Abstract

It is shown theoretically that second harmonic generation can be quasi-phase-matched by using a pump beam consisting of a forward propagating field and a counterpropagating pulse train. The counterpropagating setup can also be used for direct measurement of the coherence length of the nonlinear process which can determine the dispersion properties of the medium.

© 2008 Optical Society of America

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  1. J. A. Armstrong, N.  Bloembergen, J.  Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
    [CrossRef]
  2. 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, 2631-2654 (1992).
    [CrossRef]
  3. S.  Somekh and A.  Yariv, "Phase-matchable nonlinear optical interactions in periodic thin films," Appl. Phys. Lett. 21, 140 (1972).
    [CrossRef]
  4. C.  Tang and P.  Bey, "Phase matching in second-harmonic generation using artificial periodic structures," IEEE J. Quantum Electron 9, 9-17 (1973).
    [CrossRef]
  5. M.  Yamada, N.  Nada, M.  Saitoh, and K.  Watanabe, "First-order quasi-phase matched linbo$_3$ waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
    [CrossRef]
  6. A.  Kewitsch, M.  Segev, A.  Yariv, G.  Salamo, T.  Towe, E.  Sharp, and R.  Neurgaonkar, "Tunable quasi-phase matching using dynamic ferroelectric domain gratings induced by photorefractive space-charge fields," Appl. Phys. Lett. 64, 3068 (1994).
    [CrossRef]
  7. D.  Hum and M.  Fejer, "Quasi-phasematching," Physique 8, 180-198 (2007).
    [CrossRef]
  8. P.  Dong and A.  Kirk, "Nonlinear Frequency Conversion in Waveguide Directional Couplers," Phys. Rev. Lett. 93, 133901 (2004).
    [CrossRef] [PubMed]
  9. J.  Peatross, M.  Fedorov, and K.  Kulander, "Intensity-dependent phase-matching effects in harmonic generation," J. Opt. Soc. Am. B 12, 863-870 (1995).
    [CrossRef]
  10. P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994-1997 (1993).
    [CrossRef] [PubMed]
  11. S. L. Voronov, I.  Kohl, J. B. Madsen, J.  Simmons, N.  Terry, J.  Titensor, Q.  Wang, and J.  Peatross, "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001).
    [CrossRef] [PubMed]
  12. J.  Peatross, S.  Voronov, and I.  Prokopovich, "Selective zoning of high harmonic emission using counter-propagating light," Opt. Express 1, 114-125 (1997).
    [CrossRef] [PubMed]
  13. M.  Landreman, K.  O'Keeffe, T.  Robinson, M.  Zepf, B.  Dromey, and S.  Hooker, "Comparison of parallel and perpendicular polarized counterpropagating light for suppressing high harmonic generation," J. Opt. Soc. Am. B 24, 2421-2427 (2007).
    [CrossRef]
  14. 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, 2975-2977 (2007).
    [CrossRef] [PubMed]
  15. 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, 270-275 (2007).
    [CrossRef]
  16. A.  Lytle, X.  Zhang, P.  Arpin, O.  Cohen, M.  Murnane, and H.  Kapteyn, "Quasi-phase matching of high-order harmonic generation at high photon energies using counterpropagating pulses," Opt. Lett. 33, 174-176 (2008).
    [CrossRef] [PubMed]
  17. H.  Kapteyn, O.  Cohen, I.  Christov, and M.  Murnane, "Harnessing Attosecond Science in the Quest for Coherent X-rays," Science 317, 775-778 (2007).
    [CrossRef] [PubMed]
  18. A. L. Lytle, X.  Zhang, J.  Peatross, M. M. Murnane, H. C. Kapteyn, and O.  Cohen, "Probe of high-order harmonic generation in a hollow waveguide geometry using counterpropagating light," Phys. Rev. Lett. 98, 123904 (2007).
    [CrossRef] [PubMed]
  19. A.  Lytle, X.  Zhang, R.  Sandberg, O.  Cohen, H.  Kapteyn, and M.  Murnane, "Quasi-phase matching and characterization of high-order harmonic generation in hollow waveguides using counterpropagating light," Opt. Express 16, 6544-6566 (2008).
    [CrossRef] [PubMed]
  20. 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, 053902 (2007).
    [CrossRef] [PubMed]
  21. R.  Shelby, D.  Smith, and S.  Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  22. O.  Gayer, Z.  Sacks, E.  Galun, and A.  Arie, "Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3," Appl. Phys. B 91, 343-348 (2008).
    [CrossRef]
  23. S.  Diddams and J.  Diels, "Dispersion measurements with white-light interferometry," J. Opt. Soc. Am. B 13, 1120-1129 (1996).
    [CrossRef]
  24. A.  L'Huillier, K.  Schafer, and K.  Kulander, "Higher-order harmonic generation in xenon at 1064 nm: The role of phase matching," Phys. Rev. Lett. 66, 2200-2203 (1991).
    [CrossRef] [PubMed]
  25. R.  Myers, N.  Mukherjee, S.  Brueck et al., "Large second-order nonlinearity in poled fused silica," Opt. Lett. 16, 1732-1734 (1991).
    [CrossRef] [PubMed]
  26. M.  Arbore, O.  Marco, and M.  Fejer, "Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings," Opt. Lett. 22, 865-867 (1997).
    [CrossRef]
  27. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001).

