Abstract

A model of three-wave parametric interactions is presented, in both type I and type II noncollinear phase-matching conditions, in which the group velocities of the interacting pulses are suitably linked to each other. We consider two conditions for group velocities that are significant in the parametric generation/amplification of femtosecond pulses and determine the interaction geometry required to fulfill them in uniaxial crystals. The results are compared with those for collinear phase matching and are used to interpret the behavior of ultrabroadband parametric sources.

© 1998 Optical Society of America

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References

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  1. K. R. Wilson, V. V. Yakovlev, “Ultrafast rainbow: tunable ultrashort pulses from a solid-state kilohertz system,” J. Opt. Soc. Am. B 14, 444–448 (1997).
    [CrossRef]
  2. R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973–975 (1996).
    [CrossRef] [PubMed]
  3. R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 551–562 (1998).
    [CrossRef]
  4. P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).
  5. P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
    [CrossRef] [PubMed]
  6. P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, A. Piskarskas, “Matching of group velocities in three-wave parametric interaction with fs pulses and application to travelling-wave generators,” J. Opt. Soc. Am. B 12, 2237–2244 (1995).
    [CrossRef]
  7. P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, A. Piskarskas, “Efficient conversion of femtosecond blue pulses by travelling-wave parametric generation in non-collinear phase-matching,” Opt. Commun. 119, 327–332 (1995).
    [CrossRef]
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    [CrossRef]
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1998 (1)

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 551–562 (1998).
[CrossRef]

1997 (2)

1996 (2)

1995 (4)

P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, A. Piskarskas, “Efficient conversion of femtosecond blue pulses by travelling-wave parametric generation in non-collinear phase-matching,” Opt. Commun. 119, 327–332 (1995).
[CrossRef]

C. Radzewicz, Y. B. Band, G. W. Pearson, J. S. Krasinski, “Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening,” Opt. Commun. 117, 295–302 (1995).
[CrossRef]

P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, A. Piskarskas, “Matching of group velocities in three-wave parametric interaction with fs pulses and application to travelling-wave generators,” J. Opt. Soc. Am. B 12, 2237–2244 (1995).
[CrossRef]

1993 (1)

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Agnesi, A.

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Andreoni, A.

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 551–562 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973–975 (1996).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, A. Piskarskas, “Efficient conversion of femtosecond blue pulses by travelling-wave parametric generation in non-collinear phase-matching,” Opt. Commun. 119, 327–332 (1995).
[CrossRef]

P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, A. Piskarskas, “Matching of group velocities in three-wave parametric interaction with fs pulses and application to travelling-wave generators,” J. Opt. Soc. Am. B 12, 2237–2244 (1995).
[CrossRef]

Band, Y. B.

C. Radzewicz, Y. B. Band, G. W. Pearson, J. S. Krasinski, “Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening,” Opt. Commun. 117, 295–302 (1995).
[CrossRef]

Banfi, G. P.

P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
[CrossRef] [PubMed]

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Daido, H.

Danielius, R.

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 551–562 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973–975 (1996).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, A. Piskarskas, “Efficient conversion of femtosecond blue pulses by travelling-wave parametric generation in non-collinear phase-matching,” Opt. Commun. 119, 327–332 (1995).
[CrossRef]

P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, A. Piskarskas, “Matching of group velocities in three-wave parametric interaction with fs pulses and application to travelling-wave generators,” J. Opt. Soc. Am. B 12, 2237–2244 (1995).
[CrossRef]

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Deng, D.

Di Trapani, P.

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 551–562 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973–975 (1996).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, A. Piskarskas, “Efficient conversion of femtosecond blue pulses by travelling-wave parametric generation in non-collinear phase-matching,” Opt. Commun. 119, 327–332 (1995).
[CrossRef]

P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, A. Piskarskas, “Matching of group velocities in three-wave parametric interaction with fs pulses and application to travelling-wave generators,” J. Opt. Soc. Am. B 12, 2237–2244 (1995).
[CrossRef]

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Dubietis, A.

Foggi, P.

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 551–562 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973–975 (1996).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, A. Piskarskas, “Efficient conversion of femtosecond blue pulses by travelling-wave parametric generation in non-collinear phase-matching,” Opt. Commun. 119, 327–332 (1995).
[CrossRef]

P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, A. Piskarskas, “Matching of group velocities in three-wave parametric interaction with fs pulses and application to travelling-wave generators,” J. Opt. Soc. Am. B 12, 2237–2244 (1995).
[CrossRef]

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Kato, Y.

