## Sawtooth grating-assisted phase-matching |

Optics Express, Vol. 18, Issue 22, pp. 22686-22692 (2010)

http://dx.doi.org/10.1364/OE.18.022686

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### 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

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]

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]

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]

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]

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). [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). [CrossRef] [PubMed]

8. 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]

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]

**28**(11), 2631–2654 (1992). [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]

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]

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). [CrossRef] [PubMed]

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]

**28**(11), 2631–2654 (1992). [CrossRef]

**28**(11), 2631–2654 (1992). [CrossRef]

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]

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]

**99**(5), 053902 (2007). [CrossRef] [PubMed]

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]

17. 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]

18. 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]

**28**(11), 2631–2654 (1992). [CrossRef]

**63**(4 Pt 2), 046609 (2001). [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]

**99**(5), 053902 (2007). [CrossRef] [PubMed]

8. 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]

**104**(7), 073901 (2010). [CrossRef] [PubMed]

*finite*series of sinusoidal waves is employed for the GAPM modulation. We show that the corresponding conversion efficiency increases when more waves form the sawtooth modulation structure. Notably, 2-wave sawtooth GAPM is already significantly more efficient than the sinusoidal and square GAPM structures that were previously investigated. As an example, we propose and analyze sawtooth GAPM in high harmonic generation where the phase-shift is induced by multiple weak quasi-CW waves. We demonstrate numerically that we can approach the ideal phase-matching case in a regime where perfect phase matching is otherwise impossible.

_{0}(z), which grows linearly along propagation axis, z, and an oscillating term, ΔΦ

_{GAPM}(z), which results from the GAPM modulation. The coherent buildup of the harmonic field is given by: where

_{0}(z) = πz/L

_{C}, where L

_{C}is the coherence length in the un-modulated medium (Fig. 1a). Figures 1b and 1c show that the optimal GAPM phase-shift is a sawtooth profile with a periodicity that corresponds to two coherence lengths and a slope that corresponds to −ΔΦ

_{0}(z). The combination of the medium phase-mismatch and the sawtooth phase-shift ΔΦ

_{0}+ ΔΦ results in a step-function phase-shift that leads to a linear growth of the HHG signal, in the same fashion as in true phase matching. That is, the QPM efficiency factor of sharp sawtooth GAPM is one.

_{m}= 2/m. The N = 1 and A = 1.84 case corresponds to the sinusoidal GAPM that was investigated previously [1

**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]

**99**(5), 053902 (2007). [CrossRef] [PubMed]

_{1}, A

_{2},…A

_{N}for which the N-dimensional generalized Bessel functions attain their maxima, and thus the N-waves sawtooth GAPM is most efficient, can be calculated quasi-analytically by decomposing the N-dimensional generalized Bessel function in terms of ordinary Bessel function [19]. The optimal values of A

_{1}, A

_{2},…A

_{N}are somewhat smaller than 2/m (Fig. 1e). The QPM efficiency factor of an N-wave sawtooth GAPM implementation is shown in Fig. 1f. As shown, the QPM efficiency factor of 2-wave sawtooth GAPM is approximately 0.52, which is already larger than the QPM efficiency factor for either sinusoidal (∼0.37) or square (∼0.4) GAPM that were investigated in previous works [1

**28**(11), 2631–2654 (1992). [CrossRef]

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]

17. 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]

18. 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]

**99**(5), 053902 (2007). [CrossRef] [PubMed]

**99**(5), 053902 (2007). [CrossRef] [PubMed]

**99**(5), 053902 (2007). [CrossRef] [PubMed]

20. J. Chen, A. Suda, E. J. Takahashi, M. Nurhuda, and K. Midorikawa, “Compression of intense ultrashort laser pulses in a gas-filled planar waveguide,” Opt. Lett. **33**(24), 2992–2994 (2008). [CrossRef] [PubMed]

*E*=

_{D}*E*

_{0}

*B*(

*y*)

*F*(

*x*,

*z*,

*t*)cos(

*ωt*–

*kz*) at angular frequency ω and wave-number k = ωn/c, where c is the velocity of light in vacuum, n is the effective index of refraction, E

_{0}is the peak electric field of the driving pulse F(x,z,t) is an envelope function and B(y) is the mode of the planar waveguide [B(y = 0) = 1]. It is assumed that the focal spot in x is wide such that the beam does not experience significant diffraction in x direction. The driving pulse interferes with m = 1,2…N quasi-CW fields that are given by

_{m}are propagation angles relative to z axis and r

_{m}<<1 are field ratio parameters. Transforming into a frame moving in the forward direction at phase velocity of the driving field c/n gives E

_{D}= E

_{0}B(y)F(x,z,τ)cos(ωτ) and

*= 2*

_{m}*π*/

*k*[1 – cos(

*θ*)]. The total intensity at the peak of the pulses,

_{m}_{0}+ ΔI(z) where

_{m}r

_{m′}are neglected). A GAPM phase-shift is induced by the CW waves since the intrinsic phase of HHG is proportional to the total intensity of the driving laser [17

17. 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]

18. 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]

_{m}(E

_{0},r

_{m}) are the amplitudes of the induced GAPM phase-shift sinusoidal components.

