## Quasi-phase-matching of only even-order high harmonics |

Optics Express, Vol. 22, Issue 6, pp. 7145-7153 (2014)

http://dx.doi.org/10.1364/OE.22.007145

Acrobat PDF (1043 KB)

### Abstract

High harmonic spectrum of a quasi-monochromatic pump that interacts with isotropic media consists of only odd-order harmonics. Addition of a secondary pump, e.g. a static field or the second harmonic of the primary pump, can results with generation of both odd and even harmonics of the primary pump. We propose a method for quasi-phase matching of only the even-order harmonics of the primary pump. We formulate a theory for this process and demonstrate it numerically. We also show that it leads to attosecond pulse trains with constant carrier envelop phase and high repetition rate.

© 2014 Optical Society of America

## 1. Introduction

1. H. Kapteyn, O. Cohen, I. Christov, and M. Murnane, “Harnessing Attosecond Science in the Quest for Coherent X-rays,” Science **317**(5839), 775–778 (2007). [CrossRef] [PubMed]

2. T. Pfeifer, C. Spielmann, and G. Gerber, “Femtosecond x-ray science,” Rep. Prog. Phys. **69**(2), 443–505 (2006). [CrossRef]

3. P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science **292**(5522), 1689–1692 (2001). [CrossRef] [PubMed]

4. M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature **414**(6863), 509–513 (2001). [CrossRef] [PubMed]

5. R. I. Tobey, M. E. Siemens, O. Cohen, M. M. Murnane, H. C. Kapteyn, and K. A. Nelson, “Ultrafast extreme ultraviolet holography: dynamic monitoring of surface deformation,” Opt. Lett. **32**(3), 286–288 (2007). [CrossRef] [PubMed]

6. R. L. Sandberg, A. Paul, D. A. Raymondson, S. Hädrich, D. M. Gaudiosi, J. Holtsnider, R. I. Tobey, O. Cohen, M. M. Murnane, H. C. Kapteyn, C. Song, J. Miao, Y. Liu, and F. Salmassi, “Lensless Diffractive Imaging Using Tabletop Coherent High-Harmonic Soft-X-Ray Beams,” Phys. Rev. Lett. **99**(9), 098103 (2007). [CrossRef] [PubMed]

7. N. Ben-Tal, N. Moiseyev, and A. Beswick, “The effect of Hamiltonian symmetry on generation of odd and even harmonics,” J. Phys. B **26**(18), 3017–3024 (1993). [CrossRef]

8. E. Frumker, C. T. Hebeisen, N. Kajumba, J. B. Bertrand, H. J. Wörner, M. Spanner, D. M. Villeneuve, A. Naumov, and P. B. Corkum, “Oriented Rotational Wave-Packet Dynamics Studies via High Harmonic Generation,” Phys. Rev. Lett. **109**(11), 113901 (2012). [CrossRef] [PubMed]

7. N. Ben-Tal, N. Moiseyev, and A. Beswick, “The effect of Hamiltonian symmetry on generation of odd and even harmonics,” J. Phys. B **26**(18), 3017–3024 (1993). [CrossRef]

9. M. D. Perry and J. K. Crane, “High-order harmonic emission from mixed fields,” Phys. Rev. A **48**(6), 4051–4054 (1993). [CrossRef] [PubMed]

13. M. Kozlov, O. Kfir, A. Fleischer, A. Kaplan, T. Carmon, H. G. Schwefel, G. Bartal, and O. Cohen, “Narrow-bandwidth high-order harmonics driven by long-duration hot spots,” New J. Phys. **14**(6), 063036 (2012). [CrossRef]

14. S. Guo, S. Q. Duan, N. Yang, W. D. Ch, and W. Zhang, “Generation of even harmonics in coupled quantum dots,” Phys. Rev. A **84**(1), 015803 (2011). [CrossRef]

