Waveguide,” Appl. Phys. Lett. 29(12), 775–777 (1976).
    [CrossRef]
  4. J. Khurgin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism of efficient blue second harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57(24), 2540–2542 (1990).
    [CrossRef]
  5. A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasiphase matching conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(4 Pt 2), 046609 (2001).
    [CrossRef] [PubMed]
  6. O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-assisted phase matching in extreme nonlinear optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
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    [CrossRef]
  9. C. Serrat and J. Biegert, “All-regions tunable high harmonic enhancement by a periodic static electric field,” Phys. Rev. Lett. 104(7), 073901 (2010).
    [CrossRef] [PubMed]
  10. A. Bahabad, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching and dispersion characterization of harmonic generation in the perturbative regime using counterpropagating beams,” Opt. Express 16(20), 15923–15931 (2008).
    [CrossRef] [PubMed]
  11. T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
    [CrossRef] [PubMed]
  12. A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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  15. M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-X-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99(14), 143901 (2007).
    [CrossRef] [PubMed]
  16. O. Cohen, A. L. Lytle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, “Optimizing quasi-phase matching of high harmonic generation using counterpropagating pulse trains,” Opt. Lett. 32(20), 2975–2977 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]
  19. P. Appel, “Sur l'inversion approchée de certaines intégrales réelles et sur l'extension de Kepler et des fonctions de Bessel,” Acad. Sci., Paris, C. R. 160, 419 (1915).
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    [CrossRef]
  22. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191 (1986).
  23. M. Y. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A 54(1), 742–745 (1996).
    [CrossRef] [PubMed]
  24. A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi Phase Matching of Momentum and Energy in Nonlinear Optical Processes,” Nat. Photonics 4(8), 571 (2010).
    [CrossRef]

2010 (3)

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi Phase Matching of Momentum and Energy in Nonlinear Optical Processes,” Nat. Photonics 4(8), 571 (2010).
[CrossRef]

D. Faccio, C. Serrat, J. M. Cela, A. Farrés, P. Di Trapani, and J. Biegert, “Modulated phase matching and high-order harmonic enhancement mediated by the carrier-envelope phase,” Phys. Rev. A 81, 011803 (2010).
[CrossRef]

C. Serrat and J. Biegert, “All-regions tunable high harmonic enhancement by a periodic static electric field,” Phys. Rev. Lett. 104(7), 073901 (2010).
[CrossRef] [PubMed]

2009 (1)

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (5)

O. Cohen, A. L. Lytle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, “Optimizing quasi-phase matching of high harmonic generation using counterpropagating pulse trains,” Opt. Lett. 32(20), 2975–2977 (2007).
[CrossRef] [PubMed]

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-assisted phase matching in extreme nonlinear optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[CrossRef]

J. Seres, V. Yakovlev, J. Seres, C. Streli, P. Wobrauschek, C. Spielmann, and F. Krausz, “Coherent superposition of laser-driven soft-x-ray harmonics from successive sources,” Nat. Phys. 3(12), 878–883 (2007).
[CrossRef]

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-X-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99(14), 143901 (2007).
[CrossRef] [PubMed]

2003 (1)

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
[CrossRef] [PubMed]

2001 (1)

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasiphase matching conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(4 Pt 2), 046609 (2001).
[CrossRef] [PubMed]

2000 (1)

M. Geissler, G. Tempea, and T. Brabec, “Phase-matched high-order harmonic generation in the nonadiabatic limit,” Phys. Rev. A 62(3), 033817 (2000).
[CrossRef]

1998 (1)

Z. Chang, A. Rundquist, H. Wang, I. Christov, H. C. Kapteyn, and M. M. Murnane, “Temporal phase control of soft-x-ray harmonic emission,” Phys. Rev. A 58(1), R30–R33 (1998).
[CrossRef]

1996 (1)

M. Y. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A 54(1), 742–745 (1996).
[CrossRef] [PubMed]

1995 (1)

M. Lewenstein, P. Salières, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A 52(6), 4747–4754 (1995).
[CrossRef] [PubMed]

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

1990 (1)

J. Khurgin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism of efficient blue second harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57(24), 2540–2542 (1990).
[CrossRef]

1986 (1)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191 (1986).

