Abstract

A general technique for quasi phase matching of nonlinear frequency conversion in molecular gases through the dynamically structured nonlinear optical response of a molecular ensemble is presented. We show that the transient molecular alignment created by an ultrafast alignment pulse can be controlled to periodically modulate the strength of the nonlinear optical susceptibility in the reference frame of a fundamental pulse propagating through the aligned molecules. Simulations show efficient third harmonic generation of an ultrafast fundamental laser pulse.

© 2005 Optical Society of America

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  1. J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. A 127, 1918 (1962).
  2. S. Somekh and A. Yariv, “Phase matching by periodic modulation of the nonlinear optical properties,” Opt. Commun. 6, 301 – 304 (1972).
    [Crossref]
  3. J. D. McMullen, “Optical Parametric Interactions in Isotropic Materials Using a Phase-Corrected Stack of Nonlinear Dielectric Plates,” J. Appl. Phys. 46, 3076–3081 (1975).
    [Crossref]
  4. A. Szilagyi, A. Hordvik, and H. Schlossberg, “Quasi-Phase-Matching Technique for Efficient Optical Mixing and Frequency Doubling,” J. Appl. Phys. 47, 2025–2032 (1976).
    [Crossref]
  5. D. E. Thompson, J. D. McMullen, and D. B. Anderson, “2nd-Harmonic Generation in GaAs “Stack of Plates” Using High-Power CO2-Laser Radiation,” Appl. Phys. Lett. 29, 113–115 (1976).
    [Crossref]
  6. C. F. Dewey and L. O. Hocker, “Enhanced Nonlinear Optical Effects in Rotationally Twinned Crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
    [Crossref]
  7. D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, and Y. N. Wang, “Enhancement of 2nd-Harmonic Generation in Linbo3 Crystals with Periodic Laminar Ferroelectric Domains,” Appl. Phys. Lett. 37, 607–609 (1980).
    [Crossref]
  8. A. Feisst and P. Koidl, “Current Induced Periodic Ferroelectric Domain-Structures in Linbo3 Applied for Efficient Nonlinear Optical Frequency Mixing,” Appl. Phys. Lett. 47, 1125–1127 (1985).
    [Crossref]
  9. K. D. Singer, J. E. Sohn, and S. J. Lalama, “2nd Harmonic-Generation in Poled Polymer-Films,” Appl. Phys. Lett. 49, 248–250 (1986).
    [Crossref]
  10. D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
    [Crossref]
  11. J. F. Ward and G. H. C. New, “Optical Third Harmonic Generation in Gases by a Focused Laser Beam,” Phys. Rev. 185, 57–72 (1969).
    [Crossref]
  12. A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Lambda system,” Phys. Rev. Lett. 84, 5308–5311 (2000).
    [Crossref] [PubMed]
  13. C. Durfee, S. Backus, M. Murnane, and H. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. 22, 1565–1567 (1997).
    [Crossref]
  14. A. Rundquist, C. Durfee, S. Backus, C. Herne, Z. Chang, M. Murnane, and H. Kapteyn, “Phase-matched Generation of Coherent Soft X-rays,” Science 280, 1412–1415 (1998).
    [Crossref] [PubMed]
  15. R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
    [PubMed]
  16. B. F. Levine, C. G. Bethea, and R. A. Logan, “Phase-Matched Second-Harmonic Generation in a Liquid-Filled Waveguide,” Appl. Phys. Lett. 26, 375–377 (1975).
    [Crossref]
  17. 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, 51–54 (2003).
    [Crossref] [PubMed]
  18. B. Friedrich and D. Herschbach, “Alignment and Trapping of Molecules in Intense Laser Fields,” Phys. Rev. Lett. 74, 4623–4626 (1995).
    [Crossref] [PubMed]
  19. T. Seideman, “Rotational-Excitation and Molecular Alignment in Intense Laser Fields,” J. Chem. Phys. 103, 7887–7896 (1995).
    [Crossref]
  20. R. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, New York, 2003)
  21. R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88 (2002).
    [PubMed]
  22. H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75, 543–557 (2003).
    [Crossref]
  23. R. A. Bartels, N. L. Wagner, M. D. Baertschy, J. Wyss, M. M. Murnane, and H. C. Kapteyn, “Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases,” Opt. Lett. 28, 346–348 (2003).
    [Crossref] [PubMed]
  24. Zs. Bor and B. Ràcz “Group Velocity Dispersion in Prisms and its application to pulse compression and travelling-wave excitation,” Opt. Commun. 54 (1985)
    [Crossref]
  25. A. Rizzo and N. Rahman, “The linear and nonlinear susceptibilities of acetylene relevant for high order harmonic generation,” Laser Phys. 9, 416–421 (1999).
  26. M. Leibscher, I. S. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90 (2003).
    [Crossref] [PubMed]
  27. G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, New York, 2001).
  28. L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 8701 (2001).

