Abstract

We numerically show that it is possible to achieve adiabatic compression of femtosecond quadratic solitons in aperiodically poled lithium niobate device. Two-colored solitons of the fundamental wavelength of 1560 nm can be adiabatically shaped by using group-velocity matching schemes available in quasi-phase-matching (QPM) devices. We investigate the performance of the adiabatic compression based on two different group-velocity matching schemes: type-I (e: o + o) collinear QPM geometry and type-0 (e: e + e) non-collinear QPM geometry. Two-colored temporal solitons with pulse duration of 35 fs are generated without visible pedestals from 100-fs fundamental pulse. We also show that walking solitons with shorter pulse durations are adiabatically excited under small group-velocity mismatch condition. The walking solitons experience deceleration or acceleration during compression, depending on the sign of the group-velocity-mismatch. The demonstrated adiabatic pulse shaping is useful for generation of shorter pulses with clean temporal profiles, efficient femtosecond second harmonic generation and group-velocity control.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  4. A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, "Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications," Phys. Rep. 370, 63-235 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  13. R. Schiek, R. Iwanow, T. Pertsch, G. I. Stegeman, G. Schreiber, and W. Sohler, "One-dimensional spatial soliton families in optimally engineered quasi-phase-matched lithium niobate waveguides," Opt. Lett. 29, 596-598 (2004).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2006 (2)

2005 (2)

2004 (3)

2003 (3)

2002 (5)

2001 (1)

S. Carrasco, J. Torres, L. Torner, and F. W. Wise, "Walk-off acceptance for quadratic soliton generation," Opt. Commun. 191, 363-370 (2001).
[CrossRef]

2000 (1)

1999 (3)

1998 (2)

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, "Observation of temporal solitons in second-harmonic generation with tilted pulses," Phys. Rev. Lett. 81, 570- 573 (1998).
[CrossRef]

L. Torner, C. Clausen, and M. Fejer, "Adiabatic shaping of quadratic solitons," Opt. Lett. 23, 903-905 (1998).
[CrossRef]

1997 (3)

1996 (1)

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

1995 (2)

L. Torner, "Stationary solitary waves with second-order nonlinearities," Opt. Commun. 114, 136-140 (1995).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, "Observation of two-dimensional Spatial Solitary Waves in a Quadratic Medium," Phys. Rev. Lett. 74, 5036-5039 (1995).
[CrossRef] [PubMed]

1994 (2)

1991 (1)

1989 (1)

E. Martinez, "Achromatic phase matching for second harmonic generation of femtosecond pulses," IEEE J. Quantum Electron. 25, 2464-2468 (1989).
[CrossRef]

Ashihara, S.

Baronio, F.

Barthelemy, A.

Beckwitt, K.

Bosshard, C.

Bourliaguet, B.

Buryak, A. V.

A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, "Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications," Phys. Rep. 370, 63-235 (2002).
[CrossRef]

A. V. Buryak and Y. S. Kivshar, "Spatial optical solitons governed by quadratic nonlinearity," Opt. Lett. 19, 1612-1615(1994).
[CrossRef] [PubMed]

Caironi, D.

G. Valiulis, A. Dubietis, R. Danielius, D. Caironi, A. Visconti, and P. Di Trapani, "Temporal solitons in Chi(2) materials with tilted pulses," J. Opt. Soc. Am. B 16, 722-731 (1999).
[CrossRef]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, "Observation of temporal solitons in second-harmonic generation with tilted pulses," Phys. Rev. Lett. 81, 570- 573 (1998).
[CrossRef]

Carrasco, S.

Cerullo, G.

Cha, M.

Charbonneau-Lefort, M.

Chernikov, S.

Clausen, C.

Couderc, V.

Danielius, R.

G. Valiulis, A. Dubietis, R. Danielius, D. Caironi, A. Visconti, and P. Di Trapani, "Temporal solitons in Chi(2) materials with tilted pulses," J. Opt. Soc. Am. B 16, 722-731 (1999).
[CrossRef]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, "Observation of temporal solitons in second-harmonic generation with tilted pulses," Phys. Rev. Lett. 81, 570- 573 (1998).
[CrossRef]

De Angelis, C.

Di Trapani, P.

