Abstract

We interpret the purely spectral forward Maxwell equation with up to third-order induced polarizations for pulse propagation and interactions in quadratic nonlinear crystals. The interpreted equation, also named the nonlinear wave equation in the frequency domain, includes quadratic and cubic nonlinearities, delayed Raman effects, and anisotropic nonlinearities. The full potential of this wave equation is demonstrated by investigating simulations of solitons generated in the process of ultrafast cascaded second-harmonic generation. We show that a balance in the soliton delay can be achieved due to competition between self-steepening, Raman effects, and self-steepening-like effects from cascading originating in the group-velocity mismatch between the pump and the second harmonic. We analyze the first-order contributions, and show that this balance can be broken to create fast or slow pulses. Through further simulations we demonstrate few-cycle compressed solitons in extremely short crystals, where spectral phenomena, such as blue/red shifting, nonstationary radiation in accordance with the nonlocal phase-matching condition, and dispersive-wave generation are observed and marked, which helps improve the experimental knowledge of cascading nonlinear soliton pulse compression.

© 2013 Optical Society of America

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  1. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992).
    [CrossRef]
  2. 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).
    [CrossRef]
  3. J. Moses and F. Wise, “Soliton compression in quadratic media: high-energy few-cycle pulses with a frequency-doubling crystal,” Opt. Lett. 31, 1881–1883 (2006).
    [CrossRef]
  4. B. Zhou, A. Chong, F. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. 109, 043902 (2012).
    [CrossRef]
  5. D. Nikogosian, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).
  6. T. Kartalo?lu, K. Köprülü, O. Aytür, M. Sundheimer, and W. Risk, “Femtosecond optical parametric oscillator based on periodically poled KTiOPO4,” Opt. Lett. 23, 61–63 (1998).
    [CrossRef]
  7. M. Sundheimer, C. Bosshard, E. Van Stryland, G. Stegeman, and J. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes,” Opt. Lett. 18, 1397–1399 (1993).
    [CrossRef]
  8. S. Ashihara, T. Shimura, K. Kuroda, N. 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, 1055–1058 (2004).
    [CrossRef]
  9. 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]
  10. C. Phillips, C. Langrock, J. Pelc, M. Fejer, I. Hartl, and M. Fermann, “Supercontinuum generation in quasi-phasematched waveguides,” Opt. Express 19, 18754–18773 (2011).
    [CrossRef]
  11. J. Moses and F. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. 97, 73903 (2006).
    [CrossRef]
  12. F. Ilday, K. Beckwitt, Y. Chen, H. Lim, and F. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B 21, 376–383 (2004).
    [CrossRef]
  13. C. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to ?(2):?(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
    [CrossRef]
  14. M. Bache, O. Bang, J. Moses, and F. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. 32, 2490–2492 (2007).
    [CrossRef]
  15. M. Bache, J. Moses, and F. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B 24, 2752–2762 (2007).
    [CrossRef]
  16. M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008).
    [CrossRef]
  17. M. Kolesik, P. Townsend, and J. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. 18, 494–506(2012).
    [CrossRef]
  18. R. Bullough, P. Jack, P. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979).
    [CrossRef]
  19. A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901(4) (2001).
    [CrossRef]
  20. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841(4) (2010).
    [CrossRef]
  21. R. Boyd, Nonlinear Optics (Academic, 2003).
  22. M. Conforti, F. Baronio, and C. De Angelis, “Modelling of broadband and single cycle phenomena in anisotropic quadratic crystals,” J. Opt. Soc. Am. B 28, 1231–1237 (2011).
    [CrossRef]
  23. M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
    [CrossRef]
  24. M. Bache and F. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A 81, 053815 (2010).
    [CrossRef]
  25. P. Banks, M. Feit, and M. Perry, “High-intensity third-harmonic generation,” J. Opt. Soc. Am. B 19, 102–118 (2002).
    [CrossRef]
  26. M. Bache, H. Guo, B. Zhou, and X. Zeng, “The anisotropic Kerr nonlinear refractive index of ?-BaB2O4,” arXiv:1209.3158 [physics.optics], (2012).
  27. G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
    [CrossRef]
  28. G. Stegeman, D. Hagan, and L. Torner, “?(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]
  29. M. Bache, F. Eilenberger, and S. Minardi, “Higher-order Kerr effect and harmonic cascading in gases,” Opt. Lett. 37, 4612–4614 (2012).
    [CrossRef]
  30. 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]
  31. P. Ney, M. Fontana, A. Maillard, and K. Polgar, “Assignment of the Raman lines in single crystal barium metaborate,” J. Phys. Condens. Matter 10, 673–681 (1998).
    [CrossRef]