2003 (4)

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, 51–54 (2003).
[Crossref] [PubMed]

H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75, 543–557 (2003).
[Crossref]

M. Leibscher, I. S. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90 (2003).
[Crossref] [PubMed]

R. A. Bartels, N. L. Wagner, M. D. Baertschy, J. Wyss, M. M. Murnane, and H. C. Kapteyn, “Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases,” Opt. Lett. 28, 346–348 (2003).
[Crossref] [PubMed]

2002 (2)

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88 (2002).
[PubMed]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

2001 (1)

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 8701 (2001).

2000 (1)

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Lambda system,” Phys. Rev. Lett. 84, 5308–5311 (2000).
[Crossref] [PubMed]

1999 (1)

A. Rizzo and N. Rahman, “The linear and nonlinear susceptibilities of acetylene relevant for high order harmonic generation,” Laser Phys. 9, 416–421 (1999).

1998 (2)

A. Rundquist, C. Durfee, S. Backus, C. Herne, Z. Chang, M. Murnane, and H. Kapteyn, “Phase-matched Generation of Coherent Soft X-rays,” Science 280, 1412–1415 (1998).
[Crossref] [PubMed]

D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
[Crossref]

1997 (1)

1995 (2)

B. Friedrich and D. Herschbach, “Alignment and Trapping of Molecules in Intense Laser Fields,” Phys. Rev. Lett. 74, 4623–4626 (1995).
[Crossref] [PubMed]

T. Seideman, “Rotational-Excitation and Molecular Alignment in Intense Laser Fields,” J. Chem. Phys. 103, 7887–7896 (1995).
[Crossref]

1986 (1)

K. D. Singer, J. E. Sohn, and S. J. Lalama, “2nd Harmonic-Generation in Poled Polymer-Films,” Appl. Phys. Lett. 49, 248–250 (1986).
[Crossref]

1985 (2)

A. Feisst and P. Koidl, “Current Induced Periodic Ferroelectric Domain-Structures in Linbo3 Applied for Efficient Nonlinear Optical Frequency Mixing,” Appl. Phys. Lett. 47, 1125–1127 (1985).
[Crossref]

Zs. Bor and B. Ràcz “Group Velocity Dispersion in Prisms and its application to pulse compression and travelling-wave excitation,” Opt. Commun. 54 (1985)
[Crossref]

1980 (1)

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, and Y. N. Wang, “Enhancement of 2nd-Harmonic Generation in Linbo3 Crystals with Periodic Laminar Ferroelectric Domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[Crossref]

1976 (2)

A. Szilagyi, A. Hordvik, and H. Schlossberg, “Quasi-Phase-Matching Technique for Efficient Optical Mixing and Frequency Doubling,” J. Appl. Phys. 47, 2025–2032 (1976).
[Crossref]

D. E. Thompson, J. D. McMullen, and D. B. Anderson, “2nd-Harmonic Generation in GaAs “Stack of Plates” Using High-Power CO2-Laser Radiation,” Appl. Phys. Lett. 29, 113–115 (1976).
[Crossref]

1975 (3)

C. F. Dewey and L. O. Hocker, “Enhanced Nonlinear Optical Effects in Rotationally Twinned Crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[Crossref]

J. D. McMullen, “Optical Parametric Interactions in Isotropic Materials Using a Phase-Corrected Stack of Nonlinear Dielectric Plates,” J. Appl. Phys. 46, 3076–3081 (1975).
[Crossref]

B. F. Levine, C. G. Bethea, and R. A. Logan, “Phase-Matched Second-Harmonic Generation in a Liquid-Filled Waveguide,” Appl. Phys. Lett. 26, 375–377 (1975).
[Crossref]

1972 (1)

S. Somekh and A. Yariv, “Phase matching by periodic modulation of the nonlinear optical properties,” Opt. Commun. 6, 301 – 304 (1972).
[Crossref]

1969 (1)

J. F. Ward and G. H. C. New, “Optical Third Harmonic Generation in Gases by a Focused Laser Beam,” Phys. Rev. 185, 57–72 (1969).
[Crossref]

1962 (1)

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. A 127, 1918 (1962).

