Abstract

We study the anisotropic nature of the Kerr nonlinear response in a beta-barium borate (β-BaB2O4, BBO) nonlinear crystal. The focus is on determining the relevant χ(3) cubic tensor components that affect interaction of type I cascaded second-harmonic generation. Various experiments in the literature are analyzed and we correct the data from some of the experiments for contributions from cascading as well as for updated material parameters. We also perform an additional experimental measurement of the Kerr nonlinear tensor component responsible for self-phase modulation in cascading, and we show that the average value of 14 different measurements is considerably larger than what has been used to date. Our own measurements are consistent with this average value. We also treat data measurements for mixtures of tensor components, and by disentangling them we present for the first time a complete list that we propose as reference of the four major cubic tensor components in BBO. We finally discuss the impact of using the cubic anisotropic response in ultrafast cascading experiments in BBO.

© 2013 OSA

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  2. J. M. R. Thomas and J. P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun.4, 329–334 (1972).
    [CrossRef]
  3. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett.17, 28–30 (1992).
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  4. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron.28, 1691–1740 (1996).
    [CrossRef]
  5. X. Liu, L.-J. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascaded χ(2) : χ(2) nonlinearity,” Opt. Lett.24, 1777–1779 (1999).
    [CrossRef]
  6. K. Beckwitt, F. W. Wise, L. Qian, L. A. Walker, and E. Canto-Said, “Compensation for self-focusing by use of cascade quadratic nonlinearity,” Opt. Lett.26, 1696–1698 (2001).
    [CrossRef]
  7. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B19, 2505–2510 (2002).
    [CrossRef]
  8. F. Ö. Ilday, K. Beckwitt, Y.-F. Chen, H. Lim, and F. W. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B21, 376–383 (2004).
    [CrossRef]
  9. J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett.31, 1881–1883 (2006).
    [CrossRef] [PubMed]
  10. J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett.97, 073903 (2006).
    [CrossRef] [PubMed]
  11. J. Moses, E. Alhammali, J. M. Eichenholz, and F. W. Wise, “Efficient high-energy femtosecond pulse compression in quadratic media with flattop beams,” Opt. Lett.32, 2469–2471 (2007).
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  12. M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical Cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A82, 063806 (2010).
    [CrossRef]
  13. H. Tan, G. P. Banfi, and A. Tomaselli, “Optical frequency mixing through cascaded second-order processes in beta-barium borate,” Appl. Phys. Lett.63, 2472–2474 (1993).
    [CrossRef]
  14. F. Hache, A. Zéboulon, G. Gallot, and G. M. Gale, “Cascaded second-order effects in the femtosecond regime in β-barium borate: self-compression in a visible femtosecond optical parametric oscillator,” Opt. Lett.20, 1556–1558 (1995).
    [CrossRef] [PubMed]
  15. J. Moses, B. A. Malomed, and F. W. Wise, “Self-steepening of ultrashort optical pulses without self-phase modulation,” Phys. Rev. A76, 021802(R) (2007).
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  16. R. DeSalvo, A. A. Said, D. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron.32, 1324–1333 (1996).
    [CrossRef]
  17. M. Sheik-Bahae and M. Ebrahimzadeh, “Measurements of nonlinear refraction in the second-order χ(2) materials KTiOPO4, KNbO3, β-BaB2O4 and LiB3O5,” Opt. Commun.142, 294–298 (1997).
    [CrossRef]
  18. H. Li, F. Zhou, X. Zhang, and W. Ji, “Bound electronic kerr effect and self-focusing induced damage in second-harmonic-generation crystals,” Opt. Commun.144, 75–81 (1997).
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  19. H. P. Li, C. H. Kam, Y. L. Lam, and W. Ji, “Femtosecond Z-scan measurements of nonlinear refraction in nonlinear optical crystals,” Opt. Mater.15, 237–242 (2001).
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  20. R. Ganeev, I. Kulagin, A. Ryasnyanskii, R. Tugushev, and T. Usmanov, “The nonlinear refractive indices and nonlinear third-order susceptibilities of quadratic crystals,” Opt. Spectrosc.94, 561–568 (2003). [Opt. Spektrosk. 94, 615–623 (2003)].
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  21. P. S. Banks, M. D. Feit, and M. D. Perry, “High-intensity third-harmonic generation,” J. Opt. Soc. Am. B19, 102–118 (2002).
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  22. J. E. Midwinter and J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization,” Br. J. Appl. Phys.16, 1667–1674 (1965).
    [CrossRef]
  23. C. Wang and E. Baardsen, “Optical third harmonic generation using mode-locked and non-mode-locked lasers,” Appl. Phys. Lett.15, 396–397 (1969).
    [CrossRef]
  24. M. Sheik-Bahae, D. Hutchings, D. Hagan, and E. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron.27, 1296 –1309 (1991).
    [CrossRef]
  25. S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett.84, 1055–1057 (2004).
    [CrossRef]
  26. X. Zeng, S. Ashihara, X. Chen, T. Shimura, and K. Kuroda, “Two-color pulse compression in aperiodically-poled lithium niobate,” Opt. Commun.281, 4499–4503 (2008).
    [CrossRef]
  27. B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett.109, 043902 (2012).
    [CrossRef] [PubMed]
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  29. D. N. Nikogosyan, “Beta barium borate (BBO) - A review of its properties and applications,” Appl. Phys. A52, 359–368 (1991).
    [CrossRef]
  30. M. Bache and F. W. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A81, 053815 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  34. R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett.5, 17–19 (1964).
    [CrossRef]
  35. J. J. Wynne, “Optical third-order mixing in GaAs, Ge, Si, and InAs,” Phys. Rev.178, 1295–1303 (1969).
    [CrossRef]
  36. W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulæ,” Opt. Express18, 6613–6620 (2010).
    [CrossRef] [PubMed]
  37. M. Bache, J. Moses, and F. W. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B24, 2752–2762 (2007).
    [CrossRef]
  38. M. Sheik-Bahae, A. Said, T.-H. Wei, D. Hagan, and E. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron.26, 760–769 (1990).
    [CrossRef]
  39. T. D. Krauss and F. W. Wise, “Femtosecond measurement of nonlinear absorption and refraction in CdS, ZnSe, and ZnS,” Appl. Phys. Lett.65, 1739–1741 (1994).
    [CrossRef]
  40. A. Gnoli, L. Razzari, and M. Righini, “Z-scan measurements using high repetition rate lasers: how to manage thermal effects,” Opt. Express13, 7976–7981 (2005).
    [CrossRef] [PubMed]
  41. E. Nibbering, M. Franco, B. Prade, G. Grillon, C. L. Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun.119, 479–484 (1995).
    [CrossRef]
  42. I. Shoji, H. Nakamura, K. Ohdaira, T. Kondo, R. Ito, T. Okamoto, K. Tatsuki, and S. Kubota, “Absolute measurement of second-order nonlinear-optical coefficients of β-BaB2O4 for visible to ultraviolet second-harmonic wavelengths,” J. Opt. Soc. Am. B16, 620–624 (1999).
    [CrossRef]
  43. D. Zhang, Y. Kong, and J. Zhang, “Optical parametric properties of 532-nm-pumped beta-barium-borate near the infrared absorption edge,” Opt. Commun.184, 485–491 (2000).
    [CrossRef]
  44. R. A. Ganeev, private communication (2012).
  45. J. A. Moses, private communication (2010).
  46. H. Guo, X. Zeng, B. Zhou, and M. Bache, “Electric field modeling and self-steepening counterbalance of cascading nonlinear soliton pulse compression,” (submitted to J. Opt. Soc. Am. B), arXiv:1210.5903.
  47. C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: Cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B61, 10688–10701 (2000).
    [CrossRef]
  48. M. Sheik-Bahae, “Femtosecond kerr-lens autocorrelation,” Opt. Lett.22, 399–401 (1997).
    [CrossRef] [PubMed]
  49. Y. Fan, R. Eckardt, R. Byer, C. Chen, and A. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron.25, 1196 –1199 (1989).
    [CrossRef]
  50. N. Boling, A. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron.14, 601–608 (1978).
    [CrossRef]
  51. M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett.32, 2490–2492 (2007).
    [CrossRef] [PubMed]
  52. M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express16, 3273–3287 (2008).
    [CrossRef] [PubMed]
  53. X. Zeng, H. Guo, B. Zhou, and M. Bache, “Soliton compression to few-cycle pulses with a high quality factor by engineering cascaded quadratic nonlinearities,” Opt. Express20, 27071–27082 (2012). ArXiv:1210.5928.
    [CrossRef] [PubMed]
  54. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).
  55. A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioners guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top.199, 5–76 (2011).
    [CrossRef]
  56. M. Bache, F. Eilenberger, and S. Minardi, “Higher-order Kerr effect and harmonic cascading in gases,” Opt. Lett.37, 4612–4614 (2012).
    [PubMed]
  57. C. Chen, B. Wu, A. Jiang, and G. You, “A new-type ultraviolet SHG crystal - beta-BaB2O4,” Sci. Sin., Ser. B28, 235–243 (1985).
  58. R. S. Klein, G. E. Kugel, A. Maillard, A. Sifi, and K. Polgar, “Absolute non-linear optical coefficients measurements of BBO single crystal and determination of angular acceptance by second harmonic generation,” Opt. Mater.22, 163–169 (2003).
    [CrossRef]
  59. R. C. Eckardt and G. C. Cattela, “Characterization techniques for second-order nonlinear optical materials,” Proc. SPIE5337, 1–10 (2004).
    [CrossRef]