1976 (1)

J. P. van der Ziel, M. Ilegems, P. W. Foy, and R. M. Mikulyak, “Phase-Matched Second-Harmonic Generation in a Periodic GaAs Waveguide,” Appl. Phys. Lett. 29(12), 775–777 (1976).
[CrossRef]

1972 (1)

S. Somekh and A. Yariv, “Phase-matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972).
[CrossRef]

1915 (1)

P. Appel, “Sur l'inversion approchée de certaines intégrales réelles et sur l'extension de Kepler et des fonctions de Bessel,” Acad. Sci., Paris, C. R. 160, 419 (1915).

Ammosov, M. V.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191 (1986).

Appel, P.

P. Appel, “Sur l'inversion approchée de certaines intégrales réelles et sur l'extension de Kepler et des fonctions de Bessel,” Acad. Sci., Paris, C. R. 160, 419 (1915).

Backus, S.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
[CrossRef] [PubMed]

Bahabad, A.

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi Phase Matching of Momentum and Energy in Nonlinear Optical Processes,” Nat. Photonics 4(8), 571 (2010).
[CrossRef]

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
[CrossRef] [PubMed]

A. Bahabad, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching and dispersion characterization of harmonic generation in the perturbative regime using counterpropagating beams,” Opt. Express 16(20), 15923–15931 (2008).
[CrossRef] [PubMed]

Balakin, A. V.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasiphase matching conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(4 Pt 2), 046609 (2001).
[CrossRef] [PubMed]

Bartels, R. A.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
[CrossRef] [PubMed]

Bhargava, R. N.

J. Khurgin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism of efficient blue second harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57(24), 2540–2542 (1990).
[CrossRef]

Biegert, J.

D. Faccio, C. Serrat, J. M. Cela, A. Farrés, P. Di Trapani, and J. Biegert, “Modulated phase matching and high-order harmonic enhancement mediated by the carrier-envelope phase,” Phys. Rev. A 81, 011803 (2010).
[CrossRef]

C. Serrat and J. Biegert, “All-regions tunable high harmonic enhancement by a periodic static electric field,” Phys. Rev. Lett. 104(7), 073901 (2010).
[CrossRef] [PubMed]

Brabec, T.

M. Geissler, G. Tempea, and T. Brabec, “Phase-matched high-order harmonic generation in the nonadiabatic limit,” Phys. Rev. A 62(3), 033817 (2000).
[CrossRef]

M. Y. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A 54(1), 742–745 (1996).
[CrossRef] [PubMed]

Burnett, N.

M. Y. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A 54(1), 742–745 (1996).
[CrossRef] [PubMed]

Bushuev, V. A.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasiphase matching conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(4 Pt 2), 046609 (2001).
[CrossRef] [PubMed]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Cela, J. M.

D. Faccio, C. Serrat, J. M. Cela, A. Farrés, P. Di Trapani, and J. Biegert, “Modulated phase matching and high-order harmonic enhancement mediated by the carrier-envelope phase,” Phys. Rev. A 81, 011803 (2010).
[CrossRef]

Chang, Z.

Z. Chang, A. Rundquist, H. Wang, I. Christov, H. C. Kapteyn, and M. M. Murnane, “Temporal phase control of soft-x-ray harmonic emission,” Phys. Rev. A 58(1), R30–R33 (1998).
[CrossRef]

Chen, J.

Chen, M.-C.

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
[CrossRef] [PubMed]

Christov, I.

Z. Chang, A. Rundquist, H. Wang, I. Christov, H. C. Kapteyn, and M. M. Murnane, “Temporal phase control of soft-x-ray harmonic emission,” Phys. Rev. A 58(1), R30–R33 (1998).
[CrossRef]

Christov, I. P.

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
[CrossRef] [PubMed]

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
[CrossRef] [PubMed]

Cohen, O.

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
[CrossRef] [PubMed]

A. Bahabad, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching and dispersion characterization of harmonic generation in the perturbative regime using counterpropagating beams,” Opt. Express 16(20), 15923–15931 (2008).
[CrossRef] [PubMed]

O. Cohen, A. L. Lytle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, “Optimizing quasi-phase matching of high harmonic generation using counterpropagating pulse trains,” Opt. Lett. 32(20), 2975–2977 (2007).
[CrossRef] [PubMed]

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-assisted phase matching in extreme nonlinear optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[CrossRef]

Colak, S.

J. Khurgin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism of efficient blue second harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57(24), 2540–2542 (1990).
[CrossRef]

Delone, N. B.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191 (1986).

Di Trapani, P.

D. Faccio, C. Serrat, J. M. Cela, A. Farrés, P. Di Trapani, and J. Biegert, “Modulated phase matching and high-order harmonic enhancement mediated by the carrier-envelope phase,” Phys. Rev. A 81, 011803 (2010).
[CrossRef]

Dromey, B.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-X-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99(14), 143901 (2007).
[CrossRef] [PubMed]

Faccio, D.