A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, "Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications," Phys. Rep. 370, 63-235 (2002).
[CrossRef]

G. Valiulis, A. Dubietis, R. Danielius, D. Caironi, A. Visconti, and P. Di Trapani, "Temporal solitons in Chi(2) materials with tilted pulses," J. Opt. Soc. Am. B 16, 722-731 (1999).
[CrossRef]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, "Observation of temporal solitons in second-harmonic generation with tilted pulses," Phys. Rev. Lett. 81, 570- 573 (1998).
[CrossRef]

Dubietis, A.

G. Valiulis, A. Dubietis, R. Danielius, D. Caironi, A. Visconti, and P. Di Trapani, "Temporal solitons in Chi(2) materials with tilted pulses," J. Opt. Soc. Am. B 16, 722-731 (1999).
[CrossRef]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, "Observation of temporal solitons in second-harmonic generation with tilted pulses," Phys. Rev. Lett. 81, 570- 573 (1998).
[CrossRef]

Eger, D.

Fejer, M.

Fuerst, R.

Fujioka, N.

Gunter, P.

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, "Chi (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]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, "Observation of two-dimensional Spatial Solitary Waves in a Quadratic Medium," Phys. Rev. Lett. 74, 5036-5039 (1995).
[CrossRef] [PubMed]

Hanna, D.

Ilday, F.

Iwanow, R.

Jankovic, L.

Jundt, D.

Jundt, D. H.

Katz, M.

Kim, H.

Kitamura, K.

Kivshar, Y. S.

Kurimura, S.

Kuroda, K.

Liu, X.

F. W. Wise, L. Qian, and X. Liu, "Applications of Cascaded Quadratic Nonlinearities to Femtosecond Pulse Generation," J. Nonlinear Opt. Phys. Mater. 11, 317-338 (2002).
[CrossRef]

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

Malendevich, R.

Mamyshev, P.

Manzoni, C.

Marangoni, M.

Martinez, E.

E. Martinez, "Achromatic phase matching for second harmonic generation of femtosecond pulses," IEEE J. Quantum Electron. 25, 2464-2468 (1989).
[CrossRef]

Menyuk, C.

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, "Observation of two-dimensional Spatial Solitary Waves in a Quadratic Medium," Phys. Rev. Lett. 74, 5036-5039 (1995).
[CrossRef] [PubMed]

Moses, J.

Nan Ei Yu, K.

S. Ashihara, T. Shimura, K. Kuroda, Nan Ei Yu, S. Kurimura, K. Kitamura, Myoungsik Cha, and Takunori Taira, "Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate," Appl. Phys. Lett. 84, 1055-1057 (2004).
[CrossRef]

Nishina, J.

Ono, H.

Pertsch, T.

Piskarskas, A.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, "Observation of temporal solitons in second-harmonic generation with tilted pulses," Phys. Rev. Lett. 81, 570- 573 (1998).
[CrossRef]

Polyakov, S.

Qian, L.

F. W. Wise, L. Qian, and X. Liu, "Applications of Cascaded Quadratic Nonlinearities to Femtosecond Pulse Generation," J. Nonlinear Opt. Phys. Mater. 11, 317-338 (2002).
[CrossRef]

Qian, L. J.

Ramponi, R.

Ro, J. H.

Ross, G.

Schiek, R.

Schober, A.

Schreiber, G.

Shimura, T.

Skryabin, D.

A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, "Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications," Phys. Rep. 370, 63-235 (2002).
[CrossRef]

Small, D.

Smith, P.

Sohler, W.

Stegeman, G.

Stegeman, G. I.

R. Schiek, R. Iwanow, T. Pertsch, G. I. Stegeman, G. Schreiber, and W. Sohler, "One-dimensional spatial soliton families in optimally engineered quasi-phase-matched lithium niobate waveguides," Opt. Lett. 29, 596-598 (2004).

G. I. Stegeman, D. J. Hagan, and L. Torner, "Chi (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]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, "Observation of two-dimensional Spatial Solitary Waves in a Quadratic Medium," Phys. Rev. Lett. 74, 5036-5039 (1995).
[CrossRef] [PubMed]

Taira, T.

Torner, L.