2012 (3)

2011 (2)

C. R. Phillips, C. Langrock, J. S. Pelc, M. M. Fejer, I. Hartl, and M. E. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express19, 18754–18773 (2011).
[CrossRef] [PubMed]

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioners guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top.199, 5–76 (2011).
[CrossRef]

2010 (3)

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical Cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A82, 063806 (2010).
[CrossRef]

M. Bache and F. W. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A81, 053815 (2010).
[CrossRef]

W. Ettoumi, Y. Petit, J. Kasparian, and J.-P. Wolf, “Generalized Miller formulæ,” Opt. Express18, 6613–6620 (2010).
[CrossRef] [PubMed]

2008 (2)

M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express16, 3273–3287 (2008).
[CrossRef] [PubMed]

X. Zeng, S. Ashihara, X. Chen, T. Shimura, and K. Kuroda, “Two-color pulse compression in aperiodically-poled lithium niobate,” Opt. Commun.281, 4499–4503 (2008).
[CrossRef]

2007 (5)

2006 (2)

J. Moses and F. W. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett.31, 1881–1883 (2006).
[CrossRef] [PubMed]

J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett.97, 073903 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (3)

F. Ö. Ilday, K. Beckwitt, Y.-F. Chen, H. Lim, and F. W. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B21, 376–383 (2004).
[CrossRef]

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett.84, 1055–1057 (2004).
[CrossRef]

R. C. Eckardt and G. C. Cattela, “Characterization techniques for second-order nonlinear optical materials,” Proc. SPIE5337, 1–10 (2004).
[CrossRef]

2003 (2)

R. S. Klein, G. E. Kugel, A. Maillard, A. Sifi, and K. Polgar, “Absolute non-linear optical coefficients measurements of BBO single crystal and determination of angular acceptance by second harmonic generation,” Opt. Mater.22, 163–169 (2003).
[CrossRef]

R. Ganeev, I. Kulagin, A. Ryasnyanskii, R. Tugushev, and T. Usmanov, “The nonlinear refractive indices and nonlinear third-order susceptibilities of quadratic crystals,” Opt. Spectrosc.94, 561–568 (2003). [Opt. Spektrosk. 94, 615–623 (2003)].
[CrossRef]

2002 (2)

2001 (2)

K. Beckwitt, F. W. Wise, L. Qian, L. A. Walker, and E. Canto-Said, “Compensation for self-focusing by use of cascade quadratic nonlinearity,” Opt. Lett.26, 1696–1698 (2001).
[CrossRef]

H. P. Li, C. H. Kam, Y. L. Lam, and W. Ji, “Femtosecond Z-scan measurements of nonlinear refraction in nonlinear optical crystals,” Opt. Mater.15, 237–242 (2001).
[CrossRef]

2000 (2)

D. Zhang, Y. Kong, and J. Zhang, “Optical parametric properties of 532-nm-pumped beta-barium-borate near the infrared absorption edge,” Opt. Commun.184, 485–491 (2000).
[CrossRef]

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: Cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B61, 10688–10701 (2000).
[CrossRef]

1999 (2)

1997 (3)

M. Sheik-Bahae, “Femtosecond kerr-lens autocorrelation,” Opt. Lett.22, 399–401 (1997).
[CrossRef] [PubMed]

M. Sheik-Bahae and M. Ebrahimzadeh, “Measurements of nonlinear refraction in the second-order χ(2) materials KTiOPO4, KNbO3, β-BaB2O4 and LiB3O5,” Opt. Commun.142, 294–298 (1997).
[CrossRef]

H. Li, F. Zhou, X. Zhang, and W. Ji, “Bound electronic kerr effect and self-focusing induced damage in second-harmonic-generation crystals,” Opt. Commun.144, 75–81 (1997).
[CrossRef]

1996 (2)