D. Faccio, C. Serrat, J. M. Cela, A. Farrés, P. Di Trapani, and J. Biegert, “Modulated phase matching and high-order harmonic enhancement mediated by the carrier-envelope phase,” Phys. Rev. A 81, 011803 (2010).
[CrossRef]

Farrés, A.

D. Faccio, C. Serrat, J. M. Cela, A. Farrés, P. Di Trapani, and J. Biegert, “Modulated phase matching and high-order harmonic enhancement mediated by the carrier-envelope phase,” Phys. Rev. A 81, 011803 (2010).
[CrossRef]

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Foster, P.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-X-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99(14), 143901 (2007).
[CrossRef] [PubMed]

Foy, P. W.

J. P. van der Ziel, M. Ilegems, P. W. Foy, and R. M. Mikulyak, “Phase-Matched Second-Harmonic Generation in a Periodic GaAs Waveguide,” Appl. Phys. Lett. 29(12), 775–777 (1976).
[CrossRef]

Geissler, M.

M. Geissler, G. Tempea, and T. Brabec, “Phase-matched high-order harmonic generation in the nonadiabatic limit,” Phys. Rev. A 62(3), 033817 (2000).
[CrossRef]

Gerrity, M.

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
[CrossRef] [PubMed]

Green, H.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
[CrossRef] [PubMed]

Hooker, S. M.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-X-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99(14), 143901 (2007).
[CrossRef] [PubMed]

Ilegems, M.

J. P. van der Ziel, M. Ilegems, P. W. Foy, and R. M. Mikulyak, “Phase-Matched Second-Harmonic Generation in a Periodic GaAs Waveguide,” Appl. Phys. Lett. 29(12), 775–777 (1976).
[CrossRef]

Ivanov, M. Y.

M. Y. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A 54(1), 742–745 (1996).
[CrossRef] [PubMed]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Kapteyn, H. C.

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi Phase Matching of Momentum and Energy in Nonlinear Optical Processes,” Nat. Photonics 4(8), 571 (2010).
[CrossRef]

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
[CrossRef] [PubMed]

A. Bahabad, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching and dispersion characterization of harmonic generation in the perturbative regime using counterpropagating beams,” Opt. Express 16(20), 15923–15931 (2008).
[CrossRef] [PubMed]

O. Cohen, A. L. Lytle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, “Optimizing quasi-phase matching of high harmonic generation using counterpropagating pulse trains,” Opt. Lett. 32(20), 2975–2977 (2007).
[CrossRef] [PubMed]

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-assisted phase matching in extreme nonlinear optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[CrossRef]

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
[CrossRef] [PubMed]

Z. Chang, A. Rundquist, H. Wang, I. Christov, H. C. Kapteyn, and M. M. Murnane, “Temporal phase control of soft-x-ray harmonic emission,” Phys. Rev. A 58(1), R30–R33 (1998).
[CrossRef]

Khurgin, J.

J. Khurgin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism of efficient blue second harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57(24), 2540–2542 (1990).
[CrossRef]

Krainov, V. P.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191 (1986).

Krausz, F.

J. Seres, V. Yakovlev, J. Seres, C. Streli, P. Wobrauschek, C. Spielmann, and F. Krausz, “Coherent superposition of laser-driven soft-x-ray harmonics from successive sources,” Nat. Phys. 3(12), 878–883 (2007).
[CrossRef]

L’Huillier, A.

M. Lewenstein, P. Salières, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A 52(6), 4747–4754 (1995).
[CrossRef] [PubMed]

Landreman, M.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-X-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99(14), 143901 (2007).
[CrossRef] [PubMed]

Lewenstein, M.

M. Lewenstein, P. Salières, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A 52(6), 4747–4754 (1995).
[CrossRef] [PubMed]

Lytle, A. L.

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-assisted phase matching in extreme nonlinear optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

O. Cohen, A. L. Lytle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, “Optimizing quasi-phase matching of high harmonic generation using counterpropagating pulse trains,” Opt. Lett. 32(20), 2975–2977 (2007).
[CrossRef] [PubMed]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Mantsyzov, B. I.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasiphase matching conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(4 Pt 2), 046609 (2001).
[CrossRef] [PubMed]

Masselin, P.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasiphase matching conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(4 Pt 2), 046609 (2001).
[CrossRef] [PubMed]

Midorikawa, K.

