Abstract

We study cascaded quadratic soliton compressors and address the physical mechanisms that limit the compression. A nonlocal model is derived, and the nonlocal response is shown to have an additional oscillatory component in the nonstationary regime when the group-velocity mismatch (GVM) is strong. This inhibits efficient compression. Raman-like perturbations from the cascaded nonlinearity, competing cubic nonlinearities, higher-order dispersion, and soliton energy may also limit compression, and through realistic numerical simulations we point out when each factor becomes important. We find that it is theoretically possible to reach the single-cycle regime by compressing high-energy fs pulses for wavelengths λ=1.0-1.3 µm in a β-barium-borate crystal, and it requires that the system is in the stationary regime, where the phase mismatch is large enough to overcome the detrimental GVM effects. However, the simulations show that reaching single-cycle duration is ultimately inhibited by competing cubic nonlinearities as well as dispersive waves, that only show up when taking higher-order dispersion into account.

© 2008 Optical Society of America

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  2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
    [CrossRef]
  3. 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]
  4. 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]
  5. J. A. Moses, J. Nees, B. Hou, K.-H. Hong, G. Mourou, and F. W. Wise, "Chirped-pulse cascaded quadratic compression of 1-mJ, 35-fs pulses with low wavefront distortions," In Conference on Lasers and Electro-Optics, p. CTuS5 (Optical Society of America, 2005).
  6. 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]
  7. 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] [PubMed]
  8. 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).
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  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).
    [CrossRef]
  10. 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), arXiv:0706.1933.
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. C. B. Clausen, O. Bang, and Y. S. Kivshar, "Spatial solitons and Induced Kerr effects in quasi-phase-matched Quadratic media," Phys. Rev. Lett. 78, 4749-4752 (1997).
    [CrossRef]
  14. X. Liu, L. Qian, and F. W. Wise, "High-energy pulse compression by use of negative phase shifts produced by the cascaded |(2) : |(2) nonlinearity," Opt. Lett. 24, 1777-1779 (1999).
    [CrossRef]
  15. P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing," Phys. Rev. Lett. 87, 183902 (2001).
    [CrossRef]
  16. L. Berge, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, "Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities," Phys. Rev. E 55, 3555-3570 (1997).
    [CrossRef]
  17. F. O. 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]
  18. N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614 (2003).
    [CrossRef]
  19. W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclass. Opt. 6, s288 (2004).
    [CrossRef]
  20. J. Moses and F. W. Wise, "Controllable self-steepening of ultrashort pulses in quadratic nonlinear media," Phys. Rev. Lett. 97, 073903 (2006), see also arXiv:physics/0604170.
    [CrossRef] [PubMed]
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  22. On dimensional formR(t)=R(t/Tin)/Tin, which is independent on Tin since |a,b in Eqs. (10,12) must be replaced by the dimensional form ta,b =|a,bTin. In the frequency domain both ˜R and ˜R are dimensionless.
  23. W. Krolikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2000).
    [CrossRef]
  24. M. Bache, O. Bang, and W. Krolikowski, (2008), in preparation.
  25. The factor sa on the RHS of Eq. (17) was unfortunately lost during the proofs in Eq. (12) of Ref. [10].
  26. These experiments were actually done in the nonstationary regime according to the nonlocal theory.
  27. This is a typical experimental situation: the optimal compression point zopt scales with Neff [11], and since the nonlinear crystal length is a constant parameter one adjusts the intensity so zopt coincides with the crystal length.
  28. A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997).
    [CrossRef]
  29. W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
    [CrossRef]
  30. I. V. Shadrivov and A. A. Zharov, "Dynamics of optical spatial solitons near the interface between two quadratically nonlinear media," J. Opt. Soc. Am. B 19, 596-602 (2002).
    [CrossRef]
  31. N. Akhmediev and M. Karlsson, "Cherenkov radiation emitted by solitons in optical fibers," Phys. Rev. A 51, 2602-2607 (1995).
    [CrossRef] [PubMed]
  32. D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1708 (2003).
    [CrossRef] [PubMed]
  33. I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, "Dispersive wave generation by solitons in microstructured optical fibers," Opt. Express 12, 124-135 (2003).
    [CrossRef]
  34. K. C. Chan and M. S. F. Liu, "Short-pulse generation by higher-order soliton-effect compression: Effects of fiber characteristics," IEEE J. Quantum Electron. 31, 2226-2235 (1995).
    [CrossRef]
  35. K.-T. Chan and W.-H. Cao, "Improved soliton-effect pulse compression by combined action of negative thirdorder dispersion and Raman self-scattering in optical fibers," J. Opt. Soc. Am. B 15, 2371-2376.
  36. 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), arXiv:physics/0511244.
    [CrossRef] [PubMed]

