Abstract

Optical parametric generation is treated as a particular case of parametric amplification of two waves at frequencies ω1 and ω2 from a pump wave ω3 propagating in a nonlinear crystal. This theory describes the configuration in which initial intensities for ω1 and ω2 at the input plane of the material are equal to zero. This research is based on the theory of optical parametric fluorescence. According to this quantum-mechanical model, there is a probability for a photon ω3 to be spontaneously scattered into photons ω1 and ω2, respectively. We introduce a critical length over which the first signal and idler photons are created. This new approach allows us to take into account optical parametric fluorescence not only at the entrance of material but also over its entire interaction length. We show that this can widely modify the calculated generated intensities. This model is applied to a KTP optical parametric generator and amplifier pumped by the second harmonic (532 nm) of a Nd:YAG picosecond laser.

© 2000 Optical Society of America

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References

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  2. W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I.,” Phys. Rev. 124, 1646–1654 (1961).
    [CrossRef]
  3. T. G. Giallorenzi and C. L. Tang, “Quantum theory of spontaneous parametric scattering,” Phys. Rev. 166, 225–233 (1968).
    [CrossRef]
  4. S. E. Harris, M. K. Oshman, and R. L. Byer, “Observation of tunable optical parametric fluorescence,” Phys. Rev. Lett. 18, 732–734 (1967).
    [CrossRef]
  5. R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968).
    [CrossRef]
  6. S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
    [CrossRef]
  7. D. A. Kleinman, “Theory of optical parametric noise,” Phys. Rev. 174, 1027–1041 (1968).
    [CrossRef]
  8. J. P. Girardeau-Montaut and R. Lambert, Les Lasers et leurs Applications Médicales (Editions Médicales, Paris, 1987), Chap. 3.3.6.
  9. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 9.5, p. 134.
  10. H. Vanherzeele and J. D. Bierlein, “Magnitude of the nonlinear-optical coefficients of KTiOPO4,” Opt. Lett. 17, 982–984 (1992).
    [CrossRef] [PubMed]
  11. CASIX Crystal Guide, Crystals and Materials—Laser Accessories (1995) p. 24.
  12. R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
    [CrossRef]
  13. C. Sauteret, La conversion de fréquence, Rapport CEAR-5383 (Service de Documentation, Centre d’Etudes Nucleaires, Saclay, Gif-Sur-Yvette, France, 1987), Chap. 8, p. 68.
  14. P. P. Bey and C. L. Tang, “Plane-wave theory of parametric oscillator and coupled oscillator-upconverter,” IEEE J. Quantum Electron. QE-8, 361–369 (1972).
    [CrossRef]
  15. L. Carrion and J. P. Girardeau-Montaut, “Realization, investigation, and applications of a picosecond optical parametric generator and amplifier,” Quantum Semiclassic. Opt. 9 Editorial (1997).
  16. L. Carrion and J. P. Girardeau-Montaut, “Performance of a new picosecond KTP optical parametric generator and amplifier,” Opt. Commun. 152, 347–350 (1998).
    [CrossRef]
  17. W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
    [CrossRef]
  18. T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KiTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
    [CrossRef]
  19. B. Boulanger, J. P. Fève, G. Marnier, B. Ménaert, X. Cabirol, P. Villeval, and C. Bonnin, “Relative sign and absolute magnitude of d(2) nonlinear coefficients of KTP from second-harmonic-generation measurements,” J. Opt. Soc. Am. B 11, 750–757 (1994).
    [CrossRef]
  20. J. J. Zondy, M. Abed, and S. Khodja, “Twin-crystal walk-off compensated type-II second-harmonic generation: single-pass and cavity-enhanced experiments in KTiOPO4,” J. Opt. Soc. Am. B 11, 2368–2379 (1994).
    [CrossRef]
  21. J. F. Coutonly, P. Deprez, and A. Deffontaine, “Simple is best for real-time beam analysis,” Opt. Laser Eur. 58, 34–37 (1999).
  22. M. G. Roelofs, “Identification of Ti3+ in potassium titanyl phosphate and its possible role in laser damage,” J. Appl. Phys. 65, 4976–4982 (1989).
    [CrossRef]
  23. R. Blachman, P. F. Bordui, and M. M. Fejer, “Laser-induced photochromic damage in potassium titanyl phosphate,” Appl. Phys. Lett. 64, 1318–1320 (1994).
    [CrossRef]
  24. G. M. Loiacono, D. N. Loiacono, T. McGee, and M. Babb, “Laser damage formation in KTiOPO4 crystals: grey tracks,” J. Appl. Phys. 72, 2705–2712 (1992).
    [CrossRef]
  25. M. K. Reed, M. K. Steiner-Shepard, M. S. Armas, and D. K. Negus, “Microjoule-energy ultrafast optical parametric amplifiers,” J. Opt. Soc. Am. B 11, 2229–2236 (1995).
    [CrossRef]

