Abstract

We present a detailed study of soliton compression of ultrashort pulses based on phase-mismatched second-harmonic generation (SHG) (i.e., the cascaded quadratic nonlinearity) in bulk quadratic nonlinear media. The single-cycle propagation equations in the temporal domain including higher-order nonlinear terms are presented. The balance between the quadratic (SHG) and the cubic (Kerr) nonlinearity plays a crucial role; we define an effective soliton number—related to the difference between the SHG and the Kerr soliton numbers—and show that it has to be larger than unity for successful pulse compression to take place. This requires that the phase mismatch be below a critical level, which is high in a material where the quadratic nonlinearity dominates over the cubic Kerr nonlinearity. Through extensive numerical simulations we find dimensionless scaling laws, expressed through the effective soliton number, that control the behavior of the compressed pulses. These laws hold in the stationary regime, in which group-velocity mismatch effects are small, and they are similar to the ones observed for fiber soliton compressors. The numerical simulations indicate that clean compressed pulses below two optical cycles can be achieved in a β-barium borate crystal at appropriate wavelengths, even for picosecond input pulses.

© 2007 Optical Society of America

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  1. 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).
  2. G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, "Large nonlinear phase shifts in second-order nonlinear-optical processes," Opt. Lett. 18, 13-15 (1993).
  3. C. R. Menyuk, R. Schiek, and L. Torner, "Solitary waves due to χ(2):χ(2) cascading," J. Opt. Soc. Am. B 11, 2434-2443 (1994).
  4. X. Liu, L. 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).
  5. S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, "Soliton compression of femtosecond pulses in quadratic media," J. Opt. Soc. Am. B 19, 2505-2510 (2002).
  6. 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]
  7. X. Zeng, S. Ashihara, N. Fujioka, T. Shimura, and K. Kuroda, "Adiabatic compression of quadratic temporal solitons in aperiodic quasi-phase-matching gratings," Opt. Express 14, 9358-9370 (2006).
    [CrossRef]
  8. 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]
  9. G. Xie, D. Zhang, L. Qian, H. Zhu, and D. Tang, "Multi-stage pulse compression by use of cascaded quadratic nonlinearity," Opt. Commun. 273, 207-213 (2007).
  10. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
  11. G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2001).
  12. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, "Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers," Opt. Lett. 8, 289-291 (1983).
  13. W. J. Tomlinson, R. H. Stolen, and C. V. Shank, "Compression of optical pulses chirped by self-phase modulation in fibers," J. Opt. Soc. Am. B 1, 139-149 (1984).
  14. E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, "Optimal compression of multi-soliton pulses in optical fibers," Sov. Tech. Phys. Lett. 12, 311-313 (1986).
  15. 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. B 21, 376-383 (2004).
    [CrossRef]
  16. 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]
  17. T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
    [CrossRef]
  18. J. Moses and F. W. Wise, "Controllable self-steepening of ultrashort pulses in quadratic nonlinear media," Phys. Rev. Lett. 97, 073903 (2006).
    [CrossRef]
  19. V. Dmitriev, G. Gurzadyan, and D. Nikogosyan, Handbook of Nonlinear Optical Crystals, Vol. 64 of Springer Series in Optical Sciences (Springer, 1999).
  20. 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).
    [CrossRef]
  21. J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
    [CrossRef]
  22. C.-M. Chen and P. L. Kelley, "Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis," J. Opt. Soc. Am. B 19, 1961-1967 (2002).
  23. J. Moses, B. A. Malomed, and F. W. Wise, "Self-steepening of ultrashort optical pulses without self-phase modulation," Phys. Rev. A 76, 021802R (2007).
  24. M. Bache, H. Nielsen, J. Lægsgaard, and O. Bang, "Tuning quadratic nonlinear photonic crystal fibers for zero group-velocity mismatch," Opt. Lett. 31, 1612-1614 (2006).
    [CrossRef]
  25. S. Kumar, A. Selvarajan, and G. Anand, "Influence of Raman scattering on the cross-phase modulation in optical fibers," Opt. Commun. 102, 329-335 (1993).
    [CrossRef]
  26. K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
    [CrossRef]
  27. 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]
  28. 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]

2007 (4)

G. Xie, D. Zhang, L. Qian, H. Zhu, and D. Tang, "Multi-stage pulse compression by use of cascaded quadratic nonlinearity," Opt. Commun. 273, 207-213 (2007).

J. Moses, B. A. Malomed, and F. W. Wise, "Self-steepening of ultrashort optical pulses without self-phase modulation," Phys. Rev. A 76, 021802R (2007).

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).
[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]

2006 (5)

2004 (2)

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]

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. B 21, 376-383 (2004).
[CrossRef]

2002 (2)

1999 (1)

1997 (2)

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]

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

1996 (1)

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]

1994 (1)

1993 (2)

G. I. Stegeman, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, "Large nonlinear phase shifts in second-order nonlinear-optical processes," Opt. Lett. 18, 13-15 (1993).

S. Kumar, A. Selvarajan, and G. Anand, "Influence of Raman scattering on the cross-phase modulation in optical fibers," Opt. Commun. 102, 329-335 (1993).
[CrossRef]

1992 (1)

1989 (1)

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

1986 (1)

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, "Optimal compression of multi-soliton pulses in optical fibers," Sov. Tech. Phys. Lett. 12, 311-313 (1986).

1984 (1)

1983 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

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

Alhammali, E.

Anand, G.

S. Kumar, A. Selvarajan, and G. Anand, "Influence of Raman scattering on the cross-phase modulation in optical fibers," Opt. Commun. 102, 329-335 (1993).
[CrossRef]

Ashihara, S.

Assanto, G.

Bache, M.