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, New York, 2001).

Anderson, D. B.

D. E. Thompson, J. D. McMullen, and D. B. Anderson, “2nd-Harmonic Generation in GaAs “Stack of Plates” Using High-Power CO2-Laser Radiation,” Appl. Phys. Lett. 29, 113–115 (1976).
[Crossref]

Armstrong, J.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. A 127, 1918 (1962).

Attwood, D.

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

Averbukh, I. S.

M. Leibscher, I. S. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90 (2003).
[Crossref] [PubMed]

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, 51–54 (2003).
[Crossref] [PubMed]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 8701 (2001).

A. Rundquist, C. Durfee, S. Backus, C. Herne, Z. Chang, M. Murnane, and H. Kapteyn, “Phase-matched Generation of Coherent Soft X-rays,” Science 280, 1412–1415 (1998).
[Crossref] [PubMed]

C. Durfee, S. Backus, M. Murnane, and H. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. 22, 1565–1567 (1997).
[Crossref]

Baertschy, M.

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88 (2002).
[PubMed]

Baertschy, M. D.

Bartels, R.

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 8701 (2001).

Bartels, R. A.

R. A. Bartels, N. L. Wagner, M. D. Baertschy, J. Wyss, M. M. Murnane, and H. C. Kapteyn, “Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases,” Opt. Lett. 28, 346–348 (2003).
[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, 51–54 (2003).
[Crossref] [PubMed]

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88 (2002).
[PubMed]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

Bethea, C. G.

B. F. Levine, C. G. Bethea, and R. A. Logan, “Phase-Matched Second-Harmonic Generation in a Liquid-Filled Waveguide,” Appl. Phys. Lett. 26, 375–377 (1975).
[Crossref]

Bloembergen, N.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. A 127, 1918 (1962).

Bor, Zs.

Zs. Bor and B. Ràcz “Group Velocity Dispersion in Prisms and its application to pulse compression and travelling-wave excitation,” Opt. Commun. 54 (1985)
[Crossref]

Boyd, R.

R. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, New York, 2003)

Chang, Z.

A. Rundquist, C. Durfee, S. Backus, C. Herne, Z. Chang, M. Murnane, and H. Kapteyn, “Phase-matched Generation of Coherent Soft X-rays,” Science 280, 1412–1415 (1998).
[Crossref] [PubMed]

Christov, I. P.

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, 51–54 (2003).
[Crossref] [PubMed]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

Dewey, C. F.

C. F. Dewey and L. O. Hocker, “Enhanced Nonlinear Optical Effects in Rotationally Twinned Crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[Crossref]

Ducuing, J.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. A 127, 1918 (1962).

Durfee, C.

A. Rundquist, C. Durfee, S. Backus, C. Herne, Z. Chang, M. Murnane, and H. Kapteyn, “Phase-matched Generation of Coherent Soft X-rays,” Science 280, 1412–1415 (1998).
[Crossref] [PubMed]

C. Durfee, S. Backus, M. Murnane, and H. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. 22, 1565–1567 (1997).
[Crossref]

Durfee, C. G.

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 8701 (2001).

Feisst, A.

A. Feisst and P. Koidl, “Current Induced Periodic Ferroelectric Domain-Structures in Linbo3 Applied for Efficient Nonlinear Optical Frequency Mixing,” Appl. Phys. Lett. 47, 1125–1127 (1985).
[Crossref]

Feng, D.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, and Y. N. Wang, “Enhancement of 2nd-Harmonic Generation in Linbo3 Crystals with Periodic Laminar Ferroelectric Domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[Crossref]

Friedrich, B.