1999 (1)

J. F. Coutonly, P. Deprez, and A. Deffontaine, “Simple is best for real-time beam analysis,” Opt. Laser Eur. 58, 34–37 (1999).

1998 (1)

L. Carrion and J. P. Girardeau-Montaut, “Performance of a new picosecond KTP optical parametric generator and amplifier,” Opt. Commun. 152, 347–350 (1998).
[CrossRef]

1995 (2)

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KiTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

M. K. Reed, M. K. Steiner-Shepard, M. S. Armas, and D. K. Negus, “Microjoule-energy ultrafast optical parametric amplifiers,” J. Opt. Soc. Am. B 11, 2229–2236 (1995).
[CrossRef]

1994 (3)

1992 (2)

G. M. Loiacono, D. N. Loiacono, T. McGee, and M. Babb, “Laser damage formation in KTiOPO4 crystals: grey tracks,” J. Appl. Phys. 72, 2705–2712 (1992).
[CrossRef]

H. Vanherzeele and J. D. Bierlein, “Magnitude of the nonlinear-optical coefficients of KTiOPO4,” Opt. Lett. 17, 982–984 (1992).
[CrossRef] [PubMed]

1989 (2)

M. G. Roelofs, “Identification of Ti3+ in potassium titanyl phosphate and its possible role in laser damage,” J. Appl. Phys. 65, 4976–4982 (1989).
[CrossRef]

W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
[CrossRef]

1979 (1)

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

1972 (1)

P. P. Bey and C. L. Tang, “Plane-wave theory of parametric oscillator and coupled oscillator-upconverter,” IEEE J. Quantum Electron. QE-8, 361–369 (1972).
[CrossRef]

1969 (1)

S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
[CrossRef]

1968 (3)

D. A. Kleinman, “Theory of optical parametric noise,” Phys. Rev. 174, 1027–1041 (1968).
[CrossRef]

T. G. Giallorenzi and C. L. Tang, “Quantum theory of spontaneous parametric scattering,” Phys. Rev. 166, 225–233 (1968).
[CrossRef]

R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968).
[CrossRef]

1967 (1)

S. E. Harris, M. K. Oshman, and R. L. Byer, “Observation of tunable optical parametric fluorescence,” Phys. Rev. Lett. 18, 732–734 (1967).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

1961 (1)

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I.,” Phys. Rev. 124, 1646–1654 (1961).
[CrossRef]

Abed, M.

Armas, M. S.

M. K. Reed, M. K. Steiner-Shepard, M. S. Armas, and D. K. Negus, “Microjoule-energy ultrafast optical parametric amplifiers,” J. Opt. Soc. Am. B 11, 2229–2236 (1995).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Babb, M.

G. M. Loiacono, D. N. Loiacono, T. McGee, and M. Babb, “Laser damage formation in KTiOPO4 crystals: grey tracks,” J. Appl. Phys. 72, 2705–2712 (1992).
[CrossRef]

Baumgartner, R. A.

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

Bey, P. P.

P. P. Bey and C. L. Tang, “Plane-wave theory of parametric oscillator and coupled oscillator-upconverter,” IEEE J. Quantum Electron. QE-8, 361–369 (1972).
[CrossRef]

Bierlein, J. D.

Blachman, R.

R. Blachman, P. F. Bordui, and M. M. Fejer, “Laser-induced photochromic damage in potassium titanyl phosphate,” Appl. Phys. Lett. 64, 1318–1320 (1994).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bonnin, C.

Bordui, P. F.

