Abstract

We present a comprehensive model for the description of different types of parametric interactions, associated with simultaneous phase-matching of several optical processes: the so-called multistep parametric interactions. Our approach is based on a recently derived single-wave broadband equation that is able to describe general quadratic nonlinear optical interactions and can be solved with modest computational effort. We compare theoretical results with experiments on simultaneous second- and third-harmonic generation performed in periodically poled lithium tantalate crystals.

© 2011 Optical Society of America

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  1. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [CrossRef]
  2. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).
  3. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [CrossRef]
  4. M. Conforti, F. Baronio, and C. De Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high-efficiency conversion,” Opt. Lett. 32, 1779–1781(2007).
    [CrossRef] [PubMed]
  5. S. M. Saltiel, A. A. Sukhourukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” in Vol.  47 of Progress in Optics (Elsevier, 2005), pp. 1–73.
    [CrossRef]
  6. G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
    [CrossRef]
  7. V. Couderc, E. L. Lago, A. Barthelemy, C. De Angelis, and F. Gringoli, “Trapping of a weak probe through coupling with a two-color quadratic spatial soliton,” Opt. Commun. 203, 421–425 (2002).
    [CrossRef]
  8. A. A. Sukhorukov, T. J. Alexander, Y. S. Kivshar, and S. M. Saltiel, “Multistep cascading and fourth harmonic generation,” Phys. Lett. A 281, 34–38 (2001).
    [CrossRef]
  9. M. C. Cardakli, D. Gurkan, S. A. Havstad, A. E. Willner, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Tunable all-optical time-slot-interchange and wavelength conversion using difference-frequency-generation and optical buffers,” IEEE Photon. Technol. Lett. 14, 200–202 (2002).
    [CrossRef]
  10. M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841 (2010).
    [CrossRef]
  11. M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (2010).
    [CrossRef]
  12. S. Amiranashvili, U. Bandelow, and A. Milke, “Pade approximant for the refractive index and nonlocal envelope equations,” Opt. Commun. 283, 480–485 (2010).
    [CrossRef]
  13. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285(1997).
    [CrossRef]
  14. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
    [CrossRef]
  15. P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
    [CrossRef]
  16. A. V. Housakou and J. Herrmnann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
    [CrossRef]
  17. Y.-Q. Qin, Y.-Y. Zhu, C. Zhang, and N.-B. Ming, “Theoretical investigations of efficient cascaded third-harmonic generation in quasi-phase-matched and -mismatched configurations,” J. Opt. Soc. Am. B 20, 73–82 (2003).
    [CrossRef]
  18. A. Bruner, D. Eger, M. B. Oron, P. Blau, and M. Katz, “Temperature-dependent Sellmeier equation for the refractive index of stoichiometric lithium tantalate,” Opt. Lett. 28, 194–196(2003).
    [CrossRef] [PubMed]

2010 (4)

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

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (2010).
[CrossRef]

S. Amiranashvili, U. Bandelow, and A. Milke, “Pade approximant for the refractive index and nonlocal envelope equations,” Opt. Commun. 283, 480–485 (2010).
[CrossRef]

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

2007 (1)

2004 (1)

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

2003 (2)

2002 (2)

M. C. Cardakli, D. Gurkan, S. A. Havstad, A. E. Willner, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Tunable all-optical time-slot-interchange and wavelength conversion using difference-frequency-generation and optical buffers,” IEEE Photon. Technol. Lett. 14, 200–202 (2002).
[CrossRef]

V. Couderc, E. L. Lago, A. Barthelemy, C. De Angelis, and F. Gringoli, “Trapping of a weak probe through coupling with a two-color quadratic spatial soliton,” Opt. Commun. 203, 421–425 (2002).
[CrossRef]

2001 (3)

A. A. Sukhorukov, T. J. Alexander, Y. S. Kivshar, and S. M. Saltiel, “Multistep cascading and fourth harmonic generation,” Phys. Lett. A 281, 34–38 (2001).
[CrossRef]

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

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

1997 (1)

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

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Alexander, T. J.