2008 (3)

2007 (7)

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, 053902 (2007).
[CrossRef] [PubMed]

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

A. L. Lytle, X.  Zhang, J.  Peatross, M. M. Murnane, H. C. Kapteyn, and O.  Cohen, "Probe of high-order harmonic generation in a hollow waveguide geometry using counterpropagating light," Phys. Rev. Lett. 98, 123904 (2007).
[CrossRef] [PubMed]

M.  Landreman, K.  O'Keeffe, T.  Robinson, M.  Zepf, B.  Dromey, and S.  Hooker, "Comparison of parallel and perpendicular polarized counterpropagating light for suppressing high harmonic generation," J. Opt. Soc. Am. B 24, 2421-2427 (2007).
[CrossRef]

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, 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, 270-275 (2007).
[CrossRef]

D.  Hum and M.  Fejer, "Quasi-phasematching," Physique 8, 180-198 (2007).
[CrossRef]

2004 (1)

P.  Dong and A.  Kirk, "Nonlinear Frequency Conversion in Waveguide Directional Couplers," Phys. Rev. Lett. 93, 133901 (2004).
[CrossRef] [PubMed]

2001 (2)

S. L. Voronov, I.  Kohl, J. B. Madsen, J.  Simmons, N.  Terry, J.  Titensor, Q.  Wang, and J.  Peatross, "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001).
[CrossRef] [PubMed]

R.  Shelby, D.  Smith, and S.  Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

1997 (2)

1996 (1)

1995 (1)

1994 (1)

A.  Kewitsch, M.  Segev, A.  Yariv, G.  Salamo, T.  Towe, E.  Sharp, and R.  Neurgaonkar, "Tunable quasi-phase matching using dynamic ferroelectric domain gratings induced by photorefractive space-charge fields," Appl. Phys. Lett. 64, 3068 (1994).
[CrossRef]

1993 (2)

M.  Yamada, N.  Nada, M.  Saitoh, and K.  Watanabe, "First-order quasi-phase matched linbo$_3$ waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

P. B. Corkum, "Plasma perspective on strong field multiphoton ionization," Phys. Rev. Lett. 71, 1994-1997 (1993).
[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, 2631-2654 (1992).
[CrossRef]

1991 (2)

A.  L'Huillier, K.  Schafer, and K.  Kulander, "Higher-order harmonic generation in xenon at 1064 nm: The role of phase matching," Phys. Rev. Lett. 66, 2200-2203 (1991).
[CrossRef] [PubMed]

R.  Myers, N.  Mukherjee, S.  Brueck et al., "Large second-order nonlinearity in poled fused silica," Opt. Lett. 16, 1732-1734 (1991).
[CrossRef] [PubMed]

1973 (1)

C.  Tang and P.  Bey, "Phase matching in second-harmonic generation using artificial periodic structures," IEEE J. Quantum Electron 9, 9-17 (1973).
[CrossRef]

1972 (1)

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

1962 (1)

J. A. Armstrong, N.  Bloembergen, J.  Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Arbore, M.

Arie, A.