R. Blachman, P. F. Bordui, and M. M. Fejer, “Laser-induced photochromic damage in potassium titanyl phosphate,” Appl. Phys. Lett. 64, 1318–1320 (1994).
[CrossRef]

Bosenberg, W. R.

W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
[CrossRef]

Boulanger, B.

Byer, R. L.

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968).
[CrossRef]

S. E. Harris, M. K. Oshman, and R. L. Byer, “Observation of tunable optical parametric fluorescence,” Phys. Rev. Lett. 18, 732–734 (1967).
[CrossRef]

Cabirol, X.

Carrion, L.

L. Carrion and J. P. Girardeau-Montaut, “Performance of a new picosecond KTP optical parametric generator and amplifier,” Opt. Commun. 152, 347–350 (1998).
[CrossRef]

Coutonly, J. F.

J. F. Coutonly, P. Deprez, and A. Deffontaine, “Simple is best for real-time beam analysis,” Opt. Laser Eur. 58, 34–37 (1999).

Deffontaine, A.

J. F. Coutonly, P. Deprez, and A. Deffontaine, “Simple is best for real-time beam analysis,” Opt. Laser Eur. 58, 34–37 (1999).

Deprez, P.

J. F. Coutonly, P. Deprez, and A. Deffontaine, “Simple is best for real-time beam analysis,” Opt. Laser Eur. 58, 34–37 (1999).

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Fejer, M. M.

R. Blachman, P. F. Bordui, and M. M. Fejer, “Laser-induced photochromic damage in potassium titanyl phosphate,” Appl. Phys. Lett. 64, 1318–1320 (1994).
[CrossRef]

Fève, J. P.

Giallorenzi, T. G.

T. G. Giallorenzi and C. L. Tang, “Quantum theory of spontaneous parametric scattering,” Phys. Rev. 166, 225–233 (1968).
[CrossRef]

Girardeau-Montaut, J. P.

L. Carrion and J. P. Girardeau-Montaut, “Performance of a new picosecond KTP optical parametric generator and amplifier,” Opt. Commun. 152, 347–350 (1998).
[CrossRef]

Harris, S. E.

S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
[CrossRef]

R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968).
[CrossRef]

S. E. Harris, M. K. Oshman, and R. L. Byer, “Observation of tunable optical parametric fluorescence,” Phys. Rev. Lett. 18, 732–734 (1967).
[CrossRef]

Khodja, S.

Kleinman, D. A.

D. A. Kleinman, “Theory of optical parametric noise,” Phys. Rev. 174, 1027–1041 (1968).
[CrossRef]

Loiacono, D. N.

G. M. Loiacono, D. N. Loiacono, T. McGee, and M. Babb, “Laser damage formation in KTiOPO4 crystals: grey tracks,” J. Appl. Phys. 72, 2705–2712 (1992).
[CrossRef]

Loiacono, G. M.

G. M. Loiacono, D. N. Loiacono, T. McGee, and M. Babb, “Laser damage formation in KTiOPO4 crystals: grey tracks,” J. Appl. Phys. 72, 2705–2712 (1992).
[CrossRef]

Louisell, W. H.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I.,” Phys. Rev. 124, 1646–1654 (1961).
[CrossRef]

Marnier, G.

McGee, T.

G. M. Loiacono, D. N. Loiacono, T. McGee, and M. Babb, “Laser damage formation in KTiOPO4 crystals: grey tracks,” J. Appl. Phys. 72, 2705–2712 (1992).
[CrossRef]

Ménaert, B.

Negus, D. K.

M. K. Reed, M. K. Steiner-Shepard, M. S. Armas, and D. K. Negus, “Microjoule-energy ultrafast optical parametric amplifiers,” J. Opt. Soc. Am. B 11, 2229–2236 (1995).
[CrossRef]

Nishikawa, T.

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KiTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

Oshman, M. K.

S. E. Harris, M. K. Oshman, and R. L. Byer, “Observation of tunable optical parametric fluorescence,” Phys. Rev. Lett. 18, 732–734 (1967).
[CrossRef]

Pelouch, W. S.

W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Reed, M. K.