Krasinski, J. S.

C. Radzewicz, Y. B. Band, G. W. Pearson, J. S. Krasinski, “Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening,” Opt. Commun. 117, 295–302 (1995).
[CrossRef]

Monguzzi, M.

P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
[CrossRef] [PubMed]

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

Pearson, G. W.

C. Radzewicz, Y. B. Band, G. W. Pearson, J. S. Krasinski, “Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening,” Opt. Commun. 117, 295–302 (1995).
[CrossRef]

Piskarskas, A.

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 551–562 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973–975 (1996).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, A. Piskarskas, “Efficient conversion of femtosecond blue pulses by travelling-wave parametric generation in non-collinear phase-matching,” Opt. Commun. 119, 327–332 (1995).
[CrossRef]

P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, A. Piskarskas, “Matching of group velocities in three-wave parametric interaction with fs pulses and application to travelling-wave generators,” J. Opt. Soc. Am. B 12, 2237–2244 (1995).
[CrossRef]

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Radzewicz, C.

C. Radzewicz, Y. B. Band, G. W. Pearson, J. S. Krasinski, “Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening,” Opt. Commun. 117, 295–302 (1995).
[CrossRef]

Righini, R.

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Sa’nta, I.

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Solcia, C.

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 551–562 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973–975 (1996).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, A. Piskarskas, “Efficient conversion of femtosecond blue pulses by travelling-wave parametric generation in non-collinear phase-matching,” Opt. Commun. 119, 327–332 (1995).
[CrossRef]

P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
[CrossRef] [PubMed]

P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, A. Piskarskas, “Matching of group velocities in three-wave parametric interaction with fs pulses and application to travelling-wave generators,” J. Opt. Soc. Am. B 12, 2237–2244 (1995).
[CrossRef]

Sozzi, C.

P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
[CrossRef] [PubMed]

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Wang, H.

Wilson, K. R.

Wong, G. K. L.

Wong, K. S.

Xu, Z.

Yakovlev, V. V.

Zhang, J.

Zhang, T.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 551–562 (1998).
[CrossRef]

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

Liet. Fis. Zu. (Lith. J. Phys.) (1)

P. Di Trapani, A. Agnesi, G. P. Banfi, R. Danielius, A. Piskarskas, P. Foggi, R. Righini, I. Sa’nta, M. Monguzzi, C. Sozzi, “Widely tunable β-BaB2O4 parametric laser pumped by femtosecond Ti:sapphire laser-amplifier system,” Liet. Fis. Zu. (Lith. J. Phys.) 33, 324–327 (1993).

Opt. Commun. (2)

C. Radzewicz, Y. B. Band, G. W. Pearson, J. S. Krasinski, “Short pulse nonlinear frequency conversion without group-velocity-mismatch broadening,” Opt. Commun. 117, 295–302 (1995).
[CrossRef]

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, A. Piskarskas, “Efficient conversion of femtosecond blue pulses by travelling-wave parametric generation in non-collinear phase-matching,” Opt. Commun. 119, 327–332 (1995).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, C. Sozzi, “Group-velocity self-matching of femtosecond pulses in non-collinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
[CrossRef] [PubMed]

Other (3)

Fujian Castech Crystals Inc. , Crystals (Fujian Castech Crystals, Xihe, Fuzhou, Fujian, China, 1996).

CASIX Inc., Crystals & Materials (CASIX, Fuzhou, Fujian, China, 1996).

V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

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Figures (7)

Fig. 1
Fig. 1

Wave vectors of the three interacting noncollinear phase-matched waves.

Fig. 2
Fig. 2

Fig. 2. k s -to-k p angle θ s (upper plots) and angle α between k p and the crystal optical axis (lower plots) as a function of the signal wavelength in (a) BBO, (b) LiIO3, and (c) KDP for different pump wavelengths λ p for PM I and either GV s,par = GV p [condition (i), solid curves] or (GV s,par + GV i,par)/2 = GV p [condition (ii), dotted curves]. See the text for details.