O.  Gayer, Z.  Sacks, E.  Galun, and A.  Arie, "Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3," Appl. Phys. B 91, 343-348 (2008).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N.  Bloembergen, J.  Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Arpin, P.

Bey, P.

C.  Tang and P.  Bey, "Phase matching in second-harmonic generation using artificial periodic structures," IEEE J. Quantum Electron 9, 9-17 (1973).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N.  Bloembergen, J.  Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Brueck, S.

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, 2631-2654 (1992).
[CrossRef]

Christov, I.

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

Cohen, O.

A.  Lytle, X.  Zhang, P.  Arpin, O.  Cohen, M.  Murnane, and H.  Kapteyn, "Quasi-phase matching of high-order harmonic generation at high photon energies using counterpropagating pulses," Opt. Lett. 33, 174-176 (2008).
[CrossRef] [PubMed]

A.  Lytle, X.  Zhang, R.  Sandberg, O.  Cohen, H.  Kapteyn, and M.  Murnane, "Quasi-phase matching and characterization of high-order harmonic generation in hollow waveguides using counterpropagating light," Opt. Express 16, 6544-6566 (2008).
[CrossRef] [PubMed]

A. L. Lytle, X.  Zhang, J.  Peatross, M. M. Murnane, H. C. Kapteyn, and O.  Cohen, "Probe of high-order harmonic generation in a hollow waveguide geometry using counterpropagating light," Phys. Rev. Lett. 98, 123904 (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, 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, 270-275 (2007).
[CrossRef]

H.  Kapteyn, O.  Cohen, I.  Christov, and M.  Murnane, "Harnessing Attosecond Science in the Quest for Coherent X-rays," Science 317, 775-778 (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, 2975-2977 (2007).
[CrossRef] [PubMed]

Corkum, P. B.

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

Diddams, S.

Diels, J.

Dong, P.

P.  Dong and A.  Kirk, "Nonlinear Frequency Conversion in Waveguide Directional Couplers," Phys. Rev. Lett. 93, 133901 (2004).
[CrossRef] [PubMed]

Dromey, B.

Ducuing, J.

J. A. Armstrong, N.  Bloembergen, J.  Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Fedorov, M.

Fejer, M.

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, 2631-2654 (1992).
[CrossRef]

Galun, E.

O.  Gayer, Z.  Sacks, E.  Galun, and A.  Arie, "Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3," Appl. Phys. B 91, 343-348 (2008).
[CrossRef]

Gayer, O.

O.  Gayer, Z.  Sacks, E.  Galun, and A.  Arie, "Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3," Appl. Phys. B 91, 343-348 (2008).
[CrossRef]

Hooker, S.

Hum, D.

D.  Hum and M.  Fejer, "Quasi-phasematching," Physique 8, 180-198 (2007).
[CrossRef]

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, 2631-2654 (1992).
[CrossRef]

Kapteyn, H.

Kapteyn, H. C.

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, 270-275 (2007).
[CrossRef]

A. L. Lytle, X.  Zhang, J.  Peatross, M. M. Murnane, H. C. Kapteyn, and O.  Cohen, "Probe of high-order harmonic generation in a hollow waveguide geometry using counterpropagating light," Phys. Rev. Lett. 98, 123904 (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, 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, 053902 (2007).
[CrossRef] [PubMed]

Kewitsch, A.