R. DeSalvo, A. A. Said, D. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron.32, 1324–1333 (1996).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron.28, 1691–1740 (1996).
[CrossRef]

1995 (2)

F. Hache, A. Zéboulon, G. Gallot, and G. M. Gale, “Cascaded second-order effects in the femtosecond regime in β-barium borate: self-compression in a visible femtosecond optical parametric oscillator,” Opt. Lett.20, 1556–1558 (1995).
[CrossRef] [PubMed]

E. Nibbering, M. Franco, B. Prade, G. Grillon, C. L. Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun.119, 479–484 (1995).
[CrossRef]

1994 (1)

T. D. Krauss and F. W. Wise, “Femtosecond measurement of nonlinear absorption and refraction in CdS, ZnSe, and ZnS,” Appl. Phys. Lett.65, 1739–1741 (1994).
[CrossRef]

1993 (1)

H. Tan, G. P. Banfi, and A. Tomaselli, “Optical frequency mixing through cascaded second-order processes in beta-barium borate,” Appl. Phys. Lett.63, 2472–2474 (1993).
[CrossRef]

1992 (1)

1991 (2)

M. Sheik-Bahae, D. Hutchings, D. Hagan, and E. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron.27, 1296 –1309 (1991).
[CrossRef]

D. N. Nikogosyan, “Beta barium borate (BBO) - A review of its properties and applications,” Appl. Phys. A52, 359–368 (1991).
[CrossRef]

1990 (1)

M. Sheik-Bahae, A. Said, T.-H. Wei, D. Hagan, and E. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron.26, 760–769 (1990).
[CrossRef]

1989 (1)

Y. Fan, R. Eckardt, R. Byer, C. Chen, and A. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron.25, 1196 –1199 (1989).
[CrossRef]

1985 (1)

C. Chen, B. Wu, A. Jiang, and G. You, “A new-type ultraviolet SHG crystal - beta-BaB2O4,” Sci. Sin., Ser. B28, 235–243 (1985).

1978 (1)

N. Boling, A. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron.14, 601–608 (1978).
[CrossRef]

1972 (1)

J. M. R. Thomas and J. P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun.4, 329–334 (1972).
[CrossRef]

1969 (2)

C. Wang and E. Baardsen, “Optical third harmonic generation using mode-locked and non-mode-locked lasers,” Appl. Phys. Lett.15, 396–397 (1969).
[CrossRef]

J. J. Wynne, “Optical third-order mixing in GaAs, Ge, Si, and InAs,” Phys. Rev.178, 1295–1303 (1969).
[CrossRef]

1967 (1)

L. A. Ostrovskii, “Self-action of light in crystals,” Pisma Zh. Eksp. Teor. Fiz.5, 331–334 (1967). [JETP Lett.5, 272–275 (1967)].

1965 (1)

J. E. Midwinter and J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization,” Br. J. Appl. Phys.16, 1667–1674 (1965).
[CrossRef]

1964 (1)

R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett.5, 17–19 (1964).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).

Alhammali, E.

Ashihara, S.

X. Zeng, S. Ashihara, X. Chen, T. Shimura, and K. Kuroda, “Two-color pulse compression in aperiodically-poled lithium niobate,” Opt. Commun.281, 4499–4503 (2008).
[CrossRef]

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett.84, 1055–1057 (2004).
[CrossRef]

S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B19, 2505–2510 (2002).
[CrossRef]

Baardsen, E.

C. Wang and E. Baardsen, “Optical third harmonic generation using mode-locked and non-mode-locked lasers,” Appl. Phys. Lett.15, 396–397 (1969).
[CrossRef]

Bache, M.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett.109, 043902 (2012).
[CrossRef] [PubMed]

M. Bache, F. Eilenberger, and S. Minardi, “Higher-order Kerr effect and harmonic cascading in gases,” Opt. Lett.37, 4612–4614 (2012).
[PubMed]

X. Zeng, H. Guo, B. Zhou, and M. Bache, “Soliton compression to few-cycle pulses with a high quality factor by engineering cascaded quadratic nonlinearities,” Opt. Express20, 27071–27082 (2012). ArXiv:1210.5928.
[CrossRef] [PubMed]

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical Cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A82, 063806 (2010).
[CrossRef]

M. Bache and F. W. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A81, 053815 (2010).
[CrossRef]

M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express16, 3273–3287 (2008).
[CrossRef] [PubMed]

M. Bache, J. Moses, and F. W. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B24, 2752–2762 (2007).
[CrossRef]

M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett.32, 2490–2492 (2007).
[CrossRef] [PubMed]

H. Guo, X. Zeng, B. Zhou, and M. Bache, “Electric field modeling and self-steepening counterbalance of cascading nonlinear soliton pulse compression,” (submitted to J. Opt. Soc. Am. B), arXiv:1210.5903.

Banfi, G. P.

H. Tan, G. P. Banfi, and A. Tomaselli, “Optical frequency mixing through cascaded second-order processes in beta-barium borate,” Appl. Phys. Lett.63, 2472–2474 (1993).
[CrossRef]

Bang, O.

Banks, P. S.

Beckwitt, K.

Blanc, C. L.

E. Nibbering, M. Franco, B. Prade, G. Grillon, C. L. Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun.119, 479–484 (1995).
[CrossRef]

Boling, N.

N. Boling, A. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron.14, 601–608 (1978).
[CrossRef]

Bosshard, C.

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: Cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B61, 10688–10701 (2000).
[CrossRef]

Boulanger, B.

B. Boulanger and J. Zyss, International Tables for Crystallography (Springer, 2006), Vol. D: Physical Properties of Crystals, Chap. 1.7: Nonlinear optical properties, pp. 178–219.
[CrossRef]

Boyd, R. W.

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A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioners guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top.199, 5–76 (2011).
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Y. Fan, R. Eckardt, R. Byer, C. Chen, and A. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron.25, 1196 –1199 (1989).
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Cattela, G. C.

R. C. Eckardt and G. C. Cattela, “Characterization techniques for second-order nonlinear optical materials,” Proc. SPIE5337, 1–10 (2004).
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Cha, M.

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett.84, 1055–1057 (2004).
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Chen, C.

Y. Fan, R. Eckardt, R. Byer, C. Chen, and A. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron.25, 1196 –1199 (1989).
[CrossRef]

C. Chen, B. Wu, A. Jiang, and G. You, “A new-type ultraviolet SHG crystal - beta-BaB2O4,” Sci. Sin., Ser. B28, 235–243 (1985).

Chen, X.

X. Zeng, S. Ashihara, X. Chen, T. Shimura, and K. Kuroda, “Two-color pulse compression in aperiodically-poled lithium niobate,” Opt. Commun.281, 4499–4503 (2008).
[CrossRef]

Chen, Y.-F.