2012 (3)

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

M. Kolesik, P. Townsend, and J. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. 18, 494–506(2012).
[CrossRef]

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

2011 (3)

2010 (2)

M. Bache and F. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A 81, 053815 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841(4) (2010).
[CrossRef]

2008 (1)

2007 (2)

2006 (2)

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

J. Moses and F. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. 97, 73903 (2006).
[CrossRef]

2004 (3)

F. Ilday, K. Beckwitt, Y. Chen, H. Lim, and F. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B 21, 376–383 (2004).
[CrossRef]

S. Ashihara, T. Shimura, K. Kuroda, N. 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, 1055–1058 (2004).
[CrossRef]

M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

2002 (2)

2001 (1)

A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901(4) (2001).
[CrossRef]

1998 (2)

P. Ney, M. Fontana, A. Maillard, and K. Polgar, “Assignment of the Raman lines in single crystal barium metaborate,” J. Phys. Condens. Matter 10, 673–681 (1998).
[CrossRef]

T. Kartalo?lu, K. Köprülü, O. Aytür, M. Sundheimer, and W. Risk, “Femtosecond optical parametric oscillator based on periodically poled KTiOPO4,” Opt. Lett. 23, 61–63 (1998).
[CrossRef]

1997 (1)

1996 (1)

G. Stegeman, D. Hagan, and L. Torner, “?(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]

1994 (1)

1993 (1)

1992 (1)

1991 (1)

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]

1979 (1)

R. Bullough, P. Jack, P. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979).
[CrossRef]

Arbore, M.

Ashihara, S.

S. Ashihara, T. Shimura, K. Kuroda, N. 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, 1055–1058 (2004).
[CrossRef]

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).
[CrossRef]

Aytür, O.

Bache, M.

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

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

M. Bache and F. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A 81, 053815 (2010).
[CrossRef]

M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008).
[CrossRef]

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

M. Bache, J. Moses, and F. Wise, “Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities,” J. Opt. Soc. Am. B 24, 2752–2762 (2007).
[CrossRef]

M. Bache, H. Guo, B. Zhou, and X. Zeng, “The anisotropic Kerr nonlinear refractive index of ?-BaB2O4,” arXiv:1209.3158 [physics.optics], (2012).

Bang, O.

Banks, P.

Baronio, F.

M. Conforti, F. Baronio, and C. De Angelis, “Modelling of broadband and single cycle phenomena in anisotropic quadratic crystals,” J. Opt. Soc. Am. B 28, 1231–1237 (2011).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841(4) (2010).
[CrossRef]

Beckwitt, K.

Bierlein, J.

Bosshard, C.

Boyd, R.

R. Boyd, Nonlinear Optics (Academic, 2003).

Bullough, R.

R. Bullough, P. Jack, P. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979).
[CrossRef]

Cha, M.

S. Ashihara, T. Shimura, K. Kuroda, N. 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, 1055–1058 (2004).
[CrossRef]

Chen, Y.

Chong, A.

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

Clerici, M.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Conforti, M.

M. Conforti, F. Baronio, and C. De Angelis, “Modelling of broadband and single cycle phenomena in anisotropic quadratic crystals,” J. Opt. Soc. Am. B 28, 1231–1237 (2011).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841(4) (2010).
[CrossRef]

De Angelis, C.

M. Conforti, F. Baronio, and C. De Angelis, “Modelling of broadband and single cycle phenomena in anisotropic quadratic crystals,” J. Opt. Soc. Am. B 28, 1231–1237 (2011).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841(4) (2010).
[CrossRef]

DeSalvo, R.

DiTrapani, P.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Eilenberger, F.

Feit, M.

Fejer, M.

Fermann, M.