Fig. 3
Fig. 3

Idler-to-pump GV mismatch (GVM ip , solid curves) and idler-to-signal GV mismatch (GVM is , dotted curves) as a function of the signal wavelength for PM I and either GV s,par = GV p or (GV s,par + GV i,par)/2 = GV p , respectively, in (a) BBO, (b) LiIO3, and (c) KDP for different pump wavelengths λ p . For KDP I the idler-to-pump GV mismatch for collinearly phase-matched waves, GVM ip COLL, is also reported.

Fig. 4
Fig. 4

BBO I pumped at 395 nm. Left-hand scale: tuning curves (λ s versus α) for collinear and noncollinear PM I. Right-hand scale: time delays among the interacting pulses as a function of the pump angle α across crystals of 2- and 4-mm depth (see Refs. 12 and 7, respectively). For noncollinear PM [z = 4 mm (Ref. 7)]: idler-to-pump delay (i) when GV s,par = GV p and idler-to-signal delay (ii) when (GV s,par + GV i,par)/2 = GV p . For collinear PM [z = 2 mm, (Ref. 12)]: idler-to-signal, idler-to-pump, and pump-to-signal delays; in the α range that simultaneously allows noncollinear PM, delays (i) and (ii) are also shown, for z = 2 mm.

Fig. 5
Fig. 5

Same as Fig. 2, but for PM II and either GV s,par = GV p [condition (i), solid and dashed curves] or (GV s,par + GV i,par)/2 = GV p [condition (ii), dotted curves]. The pump angle αCOLL for collinearly phase-matched waves is also shown (lower plots) for some λ p values.

Fig. 6
Fig. 6

Same as Fig. 3, but for PM II (curve styles as in Fig. 5). For BBO II and KDP II, the idler-to-pump GV mismatch for collinearly phase-matched waves, GVM ip COLL, is also reported.

Fig. 7
Fig. 7

Noncollinear PM II in BBO pumped at 600 nm: superfluorescence cones corresponding to the two solutions in the upper plot of Fig. 5(a) with GV s,par = GV p [condition (i)]. Upper plot: k s -to-k p angles θ s (signs as in Fig. 1, solid and dashed curves as in Fig. 5); the vertical bars evidence the experimentally observed emission.4,5 Lower panel: corresponding GVM ip as a function of the pump angle α (solid and dashed curves) and GVM ip calculated for collinear PM II at the same pump angle (dotted–dashed curve; see αCOLL in Fig. 5).

Equations (28)