Chong, A.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett.109, 043902 (2012).
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A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioners guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top.199, 5–76 (2011).
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A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioners guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top.199, 5–76 (2011).
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R. DeSalvo, A. A. Said, D. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron.32, 1324–1333 (1996).
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R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett.17, 28–30 (1992).
[CrossRef] [PubMed]

Ebrahimzadeh, M.

M. Sheik-Bahae and M. Ebrahimzadeh, “Measurements of nonlinear refraction in the second-order χ(2) materials KTiOPO4, KNbO3, β-BaB2O4 and LiB3O5,” Opt. Commun.142, 294–298 (1997).
[CrossRef]

Eckardt, R.

Y. Fan, R. Eckardt, R. Byer, C. Chen, and A. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron.25, 1196 –1199 (1989).
[CrossRef]

Eckardt, R. C.

R. C. Eckardt and G. C. Cattela, “Characterization techniques for second-order nonlinear optical materials,” Proc. SPIE5337, 1–10 (2004).
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Eilenberger, F.

Ettoumi, W.

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Y. Fan, R. Eckardt, R. Byer, C. Chen, and A. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron.25, 1196 –1199 (1989).
[CrossRef]

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Fejer, M. M.

Fermann, M. E.

Franco, M.

E. Nibbering, M. Franco, B. Prade, G. Grillon, C. L. Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun.119, 479–484 (1995).
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Gallot, G.

Ganeev, R.

R. Ganeev, I. Kulagin, A. Ryasnyanskii, R. Tugushev, and T. Usmanov, “The nonlinear refractive indices and nonlinear third-order susceptibilities of quadratic crystals,” Opt. Spectrosc.94, 561–568 (2003). [Opt. Spektrosk. 94, 615–623 (2003)].
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R. A. Ganeev, private communication (2012).

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N. Boling, A. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron.14, 601–608 (1978).
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Grillon, G.

E. Nibbering, M. Franco, B. Prade, G. Grillon, C. L. Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun.119, 479–484 (1995).
[CrossRef]

Gubler, U.

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: Cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B61, 10688–10701 (2000).
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Guo, H.

X. Zeng, H. Guo, B. Zhou, and M. Bache, “Soliton compression to few-cycle pulses with a high quality factor by engineering cascaded quadratic nonlinearities,” Opt. Express20, 27071–27082 (2012). ArXiv:1210.5928.
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H. Guo, X. Zeng, B. Zhou, and M. Bache, “Electric field modeling and self-steepening counterbalance of cascading nonlinear soliton pulse compression,” (submitted to J. Opt. Soc. Am. B), arXiv:1210.5903.

Hache, F.

Hagan, D.

R. DeSalvo, A. A. Said, D. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron.32, 1324–1333 (1996).
[CrossRef]

R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett.17, 28–30 (1992).
[CrossRef] [PubMed]

M. Sheik-Bahae, D. Hutchings, D. Hagan, and E. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron.27, 1296 –1309 (1991).
[CrossRef]

M. Sheik-Bahae, A. Said, T.-H. Wei, D. Hagan, and E. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron.26, 760–769 (1990).
[CrossRef]

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron.28, 1691–1740 (1996).
[CrossRef]

Hartl, I.

Hutchings, D.

M. Sheik-Bahae, D. Hutchings, D. Hagan, and E. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron.27, 1296 –1309 (1991).
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Ilday, F. Ö.

Ito, R.

Ji, W.

H. P. Li, C. H. Kam, Y. L. Lam, and W. Ji, “Femtosecond Z-scan measurements of nonlinear refraction in nonlinear optical crystals,” Opt. Mater.15, 237–242 (2001).
[CrossRef]

H. Li, F. Zhou, X. Zhang, and W. Ji, “Bound electronic kerr effect and self-focusing induced damage in second-harmonic-generation crystals,” Opt. Commun.144, 75–81 (1997).
[CrossRef]

Jiang, A.

Y. Fan, R. Eckardt, R. Byer, C. Chen, and A. Jiang, “Barium borate optical parametric oscillator,” IEEE J. Quantum Electron.25, 1196 –1199 (1989).
[CrossRef]

C. Chen, B. Wu, A. Jiang, and G. You, “A new-type ultraviolet SHG crystal - beta-BaB2O4,” Sci. Sin., Ser. B28, 235–243 (1985).

Kaatz, P.

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: Cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B61, 10688–10701 (2000).
[CrossRef]

Kam, C. H.

H. P. Li, C. H. Kam, Y. L. Lam, and W. Ji, “Femtosecond Z-scan measurements of nonlinear refraction in nonlinear optical crystals,” Opt. Mater.15, 237–242 (2001).
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Kasparian, J.

Kitamura, K.

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett.84, 1055–1057 (2004).
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Klein, R. S.

R. S. Klein, G. E. Kugel, A. Maillard, A. Sifi, and K. Polgar, “Absolute non-linear optical coefficients measurements of BBO single crystal and determination of angular acceptance by second harmonic generation,” Opt. Mater.22, 163–169 (2003).
[CrossRef]

Kolesik, M.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioners guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top.199, 5–76 (2011).
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Kong, Y.

D. Zhang, Y. Kong, and J. Zhang, “Optical parametric properties of 532-nm-pumped beta-barium-borate near the infrared absorption edge,” Opt. Commun.184, 485–491 (2000).
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T. D. Krauss and F. W. Wise, “Femtosecond measurement of nonlinear absorption and refraction in CdS, ZnSe, and ZnS,” Appl. Phys. Lett.65, 1739–1741 (1994).
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Krolikowski, W.

Kubota, S.

Kugel, G. E.

R. S. Klein, G. E. Kugel, A. Maillard, A. Sifi, and K. Polgar, “Absolute non-linear optical coefficients measurements of BBO single crystal and determination of angular acceptance by second harmonic generation,” Opt. Mater.22, 163–169 (2003).
[CrossRef]

Kulagin, I.

R. Ganeev, I. Kulagin, A. Ryasnyanskii, R. Tugushev, and T. Usmanov, “The nonlinear refractive indices and nonlinear third-order susceptibilities of quadratic crystals,” Opt. Spectrosc.94, 561–568 (2003). [Opt. Spektrosk. 94, 615–623 (2003)].
[CrossRef]

Kurimura, S.

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett.84, 1055–1057 (2004).
[CrossRef]

Kuroda, K.

X. Zeng, S. Ashihara, X. Chen, T. Shimura, and K. Kuroda, “Two-color pulse compression in aperiodically-poled lithium niobate,” Opt. Commun.281, 4499–4503 (2008).
[CrossRef]

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett.84, 1055–1057 (2004).
[CrossRef]

S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B19, 2505–2510 (2002).
[CrossRef]

Lam, Y. L.