2007

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).
[CrossRef]

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] [PubMed]

2006

2004

F. O. 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]

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]

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclass. Opt. 6, s288 (2004).
[CrossRef]

2003

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614 (2003).
[CrossRef]

I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, "Dispersive wave generation by solitons in microstructured optical fibers," Opt. Express 12, 124-135 (2003).
[CrossRef]

2002

2001

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing," Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

2000

W. Krolikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2000).
[CrossRef]

1999

1997

A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997).
[CrossRef]

L. Berge, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, "Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities," Phys. Rev. E 55, 3555-3570 (1997).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, "Spatial solitons and Induced Kerr effects in quasi-phase-matched Quadratic media," Phys. Rev. Lett. 78, 4749-4752 (1997).
[CrossRef]

1995

N. Akhmediev and M. Karlsson, "Cherenkov radiation emitted by solitons in optical fibers," Phys. Rev. A 51, 2602-2607 (1995).
[CrossRef] [PubMed]

K. C. Chan and M. S. F. Liu, "Short-pulse generation by higher-order soliton-effect compression: Effects of fiber characteristics," IEEE J. Quantum Electron. 31, 2226-2235 (1995).
[CrossRef]

1992

1980

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Akhmediev, N.

N. Akhmediev and M. Karlsson, "Cherenkov radiation emitted by solitons in optical fibers," Phys. Rev. A 51, 2602-2607 (1995).
[CrossRef] [PubMed]

Alhammali, E.

Ashihara, S.

Bache, M.

Bang, O.

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), arXiv:physics/0511244.
[CrossRef] [PubMed]

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclass. Opt. 6, s288 (2004).
[CrossRef]

N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614 (2003).
[CrossRef]

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

W. Krolikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2000).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, "Spatial solitons and Induced Kerr effects in quasi-phase-matched Quadratic media," Phys. Rev. Lett. 78, 4749-4752 (1997).
[CrossRef]

Bramati, A.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing," Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Cao, W.-H.

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]

Chan, K. C.

K. C. Chan and M. S. F. Liu, "Short-pulse generation by higher-order soliton-effect compression: Effects of fiber characteristics," IEEE J. Quantum Electron. 31, 2226-2235 (1995).
[CrossRef]

Chan, K.-T.

Chinaglia, W.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing," Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Clausen, C. B.

C. B. Clausen, O. Bang, and Y. S. Kivshar, "Spatial solitons and Induced Kerr effects in quasi-phase-matched Quadratic media," Phys. Rev. Lett. 78, 4749-4752 (1997).
[CrossRef]

Conti, C.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing," Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Cristiani, I.

Degiorgio, V.

DeSalvo, R.

Di Trapani, P.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing," Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Edmundson, D.

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclass. Opt. 6, s288 (2004).
[CrossRef]

Eichenholz, J. M.

Fujioka, N.

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Hagan, D.

Karlsson, M.

N. Akhmediev and M. Karlsson, "Cherenkov radiation emitted by solitons in optical fibers," Phys. Rev. A 51, 2602-2607 (1995).
[CrossRef] [PubMed]

Kilius, J.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing," Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

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]

Kivshar, Y. S.

C. B. Clausen, O. Bang, and Y. S. Kivshar, "Spatial solitons and Induced Kerr effects in quasi-phase-matched Quadratic media," Phys. Rev. Lett. 78, 4749-4752 (1997).
[CrossRef]

Knight, J. C.

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Krolikowski, W.

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclass. Opt. 6, s288 (2004).
[CrossRef]

N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614 (2003).
[CrossRef]

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

W. Krolikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2000).
[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.

Liu, M. S. F.

K. C. Chan and M. S. F. Liu, "Short-pulse generation by higher-order soliton-effect compression: Effects of fiber characteristics," IEEE J. Quantum Electron. 31, 2226-2235 (1995).
[CrossRef]

Liu, X.

Luan, F.

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Minardi, S.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing," Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Moses, J.

Neshev, D.

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclass. Opt. 6, s288 (2004).
[CrossRef]

N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614 (2003).
[CrossRef]

Nielsen, H.

Nikolov, N.

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclass. Opt. 6, s288 (2004).
[CrossRef]

Nikolov, N. I.

N. I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614 (2003).
[CrossRef]

Nishina, J.

O, F.

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).
[CrossRef]

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

Rasmussen, J.

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclass. Opt. 6, s288 (2004).
[CrossRef]

Rasmussen, J. J.