M. K. Reed, M. K. Steiner-Shepard, M. S. Armas, and D. K. Negus, “Microjoule-energy ultrafast optical parametric amplifiers,” J. Opt. Soc. Am. B 11, 2229–2236 (1995).
[CrossRef]

Roelofs, M. G.

M. G. Roelofs, “Identification of Ti3+ in potassium titanyl phosphate and its possible role in laser damage,” J. Appl. Phys. 65, 4976–4982 (1989).
[CrossRef]

Siegman, A. E.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I.,” Phys. Rev. 124, 1646–1654 (1961).
[CrossRef]

Steiner-Shepard, M. K.

M. K. Reed, M. K. Steiner-Shepard, M. S. Armas, and D. K. Negus, “Microjoule-energy ultrafast optical parametric amplifiers,” J. Opt. Soc. Am. B 11, 2229–2236 (1995).
[CrossRef]

Tang, C. L.

W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
[CrossRef]

P. P. Bey and C. L. Tang, “Plane-wave theory of parametric oscillator and coupled oscillator-upconverter,” IEEE J. Quantum Electron. QE-8, 361–369 (1972).
[CrossRef]

T. G. Giallorenzi and C. L. Tang, “Quantum theory of spontaneous parametric scattering,” Phys. Rev. 166, 225–233 (1968).
[CrossRef]

Uesugi, N.

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KiTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

Vanherzeele, H.

Villeval, P.

Yariv, A.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I.,” Phys. Rev. 124, 1646–1654 (1961).
[CrossRef]

Zondy, J. J.

Appl. Phys. Lett. (2)

W. R. Bosenberg, W. S. Pelouch, and C. L. Tang, “High-efficiency and narrow-linewidth operation of a two-crystal β-BaB2O4 optical parametric oscillator,” Appl. Phys. Lett. 55, 1952–1954 (1989).
[CrossRef]

R. Blachman, P. F. Bordui, and M. M. Fejer, “Laser-induced photochromic damage in potassium titanyl phosphate,” Appl. Phys. Lett. 64, 1318–1320 (1994).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. P. Bey and C. L. Tang, “Plane-wave theory of parametric oscillator and coupled oscillator-upconverter,” IEEE J. Quantum Electron. QE-8, 361–369 (1972).
[CrossRef]

R. A. Baumgartner and R. L. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. QE-15, 432–444 (1979).
[CrossRef]

J. Appl. Phys. (3)

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KiTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

G. M. Loiacono, D. N. Loiacono, T. McGee, and M. Babb, “Laser damage formation in KTiOPO4 crystals: grey tracks,” J. Appl. Phys. 72, 2705–2712 (1992).
[CrossRef]

M. G. Roelofs, “Identification of Ti3+ in potassium titanyl phosphate and its possible role in laser damage,” J. Appl. Phys. 65, 4976–4982 (1989).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

L. Carrion and J. P. Girardeau-Montaut, “Performance of a new picosecond KTP optical parametric generator and amplifier,” Opt. Commun. 152, 347–350 (1998).
[CrossRef]

Opt. Laser Eur. (1)

J. F. Coutonly, P. Deprez, and A. Deffontaine, “Simple is best for real-time beam analysis,” Opt. Laser Eur. 58, 34–37 (1999).

Opt. Lett. (1)

Phys. Rev. (5)

R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes. I.,” Phys. Rev. 124, 1646–1654 (1961).
[CrossRef]

T. G. Giallorenzi and C. L. Tang, “Quantum theory of spontaneous parametric scattering,” Phys. Rev. 166, 225–233 (1968).
[CrossRef]

D. A. Kleinman, “Theory of optical parametric noise,” Phys. Rev. 174, 1027–1041 (1968).
[CrossRef]

Phys. Rev. Lett. (1)

S. E. Harris, M. K. Oshman, and R. L. Byer, “Observation of tunable optical parametric fluorescence,” Phys. Rev. Lett. 18, 732–734 (1967).
[CrossRef]

Proc. IEEE (1)

S. E. Harris, “Tunable optical parametric oscillators,” Proc. IEEE 57, 2096–2113 (1969).
[CrossRef]

Other (5)

C. Sauteret, La conversion de fréquence, Rapport CEAR-5383 (Service de Documentation, Centre d’Etudes Nucleaires, Saclay, Gif-Sur-Yvette, France, 1987), Chap. 8, p. 68.