Bang, O.

Beckwitt, K.

Blow, K. J.

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Brabec, T.

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

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).
[CrossRef]

Chen, C.-M.

Chen, Y.-F.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

DeSalvo, R.

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).

Dianov, E. M.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, "Optimal compression of multi-soliton pulses in optical fibers," Sov. Tech. Phys. Lett. 12, 311-313 (1986).

Dmitriev, V.

V. Dmitriev, G. Gurzadyan, and D. Nikogosyan, Handbook of Nonlinear Optical Crystals, Vol. 64 of Springer Series in Optical Sciences (Springer, 1999).

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

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]

Eichenholz, J. M.

Fujioka, N.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Gordon, J. P.

Gurzadyan, G.

V. Dmitriev, G. Gurzadyan, and D. Nikogosyan, Handbook of Nonlinear Optical Crystals, Vol. 64 of Springer Series in Optical Sciences (Springer, 1999).

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).

Ilday, F. Ö.

Kelley, P. L.

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).
[CrossRef]

Krausz, F.

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Kumar, S.

S. Kumar, A. Selvarajan, and G. Anand, "Influence of Raman scattering on the cross-phase modulation in optical fibers," Opt. Commun. 102, 329-335 (1993).
[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.

Lægsgaard, J.

Lim, H.

Liu, X.

Malomed, B. A.

J. Moses, B. A. Malomed, and F. W. Wise, "Self-steepening of ultrashort optical pulses without self-phase modulation," Phys. Rev. A 76, 021802R (2007).

Menyuk, C. R.

Mollenauer, L. F.

Moses, J.

Nielsen, H.

Nikogosyan, D.

V. Dmitriev, G. Gurzadyan, and D. Nikogosyan, Handbook of Nonlinear Optical Crystals, Vol. 64 of Springer Series in Optical Sciences (Springer, 1999).

Nikonova, Z. S.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, "Optimal compression of multi-soliton pulses in optical fibers," Sov. Tech. Phys. Lett. 12, 311-313 (1986).

Nishina, J.

Prokhorov, A. M.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, "Optimal compression of multi-soliton pulses in optical fibers," Sov. Tech. Phys. Lett. 12, 311-313 (1986).

Qian, L.

G. Xie, D. Zhang, L. Qian, H. Zhu, and D. Tang, "Multi-stage pulse compression by use of cascaded quadratic nonlinearity," Opt. Commun. 273, 207-213 (2007).

X. Liu, L. 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).

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]

Schiek, R.

Selvarajan, A.

S. Kumar, A. Selvarajan, and G. Anand, "Influence of Raman scattering on the cross-phase modulation in optical fibers," Opt. Commun. 102, 329-335 (1993).
[CrossRef]

Serkin, V. N.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, "Optimal compression of multi-soliton pulses in optical fibers," Sov. Tech. Phys. Lett. 12, 311-313 (1986).

Shank, C. V.

Sheik-Bahae, 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]

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, M. Sheik-Bahae, E. Van Stryland, and G. Assanto, "Large nonlinear phase shifts in second-order nonlinear-optical processes," Opt. Lett. 18, 13-15 (1993).

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).

Shimura, T.

Stegeman, G.

Stegeman, G. I.

Stolen, R. H.

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).
[CrossRef]

Tang, D.

G. Xie, D. Zhang, L. Qian, H. Zhu, and D. Tang, "Multi-stage pulse compression by use of cascaded quadratic nonlinearity," Opt. Commun. 273, 207-213 (2007).

Tomlinson, W. J.

Torner, L.

Van Stryland, E.

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).

Vanherzeele, H.

Wise, F. W.

Wood, D.

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Xie, G.

G. Xie, D. Zhang, L. Qian, H. Zhu, and D. Tang, "Multi-stage pulse compression by use of cascaded quadratic nonlinearity," Opt. Commun. 273, 207-213 (2007).

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]

Zeng, X.

Zhang, D.

G. Xie, D. Zhang, L. Qian, H. Zhu, and D. Tang, "Multi-stage pulse compression by use of cascaded quadratic nonlinearity," Opt. Commun. 273, 207-213 (2007).

Zhu, H.

G. Xie, D. Zhang, L. Qian, H. Zhu, and D. Tang, "Multi-stage pulse compression by use of cascaded quadratic nonlinearity," Opt. Commun. 273, 207-213 (2007).

Appl. Phys. Lett. (1)

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]

IEEE J. Quantum Electron. (2)

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[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]

J. Opt. Soc. Am. B (5)

Opt. Commun. (3)

S. Kumar, A. Selvarajan, and G. Anand, "Influence of Raman scattering on the cross-phase modulation in optical fibers," Opt. Commun. 102, 329-335 (1993).
[CrossRef]

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]

G. Xie, D. Zhang, L. Qian, H. Zhu, and D. Tang, "Multi-stage pulse compression by use of cascaded quadratic nonlinearity," Opt. Commun. 273, 207-213 (2007).

Opt. Express (1)

Opt. Lett. (8)

Phys. Rev. A (1)

J. Moses, B. A. Malomed, and F. W. Wise, "Self-steepening of ultrashort optical pulses without self-phase modulation," Phys. Rev. A 76, 021802R (2007).

Phys. Rev. Lett. (2)

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

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

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Sov. Tech. Phys. Lett. (1)

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and V. N. Serkin, "Optimal compression of multi-soliton pulses in optical fibers," Sov. Tech. Phys. Lett. 12, 311-313 (1986).

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

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

V. Dmitriev, G. Gurzadyan, and D. Nikogosyan, Handbook of Nonlinear Optical Crystals, Vol. 64 of Springer Series in Optical Sciences (Springer, 1999).

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