Mikulyak, R. M.

J. P. van der Ziel, M. Ilegems, P. W. Foy, and R. M. Mikulyak, “Phase-Matched Second-Harmonic Generation in a Periodic GaAs Waveguide,” Appl. Phys. Lett. 29(12), 775–777 (1976).
[CrossRef]

Mouret, G.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasiphase matching conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(4 Pt 2), 046609 (2001).
[CrossRef] [PubMed]

Murnane, M. M.

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi Phase Matching of Momentum and Energy in Nonlinear Optical Processes,” Nat. Photonics 4(8), 571 (2010).
[CrossRef]

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
[CrossRef] [PubMed]

A. Bahabad, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching and dispersion characterization of harmonic generation in the perturbative regime using counterpropagating beams,” Opt. Express 16(20), 15923–15931 (2008).
[CrossRef] [PubMed]

O. Cohen, A. L. Lytle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, “Optimizing quasi-phase matching of high harmonic generation using counterpropagating pulse trains,” Opt. Lett. 32(20), 2975–2977 (2007).
[CrossRef] [PubMed]

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-assisted phase matching in extreme nonlinear optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[CrossRef]

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
[CrossRef] [PubMed]

Z. Chang, A. Rundquist, H. Wang, I. Christov, H. C. Kapteyn, and M. M. Murnane, “Temporal phase control of soft-x-ray harmonic emission,” Phys. Rev. A 58(1), R30–R33 (1998).
[CrossRef]

Nazarkin, A.

Nold, J.

Nurhuda, M.

Ozheredov, I. A.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasiphase matching conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(4 Pt 2), 046609 (2001).
[CrossRef] [PubMed]

Paul, A.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
[CrossRef] [PubMed]

Petrov, E. V.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasiphase matching conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(4 Pt 2), 046609 (2001).
[CrossRef] [PubMed]

Popmintchev, T.

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
[CrossRef] [PubMed]

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[CrossRef]

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-assisted phase matching in extreme nonlinear optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

Ren, H.

Rundquist, A.

Z. Chang, A. Rundquist, H. Wang, I. Christov, H. C. Kapteyn, and M. M. Murnane, “Temporal phase control of soft-x-ray harmonic emission,” Phys. Rev. A 58(1), R30–R33 (1998).
[CrossRef]

Russell, P. S.

Salières, P.

M. Lewenstein, P. Salières, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A 52(6), 4747–4754 (1995).
[CrossRef] [PubMed]

Seres, J.

J. Seres, V. Yakovlev, J. Seres, C. Streli, P. Wobrauschek, C. Spielmann, and F. Krausz, “Coherent superposition of laser-driven soft-x-ray harmonics from successive sources,” Nat. Phys. 3(12), 878–883 (2007).
[CrossRef]

J. Seres, V. Yakovlev, J. Seres, C. Streli, P. Wobrauschek, C. Spielmann, and F. Krausz, “Coherent superposition of laser-driven soft-x-ray harmonics from successive sources,” Nat. Phys. 3(12), 878–883 (2007).
[CrossRef]

Serrat, C.

C. Serrat and J. Biegert, “All-regions tunable high harmonic enhancement by a periodic static electric field,” Phys. Rev. Lett. 104(7), 073901 (2010).
[CrossRef] [PubMed]

D. Faccio, C. Serrat, J. M. Cela, A. Farrés, P. Di Trapani, and J. Biegert, “Modulated phase matching and high-order harmonic enhancement mediated by the carrier-envelope phase,” Phys. Rev. A 81, 011803 (2010).
[CrossRef]

Shkurinov, A. P.

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasiphase matching conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(4 Pt 2), 046609 (2001).
[CrossRef] [PubMed]

Sidorenko, P.

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
[CrossRef] [PubMed]

Somekh, S.

S. Somekh and A. Yariv, “Phase-matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972).
[CrossRef]

Spielmann, C.

J. Seres, V. Yakovlev, J. Seres, C. Streli, P. Wobrauschek, C. Spielmann, and F. Krausz, “Coherent superposition of laser-driven soft-x-ray harmonics from successive sources,” Nat. Phys. 3(12), 878–883 (2007).
[CrossRef]

Stolzenberger, R.

J. Khurgin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism of efficient blue second harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57(24), 2540–2542 (1990).
[CrossRef]

Streli, C.

J. Seres, V. Yakovlev, J. Seres, C. Streli, P. Wobrauschek, C. Spielmann, and F. Krausz, “Coherent superposition of laser-driven soft-x-ray harmonics from successive sources,” Nat. Phys. 3(12), 878–883 (2007).
[CrossRef]

Suda, A.