L. Carrion and J. P. Girardeau-Montaut, “Realization, investigation, and applications of a picosecond optical parametric generator and amplifier,” Quantum Semiclassic. Opt. 9 Editorial (1997).

J. P. Girardeau-Montaut and R. Lambert, Les Lasers et leurs Applications Médicales (Editions Médicales, Paris, 1987), Chap. 3.3.6.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 9.5, p. 134.

CASIX Crystal Guide, Crystals and Materials—Laser Accessories (1995) p. 24.

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Figures (7)

Fig. 1
Fig. 1

Calculated critical length for a type II interaction in KTP as a function of the laser pump intensity for various idler wavelengths.

Fig. 2
Fig. 2

Schematic representation of the numerical method referred to as method (I). I3, laser pump intensity; IG, total generated intensity; subscripts F and A, photons generated by fluorescence and amplification, respectively; N, total number of steps; Lc(I3(i)), critical length calculated from the pump intensity at the ith step.

Fig. 3
Fig. 3

Signal and idler energy generated as a function of KTP crystal length for the two methods employed. The incident pump intensity is 6.85 GW/cm2. The signal and the idler wavelengths are 895 and 1310 nm, respectively.

Fig. 4
Fig. 4

Signal and idler measured energies as a function of transition length (i.e. the length beyond which OPF is not taken into account anymore). The incident pump intensity is 6.85 GW/cm2. The signal and the idler wavelengths are 895 and 1310 nm, respectively.

Fig. 5
Fig. 5

Schematic representation of the KTP OPG/OPA: D, diaphragm; BS, beam splitter; DM, dichroic mirrors; MIR, protected gold mirror; M, mirrors RMAX (maximum reflectivity) at 532 nm; Pr, BK7 prisms; P, BK7 plate.

Fig. 6
Fig. 6

Theoretical (solid curve) and experimental (dots) angle-tuning curves for amplification in the (x, z) plane of KTP as a function of the crystal orientation (angle between the surface normal and the incident pump-wave vector) and the corresponding effective nonlinear coefficient.

Fig. 7
Fig. 7

Total output energy (signal+idler) of system with two KTP crystal-versus-pump irradiance. Solid curve, theoretical result; squares, experimental points. The signal and the idler wavelengths are 895 and 1310 nm, respectively.

Equations (25)

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

Δk=k3-k1-k2=0.
EizEjEk.
H=H0+Hm=i=1,2ωi(ai+ai+12)+12Γ[a1+a2+ exp(-iω3t)+a1a2 exp(iω3t)],
Γ=116π2038πω1ω2deff2 I3n12n22n3C1/2 sinΔkl2Δkl2,
da1+dt=iω1a1++i2Γa2 exp(iω3t),
da2dt=-iω2a2-i2Γa1+ exp(-iω3t).
n1¯(t)=n1¯(t=0)cosh2Γt2+[1+n2¯(t=0)]sinh2Γt2,
n2¯(t)=n2¯(t=0)cosh2Γt2+[1+n1¯(t=0)]sinh2Γt2.
n1¯(t)=n2¯(t)=sinh2Γt2.
n1¯(l)=n2¯(l)=sinh2Γl2C.
n1¯(lc)=n2¯(lc)=1.
lc=2CΓn3ln(1+2).
E1z+n1CE1t+σE1=-i ω1deffn1CE2*E3 exp(-iΔkz),
E2z+n2CE2t+σE2=-i ω2deffn2CE1*E3 exp(-iΔkz),
E3z+n3CE3t+σE3=-i ω3deffn3CE1E2 exp(iΔkz),
du1dξ=η1u1-u2u3 sin θ,
du2dξ=η2u2-u1u3 sin θ,
du3dξ=η3u3+u1u2 sin θ,
dθdξ=K+u1u2u3-u2u3u1-u1u3u2cos θ.
ui=IiωiP1/2.
ξ=4π3/2deffP1/2(0λ1λ2λ3n1n2n3)1/2z,
P=I1(0)+I2(0)+I3(0).
K=(Δk/ξ)z,
ηi=0λini4πPξ21/2σ.
reff=rTPTPC1/2.

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