10. J. Mauritsson, P. Johnsson, E. Gustafsson, A. L’Huillier, K. J. Schafer, and M. B. Gaarde, “Attosecond Pulse Trains Generated Using Two Color Laser Fields,” Phys. Rev. Lett. **97**(1), 013001 (2006). [CrossRef] [PubMed]

11. W. Hong, P. Lu, P. Lan, Q. Zhang, and X. Wang, “Few-cycle attosecond pulses with stabilized-carrier-envelope phase in the presence of a strong terahertz field,” Opt. Express **17**(7), 5139–5146 (2009). [CrossRef] [PubMed]

15. 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]

24. L. Z. Liu, K. OKeeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using polarization beating in optical waveguides,” Phys. Rev. A **85**(5), 053823 (2012). [CrossRef]

16. 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]

19. 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]

22. 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**(1), 011803 (2010). [CrossRef]

_{2}or terahertz pulses. Finally, we show that the generated APT exhibits constant CEP and that it consists of two pulses per pump cycle.

## 2. Symmetry of HHG driven by a bi-chromatic driver

7. N. Ben-Tal, N. Moiseyev, and A. Beswick, “The effect of Hamiltonian symmetry on generation of odd and even harmonics,” J. Phys. B **26**(18), 3017–3024 (1993). [CrossRef]

_{D}, at angular frequency ω

_{0}= 2π/T, where T is the optical cycle, is half-wave symmetric: E

_{d}(t + T/2) = -E

_{d}(t), hence the harmonics field, E

_{HHG}, exhibits the same symmetry: E

_{HHG}(t + T/2) = -E

_{HHG}(t). The spectrum of this field consists of only odd harmonics of ω

_{0}because symmetry dictates that even Fourier components of half-wave symmetric functions are zero. The HHG spectrum can include even-order harmonics if a secondary field breaks the half-wave symmetry. This concept was implemented in many experiments where HHG was driven by bi-chromatic drivers that consist of a strong pump and its second harmonic [9

9. M. D. Perry and J. K. Crane, “High-order harmonic emission from mixed fields,” Phys. Rev. A **48**(6), 4051–4054 (1993). [CrossRef] [PubMed]

10. J. Mauritsson, P. Johnsson, E. Gustafsson, A. L’Huillier, K. J. Schafer, and M. B. Gaarde, “Attosecond Pulse Trains Generated Using Two Color Laser Fields,” Phys. Rev. Lett. **97**(1), 013001 (2006). [CrossRef] [PubMed]

_{0}when a weak static field (or a very long-wavelength field) is added to the main strong pump [11

11. W. Hong, P. Lu, P. Lan, Q. Zhang, and X. Wang, “Few-cycle attosecond pulses with stabilized-carrier-envelope phase in the presence of a strong terahertz field,” Opt. Express **17**(7), 5139–5146 (2009). [CrossRef] [PubMed]

13. M. Kozlov, O. Kfir, A. Fleischer, A. Kaplan, T. Carmon, H. G. Schwefel, G. Bartal, and O. Cohen, “Narrow-bandwidth high-order harmonics driven by long-duration hot spots,” New J. Phys. **14**(6), 063036 (2012). [CrossRef]

_{BC}= A

_{0}cos(ω

_{0}t + φ

_{0}) + A

_{1}cos(ω

_{1}t + φ

_{0}+ Δφ) where ω

_{0}= 2π/T

_{0}and ω

_{1}= 2π/T

_{1}are angular optical frequencies, T

_{0}and T

_{1}are optical cycles, A

_{0}and A

_{1}are real amplitudes, φ

_{0}is a global phase, and Δφ is the relative phase between the two components. We compare between the harmonic fields driven by the bi-chromatic fields with the two following relative phases: Δφ

_{a}= 0 and Δφ

_{b}= π(1-ω

_{1}/ω

_{0}). We assign the generated harmonic fields by

_{BC}(t,Δφ = Δφ

_{a}) = -E

_{BC}(t + T

_{0}/2,Δφ = Δφ

_{b}). The harmonics fields also conform to this symmetry, henceInserting the Fourier decomposition of the emitted harmonic field,