A.  Kewitsch, M.  Segev, A.  Yariv, G.  Salamo, T.  Towe, E.  Sharp, and R.  Neurgaonkar, "Tunable quasi-phase matching using dynamic ferroelectric domain gratings induced by photorefractive space-charge fields," Appl. Phys. Lett. 64, 3068 (1994).
[CrossRef]

Kirk, A.

P.  Dong and A.  Kirk, "Nonlinear Frequency Conversion in Waveguide Directional Couplers," Phys. Rev. Lett. 93, 133901 (2004).
[CrossRef] [PubMed]

Kohl, I.

S. L. Voronov, I.  Kohl, J. B. Madsen, J.  Simmons, N.  Terry, J.  Titensor, Q.  Wang, and J.  Peatross, "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001).
[CrossRef] [PubMed]

Kulander, K.

J.  Peatross, M.  Fedorov, and K.  Kulander, "Intensity-dependent phase-matching effects in harmonic generation," J. Opt. Soc. Am. B 12, 863-870 (1995).
[CrossRef]

A.  L'Huillier, K.  Schafer, and K.  Kulander, "Higher-order harmonic generation in xenon at 1064 nm: The role of phase matching," Phys. Rev. Lett. 66, 2200-2203 (1991).
[CrossRef] [PubMed]

Landreman, M.

L'Huillier, A.

A.  L'Huillier, K.  Schafer, and K.  Kulander, "Higher-order harmonic generation in xenon at 1064 nm: The role of phase matching," Phys. Rev. Lett. 66, 2200-2203 (1991).
[CrossRef] [PubMed]

Lytle, A.

Lytle, A. L.

A. L. Lytle, X.  Zhang, J.  Peatross, M. M. Murnane, H. C. Kapteyn, and O.  Cohen, "Probe of high-order harmonic generation in a hollow waveguide geometry using counterpropagating light," Phys. Rev. Lett. 98, 123904 (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, 270-275 (2007).
[CrossRef]

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, 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, 053902 (2007).
[CrossRef] [PubMed]

Madsen, J. B.

S. L. Voronov, I.  Kohl, J. B. Madsen, J.  Simmons, N.  Terry, J.  Titensor, Q.  Wang, and J.  Peatross, "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001).
[CrossRef] [PubMed]

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, 2631-2654 (1992).
[CrossRef]

Marco, O.

Mukherjee, N.

Murnane, M.

Murnane, M. M.

A. L. Lytle, X.  Zhang, J.  Peatross, M. M. Murnane, H. C. Kapteyn, and O.  Cohen, "Probe of high-order harmonic generation in a hollow waveguide geometry using counterpropagating light," Phys. Rev. Lett. 98, 123904 (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, 270-275 (2007).
[CrossRef]

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, 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, 053902 (2007).
[CrossRef] [PubMed]

Myers, R.

Nada, N.

M.  Yamada, N.  Nada, M.  Saitoh, and K.  Watanabe, "First-order quasi-phase matched linbo$_3$ waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

Neurgaonkar, R.

A.  Kewitsch, M.  Segev, A.  Yariv, G.  Salamo, T.  Towe, E.  Sharp, and R.  Neurgaonkar, "Tunable quasi-phase matching using dynamic ferroelectric domain gratings induced by photorefractive space-charge fields," Appl. Phys. Lett. 64, 3068 (1994).
[CrossRef]

O'Keeffe, K.

Peatross, J.

A. L. Lytle, X.  Zhang, J.  Peatross, M. M. Murnane, H. C. Kapteyn, and O.  Cohen, "Probe of high-order harmonic generation in a hollow waveguide geometry using counterpropagating light," Phys. Rev. Lett. 98, 123904 (2007).
[CrossRef] [PubMed]

S. L. Voronov, I.  Kohl, J. B. Madsen, J.  Simmons, N.  Terry, J.  Titensor, Q.  Wang, and J.  Peatross, "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001).
[CrossRef] [PubMed]

J.  Peatross, S.  Voronov, and I.  Prokopovich, "Selective zoning of high harmonic emission using counter-propagating light," Opt. Express 1, 114-125 (1997).
[CrossRef] [PubMed]

J.  Peatross, M.  Fedorov, and K.  Kulander, "Intensity-dependent phase-matching effects in harmonic generation," J. Opt. Soc. Am. B 12, 863-870 (1995).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N.  Bloembergen, J.  Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Popmintchev, T.