B. Friedrich and D. Herschbach, “Alignment and Trapping of Molecules in Intense Laser Fields,” Phys. Rev. Lett. 74, 4623–4626 (1995).
[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, 51–54 (2003).
[Crossref] [PubMed]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

Greene, C. H.

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88 (2002).
[PubMed]

Harris, S. E.

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Lambda system,” Phys. Rev. Lett. 84, 5308–5311 (2000).
[Crossref] [PubMed]

Herne, C.

A. Rundquist, C. Durfee, S. Backus, C. Herne, Z. Chang, M. Murnane, and H. Kapteyn, “Phase-matched Generation of Coherent Soft X-rays,” Science 280, 1412–1415 (1998).
[Crossref] [PubMed]

Herschbach, D.

B. Friedrich and D. Herschbach, “Alignment and Trapping of Molecules in Intense Laser Fields,” Phys. Rev. Lett. 74, 4623–4626 (1995).
[Crossref] [PubMed]

Hocker, L. O.

C. F. Dewey and L. O. Hocker, “Enhanced Nonlinear Optical Effects in Rotationally Twinned Crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[Crossref]

Hong, J. F.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, and Y. N. Wang, “Enhancement of 2nd-Harmonic Generation in Linbo3 Crystals with Periodic Laminar Ferroelectric Domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[Crossref]

Hordvik, A.

A. Szilagyi, A. Hordvik, and H. Schlossberg, “Quasi-Phase-Matching Technique for Efficient Optical Mixing and Frequency Doubling,” J. Appl. Phys. 47, 2025–2032 (1976).
[Crossref]

Jacobsen, C.

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

Kapteyn, H.

A. Rundquist, C. Durfee, S. Backus, C. Herne, Z. Chang, M. Murnane, and H. Kapteyn, “Phase-matched Generation of Coherent Soft X-rays,” Science 280, 1412–1415 (1998).
[Crossref] [PubMed]

C. Durfee, S. Backus, M. Murnane, and H. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. 22, 1565–1567 (1997).
[Crossref]

Kapteyn, H. C.

R. A. Bartels, N. L. Wagner, M. D. Baertschy, J. Wyss, M. M. Murnane, and H. C. Kapteyn, “Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases,” Opt. Lett. 28, 346–348 (2003).
[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, 51–54 (2003).
[Crossref] [PubMed]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88 (2002).
[PubMed]

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 8701 (2001).

King, T. A.

D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
[Crossref]

Koidl, P.

A. Feisst and P. Koidl, “Current Induced Periodic Ferroelectric Domain-Structures in Linbo3 Applied for Efficient Nonlinear Optical Frequency Mixing,” Appl. Phys. Lett. 47, 1125–1127 (1985).
[Crossref]

Lalama, S. J.

K. D. Singer, J. E. Sohn, and S. J. Lalama, “2nd Harmonic-Generation in Poled Polymer-Films,” Appl. Phys. Lett. 49, 248–250 (1986).
[Crossref]

Leibscher, M.

M. Leibscher, I. S. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90 (2003).
[Crossref] [PubMed]

Levine, B. F.

B. F. Levine, C. G. Bethea, and R. A. Logan, “Phase-Matched Second-Harmonic Generation in a Liquid-Filled Waveguide,” Appl. Phys. Lett. 26, 375–377 (1975).
[Crossref]

Liu, Y. W.

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

Logan, R. A.

B. F. Levine, C. G. Bethea, and R. A. Logan, “Phase-Matched Second-Harmonic Generation in a Liquid-Filled Waveguide,” Appl. Phys. Lett. 26, 375–377 (1975).
[Crossref]

Manuszak, D.

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Lambda system,” Phys. Rev. Lett. 84, 5308–5311 (2000).
[Crossref] [PubMed]

McMullen, J. D.

D. E. Thompson, J. D. McMullen, and D. B. Anderson, “2nd-Harmonic Generation in GaAs “Stack of Plates” Using High-Power CO2-Laser Radiation,” Appl. Phys. Lett. 29, 113–115 (1976).
[Crossref]

J. D. McMullen, “Optical Parametric Interactions in Isotropic Materials Using a Phase-Corrected Stack of Nonlinear Dielectric Plates,” J. Appl. Phys. 46, 3076–3081 (1975).
[Crossref]

Merriam, A. J.