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ω s + ω i = ω p ,
k p = k s   cos   θ s + k i   cos   θ i ,
k s   sin   θ s = k i   sin   θ i ,
k j = ω j / c n j ,   j = p , s , i .
n p = cos 2   α n o 2 ω p + sin 2   α n e 2 ω p - 1 / 2 ,
n s = n o ω s ,
n i = n o ω i
n i = cos 2 θ i + α n o 2 ω i + sin 2 θ i + α n e 2 ω i - 1 / 2
GV j = c n j + ω j d n j / d ω ω = ω j ,   j = p , s , i ,
cos   θ s n o ω s + ω s d n o / d ω ω = ω s = 1 n p + ω p d n p / d ω ω = ω p ,
cos   θ s n o ω s + ω s d n o / d ω ω = ω s
+ cos   θ i n o ω i + ω i d n o / d ω ω = ω i = 2 n p + ω p d n p / d ω ω = ω p .
θ s = cos - 1 n o ω p 2 + ω p 2 d n o d ω ω = ω p / 2 n p + ω p d n p d ω ω = ω p .
GVM ip = GV i , par - 1 - GV p - 1 ,
GVM is = GV i , par - 1 - GV s , par - 1 ,
ω p 2 n p 2 + ω s 2 n s 2 - ω i 2 n i 2 2 ω p ω s n s × 1 + ω p 2 n p 2 Q   cos 2   α + d n e d ω p n e 3 ω p = n o ω s + ω s d n o d ω s ,
Q = d n o d ω p n o 3 ω p - d n e d ω p n e 3 ω p
d n o , e d ω j = d n o , e d ω ω = ω j .
D = 1 n o 2 ω p - 1 n e 2 ω p
D ω s 2 n o 2 ω s - ω i 2 n o 2 ω i D ω p + Q - 2 D 2 ω s n o ω s n o ω s + ω s d n o d ω s cos 4   α + ω p 2 D ω p + Q + ω s 2 n o 2 ω s - ω i 2 n o 2 ω i n e 2 ω p 2 D ω p + Q + D ω s 2 n o 2 ω s - ω i 2 n o 2 ω i d n e d ω p n e 3 ω p - 4 D n e 2 ω p   ω s n o ω s n o ω s + ω s d n o d ω s cos 2   α + ω p 2 d n e d ω p n e 3 ω p + ω p n e 2 ω p + ω s 2 n o 2 ω s - ω i 2 n o 2 ω i n e 2 ω p d n e d ω p n e 3 ω p + - 2 ω s n o ω s n o ω s + ω s d n o d ω s + ω s 2 n o 2 ω s - ω i 2 n o 2 ω i ω p n e 4 ω p = 0 .
D ω s 2 n o 2 ω s - ω i 2 n o 2 ω i M D ω p + Q - 4 D 2 cos 4   α + ω p 2 N D ω p + Q + ω s 2 n o 2 ω s - ω i 2 n o 2 ω i n e 2 ω p   M 2 D ω p + Q + D ω s 2 n o 2 ω s - ω i 2 n o 2 ω i M   d n e d ω p n e 3 ω p - 8 D n e 2 ω p cos 2   α + ω p 2 N   d n e d ω p n e 3 ω p + ω p n e 2 ω p   N + ω s 2 n o 2 ω s - ω i 2 n o 2 ω i n e 2 ω p × M   d n e d ω p n e 3 ω p + - 4 + ω s 2 n o 2 ω s - ω i 2 n o 2 ω i ω p   M n e 4 ω p = 0 ,
M = 1 ω s n s n o ω s + ω s d n o d ω ω = ω s - 1 ω i n i n o ω i + ω i d n o d ω ω = ω i ,
N = 1 ω s n s n o ω s + ω s d n o d ω ω = ω s + 1 ω i n i n o ω i + ω i d n o d ω ω = ω i .
A ω p 2 n p 2 - ω s 2 n s 2 + ω i 2 n i 2 2 + C 4 ω i 2 n i 2 ω p 2 n p 2 - D ω i 2 = B ω p 2 n p 2 - ω s 2 n s 2 + ω i 2 n i 2 4 ω i 2 n i 2 ω p 2 n p 2 - ω p 2 n p 2 - ω s 2 n s 2 + ω i 2 n i 2 2 1 / 2 ,
A = n e 2 ω i D i cos 2   α - sin 2   α , B = 2 n e 2 ω i D i sin   α   cos   α , C = n e 2 ω i D i sin 2   α + 1 , D = n e 2 ω i ,
D i = 1 n o 2 ω i - 1 n e 2 ω i .
D i 2 n e 4 ω i ω p 2 n p 2 - ω s 2 n s 2 + ω i 2 n i 2 4 + 4 ω i 2 ω p 2 n p 2 a n p 2 + b n i 2 - d n p 2 + c ω p 2 n p 2 - ω s 2 n s 2 + ω i 2 n i 2 2 + 16 g n p 4 + h n p 2 + l ω i 4 n i 4 ω p 4 n p 4 - 32 e n p 2 + f × ω i 4 n i 2 ω p 4 n p 4 + 16 n e 4 ω i ω i 4 ω p 4 n p 4 = 0 ,
a = 2 n e 2 ω i D i D n e 2 ω i D i + 2 , b = - 2   n e 2 ω i n e 2 ω p D i D n e 2 ω i D i + 2 - 2 n e 2 ω i D i n e 2 ω i D i + 1 , c = - 4   n e 4 ω i n e 2 ω p D i D - 2 n e 4 ω i D i , d = 4 n e 4 ω i D i D , e = - n e 4 ω i D i D , f = n e 4 ω i D i + n e 2 ω i + n e 4 ω i n e 2 ω p D i D , g = n e 4 ω i D i 2 D 2 , h = - 2   n e 4 ω i n e 2 ω p D i 2 D 2 - 2 n e 2 ω i D i D n e 2 ω i D i + 1 , l = n e 2 ω i D i + 1 2 + 2   n e 2 ω i n e 2 ω p D i D n e 2 ω i D i + 1 + n e 4 ω i n e 4 ω p D i 2 D 2 .

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