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Russell, P. S. J.

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Shadrivov, I. V.

Sheik-Bahae, M.

Shimura, T.

Skryabin, D. V.

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997).
[CrossRef]

Stegeman, G.

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[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).
[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).
[CrossRef]

Tartara, L.

Tediosi, R.

Trillo, S.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing," Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Valiulis, G.

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, "Focusing versus defocusing nonlinearities due to parametric wave mixing," Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Van Stryland, E. W.

Vanherzeele, H.

Wise, F. W.

Wyller, J.

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W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[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).
[CrossRef]

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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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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).
[CrossRef]

Zharov, A. A.

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).
[CrossRef]

Appl. Phys. Lett.

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.

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[CrossRef]

J. Opt. B: Quantum Semiclass. Opt.

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclass. Opt. 6, s288 (2004).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

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).
[CrossRef]

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W. Krolikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2000).
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[CrossRef]

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

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The factor sa on the RHS of Eq. (17) was unfortunately lost during the proofs in Eq. (12) of Ref. [10].

These experiments were actually done in the nonstationary regime according to the nonlocal theory.

This is a typical experimental situation: the optimal compression point zopt scales with Neff [11], and since the nonlinear crystal length is a constant parameter one adjusts the intensity so zopt coincides with the crystal length.

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), arXiv:0706.1933.
[CrossRef] [PubMed]

M. Bache, J. Moses, and F. W. Wise, "Scaling laws for soliton pulse compression by cascaded quadratic nonlinearities," J. Opt. Soc. Am. B 24, 2752-2762 (2007), arXiv:0706.1507.
[CrossRef]

J. A. Moses, J. Nees, B. Hou, K.-H. Hong, G. Mourou, and F. W. Wise, "Chirped-pulse cascaded quadratic compression of 1-mJ, 35-fs pulses with low wavefront distortions," In Conference on Lasers and Electro-Optics, p. CTuS5 (Optical Society of America, 2005).

J. Moses and F. W. Wise, "Controllable self-steepening of ultrashort pulses in quadratic nonlinear media," Phys. Rev. Lett. 97, 073903 (2006), see also arXiv:physics/0604170.
[CrossRef] [PubMed]

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On dimensional formR(t)=R(t/Tin)/Tin, which is independent on Tin since |a,b in Eqs. (10,12) must be replaced by the dimensional form ta,b =|a,bTin. In the frequency domain both ˜R and ˜R are dimensionless.

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Figures (5)

Fig. 1.
Fig. 1.

Numerical simulations of soliton compression of a λ 1=1064 nm 200 fs FWHM pulse in a β -barium-borate crystal with a soliton number of Neff =8. (a) and (b) Temporal and spectral components of the FW |U 1|2=I 1/I in in the stationary regime (Δk=50 mm-1, I in=59 GW/cm2). The pulse is compressed to 6 fs at the optimal compression point [z=z opt, dashed line, cuts in (e) and (f)]. (c) and (d) In the nonstationary regime (Δk=30 mm-1, I in=29 GW/cm2), a 17 fs pulse compressed pulse with trailing oscillations is observed.

Fig. 2.
Fig. 2.

The dimensional nonlocal response functions [22] in the (a,b) stationary regime (sb =+1) and (c,d) nonstationary regime (sb =-1) as calculated for the simulations in Fig. 1. The cusp at t=0 of |R ±(t)| is typical for a Lorentzian response. The spectral content of U 2 1=sech2(t/T in) having 100, 20, and 10 fs FWHM duration is shown in (b,d).

Fig. 3.
Fig. 3.

Data from numerical simulations of the full SEWA Eqs. (1) using the same parameters as in Fig. 1 and varying Δk. (a) The FW duration Δt optt FWHM opt/1.76 at z=z opt is shown both for the full SEWA model (1), and when neglecting the Kerr XPM terms. The lines show the nonlocal time scales ta,b =T inτ a,b , the characteristic Raman-like time T R,SHG=2|d 12|/Δk, and the predicted Δt opt from the scaling laws [11] as well as the predicted Δt corr opt when correcting for XPM effects on N eff. tb as calculated using only up to second-order dispersion (md =2, gray curve) is also shown. The right ordinate shows time normalized to the single-cycle pulse duration t sc=2.0 fs. Note that below Δk=10 mm-1 the cascading limit breaks down [12]. (b) The SHG and Kerr soliton numbers required to have N eff=8 fixed, achieved by adjusting I in. The corrected effective soliton number due to XPM effects N corr eff is also shown.

Fig. 4.
Fig. 4.