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, 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, 053902 (2007).
[CrossRef] [PubMed]

Prokopovich, I.

Robinson, T.

Sacks, Z.

O.  Gayer, Z.  Sacks, E.  Galun, and A.  Arie, "Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3," Appl. Phys. B 91, 343-348 (2008).
[CrossRef]

Saitoh, M.

M.  Yamada, N.  Nada, M.  Saitoh, and K.  Watanabe, "First-order quasi-phase matched linbo$_3$ waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

Salamo, G.

A.  Kewitsch, M.  Segev, A.  Yariv, G.  Salamo, T.  Towe, E.  Sharp, and R.  Neurgaonkar, "Tunable quasi-phase matching using dynamic ferroelectric domain gratings induced by photorefractive space-charge fields," Appl. Phys. Lett. 64, 3068 (1994).
[CrossRef]

Sandberg, R.

Schafer, K.

A.  L'Huillier, K.  Schafer, and K.  Kulander, "Higher-order harmonic generation in xenon at 1064 nm: The role of phase matching," Phys. Rev. Lett. 66, 2200-2203 (1991).
[CrossRef] [PubMed]

Schultz, S.

R.  Shelby, D.  Smith, and S.  Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Segev, M.

A.  Kewitsch, M.  Segev, A.  Yariv, G.  Salamo, T.  Towe, E.  Sharp, and R.  Neurgaonkar, "Tunable quasi-phase matching using dynamic ferroelectric domain gratings induced by photorefractive space-charge fields," Appl. Phys. Lett. 64, 3068 (1994).
[CrossRef]

Sharp, E.

A.  Kewitsch, M.  Segev, A.  Yariv, G.  Salamo, T.  Towe, E.  Sharp, and R.  Neurgaonkar, "Tunable quasi-phase matching using dynamic ferroelectric domain gratings induced by photorefractive space-charge fields," Appl. Phys. Lett. 64, 3068 (1994).
[CrossRef]

Shelby, R.

R.  Shelby, D.  Smith, and S.  Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Simmons, J.

S. L. Voronov, I.  Kohl, J. B. Madsen, J.  Simmons, N.  Terry, J.  Titensor, Q.  Wang, and J.  Peatross, "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001).
[CrossRef] [PubMed]

Smith, D.

R.  Shelby, D.  Smith, and S.  Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Somekh, S.

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

Tang, C.

C.  Tang and P.  Bey, "Phase matching in second-harmonic generation using artificial periodic structures," IEEE J. Quantum Electron 9, 9-17 (1973).
[CrossRef]

Terry, N.

S. L. Voronov, I.  Kohl, J. B. Madsen, J.  Simmons, N.  Terry, J.  Titensor, Q.  Wang, and J.  Peatross, "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001).
[CrossRef] [PubMed]

Titensor, J.

S. L. Voronov, I.  Kohl, J. B. Madsen, J.  Simmons, N.  Terry, J.  Titensor, Q.  Wang, and J.  Peatross, "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001).
[CrossRef] [PubMed]

Towe, T.

A.  Kewitsch, M.  Segev, A.  Yariv, G.  Salamo, T.  Towe, E.  Sharp, and R.  Neurgaonkar, "Tunable quasi-phase matching using dynamic ferroelectric domain gratings induced by photorefractive space-charge fields," Appl. Phys. Lett. 64, 3068 (1994).
[CrossRef]

Voronov, S.

Voronov, S. L.

S. L. Voronov, I.  Kohl, J. B. Madsen, J.  Simmons, N.  Terry, J.  Titensor, Q.  Wang, and J.  Peatross, "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001).
[CrossRef] [PubMed]

Wang, Q.