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Lambda system,” Phys. Rev. Lett. 84, 5308–5311 (2000).
[Crossref] [PubMed]

Ming, N. B.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, and Y. N. Wang, “Enhancement of 2nd-Harmonic Generation in Linbo3 Crystals with Periodic Laminar Ferroelectric Domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[Crossref]

Misoguti, L.

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 8701 (2001).

Murnane, M.

A. Rundquist, C. Durfee, S. Backus, C. Herne, Z. Chang, M. Murnane, and H. Kapteyn, “Phase-matched Generation of Coherent Soft X-rays,” Science 280, 1412–1415 (1998).
[Crossref] [PubMed]

C. Durfee, S. Backus, M. Murnane, and H. Kapteyn, “Ultrabroadband phase-matched optical parametric generation in the ultraviolet by use of guided waves,” Opt. Lett. 22, 1565–1567 (1997).
[Crossref]

Murnane, M. M.

R. A. Bartels, N. L. Wagner, M. D. Baertschy, J. Wyss, M. M. Murnane, and H. C. Kapteyn, “Phase-matching conditions for nonlinear frequency conversion by use of aligned molecular gases,” Opt. Lett. 28, 346–348 (2003).
[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, 51–54 (2003).
[Crossref] [PubMed]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88 (2002).
[PubMed]

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 8701 (2001).

New, G. H. C.

J. F. Ward and G. H. C. New, “Optical Third Harmonic Generation in Gases by a Focused Laser Beam,” Phys. Rev. 185, 57–72 (1969).
[Crossref]

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, 51–54 (2003).
[Crossref] [PubMed]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

Pershan, P.

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. A 127, 1918 (1962).

Rabitz, H.

M. Leibscher, I. S. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90 (2003).
[Crossref] [PubMed]

Ràcz, B.

Zs. Bor and B. Ràcz “Group Velocity Dispersion in Prisms and its application to pulse compression and travelling-wave excitation,” Opt. Commun. 54 (1985)
[Crossref]

Rahman, N.

A. Rizzo and N. Rahman, “The linear and nonlinear susceptibilities of acetylene relevant for high order harmonic generation,” Laser Phys. 9, 416–421 (1999).

Rizzo, A.

A. Rizzo and N. Rahman, “The linear and nonlinear susceptibilities of acetylene relevant for high order harmonic generation,” Laser Phys. 9, 416–421 (1999).

Rundquist, A.

A. Rundquist, C. Durfee, S. Backus, C. Herne, Z. Chang, M. Murnane, and H. Kapteyn, “Phase-matched Generation of Coherent Soft X-rays,” Science 280, 1412–1415 (1998).
[Crossref] [PubMed]

Schlossberg, H.

A. Szilagyi, A. Hordvik, and H. Schlossberg, “Quasi-Phase-Matching Technique for Efficient Optical Mixing and Frequency Doubling,” J. Appl. Phys. 47, 2025–2032 (1976).
[Crossref]

Seideman, T.

H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75, 543–557 (2003).
[Crossref]

T. Seideman, “Rotational-Excitation and Molecular Alignment in Intense Laser Fields,” J. Chem. Phys. 103, 7887–7896 (1995).
[Crossref]

Sharpe, S. J.

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Lambda system,” Phys. Rev. Lett. 84, 5308–5311 (2000).
[Crossref] [PubMed]

Shverdin, M.

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Lambda system,” Phys. Rev. Lett. 84, 5308–5311 (2000).
[Crossref] [PubMed]

Singer, K. D.

K. D. Singer, J. E. Sohn, and S. J. Lalama, “2nd Harmonic-Generation in Poled Polymer-Films,” Appl. Phys. Lett. 49, 248–250 (1986).
[Crossref]

Sohn, J. E.

K. D. Singer, J. E. Sohn, and S. J. Lalama, “2nd Harmonic-Generation in Poled Polymer-Films,” Appl. Phys. Lett. 49, 248–250 (1986).
[Crossref]

Somekh, S.

S. Somekh and A. Yariv, “Phase matching by periodic modulation of the nonlinear optical properties,” Opt. Commun. 6, 301 – 304 (1972).
[Crossref]

Stapelfeldt, H.