_{1}/ω

_{0}) phase-shift of the relative phase, while at the same time, the sign of the even-order harmonics is flipped. This feature is the source for our proposal for QPM of only even-order harmonics. Here, we explore numerically two specific configurations for the bi-chromatic drivers where in both cases the strong pump corresponds to a ti:sapphire laser pulse with central frequency ω

_{0}= 2.3 × 10

^{15}Hz. In the first case, the secondary driver is at much smaller frequency than the pump ω

_{1}<<ω

_{0}(e.g. terahertz or CO

_{2}laser) such that within the pulse-duration of the strong pulse, the field is approximately constant. Numerically, we use a static field for this case. In the second case, the second driver is the second harmonic of the strong pump. We get Δφ

_{b}= π for both cases which corresponds to a change in the sign of the static or second harmonic fields.

_{0}= 1.3 and Δ = 1 in atomic units and x is the polarization direction of the laser pulse. The ground state ionization energy of this symmetric potential is I

_{p}= 21 eV, corresponding, for example, to neutral neon and singly-ionized xenon ion. Initially, i.e. in the leading edge of the driver pulses, the electron fully populates the ground state. The polarization is calculated by:

_{HHG}[2

2. T. Pfeifer, C. Spielmann, and G. Gerber, “Femtosecond x-ray science,” Rep. Prog. Phys. **69**(2), 443–505 (2006). [CrossRef]

_{.}In the first case, the bi-chromatic drivers are

_{DC}is the amplitude of the static field. The cutoff frequency of the HHG spectrum corresponds to the 87th harmonic of the strong pump. Figures 1(a)-1(c) display the emitted phase of several harmonics order as a function of the static field. As expected from our symmetry feature, the phases of even-order harmonics, both at the cutoff and plateau spectral regions, are flipped by π when the static field changes sign. The phases of odd harmonics, on the other hand, do not exhibit such a flip (Fig. 1(c)). Figures 1(d)-1(f) show the intensity of the harmonics as a function of the static field. As shown, the strength of the even harmonics at E

_{DC}~2 × 10

^{6}V/cm is comparable to the strength of odd harmonics without static field. In the second case, we used a bi-chromatic driver of

## 3. QPM of only even-order harmonics

_{0}(primary field) and ω

_{1}(secondary field). We employ this symmetry feature for QPM of only the even-order harmonics. The method is based on the following concept: The setting is engineered such that the relative phase between the primary and secondary fields is shifted by Δφ

_{b}= π(1-ω

_{1}/ω

_{0}) every propagation distance that corresponds to the coherence length of a q-order harmonic, L

_{C.}(The coherence length is calculated when only the primary field is present because the secondary field is relatively weak; hence it approximately does not change the plasma density which is the main source for the phase mismatch). For large q, the coherence lengths for consecutive odd and even harmonic are very similar. But, there is also additional phase that results from the presence of the weak field. This additional phase is described by Eq. (2). For an even q harmonic, the fields emitted at propagation distances z and z + L

_{C}interfere constructively because the π phase-shift due to the phase-mismatch is canceled by the π phase of Eq. (2). On the other hand, odd-order harmonics that are generated in z and z + L

_{C}interfere destructively because they experience π phase-shift due to the phase mismatch and 0 phase shift due to the symmetry feature of Eq. (2). Thus, only the even order harmonics experience QPM in such a setting.

18. 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]

_{DC}:

_{DC}) = -g(z). This scheme can be implemented experimentally using the setup proposed in Ref. 21

21. 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]

25. M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light Propagation in Field-Ionizing Media: Extreme Nonlinear Optics,” Phys. Rev. Lett. **83**(15), 2930–2933 (1999). [CrossRef]

_{e}, takes into account the pre-formed plasma and the ionization that is calculated by using the ADK model [26]. The high-order polarization, P

_{HHG}, is calculated through numerical calculation of the 1D TDSE under the influence of the total field E

_{0}+ E

_{DC}. The generation and evolution of the HHG field up to a constant factor (which is associated with the gas density and is unimportant in our case because the gas density is constant), E

