Abstract

We report a semianalytical solution describing the type I second-order nonlinear interaction of the fundamental and the second-harmonic fields in a nonlinear crystal, which accounts for the phase- and group-velocity mismatch of the interacting pulses. The method uses a series-development solution of the propagation equations in respect to the second-harmonic conversion efficiency. The method describes the self-phase and self-amplitude modulation experienced by the fundamental pulse in single- and double-pass (i.e., reinjecting into the nonlinear crystal the outgoing pulses) interaction geometries, following better with respect to a numerical analysis, the dependence from the propagation parameters such as the crystal length, the pulse duration, and the phase- and group-velocity mismatch. It appears that it is possible to obtain an efficient self-phase modulation on the fundamental field even in nonstationary conditions. This paper describes the advantages of a double-pass configuration, which, for a given crystal length, allows a stronger nonlinear phase modulation of the fundamental field and minimizes its losses toward the second harmonic.

© 1998 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  34. G. Toci, M. Vannini, and R. Salimbeni, “Temporal dynamic of the non-linear mirror: an analytical description,” Opt. Commun. (to be published).
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1995 (7)

1994 (3)

1993 (3)

1992 (2)

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Varhenzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28 (1992).
[Crossref] [PubMed]

K. A. Stankov, V. P. Tzolov, and M. G. Mirkov, “Compensation of group-velocity mismatch in the frequency-doubling modelocker,” Appl. Phys. B 54, 303 (1992).
[Crossref]

1991 (2)

K. A. Stankov, V. P. Tzolov, and M. G. Mirkov, “Frequency-doubling mode locker: the influence of group-velocity mismatch,” Opt. Lett. 16, 1119 (1991).
[Crossref] [PubMed]

S. P. Velsko, M. Webb, L. Davis, and C. Huang, “Phase-matched harmonic generation in lithium triborate,” IEEE J. Quantum Electron. 27, 2182 (1991).
[Crossref]

1990 (3)

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P,BaB2O4,LiIO3,MgO:LiNbO3 and KTP measured by phase-matched second harmonic generation,” IEEE J. Quantum Electron. QE-26, 922 (1990).
[Crossref]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, “Phase modulation in second-order non-linear-optical processes,” Phys. Rev. A 42, 4085 (1990).
[Crossref] [PubMed]

1989 (1)

1988 (2)

K. A. Stankov and J. Jethwa, “A new mode-locking technique using a nonlinear mirror,” Opt. Commun. 66, 41 (1988).
[Crossref]

J. T. Manassah, “Effects of velocity dispersion on a generated second harmonic signal,” Appl. Opt. 27, 4365 (1988).
[Crossref] [PubMed]

1986 (1)

K. Kato, “Second harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron. QE-22, 1013 (1986).
[Crossref]

1984 (1)

R. C. Eckardt and J. Reintjes, “Phase-matching limitation of high efficiency second harmonic generation,” IEEE J. Quantum Electron. 20, 1178 (1984).
[Crossref]

1982 (1)

G. C. Ghosh and G. C. Bhar, “Temperature dispersion in ADP, KDP, KD*P for nonlinear devices,” IEEE J. Quantum Electron. QE-18, 143 (1982).
[Crossref]

1974 (1)

Assanto, G.

Bakker, H. J.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, “Phase modulation in second-order non-linear-optical processes,” Phys. Rev. A 42, 4085 (1990).
[Crossref] [PubMed]

H. J. Bakker, P. C. M. Planken, and H. G. Muller, “Numerical calculation of optical frequency-conversion processes: a new approach,” J. Opt. Soc. Am. B 6, 1665 (1989).
[Crossref]

Berman, B.

Bhar, G. C.

G. C. Ghosh and G. C. Bhar, “Temperature dispersion in ADP, KDP, KD*P for nonlinear devices,” IEEE J. Quantum Electron. QE-18, 143 (1982).
[Crossref]

Bierlein, J. D.

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992), and references therein.

Bossard, Ch.

Buchvarov, I.

I. Buchvarov, G. Christov, and S. Saltiel, “Transient behavior of frequency doubling mode-locker. Numerical analysis,” Opt. Commun. 107, 281 (1994).
[Crossref]

Byer, R. L.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P,BaB2O4,LiIO3,MgO:LiNbO3 and KTP measured by phase-matched second harmonic generation,” IEEE J. Quantum Electron. QE-26, 922 (1990).
[Crossref]

Cerullo, G.

Christov, G.

I. Buchvarov, G. Christov, and S. Saltiel, “Transient behavior of frequency doubling mode-locker. Numerical analysis,” Opt. Commun. 107, 281 (1994).
[Crossref]

Danailov, M. B.

Danielius, R.

Davis, L.

S. P. Velsko, M. Webb, L. Davis, and C. Huang, “Phase-matched harmonic generation in lithium triborate,” IEEE J. Quantum Electron. 27, 2182 (1991).
[Crossref]

De Silvestri, S.

DeSalvo, R.

Dienes, A.

Dubietis, A.

Eckardt, R. C.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P,BaB2O4,LiIO3,MgO:LiNbO3 and KTP measured by phase-matched second harmonic generation,” IEEE J. Quantum Electron. QE-26, 922 (1990).
[Crossref]

R. C. Eckardt and J. Reintjes, “Phase-matching limitation of high efficiency second harmonic generation,” IEEE J. Quantum Electron. 20, 1178 (1984).
[Crossref]

Fan, Y. X.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P,BaB2O4,LiIO3,MgO:LiNbO3 and KTP measured by phase-matched second harmonic generation,” IEEE J. Quantum Electron. QE-26, 922 (1990).
[Crossref]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Wetterling, Numerical Recipes (Cambridge University, Cambridge, England, 1989).