H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75, 543–557 (2003).
[Crossref]

Szilagyi, A.

A. Szilagyi, A. Hordvik, and H. Schlossberg, “Quasi-Phase-Matching Technique for Efficient Optical Mixing and Frequency Doubling,” J. Appl. Phys. 47, 2025–2032 (1976).
[Crossref]

Thompson, D. E.

D. E. Thompson, J. D. McMullen, and D. B. Anderson, “2nd-Harmonic Generation in GaAs “Stack of Plates” Using High-Power CO2-Laser Radiation,” Appl. Phys. Lett. 29, 113–115 (1976).
[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, 51–54 (2003).
[Crossref] [PubMed]

Wagner, N.

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88 (2002).
[PubMed]

Wagner, N. L.

Wang, Y. N.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, and Y. N. Wang, “Enhancement of 2nd-Harmonic Generation in Linbo3 Crystals with Periodic Laminar Ferroelectric Domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[Crossref]

Ward, J. F.

J. F. Ward and G. H. C. New, “Optical Third Harmonic Generation in Gases by a Focused Laser Beam,” Phys. Rev. 185, 57–72 (1969).
[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, 51–54 (2003).
[Crossref] [PubMed]

Weinacht, T. C.

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88 (2002).
[PubMed]

West, D. P.

D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
[Crossref]

Williams, D. L.

D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
[Crossref]

Wyss, J.

Yang, Y. S.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, and Y. N. Wang, “Enhancement of 2nd-Harmonic Generation in Linbo3 Crystals with Periodic Laminar Ferroelectric Domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[Crossref]

Yang, Z.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, and Y. N. Wang, “Enhancement of 2nd-Harmonic Generation in Linbo3 Crystals with Periodic Laminar Ferroelectric Domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[Crossref]

Yariv, A.

S. Somekh and A. Yariv, “Phase matching by periodic modulation of the nonlinear optical properties,” Opt. Commun. 6, 301 – 304 (1972).
[Crossref]

Yin, G. Y.

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Lambda system,” Phys. Rev. Lett. 84, 5308–5311 (2000).
[Crossref] [PubMed]

Zhu, J. S.

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, and Y. N. Wang, “Enhancement of 2nd-Harmonic Generation in Linbo3 Crystals with Periodic Laminar Ferroelectric Domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[Crossref]

Appl. Phys. Lett. (6)

D. E. Thompson, J. D. McMullen, and D. B. Anderson, “2nd-Harmonic Generation in GaAs “Stack of Plates” Using High-Power CO2-Laser Radiation,” Appl. Phys. Lett. 29, 113–115 (1976).
[Crossref]

C. F. Dewey and L. O. Hocker, “Enhanced Nonlinear Optical Effects in Rotationally Twinned Crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[Crossref]

D. Feng, N. B. Ming, J. F. Hong, Y. S. Yang, J. S. Zhu, Z. Yang, and Y. N. Wang, “Enhancement of 2nd-Harmonic Generation in Linbo3 Crystals with Periodic Laminar Ferroelectric Domains,” Appl. Phys. Lett. 37, 607–609 (1980).
[Crossref]

A. Feisst and P. Koidl, “Current Induced Periodic Ferroelectric Domain-Structures in Linbo3 Applied for Efficient Nonlinear Optical Frequency Mixing,” Appl. Phys. Lett. 47, 1125–1127 (1985).
[Crossref]

K. D. Singer, J. E. Sohn, and S. J. Lalama, “2nd Harmonic-Generation in Poled Polymer-Films,” Appl. Phys. Lett. 49, 248–250 (1986).
[Crossref]

B. F. Levine, C. G. Bethea, and R. A. Logan, “Phase-Matched Second-Harmonic Generation in a Liquid-Filled Waveguide,” Appl. Phys. Lett. 26, 375–377 (1975).
[Crossref]

J. Appl. Phys. (2)

J. D. McMullen, “Optical Parametric Interactions in Isotropic Materials Using a Phase-Corrected Stack of Nonlinear Dielectric Plates,” J. Appl. Phys. 46, 3076–3081 (1975).
[Crossref]

A. Szilagyi, A. Hordvik, and H. Schlossberg, “Quasi-Phase-Matching Technique for Efficient Optical Mixing and Frequency Doubling,” J. Appl. Phys. 47, 2025–2032 (1976).
[Crossref]

J. Chem. Phys. (1)

T. Seideman, “Rotational-Excitation and Molecular Alignment in Intense Laser Fields,” J. Chem. Phys. 103, 7887–7896 (1995).
[Crossref]

Laser Phys. (1)

A. Rizzo and N. Rahman, “The linear and nonlinear susceptibilities of acetylene relevant for high order harmonic generation,” Laser Phys. 9, 416–421 (1999).