H. P. Li, C. H. Kam, Y. L. Lam, and W. Ji, “Femtosecond Z-scan measurements of nonlinear refraction in nonlinear optical crystals,” Opt. Mater.15, 237–242 (2001).
[CrossRef]

Langrock, C.

Li, H.

H. Li, F. Zhou, X. Zhang, and W. Ji, “Bound electronic kerr effect and self-focusing induced damage in second-harmonic-generation crystals,” Opt. Commun.144, 75–81 (1997).
[CrossRef]

Li, H. P.

H. P. Li, C. H. Kam, Y. L. Lam, and W. Ji, “Femtosecond Z-scan measurements of nonlinear refraction in nonlinear optical crystals,” Opt. Mater.15, 237–242 (2001).
[CrossRef]

Lim, H.

Liu, X.

Maillard, A.

R. S. Klein, G. E. Kugel, A. Maillard, A. Sifi, and K. Polgar, “Absolute non-linear optical coefficients measurements of BBO single crystal and determination of angular acceptance by second harmonic generation,” Opt. Mater.22, 163–169 (2003).
[CrossRef]

Majus, D.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioners guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top.199, 5–76 (2011).
[CrossRef]

Malomed, B. A.

J. Moses, B. A. Malomed, and F. W. Wise, “Self-steepening of ultrashort optical pulses without self-phase modulation,” Phys. Rev. A76, 021802(R) (2007).
[CrossRef]

Mazerant, W.

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: Cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B61, 10688–10701 (2000).
[CrossRef]

Meier, U.

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: Cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B61, 10688–10701 (2000).
[CrossRef]

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J. E. Midwinter and J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization,” Br. J. Appl. Phys.16, 1667–1674 (1965).
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R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett.5, 17–19 (1964).
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Moses, J.

Moses, J. A.

J. A. Moses, private communication (2010).

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E. Nibbering, M. Franco, B. Prade, G. Grillon, C. L. Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun.119, 479–484 (1995).
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Nakamura, H.

Nibbering, E.

E. Nibbering, M. Franco, B. Prade, G. Grillon, C. L. Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun.119, 479–484 (1995).
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D. N. Nikogosyan, “Beta barium borate (BBO) - A review of its properties and applications,” Appl. Phys. A52, 359–368 (1991).
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Ohdaira, K.

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N. Boling, A. Glass, and A. Owyoung, “Empirical relationships for predicting nonlinear refractive index changes in optical solids,” IEEE J. Quantum Electron.14, 601–608 (1978).
[CrossRef]

Pelc, J. S.

Perry, M. D.

Petit, Y.

Phillips, C. R.

Polgar, K.

R. S. Klein, G. E. Kugel, A. Maillard, A. Sifi, and K. Polgar, “Absolute non-linear optical coefficients measurements of BBO single crystal and determination of angular acceptance by second harmonic generation,” Opt. Mater.22, 163–169 (2003).
[CrossRef]

Prade, B.

E. Nibbering, M. Franco, B. Prade, G. Grillon, C. L. Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun.119, 479–484 (1995).
[CrossRef]

Qian, L.

Qian, L.-J.

Ramírez-Góngora, O. de J.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioners guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top.199, 5–76 (2011).
[CrossRef]

Razzari, L.

Righini, M.

Ryasnyanskii, A.

R. Ganeev, I. Kulagin, A. Ryasnyanskii, R. Tugushev, and T. Usmanov, “The nonlinear refractive indices and nonlinear third-order susceptibilities of quadratic crystals,” Opt. Spectrosc.94, 561–568 (2003). [Opt. Spektrosk. 94, 615–623 (2003)].
[CrossRef]

Said, A.

M. Sheik-Bahae, A. Said, T.-H. Wei, D. Hagan, and E. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron.26, 760–769 (1990).
[CrossRef]

Said, A. A.

R. DeSalvo, A. A. Said, D. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron.32, 1324–1333 (1996).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, “Femtosecond kerr-lens autocorrelation,” Opt. Lett.22, 399–401 (1997).
[CrossRef] [PubMed]

M. Sheik-Bahae and M. Ebrahimzadeh, “Measurements of nonlinear refraction in the second-order χ(2) materials KTiOPO4, KNbO3, β-BaB2O4 and LiB3O5,” Opt. Commun.142, 294–298 (1997).
[CrossRef]

R. DeSalvo, A. A. Said, D. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron.32, 1324–1333 (1996).
[CrossRef]

R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett.17, 28–30 (1992).
[CrossRef] [PubMed]

M. Sheik-Bahae, D. Hutchings, D. Hagan, and E. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron.27, 1296 –1309 (1991).
[CrossRef]

M. Sheik-Bahae, A. Said, T.-H. Wei, D. Hagan, and E. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron.26, 760–769 (1990).
[CrossRef]

Shimura, T.

X. Zeng, S. Ashihara, X. Chen, T. Shimura, and K. Kuroda, “Two-color pulse compression in aperiodically-poled lithium niobate,” Opt. Commun.281, 4499–4503 (2008).
[CrossRef]

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett.84, 1055–1057 (2004).
[CrossRef]

S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, “Soliton compression of femtosecond pulses in quadratic media,” J. Opt. Soc. Am. B19, 2505–2510 (2002).
[CrossRef]

Shoji, I.

Sifi, A.

R. S. Klein, G. E. Kugel, A. Maillard, A. Sifi, and K. Polgar, “Absolute non-linear optical coefficients measurements of BBO single crystal and determination of angular acceptance by second harmonic generation,” Opt. Mater.22, 163–169 (2003).
[CrossRef]

Stegeman, G.

Stegeman, G. I.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron.28, 1691–1740 (1996).
[CrossRef]

Taira, T.

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett.84, 1055–1057 (2004).
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J. M. R. Thomas and J. P. E. Taran, “Pulse distortions in mismatched second harmonic generation,” Opt. Commun.4, 329–334 (1972).
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H. Tan, G. P. Banfi, and A. Tomaselli, “Optical frequency mixing through cascaded second-order processes in beta-barium borate,” Appl. Phys. Lett.63, 2472–2474 (1993).
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Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron.28, 1691–1740 (1996).
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Tugushev, R.

R. Ganeev, I. Kulagin, A. Ryasnyanskii, R. Tugushev, and T. Usmanov, “The nonlinear refractive indices and nonlinear third-order susceptibilities of quadratic crystals,” Opt. Spectrosc.94, 561–568 (2003). [Opt. Spektrosk. 94, 615–623 (2003)].
[CrossRef]

Usmanov, T.