Data from the numerical results in Fig. 3 for selected values of Δk: (a) |U 1|2 at z=z opt versus time, (b) the corresponding FW and (c) SH wavelength spectra. Only up to λ=3.5 µm is shown in (b) since this is the edge of the transparency window of BBO [21]. (d) The wavelength of the red-shifted peaks in the nonstationary regime, comparing numerical calculations (symbols) with the predictions of the nonlocal theory (lines).

Fig. 5.
Fig. 5.

Results of pulse compression simulations as in Fig. 3, taking Δk=60 mm-1 and varying N eff. (a) and (b) show I 1,opt/I in and Δt opt when using exact dispersion (md =∞) and when including up to TOD (md =3), as well as the same simulations without competing Kerr nonlinearities. The orange curves are the predicted values from the scaling laws [11], corrected for XPM. (c) and (d) show FWtime and spectral profiles for N eff=17.

Equations (21)

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( i ξ + D ̂ 1 ) U 1 + Δ k 1 2 N SHG S ̂ 1 U 1 * U 2 e i Δ k ξ + N Kerr 2 S ̂ 1 U 1 ( U 1 2 + B n ¯ U 2 2 ) = 0 ,
( i ξ i d 12 τ + D ̂ 2 , eff ) U 2 + Δ k 1 2 N SHG S ̂ 2 U 1 2 e i Δ k ξ
+ 2 n ¯ 2 N Kerr 2 S ̂ 2 U 2 ( U 2 2 + B n ¯ 1 U 1 2 ) = 0 .
N SHG 2 = L D , 1 in 2 ω 1 2 d eff 2 c 2 n 1 n 2 Δ k , N Kerr 2 = L D , 1 n Kerr , 1 in 2 ω 1 c
[ i ξ 1 2 2 τ 2 ] U 1 + N Kerr 2 U 1 U 1 2 N SHG 2 U 1 * d s R ± ( s ) U 1 2 ( ξ , τ s ) = 0 .
U 2 ( ξ , τ ) = ϕ 2 ( τ ) exp ( i Δ k ξ )
δ 2 ( 2 ) d 2 ϕ 2 d τ 2 + i d 12 d ϕ 2 d τ Δ k ϕ 2 = Δ k 1 2 N SHG U 1 2
ϕ ˜ 2 ( Ω ) = ( 2 π Δ k ) 1 2 N SHG R ˜ ( Ω ) [ U 1 2 ] ( Ω ) , R ˜ ( Ω ) Δ k ( 2 π ) 1 2 δ 2 ( 2 ) Ω 2 d 12 Ω + Δ k
ϕ 2 ( τ ) = N SHG Δ k d s R ( s ) U 1 2 ( ξ , τ s )
R ˜ ( Ω ) = ( 2 π ) 1 2 Ω a 2 + s b Ω b 2 ( Ω Ω a ) 2 + s b Ω b 2
Ω a = d 12 2 δ 2 ( 2 ) , Ω b = Δ k δ 2 ( 2 ) Ω a 2 1 2
s a = sgn [ Ω a ] , s b = sgn [ Δ k δ 2 ( 2 ) Ω a 2 ]
R + ( τ ) = τ a 2 + τ b 2 2 τ a 2 τ b exp ( is a τ τ a ) exp ( τ τ b )
τ a = Ω a 1 = 2 δ 2 ( 2 ) d 12 , τ b = Ω b 1 = Δ k δ 2 ( 2 ) Ω a 2 1 2
R ( τ ) = τ a 2 τ b 2 2 τ a 2 τ b exp ( is a τ τ a ) sin ( τ τ b )
Δ k sr = d 12 2 2 k 2 ( 2 )
R ˜ ( Ω ) F U 1 2 ( Ω ) [ R ˜ ( Ω = 0 ) + Ω d R ˜ | Ω = 0 ] U 1 2 ( Ω )
ρ ( τ , U 1 ) = i sgn ( τ ) π 2 τ a 2 τ b 2 2 τ a 2 τ b [ e i τ Ω + [ U 1 2 ] ( Ω + ) e i τ Ω [ U 1 2 ] ( Ω ) ]
d s R ( s ) U 1 2 ( ξ , τ s ) U 1 2 + is a τ R , SHG U 1 U 1 τ + 1 s b 2 ρ ( τ , U 1 )
[ i ξ 1 2 2 τ 2 ] U 1 N eff 2 U 1 U 1 2 = N SHG 2 [ is a τ R , SHG U 1 2 U 1 τ + 1 s b 2 U 1 * ρ ( τ , U 1 ) ]
T R , SHG = τ R , SHG T in = 2 d 12 Δ k

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