_{HHG}, is described by:Figure 3(a) shows the HHG spectrum after propagation distance of 0.5 mm with gas pressure of 25 torr when d

_{DC}= 18 µm which corresponds to the coherence length of the 88th cutoff harmonic. For comparison, the generated spectrum with constant static field is also presented. A clear QPM enhancement is obtained around the 88th harmonic when the static field flips sign periodically. Figure 3(b) shows the coherent buildup of the 88th and 87th harmonic fields, showing clearly that the even harmonic experience a QPM enhancement while the odd harmonic suffers from phase-mismatch. Notably, the QPM efficiency of the 88th harmonic is 0.27, which is relatively high for QPM in HHG [27

27. 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]

_{DC}= 28 µm which corresponds to the coherence length of the 70th plateau harmonic. Clear QPM enhancement is obtained around the 72th harmonic. Figure 3(d) shows the coherent buildup of the 70th and 71th harmonic fields, showing again that the even harmonic experience a QPM enhancement (with 0.23 QPM efficiency) while the odd harmonic suffers from phase-mismatch.

_{QPM}(t) that corresponds to the red spectrum in Fig. 3(a) in the spectral region 83 ± 5 harmonics. Figure 4(b) shows the average of E

_{QPM}and its T

_{0}/2 time-delayed, showing that this APT has a stable CEP. Notably, the temporal distance between consecutive pulses is T

_{0}/2. That is, in contrast to previous methods [10

10. J. Mauritsson, P. Johnsson, E. Gustafsson, A. L’Huillier, K. J. Schafer, and M. B. Gaarde, “Attosecond Pulse Trains Generated Using Two Color Laser Fields,” Phys. Rev. Lett. **97**(1), 013001 (2006). [CrossRef] [PubMed]

11. W. Hong, P. Lu, P. Lan, Q. Zhang, and X. Wang, “Few-cycle attosecond pulses with stabilized-carrier-envelope phase in the presence of a strong terahertz field,” Opt. Express **17**(7), 5139–5146 (2009). [CrossRef] [PubMed]

_{SA}(SA stands for single atom) which corresponds to the APT generated by the same strong pump beam, but without propagation and without the static field. The average of E

_{SA}and its T

_{0}/2 time-delayed show that consecutive pulses have opposite phases (Fig. 4(d)).

28. D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, “High-Order Harmonic Generation from Ions in a Capillary Discharge,” Phys. Rev. Lett. **96**(20), 203001 (2006). [CrossRef] [PubMed]

20. P. Sidorenko, M. Kozlov, A. Bahabad, T. Popmintchev, M. Murnane, H. Kapteyn, and O. Cohen, “Sawtooth grating-assisted phase-matching,” Opt. Express **18**(22), 22686–22692 (2010). [CrossRef] [PubMed]

_{C}. The incident beam in our simulation is E

_{BC}= E

_{0}+ E

_{1}where E

_{0}is the same as in the previous section and

_{0}and -Δn in the region around 2ω

_{0}. The third term in Eq. (5) gives rise to the assumed dispersion, only. Figure 5(a) shows the HHG spectrum when Δn = 8.7 × 10

^{−3}(L

_{π}= 46 µm) and after propagation distance of 1 mm. For compression, the generated spectrum when Δn = 0 is also presented. A clear QPM enhancement is obtained around the 86th harmonic. Figure 5(b) shows the coherent buildup of the 86th and 85th harmonic fields, showing clearly that the even harmonic experience a QPM enhancement (QPM efficiency is 0.27) while the odd harmonic suffers from phase-mismatch. Figure 5(c) shows the HHG spectra when Δn = 7 × 10

^{−3}(L

_{π}= 57µm) and, for compression also the Δn = 0 case. A clear QPM enhancement is obtained around the 70th harmonic. Figure 5(d) shows the coherent buildup of the 70th and 71th harmonic fields, showing that the even harmonic experience a QPM enhancement (QPM efficiency is 0.14) while the odd harmonic suffers from phase-mismatch.