A. A. Sukhorukov, T. J. Alexander, Y. S. Kivshar, and S. M. Saltiel, “Multistep cascading and fourth harmonic generation,” Phys. Lett. A 281, 34–38 (2001).
[CrossRef]

Amiranashvili, S.

S. Amiranashvili, U. Bandelow, and A. Milke, “Pade approximant for the refractive index and nonlocal envelope equations,” Opt. Commun. 283, 480–485 (2010).
[CrossRef]

Bandelow, U.

S. Amiranashvili, U. Bandelow, and A. Milke, “Pade approximant for the refractive index and nonlocal envelope equations,” Opt. Commun. 283, 480–485 (2010).
[CrossRef]

Baronio, F.

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (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 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high-efficiency conversion,” Opt. Lett. 32, 1779–1781(2007).
[CrossRef] [PubMed]

Barthelemy, A.

V. Couderc, E. L. Lago, A. Barthelemy, C. De Angelis, and F. Gringoli, “Trapping of a weak probe through coupling with a two-color quadratic spatial soliton,” Opt. Commun. 203, 421–425 (2002).
[CrossRef]

Blau, P.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

Brabec, T.

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

Brener, I.

M. C. Cardakli, D. Gurkan, S. A. Havstad, A. E. Willner, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Tunable all-optical time-slot-interchange and wavelength conversion using difference-frequency-generation and optical buffers,” IEEE Photon. Technol. Lett. 14, 200–202 (2002).
[CrossRef]

Bruner, A.

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Cardakli, M. C.

M. C. Cardakli, D. Gurkan, S. A. Havstad, A. E. Willner, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Tunable all-optical time-slot-interchange and wavelength conversion using difference-frequency-generation and optical buffers,” IEEE Photon. Technol. Lett. 14, 200–202 (2002).
[CrossRef]

Conforti, M.

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

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high-efficiency conversion,” Opt. Lett. 32, 1779–1781(2007).
[CrossRef] [PubMed]

Couderc, V.

V. Couderc, E. L. Lago, A. Barthelemy, C. De Angelis, and F. Gringoli, “Trapping of a weak probe through coupling with a two-color quadratic spatial soliton,” Opt. Commun. 203, 421–425 (2002).
[CrossRef]

De Angelis, C.

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

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high-efficiency conversion,” Opt. Lett. 32, 1779–1781(2007).
[CrossRef] [PubMed]

V. Couderc, E. L. Lago, A. Barthelemy, C. De Angelis, and F. Gringoli, “Trapping of a weak probe through coupling with a two-color quadratic spatial soliton,” Opt. Commun. 203, 421–425 (2002).
[CrossRef]

Eger, D.

Fejer, M. M.

M. C. Cardakli, D. Gurkan, S. A. Havstad, A. E. Willner, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Tunable all-optical time-slot-interchange and wavelength conversion using difference-frequency-generation and optical buffers,” IEEE Photon. Technol. Lett. 14, 200–202 (2002).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Gringoli, F.

V. Couderc, E. L. Lago, A. Barthelemy, C. De Angelis, and F. Gringoli, “Trapping of a weak probe through coupling with a two-color quadratic spatial soliton,” Opt. Commun. 203, 421–425 (2002).
[CrossRef]

Gurkan, D.

M. C. Cardakli, D. Gurkan, S. A. Havstad, A. E. Willner, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Tunable all-optical time-slot-interchange and wavelength conversion using difference-frequency-generation and optical buffers,” IEEE Photon. Technol. Lett. 14, 200–202 (2002).
[CrossRef]

Havstad, S. A.

M. C. Cardakli, D. Gurkan, S. A. Havstad, A. E. Willner, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Tunable all-optical time-slot-interchange and wavelength conversion using difference-frequency-generation and optical buffers,” IEEE Photon. Technol. Lett. 14, 200–202 (2002).
[CrossRef]

He, J. L.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Herrmnann, J.

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

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Housakou, A. V.

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

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Katz, M.

Kinsler, P.

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

Kivshar, Y. S.

