Abstract

We propose an efficient approach to improve few-cycle soliton compression with cascaded quadratic nonlinearities by using an engineered multi-section structure of the nonlinear crystal. By exploiting engineering of the cascaded quadratic nonlinearities, in each section soliton compression with a low effective order is realized, and high-quality few-cycle pulses with large compression factors are feasible. Each subsequent section is designed so that the compressed pulse exiting the previous section experiences an overall effective self-defocusing cubic nonlinearity corresponding to a modest soliton order, which is kept larger than unity to ensure further compression. This is done by increasing the cascaded quadratic nonlinearity in the new section with an engineered reduced residual phase mismatch. The low soliton orders in each section ensure excellent pulse quality and high efficiency. Numerical results show that compressed pulses with less than three-cycle duration can be achieved even when the compression factor is very large, and in contrast to standard soliton compression, these compressed pulses have minimal pedestal and high quality factor.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2012 (2)

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Few-cycle solitons in short strongly phase-mismatched frequency conversion crystals,” Phys. Rev. Lett.109, 043902 (2012), arXiv:1109.4261.
[CrossRef] [PubMed]

R. Schiek and T. Pertsch, “Absolute measurement of the quadratic nonlinear susceptibility of lithium niobate in waveguides,” Opt. Mater. Express2, 126–139 (2012).
[CrossRef]

2011 (3)

2010 (1)

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

2009 (1)

2008 (3)

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]

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B91, 343–348 (2008).
[CrossRef]

2007 (3)

2006 (4)

X. Zeng, S. Ashihara, N. Fujioka, T. Shimura, and K. Kuroda, “Adiabatic compression of quadratic temporal solitons in aperiodic quasi-phase-matching gratings,” Opt. Express14, 9358–9370 (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). See also arXiv:physics/0604170.
[CrossRef] [PubMed]

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]

C. Myatt, N. Traggis, and K. L. Dessau, “Optical fabrication: Optical contacting grows more robust,” Laser Focus World42, 95–98 (2006).

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]

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B79, 673–677 (2004).
[CrossRef]

2002 (1)

1999 (1)

1996 (1)

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett.68, 2793–2795 (1996).
[CrossRef]

Arie, A.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B91, 343–348 (2008).
[CrossRef]

Ashihara, S.

X. Zeng, S. Ashihara, Z. Wang, T. Wang, Y. Chen, and M. Cha, “Excitation of two-colored temporal solitons in a segmented quasi-phase-matching structure,” Opt. Express17, 16877–16884 (2009).
[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]

X. Zeng, S. Ashihara, N. Fujioka, T. Shimura, and K. Kuroda, “Adiabatic compression of quadratic temporal solitons in aperiodic quasi-phase-matching gratings,” Opt. Express14, 9358–9370 (2006).
[CrossRef] [PubMed]

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]

Bache, M.

Bang, O.

Biegert, J.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B79, 673–677 (2004).
[CrossRef]

Breitkopf, S.

Cha, M.

X. Zeng, S. Ashihara, Z. Wang, T. Wang, Y. Chen, and M. Cha, “Excitation of two-colored temporal solitons in a segmented quasi-phase-matching structure,” Opt. Express17, 16877–16884 (2009).
[CrossRef] [PubMed]

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, 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.

Chong, A.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Few-cycle solitons in short strongly phase-mismatched frequency conversion crystals,” Phys. Rev. Lett.109, 043902 (2012), arXiv:1109.4261.
[CrossRef] [PubMed]

Couairon, A.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B79, 673–677 (2004).
[CrossRef]

De Silvestri, S.

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett.68, 2793–2795 (1996).
[CrossRef]

Dessau, K. L.

C. Myatt, N. Traggis, and K. L. Dessau, “Optical fabrication: Optical contacting grows more robust,” Laser Focus World42, 95–98 (2006).

Fejer, M. M.

Fermann, M. E.

Fujioka, N.

Galun, E.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B91, 343–348 (2008).
[CrossRef]

Gayer, O.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B91, 343–348 (2008).
[CrossRef]

Hartl, I.

Hauri, C. P.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B79, 673–677 (2004).
[CrossRef]

Heinrich, A.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B79, 673–677 (2004).
[CrossRef]

Helbing, F. W.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B79, 673–677 (2004).
[CrossRef]

Keller, U.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B79, 673–677 (2004).
[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]

Klenke, A.

Kornelis, W.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B79, 673–677 (2004).
[CrossRef]

Krolikowski, W.

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]

X. Zeng, S. Ashihara, N. Fujioka, T. Shimura, and K. Kuroda, “Adiabatic compression of quadratic temporal solitons in aperiodic quasi-phase-matching gratings,” Opt. Express14, 9358–9370 (2006).
[CrossRef] [PubMed]

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]

Langrock, C.