## 4. Conclusions

## Acknowledgment

## References and links

1. | H. Kapteyn, O. Cohen, I. Christov, and M. Murnane, “Harnessing Attosecond Science in the Quest for Coherent X-rays,” Science |

2. | T. Pfeifer, C. Spielmann, and G. Gerber, “Femtosecond x-ray science,” Rep. Prog. Phys. |

3. | P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, and P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science |

4. | M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature |

5. | R. I. Tobey, M. E. Siemens, O. Cohen, M. M. Murnane, H. C. Kapteyn, and K. A. Nelson, “Ultrafast extreme ultraviolet holography: dynamic monitoring of surface deformation,” Opt. Lett. |

6. | R. L. Sandberg, A. Paul, D. A. Raymondson, S. Hädrich, D. M. Gaudiosi, J. Holtsnider, R. I. Tobey, O. Cohen, M. M. Murnane, H. C. Kapteyn, C. Song, J. Miao, Y. Liu, and F. Salmassi, “Lensless Diffractive Imaging Using Tabletop Coherent High-Harmonic Soft-X-Ray Beams,” Phys. Rev. Lett. |

7. | N. Ben-Tal, N. Moiseyev, and A. Beswick, “The effect of Hamiltonian symmetry on generation of odd and even harmonics,” J. Phys. B |

8. | E. Frumker, C. T. Hebeisen, N. Kajumba, J. B. Bertrand, H. J. Wörner, M. Spanner, D. M. Villeneuve, A. Naumov, and P. B. Corkum, “Oriented Rotational Wave-Packet Dynamics Studies via High Harmonic Generation,” Phys. Rev. Lett. |

9. | M. D. Perry and J. K. Crane, “High-order harmonic emission from mixed fields,” Phys. Rev. A |

10. | J. Mauritsson, P. Johnsson, E. Gustafsson, A. L’Huillier, K. J. Schafer, and M. B. Gaarde, “Attosecond Pulse Trains Generated Using Two Color Laser Fields,” Phys. Rev. Lett. |

11. | W. Hong, P. Lu, P. Lan, Q. Zhang, and X. Wang, “Few-cycle attosecond pulses with stabilized-carrier-envelope phase in the presence of a strong terahertz field,” Opt. Express |

12. | S. Odžak and D. B. Milošević, “High-order harmonic generation in the presence of a static electric field,” Phys. Rev. A |

13. | M. Kozlov, O. Kfir, A. Fleischer, A. Kaplan, T. Carmon, H. G. Schwefel, G. Bartal, and O. Cohen, “Narrow-bandwidth high-order harmonics driven by long-duration hot spots,” New J. Phys. |

14. | S. Guo, S. Q. Duan, N. Yang, W. D. Ch, and W. Zhang, “Generation of even harmonics in coupled quantum dots,” Phys. Rev. A |

15. | 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 |

16. | 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. |

17. | J. Seres, V. S. Yakovlev, E. Seres, Ch. Streli, P. Wobrauschek, Ch. Spielmann, and F. Krausz, “Coherent superposition of laser-driven soft-X-ray harmonics from successive sources,” Nat. Phys. |

18. | 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. |

19. | 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. |

20. | P. Sidorenko, M. Kozlov, A. Bahabad, T. Popmintchev, M. Murnane, H. Kapteyn, and O. Cohen, “Sawtooth grating-assisted phase-matching,” Opt. Express |

21. | C. Serrat and J. Biegert, “All-Regions Tunable High Harmonic Enhancement by a Periodic Static Electric field,” Phys. Rev. Lett. |

22. | 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 |

23. | L. Z. Liu, K. O’Keeffe, and S. M. Hooker, “Optical rotation quasi-phase-matching for circularly polarized high harmonic generation,” Opt. Lett. |

24. | L. Z. Liu, K. OKeeffe, and S. M. Hooker, “Quasi-phase-matching of high-order-harmonic generation using polarization beating in optical waveguides,” Phys. Rev. A |