A. A. Sukhorukov, T. J. Alexander, Y. S. Kivshar, and S. M. Saltiel, “Multistep cascading and fourth harmonic generation,” Phys. Lett. A 281, 34–38 (2001).
[CrossRef]

S. M. Saltiel, A. A. Sukhourukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” in Vol.  47 of Progress in Optics (Elsevier, 2005), pp. 1–73.
[CrossRef]

Kolesik, M.

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

Krausz, F.

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

Lago, E. L.

V. Couderc, E. L. Lago, A. Barthelemy, C. De Angelis, and F. Gringoli, “Trapping of a weak probe through coupling with a two-color quadratic spatial soliton,” Opt. Commun. 203, 421–425 (2002).
[CrossRef]

Liu, Z. W.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Luo, G. Z.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Milke, A.

S. Amiranashvili, U. Bandelow, and A. Milke, “Pade approximant for the refractive index and nonlocal envelope equations,” Opt. Commun. 283, 480–485 (2010).
[CrossRef]

Ming, N. B.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Ming, N.-B.

Moloney, J. V.

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

Oron, M. B.

Parameswaran, K. R.

M. C. Cardakli, D. Gurkan, S. A. Havstad, A. E. Willner, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Tunable all-optical time-slot-interchange and wavelength conversion using difference-frequency-generation and optical buffers,” IEEE Photon. Technol. Lett. 14, 200–202 (2002).
[CrossRef]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Qin, Y.-Q.

Saltiel, S. M.

A. A. Sukhorukov, T. J. Alexander, Y. S. Kivshar, and S. M. Saltiel, “Multistep cascading and fourth harmonic generation,” Phys. Lett. A 281, 34–38 (2001).
[CrossRef]

S. M. Saltiel, A. A. Sukhourukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” in Vol.  47 of Progress in Optics (Elsevier, 2005), pp. 1–73.
[CrossRef]

Sukhorukov, A. A.

A. A. Sukhorukov, T. J. Alexander, Y. S. Kivshar, and S. M. Saltiel, “Multistep cascading and fourth harmonic generation,” Phys. Lett. A 281, 34–38 (2001).
[CrossRef]

Sukhourukov, A. A.

S. M. Saltiel, A. A. Sukhourukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” in Vol.  47 of Progress in Optics (Elsevier, 2005), pp. 1–73.
[CrossRef]

Wang, H. T.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Willner, A. E.

M. C. Cardakli, D. Gurkan, S. A. Havstad, A. E. Willner, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Tunable all-optical time-slot-interchange and wavelength conversion using difference-frequency-generation and optical buffers,” IEEE Photon. Technol. Lett. 14, 200–202 (2002).
[CrossRef]

Zhang, C.

Y.-Q. Qin, Y.-Y. Zhu, C. Zhang, and N.-B. Ming, “Theoretical investigations of efficient cascaded third-harmonic generation in quasi-phase-matched and -mismatched configurations,” J. Opt. Soc. Am. B 20, 73–82 (2003).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Zhu, S. N.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Zhu, Y. Y.

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

Zhu, Y.-Y.

Appl. Phys. Lett. (1)

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, “Simultaneously efficient blue and red light generations in a periodically poled LiTaO3,” Appl. Phys. Lett. 78, 3006–3008 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

IEEE Photon. J. (1)

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (2010).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. C. Cardakli, D. Gurkan, S. A. Havstad, A. E. Willner, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Tunable all-optical time-slot-interchange and wavelength conversion using difference-frequency-generation and optical buffers,” IEEE Photon. Technol. Lett. 14, 200–202 (2002).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

S. Amiranashvili, U. Bandelow, and A. Milke, “Pade approximant for the refractive index and nonlocal envelope equations,” Opt. Commun. 283, 480–485 (2010).
[CrossRef]

V. Couderc, E. L. Lago, A. Barthelemy, C. De Angelis, and F. Gringoli, “Trapping of a weak probe through coupling with a two-color quadratic spatial soliton,” Opt. Commun. 203, 421–425 (2002).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

A. A. Sukhorukov, T. J. Alexander, Y. S. Kivshar, and S. M. Saltiel, “Multistep cascading and fourth harmonic generation,” Phys. Lett. A 281, 34–38 (2001).
[CrossRef]