Limpert, J.

Liu, X.

Moses, J.

Myatt, C.

C. Myatt, N. Traggis, and K. L. Dessau, “Optical fabrication: Optical contacting grows more robust,” Laser Focus World42, 95–98 (2006).

Mysyrowicz, A.

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B79, 673–677 (2004).
[CrossRef]

Nishina, J.

Nisoli, M.

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett.68, 2793–2795 (1996).
[CrossRef]

Pelc, J. S.

Pertsch, T.

Phillips, C. R.

Plötner, M.

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 χ(2) : χ(2) nonlinearity,” Opt. Lett.24, 1777–1779 (1999).
[CrossRef]

Sacks, Z.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B91, 343–348 (2008).
[CrossRef]

Schiek, R.

Seise, E.

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]

X. Zeng, S. Ashihara, N. Fujioka, T. Shimura, and K. Kuroda, “Adiabatic compression of quadratic temporal solitons in aperiodic quasi-phase-matching gratings,” Opt. Express14, 9358–9370 (2006).
[CrossRef] [PubMed]

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]

Svelto, O.

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett.68, 2793–2795 (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).
[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]

Traggis, N.

C. Myatt, N. Traggis, and K. L. Dessau, “Optical fabrication: Optical contacting grows more robust,” Laser Focus World42, 95–98 (2006).

Tünnermann, A.

Wang, T.

Wang, Z.

Wise, F. W.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Few-cycle solitons in short strongly phase-mismatched frequency conversion crystals,” Phys. Rev. Lett.109, 043902 (2012), arXiv:1109.4261.
[CrossRef] [PubMed]

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical Cherenkov radiation by cascaded nonlinear interaction: an efficient source of energetic few-cycle near- to mid-IR pulses,” Opt. Express19, 22557–22562 (2011).
[CrossRef] [PubMed]

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

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). See also arXiv:physics/0604170.
[CrossRef] [PubMed]

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).
[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]

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).
[CrossRef]

Zhou, B. B.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Few-cycle solitons in short strongly phase-mismatched frequency conversion crystals,” Phys. Rev. Lett.109, 043902 (2012), arXiv:1109.4261.
[CrossRef] [PubMed]

M. Bache, O. Bang, B. B. Zhou, J. Moses, and F. W. Wise, “Optical Cherenkov radiation by cascaded nonlinear interaction: an efficient source of energetic few-cycle near- to mid-IR pulses,” Opt. Express19, 22557–22562 (2011).
[CrossRef] [PubMed]

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

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. B (2)

C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B79, 673–677 (2004).
[CrossRef]

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B91, 343–348 (2008).
[CrossRef]

Appl. Phys. Lett. (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]

M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett.68, 2793–2795 (1996).
[CrossRef]

J. Opt. Soc. Am. B (2)

Laser Focus World (1)

C. Myatt, N. Traggis, and K. L. Dessau, “Optical fabrication: Optical contacting grows more robust,” Laser Focus World42, 95–98 (2006).

Opt. Commun. (2)

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]

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]

Opt. Express (5)

Opt. Lett. (4)

Opt. Mater. Express (1)

Phys. Rev. A (1)

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

Phys. Rev. Lett. (2)

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]

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Few-cycle solitons in short strongly phase-mismatched frequency conversion crystals,” Phys. Rev. Lett.109, 043902 (2012), arXiv:1109.4261.
[CrossRef] [PubMed]

Other (3)

M. Bache and R. Schiek, “Review of measurements of Kerr nonlinearities in lithium niobate: the role of the delayed Raman response,” arXiv:1211.1721 (2012), http://arxiv.org/abs/1211.1721 .

H. Guo, X. Zeng, B. B. Zhou, and M. Bache, “Electric-field modeling and self-steepening counterbalance of cascading nonlinear soliton pulse compression,” arXiv:1210.5903 (2012), http://arxiv.org/abs/1210.5903 .

M. Bache, H. Guo, B. B. Zhou, and X. Zeng, “The anisotropic Kerr nonlinear refractive index of the beta-barium borate (β-BaB2O4) nonlinear crystal,” arXiv:1209.3158 (2012), http://arxiv.org/abs/1209.3158 .

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Figures (7)

Fig. 1
Fig. 1

A schematic of multi-section structure of nonlinear crystals. (a) Cascaded nonlinearity increases and (b) the effective phase mismatch deceases upon propagation by controlling the local domain period Λj in (c) QPM grating or the angle θj between the FW input and the optical z axis of the crystal in (d) bonded BBO crystals.