Takahashi, E. J.

Tempea, G.

M. Geissler, G. Tempea, and T. Brabec, “Phase-matched high-order harmonic generation in the nonadiabatic limit,” Phys. Rev. A 62(3), 033817 (2000).
[CrossRef]

Tobey, R.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
[CrossRef] [PubMed]

van der Ziel, J. P.

J. P. van der Ziel, M. Ilegems, P. W. Foy, and R. M. Mikulyak, “Phase-Matched Second-Harmonic Generation in a Periodic GaAs Waveguide,” Appl. Phys. Lett. 29(12), 775–777 (1976).
[CrossRef]

Wang, H.

Z. Chang, A. Rundquist, H. Wang, I. Christov, H. C. Kapteyn, and M. M. Murnane, “Temporal phase control of soft-x-ray harmonic emission,” Phys. Rev. A 58(1), R30–R33 (1998).
[CrossRef]

Weiman, S.

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
[CrossRef] [PubMed]

Wobrauschek, P.

J. Seres, V. Yakovlev, J. Seres, C. Streli, P. Wobrauschek, C. Spielmann, and F. Krausz, “Coherent superposition of laser-driven soft-x-ray harmonics from successive sources,” Nat. Phys. 3(12), 878–883 (2007).
[CrossRef]

Yakovlev, V.

J. Seres, V. Yakovlev, J. Seres, C. Streli, P. Wobrauschek, C. Spielmann, and F. Krausz, “Coherent superposition of laser-driven soft-x-ray harmonics from successive sources,” Nat. Phys. 3(12), 878–883 (2007).
[CrossRef]

Yariv, A.

S. Somekh and A. Yariv, “Phase-matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972).
[CrossRef]

Zepf, M.

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-X-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99(14), 143901 (2007).
[CrossRef] [PubMed]

Zhang, X.

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[CrossRef]

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-assisted phase matching in extreme nonlinear optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

O. Cohen, A. L. Lytle, X. Zhang, M. M. Murnane, and H. C. Kapteyn, “Optimizing quasi-phase matching of high harmonic generation using counterpropagating pulse trains,” Opt. Lett. 32(20), 2975–2977 (2007).
[CrossRef] [PubMed]

Zhou, X.

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[CrossRef]

Acad. Sci., Paris, C. R. (1)

P. Appel, “Sur l'inversion approchée de certaines intégrales réelles et sur l'extension de Kepler et des fonctions de Bessel,” Acad. Sci., Paris, C. R. 160, 419 (1915).

Appl. Phys. Lett. (3)

S. Somekh and A. Yariv, “Phase-matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972).
[CrossRef]

J. P. van der Ziel, M. Ilegems, P. W. Foy, and R. M. Mikulyak, “Phase-Matched Second-Harmonic Generation in a Periodic GaAs Waveguide,” Appl. Phys. Lett. 29(12), 775–777 (1976).
[CrossRef]

J. Khurgin, S. Colak, R. Stolzenberger, and R. N. Bhargava, “Mechanism of efficient blue second harmonic generation in periodically segmented waveguides,” Appl. Phys. Lett. 57(24), 2540–2542 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992).
[CrossRef]

Nat. Photonics (1)

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi Phase Matching of Momentum and Energy in Nonlinear Optical Processes,” Nat. Photonics 4(8), 571 (2010).
[CrossRef]

Nat. Phys. (2)

X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of high-harmonic generation using counterpropagating light,” Nat. Phys. 3(4), 270–275 (2007).
[CrossRef]

J. Seres, V. Yakovlev, J. Seres, C. Streli, P. Wobrauschek, C. Spielmann, and F. Krausz, “Coherent superposition of laser-driven soft-x-ray harmonics from successive sources,” Nat. Phys. 3(12), 878–883 (2007).
[CrossRef]

Nature (1)

A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M. Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation of coherent extreme-ultraviolet light,” Nature 421(6918), 51–54 (2003).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (5)

M. Y. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A 54(1), 742–745 (1996).
[CrossRef] [PubMed]

D. Faccio, C. Serrat, J. M. Cela, A. Farrés, P. Di Trapani, and J. Biegert, “Modulated phase matching and high-order harmonic enhancement mediated by the carrier-envelope phase,” Phys. Rev. A 81, 011803 (2010).
[CrossRef]

M. Geissler, G. Tempea, and T. Brabec, “Phase-matched high-order harmonic generation in the nonadiabatic limit,” Phys. Rev. A 62(3), 033817 (2000).
[CrossRef]