Fontana, M.

P. Ney, M. Fontana, A. Maillard, and K. Polgar, “Assignment of the Raman lines in single crystal barium metaborate,” J. Phys. Condens. Matter 10, 673–681 (1998).
[CrossRef]

Guo, H.

M. Bache, H. Guo, B. Zhou, and X. Zeng, “The anisotropic Kerr nonlinear refractive index of ?-BaB2O4,” arXiv:1209.3158 [physics.optics], (2012).

Hagan, D.

G. Stegeman, D. Hagan, and L. Torner, “?(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]

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

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]

Hartl, I.

Herrmann, J.

A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901(4) (2001).
[CrossRef]

Husakou, A.

A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901(4) (2001).
[CrossRef]

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).
[CrossRef]

Ilday, F.

Jack, P.

R. Bullough, P. Jack, P. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979).
[CrossRef]

Jedrkiewicz, O.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Jukna, V.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Kartaloglu, T.

Kitamura, K.

S. Ashihara, T. Shimura, K. Kuroda, N. 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, 1055–1058 (2004).
[CrossRef]

Kitchenside, P.

R. Bullough, P. Jack, P. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979).
[CrossRef]

Kolesik, M.

M. Kolesik, P. Townsend, and J. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. 18, 494–506(2012).
[CrossRef]

M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

Köprülü, K.

Krolikowski, W.

Kurimura, S.

S. Ashihara, T. Shimura, K. Kuroda, N. 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, 1055–1058 (2004).
[CrossRef]

Kuroda, K.

S. Ashihara, T. Shimura, K. Kuroda, N. 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, 1055–1058 (2004).
[CrossRef]

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).
[CrossRef]

Langrock, C.

Lim, H.

Maillard, A.

P. Ney, M. Fontana, A. Maillard, and K. Polgar, “Assignment of the Raman lines in single crystal barium metaborate,” J. Phys. Condens. Matter 10, 673–681 (1998).
[CrossRef]

Marco, O.

Menyuk, C.

Minardi, S.

Moloney, J.

M. Kolesik, P. Townsend, and J. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. 18, 494–506(2012).
[CrossRef]

M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

Moses, J.

Ney, P.

P. Ney, M. Fontana, A. Maillard, and K. Polgar, “Assignment of the Raman lines in single crystal barium metaborate,” J. Phys. Condens. Matter 10, 673–681 (1998).
[CrossRef]

Nikogosian, D.

D. Nikogosian, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).

Nishina, J.

Pelc, J.

Perry, M.

Phillips, C.

Polgar, K.

P. Ney, M. Fontana, A. Maillard, and K. Polgar, “Assignment of the Raman lines in single crystal barium metaborate,” J. Phys. Condens. Matter 10, 673–681 (1998).
[CrossRef]

Risk, W.

Rubino, E.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Saunders, R.

R. Bullough, P. Jack, P. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979).
[CrossRef]

Schiek, R.

Sheik-Bahae, M.

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

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]

Shimura, T.

S. Ashihara, T. Shimura, K. Kuroda, N. 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, 1055–1058 (2004).
[CrossRef]

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).
[CrossRef]

Stegeman, G.

Sundheimer, M.

Taira, T.

S. Ashihara, T. Shimura, K. Kuroda, N. 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, 1055–1058 (2004).
[CrossRef]

Torner, L.

G. Stegeman, D. Hagan, and L. Torner, “?(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]

C. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to ?(2):?(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994).
[CrossRef]

Townsend, P.

M. Kolesik, P. Townsend, and J. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. 18, 494–506(2012).
[CrossRef]

Valiulis, G.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Van Stryland, E.

Vanherzeele, H.

Wise, F.

Yu, N.

S. Ashihara, T. Shimura, K. Kuroda, N. 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, 1055–1058 (2004).
[CrossRef]

Zeng, X.

M. Bache, H. Guo, B. Zhou, and X. Zeng, “The anisotropic Kerr nonlinear refractive index of ?-BaB2O4,” arXiv:1209.3158 [physics.optics], (2012).

Zhou, B.

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

M. Bache, H. Guo, B. Zhou, and X. Zeng, “The anisotropic Kerr nonlinear refractive index of ?-BaB2O4,” arXiv:1209.3158 [physics.optics], (2012).