S. L. Voronov, I.  Kohl, J. B. Madsen, J.  Simmons, N.  Terry, J.  Titensor, Q.  Wang, and J.  Peatross, "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (2001).
[CrossRef] [PubMed]

Watanabe, K.

M.  Yamada, N.  Nada, M.  Saitoh, and K.  Watanabe, "First-order quasi-phase matched linbo$_3$ waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

Yamada, M.

M.  Yamada, N.  Nada, M.  Saitoh, and K.  Watanabe, "First-order quasi-phase matched linbo$_3$ waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

Yariv, A.

A.  Kewitsch, M.  Segev, A.  Yariv, G.  Salamo, T.  Towe, E.  Sharp, and R.  Neurgaonkar, "Tunable quasi-phase matching using dynamic ferroelectric domain gratings induced by photorefractive space-charge fields," Appl. Phys. Lett. 64, 3068 (1994).
[CrossRef]

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

Zepf, M.

Zhang, X.

A.  Lytle, X.  Zhang, P.  Arpin, O.  Cohen, M.  Murnane, and H.  Kapteyn, "Quasi-phase matching of high-order harmonic generation at high photon energies using counterpropagating pulses," Opt. Lett. 33, 174-176 (2008).
[CrossRef] [PubMed]

A.  Lytle, X.  Zhang, R.  Sandberg, O.  Cohen, H.  Kapteyn, and M.  Murnane, "Quasi-phase matching and characterization of high-order harmonic generation in hollow waveguides using counterpropagating light," Opt. Express 16, 6544-6566 (2008).
[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, 053902 (2007).
[CrossRef] [PubMed]

A. L. Lytle, X.  Zhang, J.  Peatross, M. M. Murnane, H. C. Kapteyn, and O.  Cohen, "Probe of high-order harmonic generation in a hollow waveguide geometry using counterpropagating light," Phys. Rev. Lett. 98, 123904 (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, 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, 270-275 (2007).
[CrossRef]

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, 270-275 (2007).
[CrossRef]

Appl. Phys. B (1)

O.  Gayer, Z.  Sacks, E.  Galun, and A.  Arie, "Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3," Appl. Phys. B 91, 343-348 (2008).
[CrossRef]

Appl. Phys. Lett. (3)

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

M.  Yamada, N.  Nada, M.  Saitoh, and K.  Watanabe, "First-order quasi-phase matched linbo$_3$ waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

A.  Kewitsch, M.  Segev, A.  Yariv, G.  Salamo, T.  Towe, E.  Sharp, and R.  Neurgaonkar, "Tunable quasi-phase matching using dynamic ferroelectric domain gratings induced by photorefractive space-charge fields," Appl. Phys. Lett. 64, 3068 (1994).
[CrossRef]

IEEE J. Quantum Electron (2)

C.  Tang and P.  Bey, "Phase matching in second-harmonic generation using artificial periodic structures," IEEE J. Quantum Electron 9, 9-17 (1973).
[CrossRef]

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, 2631-2654 (1992).
[CrossRef]

J. Opt. Soc. Am. B (3)

Nat. Phys. (1)

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, 270-275 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. (1)

J. A. Armstrong, N.  Bloembergen, J.  Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Phys. Rev. Lett. (6)

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

S. L. Voronov, I.  Kohl, J. B. Madsen, J.  Simmons, N.  Terry, J.  Titensor, Q.  Wang, and J.  Peatross, "Control of laser high-harmonic generation with counterpropagating light," Phys. Rev. Lett. 87, 133902 (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, 053902 (2007).
[CrossRef] [PubMed]

P.  Dong and A.  Kirk, "Nonlinear Frequency Conversion in Waveguide Directional Couplers," Phys. Rev. Lett. 93, 133901 (2004).
[CrossRef] [PubMed]