H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger, and M. Katz, "Quadratic spatial solitons in periodically poled KTiOPO4," Opt. Lett. 28, 640-642 (2003).
[CrossRef] [PubMed]

L. Torner and A. Barthelemy, "Quadratic solitons: recent developments," IEEE J. Quantum Electron. 39, 22-30 (2003).
[CrossRef]

S. Carrasco, J. Torres, L. Torner, and F. W. Wise, "Walk-off acceptance for quadratic soliton generation," Opt. Commun. 191, 363-370 (2001).
[CrossRef]

S. Carrasco, J. Torres, L. Torner, and R. Schiek, "Engineerable generation of quadratic solitons in synthetic phase matching," Opt. Lett. 25, 1273-1275 (2000).
[CrossRef]

L. Torner, C. Clausen, and M. Fejer, "Adiabatic shaping of quadratic solitons," Opt. Lett. 23, 903-905 (1998).
[CrossRef]

L. Torner and G. Stegeman, "Soliton evolution in quasi-phase-matched second-harmonic generation," J. Opt. Soc. Am. B 14, 3127- 3133 (1997).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, "Chi (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]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, "Observation of two-dimensional Spatial Solitary Waves in a Quadratic Medium," Phys. Rev. Lett. 74, 5036-5039 (1995).
[CrossRef] [PubMed]

L. Torner, "Stationary solitary waves with second-order nonlinearities," Opt. Commun. 114, 136-140 (1995).
[CrossRef]

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

Torres, J.

S. Carrasco, J. Torres, L. Torner, and F. W. Wise, "Walk-off acceptance for quadratic soliton generation," Opt. Commun. 191, 363-370 (2001).
[CrossRef]

S. Carrasco, J. Torres, L. Torner, and R. Schiek, "Engineerable generation of quadratic solitons in synthetic phase matching," Opt. Lett. 25, 1273-1275 (2000).
[CrossRef]

Torruellas, W. E.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, "Observation of two-dimensional Spatial Solitary Waves in a Quadratic Medium," Phys. Rev. Lett. 74, 5036-5039 (1995).
[CrossRef] [PubMed]

Trillo, S.

A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, "Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications," Phys. Rep. 370, 63-235 (2002).
[CrossRef]

Valiulis, G.

G. Valiulis, A. Dubietis, R. Danielius, D. Caironi, A. Visconti, and P. Di Trapani, "Temporal solitons in Chi(2) materials with tilted pulses," J. Opt. Soc. Am. B 16, 722-731 (1999).
[CrossRef]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, "Observation of temporal solitons in second-harmonic generation with tilted pulses," Phys. Rev. Lett. 81, 570- 573 (1998).
[CrossRef]

VanStryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, "Observation of two-dimensional Spatial Solitary Waves in a Quadratic Medium," Phys. Rev. Lett. 74, 5036-5039 (1995).
[CrossRef] [PubMed]

Visconti, A.

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, "Observation of two-dimensional Spatial Solitary Waves in a Quadratic Medium," Phys. Rev. Lett. 74, 5036-5039 (1995).
[CrossRef] [PubMed]

Wise, F.

Wise, F. W.

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]

F. W. Wise, L. Qian, and X. Liu, "Applications of Cascaded Quadratic Nonlinearities to Femtosecond Pulse Generation," J. Nonlinear Opt. Phys. Mater. 11, 317-338 (2002).
[CrossRef]

S. Carrasco, J. Torres, L. Torner, and F. W. Wise, "Walk-off acceptance for quadratic soliton generation," Opt. Commun. 191, 363-370 (2001).
[CrossRef]

Yu, N. E.

Zelmon, D.

Appl. Phys. Lett. (1)

S. Ashihara, T. Shimura, K. Kuroda, Nan Ei Yu, S. Kurimura, K. Kitamura, Myoungsik Cha, and Takunori 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)

L. Torner and A. Barthelemy, "Quadratic solitons: recent developments," IEEE J. Quantum Electron. 39, 22-30 (2003).
[CrossRef]

E. Martinez, "Achromatic phase matching for second harmonic generation of femtosecond pulses," IEEE J. Quantum Electron. 25, 2464-2468 (1989).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

F. W. Wise, L. Qian, and X. Liu, "Applications of Cascaded Quadratic Nonlinearities to Femtosecond Pulse Generation," J. Nonlinear Opt. Phys. Mater. 11, 317-338 (2002).
[CrossRef]

J. Opt. Soc. Am. B (9)

L. Torner and G. Stegeman, "Soliton evolution in quasi-phase-matched second-harmonic generation," J. Opt. Soc. Am. B 14, 3127- 3133 (1997).
[CrossRef]

D. Zelmon, D. Small, and D. Jundt, "Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol. magnesium oxide doped lithium niobate," J. Opt. Soc. Am. B 14, 3319-3322 (1997).
[CrossRef]

G. Valiulis, A. Dubietis, R. Danielius, D. Caironi, A. Visconti, and P. Di Trapani, "Temporal solitons in Chi(2) materials with tilted pulses," J. Opt. Soc. Am. B 16, 722-731 (1999).
[CrossRef]