Phys. Rev. A (2)

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

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (3)

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

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

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Other (2)

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

S. M. Saltiel, A. A. Sukhourukov, and Y. S. Kivshar, “Multistep parametric processes in nonlinear optics,” in Vol.  47 of Progress in Optics (Elsevier, 2005), pp. 1–73.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Propagation of a femtosecond pulse in a PPSLT crystal. Evolution of the field amplitude | A | from numerical solution of Eq. (1). The initial pulse has Gaussian shape, and the parameters are T = 120 fs , I = 20 GW / cm 2 , λ in = 1400 nm , λ 0 = 2 π c / ω 0 = 700 nm , d 33 = χ LT ( 2 ) / 2 = 10.6 pm / V .

Fig. 2
Fig. 2

Comparison between coupled wave solution (dashed curves) and | A | filtered around fundamental, SH, and TH (solid curves).

Fig. 3
Fig. 3

(a) Experimental and (b) numerical SH and TH spectra at the crystal output for several poling periods. The inset in (b) shows a close-up comparison between experiment and numerics at the SH and TH.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

A z + i D A = i G ( z ) χ ( 2 ) ω 0 4 n 0 c ( 1 i ω 0 t ) [ A 2 e i ω 0 t i β 0 z + 2 | A | 2 e i ω 0 t + i β 0 z ] ,
A ( z , t ) = A 1 ( z , t ) e i ( Ω ω 0 ) t i ( q 1 β 0 ) z + A 2 ( z , t ) e i ( 2 Ω ω 0 ) t i ( q 2 β 0 ) z + A 3 ( z , t ) e i ( 3 Ω ω 0 ) t i ( q 3 β 0 ) z ,
A ^ ( ω ) z + i [ k ( ω + ω 0 ) k 0 ] A ^ ( ω ) = NL ,
m = 1 3 [ A ^ m z ( ω ( m Ω ω 0 ) ) + i [ k ( ω + ω 0 ) q m ] A ^ m ( ω ( m Ω ω 0 ) ) ] e i ( q m β 0 ) z = NL ,
m = 1 3 [ A m z + i D m A m ] e i ( m Ω ω 0 ) i ( q m β 0 ) z = i G ( z ) χ ( 2 ) ω 0 4 n 0 c ( 1 i ω 0 t ) [ A 1 2 e i ( 2 Ω ω 0 ) t i ( 2 q 1 β 0 ) z + 2 A 1 A 2 e i ( 3 Ω ω 0 ) t i ( q 1 + q 2 β 0 ) z + 2 A 2 A 1 * e i ( Ω ω 0 ) t i ( q 2 q 1 β 0 ) z + 2 A 3 A 2 * e i ( Ω ω 0 ) t i ( q 3 q 1 β 0 ) z + 2 A 3 A 1 * e i ( 2 Ω ω 0 ) t i ( q 3 q 1 β 0 ) z + R ] ,
A 1 z + i D 1 A 1 = i G ( z ) χ ( 2 ) Ω 2 n 0 c [ A 2 A 1 * e i Δ k 1 z + A 3 A 2 * e i Δ k 2 z ] , A 2 z + i D 2 A 2 = i G ( z ) χ ( 2 ) Ω 2 n 0 c [ A 1 2 e i Δ k 1 z + 2 A 3 A 1 * e i Δ k 2 z ] , A 3 z + i D 3 A 3 = i G ( z ) 3 χ ( 2 ) Ω 2 n 0 c [ A 1 A 2 * e i Δ k 2 z ] ,
A 1 z + i D ˜ 1 A 1 = i σ 1 A 2 A 1 * e i δ k 1 z i σ 3 A 3 A 2 * e i δ k 2 z , A 2 z + i D ˜ 2 A 2 = i σ 2 A 1 2 e i δ k 1 z i σ 4 A 3 A 1 * e i δ k 2 z , A 3 z + i D ˜ 3 A 3 = i σ 5 A 1 A 2 * e i δ k 2 z ,

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