Appl. Phys. Lett. (1)

S. Ashihara, T. Shimura, K. Kuroda, N. 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, 1055–1058 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

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]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Kolesik, P. Townsend, and J. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. 18, 494–506(2012).
[CrossRef]

J. Opt. Soc. Am. B (6)

J. Phys. Condens. Matter (1)

P. Ney, M. Fontana, A. Maillard, and K. Polgar, “Assignment of the Raman lines in single crystal barium metaborate,” J. Phys. Condens. Matter 10, 673–681 (1998).
[CrossRef]

Opt. Express (2)

Opt. Lett. (7)

Opt. Quantum Electron. (1)

G. Stegeman, D. Hagan, and L. Torner, “?(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]

Phys. Rev. A (3)

M. Bache and F. Wise, “Type-I cascaded quadratic soliton compression in lithium niobate: compressing femtosecond pulses from high-power fiber lasers,” Phys. Rev. A 81, 053815 (2010).
[CrossRef]

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and x-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841(4) (2010).
[CrossRef]

Phys. Rev. E (1)

M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

Phys. Rev. Lett. (3)

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B. Zhou, A. Chong, F. 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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[CrossRef]

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R. Boyd, Nonlinear Optics (Academic, 2003).

M. Bache, H. Guo, B. Zhou, and X. Zeng, “The anisotropic Kerr nonlinear refractive index of ?-BaB2O4,” arXiv:1209.3158 [physics.optics], (2012).

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Figures (5)

Fig. 1.
Fig. 1.

Compression window of MgOLN cut for type 0 interaction. (a) Cascading quadratic nonlinearity (using d33=25pm/V at 1064 nm and Miller scaling to other wavelengths) and the native electronic Kerr nonlinearity from the so-called two-band model [4,30]. (b) Critical boundary of producing an overall self-defocusing nonlinearity and the stationary/nonstationary threshold of the cascading nonlocal response function. (c) Radiation in the nonstationary region, showing the resonant wavelength as a function of the effective phase mismatch, with the pump located at 1560 nm. (d) Degradation on the compression window when using a first-order QPM on MgOLN.

Fig. 2.
Fig. 2.

Simulations of NLS-like equation on the balance of the fast/slow pulses. Dispersion properties are chosen from an LN crystal. The cascading nonlinearity is Ncasc=2.60 and the material electronic Kerr nonlinearity is NKerr,el=1.96; the input pulse has a FWHM=50fs and a peak intensity Ipeak=100GW/cm2, located at 1560 nm. (a) Shock front on pulses and slow pulses driven by dispersion. (b) Slow pulses in QPM crystal without Raman effects, in which Δkeff=129.5mm1. (c) Fast pulses driven by 50% Raman effects in a bulk crystal with Δk=319.5mm1. (d) Balanced pulses generated in a QPM crystal with 50% Raman effects. The inset figures correspond to the spectrum evolutions, the temporal profiles, and the spectra at the first compression point of the pulses (position marked by the white dashed line).

Fig. 3.
Fig. 3.

Simulation of cascading nonlinear soliton pulse compression in BBO cut for type I interaction at 1030 nm. The rotation (θ,φ)=(20.5°,90.0°), under which Δkc=78.1mm1 and Δksr=46.1mm1. Native phase mismatch Δk=55.4mm1. The launched pulse has FWHM=200fs, peak intensity Iin=100GW/cm2, ncasc(2)=7.86×1020m2/W, nKerr,el|o:ooo=5.67×1020m2/W, Neff=6.00. (a) Electric field of the first stage soliton (position marked by the white dashed line). (b) Temporal evolution of the FW. (c), (d) Spectral evolutions of both o-pol and e-pol pulses.

Fig. 4.
Fig. 4.