S. Chernikov and P. Mamyshev, "Femtosecond soliton propagation in fibers with slowly decreasing dispersion," J. Opt. Soc. Am. B 8, 1633-1641(1991).
[CrossRef]

C. Menyuk, R. Schiek, and L. Torner, "Solitary waves due to Chi (2):Chi (2) cascading," J. Opt. Soc. Am. B 11, 2434-2443 (1994).
[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]

S. Ashihara, T. Shimura, and K. Kuroda, "Group-velocity matched second-harmonic generation in tilted quasi-phase-matched gratings," J. Opt. Soc. Am. B 20, 853-856 (2003).
[CrossRef]

N. Fujioka, S. Ashihara, H. Ono, T. Shimura, and K. Kuroda, "Group-velocity-matched noncollinear second-harmonic generation in quasi-phase matching," J. Opt. Soc. Am. B 22, 1283-1289 (2005).
[CrossRef]

A. Schober, M. Charbonneau-Lefort, and M. Fejer, "Broadband quasi-phase-matched second-harmonic generation of ultrashort optical pulses with spectral angular dispersion," J. Opt. Soc. Am. B 22, 1699-1713 (2005).
[CrossRef]

Opt. Commun. (2)

S. Carrasco, J. Torres, L. Torner, and F. W. Wise, "Walk-off acceptance for quadratic soliton generation," Opt. Commun. 191, 363-370 (2001).
[CrossRef]

L. Torner, "Stationary solitary waves with second-order nonlinearities," Opt. Commun. 114, 136-140 (1995).
[CrossRef]

Opt. Lett. (13)

D. H. Jundt, "Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate," Opt. Lett. 22, 1553-1555 (1997).
[CrossRef]

L. Torner, C. Clausen, and M. Fejer, "Adiabatic shaping of quadratic solitons," Opt. Lett. 23, 903-905 (1998).
[CrossRef]

B. Bourliaguet, V. Couderc, A. Barthelemy, G. Ross, P. Smith, D. Hanna, and C. De Angelis, "Observation of quadratic spatial solitons in periodically poled lithium niobate," Opt. Lett. 24, 1410-1412 (1999).
[CrossRef]

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

S. Carrasco, J. Torres, L. Torner, and R. Schiek, "Engineerable generation of quadratic solitons in synthetic phase matching," Opt. Lett. 25, 1273-1275 (2000).
[CrossRef]

R. Malendevich, L. Jankovic, S. Polyakov, R. Fuerst, G. Stegeman, C. Bosshard, and P. Gunter, "Two-dimensional type I quadratic spatial solitons in KNbO3 near noncritical phase matching," Opt. Lett. 27, 631-633 (2002).
[CrossRef]

N. E. Yu, J. H. Ro, M. Cha, S. Kurimura, and T. Taira, "Broadband quasi-phase-matched second-harmonic generation in MgO-doped periodically poled LiNbO3 at the communications band," Opt. Lett. 27, 1046-1048 (2002).
[CrossRef]

M. Marangoni, C. Manzoni, R. Ramponi, G. Cerullo, F. Baronio, C. De Angelis, and K. Kitamura, "Group-velocity control by quadratic nonlinear interactions," Opt. Lett. 31, 534-536 (2006).
[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]

R. Schiek, R. Iwanow, T. Pertsch, G. I. Stegeman, G. Schreiber, and W. Sohler, "One-dimensional spatial soliton families in optimally engineered quasi-phase-matched lithium niobate waveguides," Opt. Lett. 29, 596-598 (2004).

K. Beckwitt, F. Ilday, and F. Wise, "Frequency shifting with local nonlinearity management in nonuniformly poled quadratic nonlinear materials," Opt. Lett. 29, 763-765 (2004).
[CrossRef] [PubMed]

H. Kim, L. Jankovic, G. Stegeman, S. Carrasco, L. Torner, D. Eger, and M. Katz, "Quadratic spatial solitons in periodically poled KTiOPO4," Opt. Lett. 28, 640-642 (2003).
[CrossRef] [PubMed]

A. V. Buryak and Y. S. Kivshar, "Spatial optical solitons governed by quadratic nonlinearity," Opt. Lett. 19, 1612-1615(1994).
[CrossRef] [PubMed]

Opt. Quantum. Electron. (1)