Fig. 2
Fig. 2

Numerical simulation of compressed pulses of Tin = 250 fs FWHM at the pump wavelength of 1.56 μm in one- and two-section QPM structures. FW temporal evolutions of (a) N1,eff = 10, ΔkQPM = 150 mm−1 and (b) N1,eff = 3, ΔkQPM = 180 mm−1 in one-section QPM structure; (c) and (d) FW temporal and spectral evolution in two-section QPM structure with N1,eff = 3, Δk1,QPM = 180 mm−1 (N2,eff = 1.6, Δk2,QPM = 140 mm−1) in the first (second) QPM. (e) and (f) Normalized intensities of the input, the optimal compression positions [white dashed lines, cuts in (a), (b) and (c)]. (g) Pulse durations and the ratio of peak intensities along propagation in two-section QPM.

Fig. 3
Fig. 3

(a) Optimal compressed pulse intensities for selected values of the effective phase mismatch Δk2,QPM of the second QPM and (b) pulse durations and peak intensities versus Δk2,QPM. (c) Wavelength dependence of Δk0 from material dispersion, the resonant threshold Δ k sr ( 2 ), the upper limit of Δkc,LN for soliton in unpoled LN. (d) Compression windows versus the wavelength between the upper limit of Δkc,QPM (black line), the resonant threshold Δksr (red line) including full SH dispersion compression and the breakeven QPM value Δkeven,QPM (purple line) to achieve the same n casc I in unpoled LN.

Fig. 4
Fig. 4

Results of numerical simulations showing the optimum compression parameters versus Neff in multi-section QPM gratings for (a) the minimum pulse duration, (b) quality factor. The filled mark with round symbols are the compression results in one section QPM with good quality factor (Qc > 0.4) and the square, star and diamond symbols represents the results in multi-section QPM (two, three and four) with high quality factor while durations are close to the empirical limit of 13 fs (green dash line). In (a), the arrow shows stronger pulse compression with the decrease of N eff c and the black dash line is fit to the minimum duration available in one-section QPM gratings. The green dashed line in (a) denotes an empirical limit of 13 fs that was observed in the simulations.

Fig. 5
Fig. 5

Temporal evolutions of FW pulse (250 fs FWHM) in two-section QPM structures (N1,eff = 5): the length of first-section QPM is (a) 39.2 mm (b) 44.5 mm (optimal compression position) and (c) 48.5 mm. (d) Peak intensifies of FW along propagation as function of the length of first QPM. (e) Comparison of the final output pulses with 10% fluctuation of input intensities in same QPM structure (L1 = 44.5 mm and L2 = 3 mm).

Fig. 6
Fig. 6

Numerical results of temporal evolution of FW pulse (250 fs FWHM) including Raman contribution, (a) one-section (b) two-section QPM when fR = 0.14 and (c) one-section QPM when fR = 0.50.

Fig. 7
Fig. 7

Evolution of the temporal profiles of FW pulses (Tin = 200 fs FWHM) in BBO crystals. Propagation in one section (a) Neff = 3.5, Δkeff = 75 mm−1 (b) Neff = 8, Δkeff = 60 mm−1; (c) Δk1 = 75 mm−1 (N1,eff = 3.5) and Δk2 = 58 mm−1 in two-section BBO crystal (L1 = 93.9 mm and L2 = 4.5 mm); (d) pulse intensities at the optimal compression positions (dashed lines) in one-section (FW1) and two-section (FW2,1: first compression; FW2,2: second compression) BBO crystals.

Equations (8)

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( i z + D ^ 1 ) E 1 + ρ 1 ( z ) E 1 * E 2 e i Δ k 0 z + σ 1 [ | E 1 | 2 E 1 + η | E 2 | 2 E 1 ] = 0
( i z i d 12 t + D ^ 2 ) E 2 + ρ 2 ( z ) E 1 2 e i Δ k 0 z + σ 2 [ | E 2 | 2 E 2 + η | E 1 | 2 E 2 ] = 0
( i ξ + 1 2 2 τ 2 ) U 1 ( N casc 2 N Kerr 2 ) U 1 | U 1 | 2 = i N SHG 2 2 δ Δ β | U 1 | 2 U 1 ( τ ) τ
R ( Ω ) Δ k QPM / Δ k n l ( Ω )
Δ k n l ( Ω ) = D ^ 2 ( Ω ) d 12 Ω + Δ k QPM = k 2 ( ω 2 + Ω ) 2 k 1 ( ω 1 ) 2 π Λ k 1 ( 1 ) Ω
N eff ( XPM ) 2 = N casc 2 N Kerr ( SPM ) 2 N Kerr ( XPM ) 2 = N eff 2 N Kerr ( XPM ) 2
N Kerr ( XPM ) 2 = N Kerr 2 ( 1 + B n 1 N casc 2 / n 2 Δ k eff )
N eff N eff c = 1 + Φ in 2 / Φ c 2

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