R. Ganeev, I. Kulagin, A. Ryasnyanskii, R. Tugushev, and T. Usmanov, “The nonlinear refractive indices and nonlinear third-order susceptibilities of quadratic crystals,” Opt. Spectrosc.94, 561–568 (2003). [Opt. Spektrosk. 94, 615–623 (2003)].
[CrossRef]

Van Stryland, E.

M. Sheik-Bahae, D. Hutchings, D. Hagan, and E. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron.27, 1296 –1309 (1991).
[CrossRef]

M. Sheik-Bahae, A. Said, T.-H. Wei, D. Hagan, and E. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron.26, 760–769 (1990).
[CrossRef]

Van Stryland, E. W.

R. DeSalvo, A. A. Said, D. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, “Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids,” IEEE J. Quantum Electron.32, 1324–1333 (1996).
[CrossRef]

R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett.17, 28–30 (1992).
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Vanherzeele, H.

Walker, L. A.

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[CrossRef]

Wei, T.-H.

M. Sheik-Bahae, A. Said, T.-H. Wei, D. Hagan, and E. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron.26, 760–769 (1990).
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Wise, F. W.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett.109, 043902 (2012).
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M. Bache and F. W. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: Compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A81, 053815 (2010).
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M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical Cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A82, 063806 (2010).
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M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express16, 3273–3287 (2008).
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M. Bache, J. Moses, and F. W. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B24, 2752–2762 (2007).
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M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett.32, 2490–2492 (2007).
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J. Moses, B. A. Malomed, and F. W. Wise, “Self-steepening of ultrashort optical pulses without self-phase modulation,” Phys. Rev. A76, 021802(R) (2007).
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J. Moses, E. Alhammali, J. M. Eichenholz, and F. W. Wise, “Efficient high-energy femtosecond pulse compression in quadratic media with flattop beams,” Opt. Lett.32, 2469–2471 (2007).
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[CrossRef] [PubMed]

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[CrossRef]

K. Beckwitt, F. W. Wise, L. Qian, L. A. Walker, and E. Canto-Said, “Compensation for self-focusing by use of cascade quadratic nonlinearity,” Opt. Lett.26, 1696–1698 (2001).
[CrossRef]

X. Liu, L.-J. Qian, and F. W. Wise, “High-energy pulse compression by use of negative phase shifts produced by the cascaded χ(2) : χ(2) nonlinearity,” Opt. Lett.24, 1777–1779 (1999).
[CrossRef]

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Wu, B.

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J. J. Wynne, “Optical third-order mixing in GaAs, Ge, Si, and InAs,” Phys. Rev.178, 1295–1303 (1969).
[CrossRef]

You, G.

C. Chen, B. Wu, A. Jiang, and G. You, “A new-type ultraviolet SHG crystal - beta-BaB2O4,” Sci. Sin., Ser. B28, 235–243 (1985).

Yu, N. E.

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, “Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate,” Appl. Phys. Lett.84, 1055–1057 (2004).
[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

Zhou, B.

X. Zeng, H. Guo, B. Zhou, and M. Bache, “Soliton compression to few-cycle pulses with a high quality factor by engineering cascaded quadratic nonlinearities,” Opt. Express20, 27071–27082 (2012). ArXiv:1210.5928.
[CrossRef] [PubMed]

H. Guo, X. Zeng, B. Zhou, and M. Bache, “Electric field modeling and self-steepening counterbalance of cascading nonlinear soliton pulse compression,” (submitted to J. Opt. Soc. Am. B), arXiv:1210.5903.

Zhou, B. B.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett.109, 043902 (2012).
[CrossRef] [PubMed]

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical Cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A82, 063806 (2010).
[CrossRef]

Zhou, F.

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[CrossRef]

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[PubMed]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical Cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A82, 063806 (2010).
[CrossRef]

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[CrossRef]

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J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett.97, 073903 (2006).
[CrossRef] [PubMed]

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett.109, 043902 (2012).
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Other (5)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).

R. A. Ganeev, private communication (2012).

J. A. Moses, private communication (2010).

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2007).

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

(a) The definition of the crystal coordinate system XYZ relative to the beam propagation direction k. (b) Top view of the optimal crystal cut for type I ooe SHG in BBO, which has ϕ = −π/2 and θ = θc for perpendicular incidence of an o-polarized FW beam, and the e-polarized SH is generated through type I ooe SHG. The specific value of the cut angle θc depends on the wavelength and the desired application. Angle-tuning the crystal in the paper plane will change the interaction angle θ. Capital letters XYZ are traditionally used to distinguish the crystal coordinate system from the beam coordinate system xyz that has its origin in the k-vector propagation direction.

Fig. 2
Fig. 2

(a) Spectral bandwidth at 10 dB below the peak of the experimentally recorded spectra vs. Δk for two different intensities. (b) Similar data from numerical simulations of the coupled SEWA equations [9, 37] (using the anisotropic quadratic and cubic nonlinear coefficients in Appendix C and Table 1, respectively). The input conditions were similar to the experiment (Gaussian pulse with 39.3 nm bandwidth and a group-delay dispersion of −330 fs2). (c) and (d) show the spectra on a linear and log scale right at the transition (Δk = 88 mm−1), where the total SPM is near zero. (e) Various spectra for I = 200 GW/cm2.

Fig. 3
Fig. 3

Summary of the experimental data for n 2 , Kerr I-values from the literature corresponding to the c11 nonlinear susceptibility coefficient ( n 2 , Kerr I = 3 c 11 / 4 n 1 2 ε 0 c). The plotted values are the ones reported in Sec. 4.2, i.e. the data values do not necessarily correspond to the ones reported in the literature. References: Tan et al. 1993: [13]; Hache et al. 1995: [14]; DeSalvo et al. 1996 [16]; Li et al. 1997 [18]; Li et al. 2001 [19]; Ganeev et al. 2003 [20]; Moses et al. 2007 [15]. The theoretically predicted electronic nonlinearity is calculated with the 2-band model [24]. The average value curve was calculated through a weighted mean of the Miller’s delta from all data, except the UV measurements below 400 nm, and the shaded areas denoted “σ” and “2σ” represent one and two standard deviations, respectively.