G. I. Stegeman, D. J. Hagan, and L. Torner, "Chi (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. Rep. (1)

A. V. Buryak, P. Di Trapani, D. Skryabin, and S. Trillo, "Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications," Phys. Rep. 370, 63-235 (2002).
[CrossRef]

Phys. Rev. Lett. (2)

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, "Observation of two-dimensional Spatial Solitary Waves in a Quadratic Medium," Phys. Rev. Lett. 74, 5036-5039 (1995).
[CrossRef] [PubMed]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, "Observation of temporal solitons in second-harmonic generation with tilted pulses," Phys. Rev. Lett. 81, 570- 573 (1998).
[CrossRef]

Other (2)

J. C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, New York, 1996).

G. P. Agrawal, Nonlinear Fiber Optics, 3rd Edition, (Academic Press).

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Figures (10)

Fig. 1.
Fig. 1.

Schematic of adiabatic compression process in the aperiodic QPM grating. The effective wave-vector mismatch Δk(z)=k2-2k1-2π/Λ(z) varies along the propagation direction.

Fig. 2.
Fig. 2.

Normalized energy ratio of FF and SH along the linearly chirped QPM grating. The effective wave-vector mismatch Δk(z) varies from 8.15 mm–1 to 0.41 mm–1. Input FF: transform-limited pulse (100 fs, 20 GW/cm-2) at the wavelength of 1560 nm;

Fig. 3.
Fig. 3.

Intensity profiles of the compressed (a) FF and (b) SH pulses in linearly chirped grating (100 mm, solid line) and periodic grating (10 mm, dashed line) at Δk=0.35 mm-1. Phase profiles of the compressed (a) FF and (b) SH pulses for linearly chirped grating are also shown, as well as evolution of the pulse duration of the FF and SH pulses for (c) linearly chirped and (d) periodic QPM gratings.

Fig. 4.
Fig. 4.

Quality factors of the linearly chirped and periodic QPM gratings as a function of propagation length

Fig. 5.
Fig. 5.

Evolution of FF and SH pulses. (a) FF and (b) SH pulses in linearly chirped QPM; (c) FF and (d) SH pulses in periodic QPM at Δk=0.35 mm-1.

Fig. 6.
Fig. 6.

(a). Schematic of SHG interactions in aperiodic noncollinear QPM grating and (b) the corresponding wave-vector diagram; Δk=k 2z-2k 1-Kp is the effective wave-vector mismatch; α, β and ρ are the propagation angle with respect to the grating vector K, the walk off angle, and the pulse front tilt angle ρ.

Fig. 7.
Fig. 7.

Evolution of (a) normalized energy ratio and (b) pulse duration of FF (solid line) and SH (dashed line) pulses in the noncollinear QPM geometry from the results of 2D BPM (black lines) and 1D BPM (red lines). The effective wave-vector mismatch Δkz of the engineered QPM grating varies from -40 mm–1 to 5 mm–1 with a linear variation (R=0.3).

Fig. 8.
Fig. 8.

(a). Temporal profiles at the spatial centre and (b) time-integrated spatial profiles of FF (solid-black line) and SH (dashed-red line) pulses. Dotted-green lines indicate the input profiles.

Fig. 9.
Fig. 9.

Evolutions of normalized intensity profiles of (a) FF and (b) SH pulses in the presence of GV mismatch (δ=0.025).

Fig. 10.
Fig. 10.

Dependence of (a) FWHM of FF and SH and (b) quality factor and walking time on the normalized group velocity mismatch δ.

Equations (5)

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z E 1 + i ( k " 1 2 ) t t 2 E 1 = i ρ 1 ( z ) E 1 * E 2 exp ( i Δ k 0 z ) + i σ 1 [ E 1 2 E 1 + 2 E 2 2 E 1 ]
z E 2 + i ( k ' 2 2 ) t t 2 E 2 = i ρ 2 ( z ) E 1 E 1 exp ( i Δ k 0 z ) + i σ 2 [ E 2 2 E 2 + 2 E 1 2 E 2 ]
L 1 E 1 = i ρ 1 E 1 * E 2 exp ( i Δ k z ) i σ 1 [ E 1 2 E 1 + 2 E 2 2 E 1 ]
L 2 E 2 = δ · t E 2 η · y E 2 i ρ 2 E 1 E 1 exp ( i Δ k z ) i σ 2 [ E 2 2 E 2 + 2 E 1 2 E 2 ]
GVD eff = λ 3 ( d α d λ ) 2 ( 2 π c 2 ) 2 ,

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