_{L}= 2×10

^{15}W/cm

^{2}, propagating in a medium that consists of pre-ionized Ar ions (second ionization potential is I

_{p}= 21 eV) at pressure P = 15 torr. In addition, multiple N weak quasi-CW laser fields at frequency ω propagate at angles θ

_{m}with respect to z. The quasi-CW fields are weak and therefore propagate linearly in the medium. The propagation of the strong driving pulse, E

_{D}(z,t,x = 0,y = 0), and harmonic fields are calculated using the model in Ref. 21

21. 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]

_{e}, takes into account the pre-formed plasma and the ionization that is calculated by using the ADK model [22]. The total optical field,

_{HHG}, is described by

_{C}= 7.4 μm. The propagation angles of the quasi-CW beams, θ

_{m}, for m = 1,2,3…18 (waves with m>18 are evanescent and therefore were not included in the calculation) are determined by the condition 2L

_{C}/m = Λ

_{m}= 2π/[k(1–cos(θ

_{m}))] which is obtained by corresponding between Eqs. (4) and (2). The angels of the first three quasi-CW waves are: θ

_{1}= 18.9°, θ

_{2}= 26.8°, and θ

_{3}= 33.1°. Following the scheme in Ref. 6

**99**(5), 053902 (2007). [CrossRef] [PubMed]

**99**(5), 053902 (2007). [CrossRef] [PubMed]

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]

^{th}harmonic. In this case, A(r) becomes somewhat nonlinear. The field ratio parameters, r

_{m}, and therefore also the peak intensities of the quasi-CW waves are determined by matching the calculated A(r) (Fig. 3a) with the optimal values of A

_{m}(Fig. 1e). The field ratio parameters and peak intensities of the first three quasi-CW waves are: r

_{1}= 2.2 × 10

^{−5}, r

_{2}= 5.2 × 10

^{−6}, r

_{3}= 2.6 × 10

^{−6}, I

_{1}= 4.4 × 10

^{10}W/cm

^{2}, I

_{2}= 1.1 × 10

^{10}W/cm

^{2}, and I

_{3}= 5.2 × 10

^{9}W/cm

^{2}. The total intensity of the eighteen quasi-CW waves is I

_{CW}= 5.7 × 10

^{10}W/cm

^{2}- which is 3.5 × 10

^{5}times smaller than the intensity of the driving laser.

_{C}are somewhat smaller than the theoretical QPM efficiency factors that are presented in Fig. 1f. The small reduction results from the fact that the parameters for the quasi-CW waves were derived from a simple model which neglects the variation of the optical field during the pulse. For example, in the case of 18-wave sawtooth GAPM, the numerically calculated normalized signal at z = 10L

_{C}and theoretical QPM efficiency factor are 0.88 and 0.93, respectively.

## Conclusion

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]

## Acknowledgments

## References and links

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. |

2. | S. Somekh and A. Yariv, “Phase-matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. |

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. |

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. |

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. |

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. |

7. | H. Ren, A. Nazarkin, J. Nold, and P. S. Russell, “Quasi-phase-matched high harmonic generation in hollow core photonic crystal fibers,” Opt. Express |

8. | 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 |

9. | C. Serrat and J. Biegert, “All-regions tunable high harmonic enhancement by a periodic static electric field,” Phys. Rev. Lett. |

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 |

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. |

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 |

13. | 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. |

14. | 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. |

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. |

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. |

17. | M. Lewenstein, P. Salières, and A. L’Huillier, “Phase of the atomic polarization in high-order harmonic generation,” Phys. Rev. A |

18. | 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 |

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. |

20. | J. Chen, A. Suda, E. J. Takahashi, M. Nurhuda, and K. Midorikawa, “Compression of intense ultrashort laser pulses in a gas-filled planar waveguide,” Opt. Lett. |

21. | M. Geissler, G. Tempea, and T. Brabec, “Phase-matched high-order harmonic generation in the nonadiabatic limit,” Phys. Rev. A |

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 |

23. | M. Y. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A |

24. | A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi Phase Matching of Momentum and Energy in Nonlinear Optical Processes,” Nat. Photonics |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: August 24, 2010

Revised Manuscript: October 6, 2010

Manuscript Accepted: October 7, 2010

Published: October 11, 2010

**Citation**

Pavel Sidorenko, Maxim Kozlov, Alon Bahabad, Tenio Popmintchev, Margaret Murnane, Henry Kapteyn, and Oren Cohen, "Sawtooth grating-assisted phase-matching," Opt. Express **18**, 22686-22692 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-22686

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### References

- 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]
- 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]
- 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]
- 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]
- H. Ren, A. Nazarkin, J. Nold, and P. S. Russell, “Quasi-phase-matched high harmonic generation in hollow core photonic crystal fibers,” Opt. Express 16(21), 17052–17059 (2008). [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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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).
- J. Chen, A. Suda, E. J. Takahashi, M. Nurhuda, and K. Midorikawa, “Compression of intense ultrashort laser pulses in a gas-filled planar waveguide,” Opt. Lett. 33(24), 2992–2994 (2008). [CrossRef] [PubMed]
- 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. 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).
- 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]
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