Fig. 4
Fig. 4

(a) Compression diagram for BBO type I cascaded SHG. In order to excite solitons Δk must be kept below the Kerr limit (red line). Optimal compression occurs when the cascaded nonlinearities dominate over GVM effects (Δk > Δksr, above the black line). Note that Δksr is calculated for the full dispersion case, and that for λ1 > 1.49 μm the FW GVD becomes anomalous. We have also indicated the operation wavelengths of Cr:forsterite, Yb and Ti:Sapphire based amplifiers. The Kerr limit employs Miller’s rule to calculate the nonlinear quadratic and cubic susceptibilities at other wavelengths, case (1) corresponds to the ‘old’ Kerr value n 2 , Kerr I ( ω 1 ; ω 1 ) = 3.65 10 20 m 2 / W @ 850 nm (taken from [17]) and case (2) corresponds to the Kerr value proposed in this work Δ1111 = 52.3 m2/V2, corresponding to n 2 , Kerr I ( ω 1 ; ω 1 ) = 5.06 10 20 m 2 / W @ 850 nm (b) Numerical simulation of the case marked with ‘x’ in (a): a 50 fs@1030 nm Iin = 500 GW/cm2 pulse propagating in a 25 mm BBO crystal with θ = 19.1°, ρ = 2.8° and ϕ = −90° (Δk = 80 mm−1). The simulations used the plane-wave SVEA equations [(Eqs. (50)(51)] including full dispersion and extended to include self-steepening, and case (1) assumes an isotropic Kerr nonlinearity and n 2 , Kerr I ( ω 1 ; ω 1 ) = 3.65 10 20 m 2 / W @ 850 nm, while (2) uses the anisotropic coefficients of Table 2, and Eq. (30) to calculate the nonlinear coefficients at 1030 nm.

Fig. 5
Fig. 5

Estimating the operating parameters vs. FW wavelength for cascaded SHG in BBO. (a) The θ-range for which Δk > 0 is achieved as well as | n 2 , casc I | > n 2 , Kerr I ( ω 1 ; ω 1 ). (b) The XPM term χ eff ( 3 ) ( ω 1 ; ω 2 ) and SH SPM term χ eff ( 3 ) ( ω 2 ; ω 2 ), both normalized to the FW SPM term χ eff ( 3 ) ( ω 1 ; ω 1 ). The range indicated corresponds to the θ range in (a). The dashed lines indicate the isotropic limit. All nonlinear coefficients are scaled to other wavelengths using Miller’s rule, and we used the SHG nonlinearities at 1064 nm from Appendix C as well as the cubic nonlinear parameters listed in Sec. 5. We also took ϕ = −π/2.

Tables (2)

Tables Icon

Table 1 Summary of the literature measurements of the cubic nonlinearities. The column (A) reports the original data and (B) our updated values, if any.

Tables Icon

Table 2 Proposed nonlinear susceptibilities for the BBO anisotropic Kerr nonlinearity. Note the c16 value is deduced from the c11 and c10 coefficients, and the c33 value is deduced from the c11, c10 and c16 coefficients. The Miller’s delta Δijkl are calculated from Eq. (30).

Equations (56)