Simulation of cascading nonlinear soliton pulse compression in MgOLN cut for type 0 interaction at 1300 nm. The rotation (θ,φ)=(90°,90°), i.e., X-cut, under which Δkc=890.4mm1 and Δksr=283.3mm1. Native phase mismatch Δk=501.5mm1. The launched pulse has FWHM=50fs, peak intensity Iin=100GW/cm2, ncasc(2)=40.05×1020m2/W, nKerr,el=22.56×1020m2/W, Ncasc2NKerr,el2=2.05. (a) Electric field of the first stage soliton (position marked by the white dashed line). (b) Temporal evolution of the FW. (c) Spectrum of the first stage soliton. (d) Spectral evolution of the e-pol pulse with obvious red shifting of the FW and blue shifting of the D-wave.

Fig. 5.
Fig. 5.

Simulation of cascading nonlinear soliton pulse compression in PPMgOLN cut for type 0 interaction at 1300 nm. The poling period is 26 μm. The rotation (θ,φ)=(90°,90°), under which Δkc,QPM=360.9mm1 and Δksr=283.3mm1. The effective phase mismatch Δkeff=259.8mm1. The launched pulse has FWHM=50fs, peak intensity Iin=100GW/cm2, ncasc,QPM(2)=31.32×1020m2/W, nKerr,el=22.56×1020m2/W, Ncasc,QPM2NKerr,el2=1.45. (a) Electric field of the pulse (position marked by the white dash line). (b) Temporal evolution of the FW. (c) Spectrum of the pulse. (d) Spectral evolution of the pulse with resonant SH radiation and strong dispersive wave.

Tables (1)

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Table 1. Nonlinear Parameters Used for Simulations

Equations (13)

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E˜z+ik(ω)E˜=iω2μ02k(ω)P˜NL.
P˜j(2)=ε0α1,α2(χ¯j;α1α2(2)F[Eα1Eα2]),
P˜j(3)=ε0α1,α2,α3{χ¯j;α1α2α3(3)[(1fR)F[Eα1Eα2Eα3]+fRF[F1[h˜R(ω)F[Eα1Eα2]E˜α3]]}.
E˜oz+iko(ω)E˜o=iω22c2ko(ω){α1,α2(χ¯o;α1α2(2)F[Eα1Eα2])+α1,α2,α3(χ¯o;α1α2α3(3)((1fR)F[Eα1Eα2Eα3]+fR·F[F1[h˜R(ω)F[Eα1Eα2]E˜α3]))},
E˜ez+ike(ω)E˜e=iω22c2ke(ω){α1,α2(χ¯e;α1α2(2)F[Eα1Eα2])+α1,α2,α3(χ¯e;α1α2α3(3)((1fR)F[Eα1Eα2Eα3]+fR·F[F1[h˜R(ω)F[Eα1Eα2]E˜α3]))}.
(iξD1)U1+sgn(Δk)Ncasc2(1iω^1τ)U1*(τ)(1iω^2τ)dτhc(τ)U12(ττ)Ncubic2(1iω^1τ)[(1fR)|U1(τ)|2U1(τ)+fRU1(τ)dτhR(τ)|U1(ττ)|2]=0,
ncasc(2)=2ω1deff2ε0c2n12n2Δk,
Δknonlocal(Ω)=k2(ω2+Ω)2k1(ω1)Ωk1(1)(ω1)=m=2Ωm·k2(m)(ω2)m!d12Ω+Δk,
ncasc,QPM(2)=2ω1(2πdeff)2ε0c2n12n2Δkeff,
Ncasc2(1iω^1τ)U1*(τ)(1iω^2τ)dτhc(τ)U12(ττ)=Ncasc2(1iω^1τ)U1*(1iω^2τ)(U12iτcτU12)=Ncasc2[|U1|2U13iω^1|U1|2τU12iτc|U1|2τU1iω^1U12τU1*]+HOT,
Ncubic2(1iω^1τ)[(1fR)|U1(τ)|2U1(τ)+fRU1(τ)dτhR(τ)|U1(ττ)|2]=Ncubic2(1iω^1τ)[|U1|2U1τRU1τ|U1|2]=Ncubic2[|U1|2U1(2iω1+τR)|U1|2τU1(iω1+τR)U12τU1*]+HOT,
Aξ=(4Ncasc23Ncubic2ω^1+2τcNcasc2)A2Aτ,
φξ=Neff2A22τRNcubic2AAτ+(2Ncasc2Ncubic2ω^1+2τcNcasc2)A2φτ,

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