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, 51–54 (2003).
[Crossref] [PubMed]

Opt. Commun. (3)

S. Somekh and A. Yariv, “Phase matching by periodic modulation of the nonlinear optical properties,” Opt. Commun. 6, 301 – 304 (1972).
[Crossref]

D. L. Williams, D. P. West, and T. A. King, “Quasi-phase matched third harmonic generation,” Opt. Commun. 148, 208–214 (1998).
[Crossref]

Zs. Bor and B. Ràcz “Group Velocity Dispersion in Prisms and its application to pulse compression and travelling-wave excitation,” Opt. Commun. 54 (1985)
[Crossref]

Opt. Lett. (2)

Phys. Rev. (1)

J. F. Ward and G. H. C. New, “Optical Third Harmonic Generation in Gases by a Focused Laser Beam,” Phys. Rev. 185, 57–72 (1969).
[Crossref]

Phys. Rev. A (1)

J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. A 127, 1918 (1962).

Phys. Rev. Lett. (5)

B. Friedrich and D. Herschbach, “Alignment and Trapping of Molecules in Intense Laser Fields,” Phys. Rev. Lett. 74, 4623–4626 (1995).
[Crossref] [PubMed]

A. J. Merriam, S. J. Sharpe, M. Shverdin, D. Manuszak, G. Y. Yin, and S. E. Harris, “Efficient nonlinear frequency conversion in an all-resonant double-Lambda system,” Phys. Rev. Lett. 84, 5308–5311 (2000).
[Crossref] [PubMed]

M. Leibscher, I. S. Averbukh, and H. Rabitz, “Molecular alignment by trains of short laser pulses,” Phys. Rev. Lett. 90 (2003).
[Crossref] [PubMed]

R. A. Bartels, T. C. Weinacht, N. Wagner, M. Baertschy, C. H. Greene, M. M. Murnane, and H. C. Kapteyn, “Phase modulation of ultrashort light pulses using molecular rotational wave packets,” Phys. Rev. Lett. 88 (2002).
[PubMed]

L. Misoguti, S. Backus, C. G. Durfee, R. Bartels, M. M. Murnane, and H. C. Kapteyn, “Generation of broadband VUV light using third-order cascaded processes,” Phys. Rev. Lett. 8701 (2001).

Rev. Mod. Phys. (1)

H. Stapelfeldt and T. Seideman, “Colloquium: Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75, 543–557 (2003).
[Crossref]

Science (2)

A. Rundquist, C. Durfee, S. Backus, C. Herne, Z. Chang, M. Murnane, and H. Kapteyn, “Phase-matched Generation of Coherent Soft X-rays,” Science 280, 1412–1415 (1998).
[Crossref] [PubMed]

R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P. Christov, Y. W. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent light at extreme ultraviolet wavelengths,” Science 297, 376–378 (2002).
[PubMed]

Other (2)

R. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, New York, 2003)

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, New York, 2001).

Cited By

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Figures (6)

Fig. 1.
Fig. 1.

Typical molecular alignment vs. time induced in a linear molecule (shown here is the time-scale for CO 2). The figure shows that the alignment of the molecular ensemble is significantly modified only near revival events. The labels τ I , τ II , and τ III refer to times of molecular alignment, anti-alignment, and non-alignment during the transient molecular alignment.

Fig. 2.
Fig. 2.

In the dynamic QPM concept, a spatio-temporally controlled ultrafast alignment pulse propagating in the x-direction creates a spatially modulated molecular alignment that evolves in time. A properly shaped alignment pulse creates a grating in the macroscopic nonlinear optical susceptibility of the molecular gas that is stationary in the group frame of the fundamental pulse.