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n 2 , casc I = 2 ω 1 d eff 2 c 2 ε 0 n 1 2 n 2 Δ k
Δ n = n 2 , tot I I 1 = ( n 2 , casc I + n 2 , Kerr I ) I 1
d eff o o o = d 22 cos 3 ϕ
d eff o o e = d eff o e o = d 31 sin ( θ + ρ ) d 22 cos ( θ + ρ ) sin 3 ϕ
d eff o e e = d eff e e o = d 22 cos 2 ( θ + ρ ) cos 3 ϕ
d eff e e e = d 22 cos 3 ( θ + ρ ) sin 3 ϕ + 3 d 31 sin ( θ + ρ ) cos 2 ( θ + ρ ) + d 33 sin 3 ( θ + ρ )
χ eff ( 3 ) ( ω 1 ; ω 1 ) = c 11
χ eff ( 3 ) ( ω 2 ; ω 2 ) = 4 c 10 sin ( θ + ρ ) cos 3 ( θ + ρ ) sin 3 ϕ + c 11 cos 4 ( θ + ρ ) + 3 2 c 16 sin 2 ( 2 θ + 2 ρ ) + c 33 sin 4 ( θ + ρ )
χ eff ( 3 ) ( ω 1 ; ω 2 ) = 1 3 c 11 cos 2 ( θ + ρ ) + c 16 sin 2 ( θ + ρ ) + c 10 sin ( 2 θ + 2 ρ ) sin 3 ϕ
χ eff ( 3 ) ( ω 1 ; ω 1 ) = χ eff ( 3 ) ( ω 2 ; ω 2 ) = χ eff ( 3 ) ( ω 1 ; ω 2 )
= 4 c 10 sin ( θ + ρ ) cos 3 ( θ + ρ ) sin 3 ϕ + c 11 cos 4 ( θ + ρ ) + 3 2 c 16 sin 2 ( 2 θ + 2 ρ ) + c 33 sin 4 ( θ + ρ )
for μ : X 1 Y 2 Z 3 for m : X X X 1 Y Y Y 2 Z Z Z 3 Y Z Z 4 Y Y Z 5 X Z Z 6 X X Z 7 X Y Y 8 X X Y 9 X Y Z 9
Δ n = n 2 I I + n 4 I I 2 +
c 11 = 5.84 ± 0.51 10 22 m 2 / V 2 , λ = 1.064 μ m
c 11 = 4.05 ± 0.52 10 22 m 2 / V 2 , λ = 1.064 μ m
n 2 , Kerr I ( ω 1 ; ω 1 ) = 4.5 ± 1.0 10 20 m 2 / W , λ = 0.8 μ m
c 11 = 4.65 ± 0.51 10 22 m 2 / V 2 , λ = 1.064 μ m
c 11 = 5.82 ± 0.99 10 22 m 2 / V 2 , λ = 0.532 μ m
c 11 = 3.92 ± 0.68 10 22 m 2 / V 2 , λ = 0.355 μ m
c 11 = 0.26 ± 0.078 10 22 m 2 / V 2 , λ = 0.266 μ m
n 2 , Kerr I ( ω 1 ; ω 1 ) = 5.41 ± 0.90 10 20 m 2 / W , [ 1 0 0 ] , λ = 0.532 μ m
n 2 , Kerr I ( ω 1 ; ω 1 ) = 4.61 ± 0.80 10 20 m 2 / W , [ 0 1 0 ] , λ = 0.532 μ m
n 2 , Kerr I ( ω 1 ; ω 1 ) = 7.01 ± 1.01 10 20 m 2 / W , λ = 1.064 μ m
n 2 , Kerr I ( ω 1 ; ω 1 ) = 5.30 ± 0.51 10 20 m 2 / W , [ 1 0 0 ] , λ = 0.78 μ m
n 2 , Kerr I ( ω 1 ; ω 1 ) = 4.50 ± 0.51 10 20 m 2 / W , [ 0 1 0 ] , λ = 0.78 μ m
n 2 , Kerr I ( ω 1 ; ω 1 ) = 7.14 ± 2.22 10 20 m 2 / W , λ = 1.064 μ m
n 2 , Kerr I ( ω 1 ; ω 1 ) = 5.24 ± 2.41 10 20 m 2 / W , λ = 0.532 μ m
n 2 , Kerr I ( ω 1 ; ω 1 ) = 5.54 ± 0.98 10 20 m 2 / W , λ = 0.8 μ m
n 2 , Kerr I ( ω 1 ; ω 1 ) = 4.87 ± 0.44 10 20 m 2 / W , λ = 1.032 μ m
Δ i j k l = χ i j k l ( 3 ) χ i ( 1 ) χ j ( 1 ) χ k ( 1 ) χ l ( 1 ) = χ ( 3 ) ( ω i ; ω j , ω k , ω l ) χ ( 1 ) ( ω i ) χ ( 1 ) ( ω j ) χ ( 1 ) ( ω k ) χ ( 1 ) ( ω l )
Δ 1111 = 52.3 ± 7.7 × 10 24 m 2 / V 2
c 10 ( THG ) = 0.24 ± 0.04 10 22 m 2 / V 2 , λ = 1.053 μ m
c 16 ( THG ) = 1.47 ± 0.34 10 22 m 2 / V 2 , λ = 1.053 μ m
χ eff ( 3 ) ( ω 1 ; ω 1 ) = 1.28 c 10 + 0.48 c 11 + 1.27 c 16 + 0.093 c 33
c 33 = 5.35 ± 8.43 10 22 m 2 / V 2 , λ = 0.85 μ m
2 E ( r , t ) = μ 0 2 t 2 D ( r , t )
P ( r , ω ) = ε 0 χ _ ( 1 ) ( ω ) E ( r , ω ) + P NL ( r , ω ) ,
ε 0 1 P NL ( r , ω ) = χ _ ( 2 ) ( ω ) : E ( r , ω ) E ( r , ω ) + χ _ ( 3 ) ( ω ) E ( r , ω ) E ( r , ω ) E ( r , ω ) +
2 z 2 E ( z , ω ) + ω 2 c 2 ε _ ( ω ) E ( z , ω ) = ω 2 c 2 ε 0 1 P NL ( z , ω )
E ( z , t ) = 1 2 [ u 1 1 ( z , t ) e i k 1 ( ω 1 ) z i ω 1 t + u 2 2 ( z , t ) e i k 2 ( ω 2 ) z i ω 2 t + c . c . ]
P NL ( z , t ) = 1 2 [ u 1 𝒫 NL , 1 ( z , t ) e i k 1 ( ω 1 ) z i ω 1 t + u 2 𝒫 NL , 2 ( z , t ) e i k 2 ( ω 2 ) z i ω 2 t + c . c . ]
2 j ( z , ω ) z 2 + 2 i k j ( ω j ) j ( z , ω ) z + [ k j 2 ( ω ) k j 2 ( ω j ) ] j ( z , ω ) = ω 2 c 2 ε 0 1 𝒫 NL , j ( z , ω ) .
i j ( z , ω ) z + [ k j ( ω ) k j ( ω j ) ] j ( z , ω ) = ω j 2 n j ( ω j ) c ε 0 1 𝒫 NL , j ( z , ω )
[ i z + i k j ( 1 ) ( ω j ) t + D ^ j ] j ( z , t ) = ω j 2 n j ( ω j ) c ε 0 1 𝒫 NL , j ( z , t )
D ^ j = m = 2 k j ( m ) ( ω j ) i m m ! m t m
ε 0 1 𝒫 NL , 1 ( 2 ) ( z , t ) = χ eff ( 2 ) 1 * ( z , t ) 2 ( z , t ) e i Δ k z , ε 0 1 𝒫 NL , 2 ( 2 ) ( z , t ) = χ eff ( 2 ) 1 2 1 2 ( z , t ) e i Δ k z
ε 0 1 𝒫 NL , j ( 3 ) ( z , t ) = 1 4 [ 3 χ eff ( 3 ) ( ω j ; ω j ) | j ( z , t ) | 2 + 6 χ eff ( 3 ) ( ω j ; ω k ) | k ( z , t ) | 2 ] j ( z , t )
[ i z + D ^ 1 ] 1 + ω 1 d eff n 1 c 1 * e e i Δ k z + 3 ω 1 8 n 1 c [ χ eff ( 3 ) ( ω 1 ; ω 1 ) 1 | 1 | 2 + 2 χ eff ( 3 ) ( ω 1 ; ω 2 ) 1 | 2 | 2 ] = 0
[ i z i d 12 τ + D ^ 2 ] 2 + ω 2 d eff n 2 c 1 2 1 2 e i Δ k z + 3 ω 2 8 n 2 c [ χ eff ( 3 ) ( ω 2 ; ω 2 ) 2 | 2 | 2 + 2 χ eff ( 3 ) ( ω 2 ; ω 1 ) 2 | 1 | 2 ] = 0
[ i z + D ^ 1 ] A 1 + κ SHG I A 1 * A 2 e i Δ k z + ω 1 c [ n 2 , Kerr I ( ω 1 ; ω 1 ) A 1 | A 1 | 2 + 2 n 2 , Kerr I ( ω 1 ; ω 2 ) A 1 | A 2 | 2 ] = 0
[ i z i d 12 τ + D ^ 2 ] A 2 + κ SHG I A 1 2 e i Δ k z + ω 2 c [ n 2 , Kerr I ( ω 2 ; ω 2 ) A 2 | A 2 | 2 + 2 n 2 , Kerr I ( ω 1 ; ω 2 ) A 2 | A 1 | 2 ] = 0
κ SHG I = ω 1 d eff n 1 c 2 n 2 ε 0 c
n 2 , Kerr I ( ω i ; ω j ) = 3 χ eff ( 3 ) ( ω i ; ω j ) 4 n i n j ε 0 c
[ i z + D ^ 1 ] 1 + 3 ω 1 8 n 1 c [ χ casc ( 3 ) + χ eff ( 3 ) ( ω 1 ; ω 1 ) ] 1 | 1 | 2 + 5 ω 1 16 n 1 c χ casc ( 5 ) 1 | 1 | 4 = 0
χ casc ( 3 ) = 8 ω 1 d eff 2 3 c n 2 Δ k , n 2 , casc I = 2 ω 1 d eff 2 c 2 ε 0 n 1 2 n 2 Δ k
χ casc ( 5 ) = 12 ω 1 2 χ eff ( 3 ) ( ω 1 ; ω 2 ) d eff 2 5 n 2 2 c 2 Δ k 2 , n 4 , casc I = 4 ω 1 2 n Kerr I ( ω 1 ; ω 2 ) d eff 2 Δ k 2 n 1 2 n 2 ε 0 c 3

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