M. Lewenstein, P. Salières, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A 52(6), 4747–4754 (1995).
[CrossRef] [PubMed]

Z. Chang, A. Rundquist, H. Wang, I. Christov, H. C. Kapteyn, and M. M. Murnane, “Temporal phase control of soft-x-ray harmonic emission,” Phys. Rev. A 58(1), R30–R33 (1998).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

A. V. Balakin, V. A. Bushuev, B. I. Mantsyzov, I. A. Ozheredov, E. V. Petrov, A. P. Shkurinov, P. Masselin, and G. Mouret, “Enhancement of sum frequency generation near the photonic band gap edge under the quasiphase matching conditions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 63(4 Pt 2), 046609 (2001).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-assisted phase matching in extreme nonlinear optics,” Phys. Rev. Lett. 99(5), 053902 (2007).
[CrossRef] [PubMed]

M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasi-phase-matched soft-X-ray harmonic radiation from argon ions,” Phys. Rev. Lett. 99(14), 143901 (2007).
[CrossRef] [PubMed]

C. Serrat and J. Biegert, “All-regions tunable high harmonic enhancement by a periodic static electric field,” Phys. Rev. Lett. 104(7), 073901 (2010).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard X-ray regions of the spectrum,” Proc. Natl. Acad. Sci. U.S.A. 106(26), 10516–10521 (2009).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191 (1986).

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Figures (3)

Fig. 1
Fig. 1

Sawtooth GAPM. (a) Under phase-mismatch conditions, the shift between the phases of the driving and harmonic fields (ΔΦ0) grows linearly, leading to oscillations in the harmonic signal. (b) a sawtooth phase-shift (ΔΦGAPM) for correcting the phase mismatch of the nonlinear process. The sharp sawtooth (solid line) wave consists of infinite number of Fourier components. Also plotted is a smooth sawtooth wave that consists of the first three Fourier components (dashed line). (c) The combination of the medium phase-mismatch and the sawtooth phase-shift ΔΦ0 + ΔΦ results in a stairs-function phase (d) Harmonic field generated along 8LC propagation distance with N-waves sawtooth GAPM. (e) Optimal values of the amplitudes of the waves in N-waves sawtooth GAPM. Shown are the differences between their upper limits, 2/m, and the optimal values. (f) The QPM efficiency factor (intensity conversion efficiency normalized by the conversion efficiency at perfect phase-matching) of N-waves sawtooth GAPM. The conversion efficiency of 2 waves sawtooth GAPM is higher than the QPM efficiency factor of 1st order QPM where the polarization direction is flipped every coherence length (e.g., as in PPLN).

Fig. 2
Fig. 2

A scheme for optical induction of sawtooth GAPM in high harmonic generation. A strong driving laser pulse interferes with several weak Quasi-CW beams inside a planar waveguide, creating a sawtooth intensity pattern along the pulses propagation direction, z. The quasi-CW beams are produced using a mask that is perpendicular (a) or parallel (b) to the propagation direction of the driving laser.

Fig. 3
Fig. 3

Numerical results of sawtooth GAPM in HHG. (a) Amplitude of q-order harmonic phase oscillations as a function of the ratio between the peak fields of the strong laser pulse and a weak quasi–CW beam. (b) Harmonic spectral intensity with (solid red) and without (dash blue) 18-waves sawtooth GAPM. (c) Harmonic spectral intensity in the enhanced spectral range using 1-wave (solid black) and 18-waves (dash red) sawtooth GAPM (d) Harmonic signal versus propagation distance for 243-order harmonic for several phase matching conditions including: without GAPM (blue), 1-wave sawtooth GAPM (green), square GAPM (purple), 2-waves sawtooth GAPM (red), 3-waves sawtooth GAPM (brown), and 18-waves sawtooth GAPM (black). All curves are normalized by the phase-matched growth-rate (dash black).

Equations (6)

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

E H = E H 0 0 L exp [ i ( Δ Φ 0 + Δ Φ GAPM ) ] d z ,
Δ Φ GAPM = m = 1 N A m sin ( m π z / L C ) .
E H = E H 0 J 1 1 , 2 ... N ( A 1 , A 2 , ... A N ) ,
Δ Φ GAPM = m = 1 N A m sin ( 2 π z / Λ m ) ,
E D z = 1 2 c τ ω p 2 E D d τ 2 π I P E D c n e τ ,
E HHG z = 2 π c P HHG τ .

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