25. | M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light Propagation in Field-Ionizing Media: Extreme Nonlinear Optics,” Phys. Rev. Lett. |

26. | 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 |

27. | 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. |

28. | D. M. Gaudiosi, B. Reagan, T. Popmintchev, M. Grisham, M. Berrill, O. Cohen, B. C. Walker, M. M. Murnane, H. C. Kapteyn, and J. J. Rocca, “High-Order Harmonic Generation from Ions in a Capillary Discharge,” Phys. Rev. Lett. |

29. | O. Kfir, P. Sidorenko, A. Paul, T. Popmintchev, H. Kapteyn, M. Murnane, and O. Cohen, “Extended Phase-Matching of High Harmonics Driven by Focusing Light in Planar Waveguide”, Frontiers in Optic (Optical Society of America, Rochester, NY, 2012), paper LTh1H.4. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.2620) Nonlinear optics : Harmonic generation and mixing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 27, 2013

Revised Manuscript: February 28, 2014

Manuscript Accepted: February 28, 2014

Published: March 19, 2014

**Citation**

Tzvi Diskin and Oren Cohen, "Quasi-phase-matching of only even-order high harmonics," Opt. Express **22**, 7145-7153 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-7145

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### References

- H. Kapteyn, O. Cohen, I. Christov, M. Murnane, “Harnessing Attosecond Science in the Quest for Coherent X-rays,” Science 317(5839), 775–778 (2007). [CrossRef] [PubMed]
- T. Pfeifer, C. Spielmann, G. Gerber, “Femtosecond x-ray science,” Rep. Prog. Phys. 69(2), 443–505 (2006). [CrossRef]
- P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Augé, P. Balcou, H. G. Muller, P. Agostini, “Observation of a Train of Attosecond Pulses from High Harmonic Generation,” Science 292(5522), 1689–1692 (2001). [CrossRef] [PubMed]
- M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, F. Krausz, “Attosecond metrology,” Nature 414(6863), 509–513 (2001). [CrossRef] [PubMed]
- R. I. Tobey, M. E. Siemens, O. Cohen, M. M. Murnane, H. C. Kapteyn, K. A. Nelson, “Ultrafast extreme ultraviolet holography: dynamic monitoring of surface deformation,” Opt. Lett. 32(3), 286–288 (2007). [CrossRef] [PubMed]
- R. L. Sandberg, A. Paul, D. A. Raymondson, S. Hädrich, D. M. Gaudiosi, J. Holtsnider, R. I. Tobey, O. Cohen, M. M. Murnane, H. C. Kapteyn, C. Song, J. Miao, Y. Liu, F. Salmassi, “Lensless Diffractive Imaging Using Tabletop Coherent High-Harmonic Soft-X-Ray Beams,” Phys. Rev. Lett. 99(9), 098103 (2007). [CrossRef] [PubMed]
- N. Ben-Tal, N. Moiseyev, A. Beswick, “The effect of Hamiltonian symmetry on generation of odd and even harmonics,” J. Phys. B 26(18), 3017–3024 (1993). [CrossRef]
- E. Frumker, C. T. Hebeisen, N. Kajumba, J. B. Bertrand, H. J. Wörner, M. Spanner, D. M. Villeneuve, A. Naumov, P. B. Corkum, “Oriented Rotational Wave-Packet Dynamics Studies via High Harmonic Generation,” Phys. Rev. Lett. 109(11), 113901 (2012). [CrossRef] [PubMed]
- M. D. Perry, J. K. Crane, “High-order harmonic emission from mixed fields,” Phys. Rev. A 48(6), 4051–4054 (1993). [CrossRef] [PubMed]
- J. Mauritsson, P. Johnsson, E. Gustafsson, A. L’Huillier, K. J. Schafer, M. B. Gaarde, “Attosecond Pulse Trains Generated Using Two Color Laser Fields,” Phys. Rev. Lett. 97(1), 013001 (2006). [CrossRef] [PubMed]
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