A.  L'Huillier, K.  Schafer, and K.  Kulander, "Higher-order harmonic generation in xenon at 1064 nm: The role of phase matching," Phys. Rev. Lett. 66, 2200-2203 (1991).
[CrossRef] [PubMed]

A. L. Lytle, X.  Zhang, J.  Peatross, M. M. Murnane, H. C. Kapteyn, and O.  Cohen, "Probe of high-order harmonic generation in a hollow waveguide geometry using counterpropagating light," Phys. Rev. Lett. 98, 123904 (2007).
[CrossRef] [PubMed]

Physique (1)

D.  Hum and M.  Fejer, "Quasi-phasematching," Physique 8, 180-198 (2007).
[CrossRef]

Science (2)

R.  Shelby, D.  Smith, and S.  Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001).
[CrossRef] [PubMed]

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

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2001).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic for quasi phase matching SHG using a counterpropagating pulse train. Red – forward propagating CW pump beam and backward propagating pulses. Blue – second harmonic radiation.

Fig. 2.
Fig. 2.

Simulations of quasi phase matching of SHG using a counterpropagating pulse train. A forward CW beam encounters a counterpropagating pulse every other coherence length. The dashed line shows the SHG growth without counterpropagating pulses present (a) Rectangular counterpropagating pulses - each overlapping with the CW beam over one coherence length. l c =5λ/2 and E b /E f =4.4 for suppression of the accumulated nonlinear field. In bold (red) - with the SVEA. (b) Similar to (a) but with E b /E f =2.73 for enhancement of the nonlinear response. (c) Case of super Gaussian counterpropagating pulses of order 20 - each overlapping with the CW beam for 0.7 l c while l c =14λ/2 and E b /E f =8.64. (d) Similar to (c) but with l c =7λ/2 and E b /E f =5.56. (Note a.u. signifies arbitrary units).

Fig. 3.
Fig. 3.

Extracting the coherence length of an SHG process using a forward propagating super Gaussian pulse and a pair of two counterpropagating super Gaussian pulses spaced by varying intervals. The super Gaussian order of all pulses is 40. The width of all pulses is l c/2, while l c =15λ/2 and E b /E f =9. The small captions schematically depict the relative spatial pulse separation in the moving frame of the forward propagating pulse. The ± regions are one coherence length long. All (+) regions contribute in-phase to the SHG signal, and similarly for the (-) regions, while the (+) and (-) regions are out of phase with each other. (a) Separation between the counterpropagating pulses is 4l c . (b) Separation is 6l c (c) Varying the separation continuously results in a periodic signal of period 4l c . (Note a.u. signifies arbitrary units).

Equations (8)

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

E ( z , t ) = E f ( z , t ) e i ( ω t kz ) + E b ( z , t ) e i ( ω t + kz ) ,
E ( z , t ) = E f ( z , t ) + E b ( z , t ) e 2 ikz e i ( ω t kz ) ,
P NL ( 2 ) ( z , t ) = κ [ E f ( z , t ) + E b ( z , t ) e 2 ikz ] 2 e 2 i ( ω t kz ) ,
( 2 z 2 2 ik 2 ω z ) E 2 w ( z ) = κ [ E f + E ˜ b ( z ) e 2 ikz ] 2 e i Δ kz ,
E 2 w ( z = L ) = κ z = 0 L [ E f + E ˜ b ( z ) e 2 ikz ] 2 e i Δ kz dz ,
E 2 w ( z = λ 2 ) = κ E f 2 m = 0 2 ( 2 m ) ( E b E f ) m z = 0 π k e ikz · 2 m + i Δ kz dz .
E 2 w ( z = L , t ) = κ e 2 i ω t z = 0 L [ E f + E ˜ b ( z + t v g ) e 2 ikz ] 2 e i Δ kz dz ,
( z + 1 v g 2 t ) E 2 w ( z , t ) = κ [ E f ( z , t ) + E b ( z , t ) e 2 ikz ] 2 e i Δ kz ,

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