Fig. 3.
Fig. 3.

The spatio-temporal evolution of the molecular alignment in the group frame of the fundamental pulse is shown here. For an alignment pulse with no pulse-front tilt (a), the transient alignment begins at each y location simultaneously. With propagation, the delay between the alignment and fundamental pulses, τ a-f (y), increases at the rate of the fundamental pulse group velocity (b). The variation in τ a-f (y) results in an evolution of the molecular alignment experienced by the fundamental pulse with propagation (c). By matching the group-delay of the fundamental pulse and pulse-front tilt of the alignment pulse [i.e., TWE criterion] (d), a stationary alignment-fundamental pulse delay (e) and thereby a constant alignment with propagation (f) can be selected.

Fig. 4.
Fig. 4.

A nonlinear susceptibility grating in the frame of the propagating fundamental pulse can be formed by (a) spatially modulating the alignment pulse intensity and meeting the TWE criterion or by (b) modulating the pulse front delay of the alignment pulse to modulate the alignment strength experienced by the fundamental pulse.

Fig. 5.
Fig. 5.

Conversion efficiency for a 50-fs fundamental pulse propagating in a 60 μm (dashed) and 21.7 μm [GVM-compensated] (solid) diameter hollow-core fiber filled with C 2 H 2. The insets show the third harmonic pulse shapes after 2(a), 4(b), and 6(c) cm of propagation.

Fig. 6.
Fig. 6.

Normalized spectra of fundamental (dotted line) and third harmonic pulses (solid line) after 6-cm of propagation in a nonlinear grating formed through molecular alignment for 60 μm (a) and 21.7 μm [GVM-compensated] (b) diameter hollow-core fibers.

Equations (15)

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

P = N μ I χ ̿ · E ,
μ I = α IJ E J + β IJK E J E K + γ IJKL E J E K E L + ,
χ ijkl ( 3 ) = N γ IJKL ijkl ,
γ IJKL ijkl = γ IJKL 4 π a iI a jJ a kK a lL G ( ϕ , θ ) d Ω ,
G ( θ , ϕ , t ) = J 0 P ( J 0 ) M 0 = J 0 J 0 ψ J 0 M 0 ( θ , ϕ , t ) 2 ,
ψ M o J o ( θ , ϕ , t ) = J M = J J c M o M J o J Y J M ( θ , ϕ ) e ι E J t ħ .
χ zzzz ( 3 ) , THG
χ xxxx ( 3 ) , THG = χ yyyy ( 3 ) , THG = 3 χ xxyy ( 3 ) , THG
χ xxyy ( 3 ) , THG = χ xyxy ( 3 ) , THG = χ xyyx ( 3 ) , THG = χ yyxx ( 3 ) , THG = χ yxyx ( 3 ) , THG = χ yxxy ( 3 ) , THG
χ xzzx ( 3 ) , THG = χ xxzz ( 3 ) , THG = χ yyzz ( 3 ) , THG = χ yzzy ( 3 ) , THG = χ xzxz ( 3 ) , THG = χ yzyz ( 3 ) , THG
χ zxxz ( 3 ) , THG = χ zyyz ( 3 ) , THG = χ zzxx ( 3 ) , THG = χ zzyy ( 3 ) , THG = χ zxzx ( 3 ) , THG = χ zyzy ( 3 ) , THG
cos 2 θ ( t ) = J 0 P J 0 M = J 0 J 0 ψ J 0 , M | cos 2 θ | ψ J 0 , M
cos 4 θ ( t ) = J 0 P J 0 M = J 0 J 0 ψ J 0 , M | cos 4 θ | ψ J 0 , M
A 1 ( y , t ) y + 1 v g 1 A 1 ( y , t ) t i D ˜ 1 A 1 ( y , t ) = γ 1 [ 3 A 1 A 1 2 + 6 A 1 A 3 2 + 3 A 3 A 1 * A 1 * e i Δ ky ]
A 3 ( y , t ) y + 1 v g 3 A 3 ( y , t ) t i D ˜ 3 A 3 ( y , t ) = γ 3 [ 3 A 3 A 3 2 + 6 A 3 A 1 2 + A 1 3 e i Δ ky ] ,

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