Gale, G. M.

Gallot, G.

Ghosh, G. C.

G. C. Ghosh and G. C. Bhar, “Temperature dispersion in ADP, KDP, KD*P for nonlinear devices,” IEEE J. Quantum Electron. QE-18, 143 (1982).
[Crossref]

Hache, F.

Hagan, D. J.

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Varhenzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28 (1992).
[Crossref] [PubMed]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

Huang, C.

S. P. Velsko, M. Webb, L. Davis, and C. Huang, “Phase-matched harmonic generation in lithium triborate,” IEEE J. Quantum Electron. 27, 2182 (1991).
[Crossref]

Jethwa, J.

K. A. Stankov and J. Jethwa, “A new mode-locking technique using a nonlinear mirror,” Opt. Commun. 66, 41 (1988).
[Crossref]

Kato, K.

K. Kato, “Second harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron. QE-22, 1013 (1986).
[Crossref]

Knoesnen, A.

Kuipers, L.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, “Phase modulation in second-order non-linear-optical processes,” Phys. Rev. A 42, 4085 (1990).
[Crossref] [PubMed]

Lagendijk, A.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, “Phase modulation in second-order non-linear-optical processes,” Phys. Rev. A 42, 4085 (1990).
[Crossref] [PubMed]

Magni, V.

Manassah, J. T.

Masuda, H.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P,BaB2O4,LiIO3,MgO:LiNbO3 and KTP measured by phase-matched second harmonic generation,” IEEE J. Quantum Electron. QE-26, 922 (1990).
[Crossref]

McGraw, D.

Menjuk, C. R.

Miller, D. A. B.

Mirkov, M. G.

K. A. Stankov, V. P. Tzolov, and M. G. Mirkov, “Compensation of group-velocity mismatch in the frequency-doubling modelocker,” Appl. Phys. B 54, 303 (1992).
[Crossref]

K. A. Stankov, V. P. Tzolov, and M. G. Mirkov, “Frequency-doubling mode locker: the influence of group-velocity mismatch,” Opt. Lett. 16, 1119 (1991).
[Crossref] [PubMed]

Monguzzi, A.

Muller, H. G.

Pierrotet, D.

Pini, R.

G. Toci, D. McGraw, R. Pini, R. Salimbeni, and M. Vannini, “Time-dependent analysis of a parametric lens detected with a 100-fs Ti:sapphire laser,” Opt. Lett. 20, 1547 (1995).
[Crossref] [PubMed]

G. Toci, R. Pini, R. Salimbeni, S. Siano, and M. Vannini, “Optical modulators based on second order non-linear processes in non-stationary conditions,” Proceedings of the International Conference on Lasers, 1995 (STS, McLean, Va., 1996), p. 761.

G. Toci, R. Pini, R. Salimbeni, and M. Vannini, “Group velocity mismatch effects in ultrafast optical modulators based on cascaded second order nonlinearities,” in Ultrafast Processes in Spectroscopy, O. Svelto, S. De Silvestri, and G. Denardo, eds. (Plenum, New York, 1996).

Piskarkas, A.

Planken, P. C. M.

H. J. Bakker, P. C. M. Planken, L. Kuipers, and A. Lagendijk, “Phase modulation in second-order non-linear-optical processes,” Phys. Rev. A 42, 4085 (1990).
[Crossref] [PubMed]

H. J. Bakker, P. C. M. Planken, and H. G. Muller, “Numerical calculation of optical frequency-conversion processes: a new approach,” J. Opt. Soc. Am. B 6, 1665 (1989).
[Crossref]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Wetterling, Numerical Recipes (Cambridge University, Cambridge, England, 1989).

Reintjes, J.

R. C. Eckardt and J. Reintjes, “Phase-matching limitation of high efficiency second harmonic generation,” IEEE J. Quantum Electron. 20, 1178 (1984).
[Crossref]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

Salimbeni, R.

G. Toci, D. McGraw, R. Pini, R. Salimbeni, and M. Vannini, “Time-dependent analysis of a parametric lens detected with a 100-fs Ti:sapphire laser,” Opt. Lett. 20, 1547 (1995).
[Crossref] [PubMed]

G. Toci, R. Pini, R. Salimbeni, S. Siano, and M. Vannini, “Optical modulators based on second order non-linear processes in non-stationary conditions,” Proceedings of the International Conference on Lasers, 1995 (STS, McLean, Va., 1996), p. 761.

G. Toci, R. Pini, R. Salimbeni, and M. Vannini, “Group velocity mismatch effects in ultrafast optical modulators based on cascaded second order nonlinearities,” in Ultrafast Processes in Spectroscopy, O. Svelto, S. De Silvestri, and G. Denardo, eds. (Plenum, New York, 1996).

G. Toci, M. Vannini, and R. Salimbeni, “Temporal dynamic of the non-linear mirror: an analytical description,” Opt. Commun. (to be published).

Saltiel, S.

I. Buchvarov, G. Christov, and S. Saltiel, “Transient behavior of frequency doubling mode-locker. Numerical analysis,” Opt. Commun. 107, 281 (1994).
[Crossref]

Schieck, R.

Segala, D.

Sheik-Bahae, M.

Shen, Y. R.

Y. R. Shen, The Principles of the Nonlinear Optics (Wiley, New York, 1984).

Siano, S.

G. Toci, R. Pini, R. Salimbeni, S. Siano, and M. Vannini, “Optical modulators based on second order non-linear processes in non-stationary conditions,” Proceedings of the International Conference on Lasers, 1995 (STS, McLean, Va., 1996), p. 761.

Sidick, E.

Smith, S. D.

Stankov, K. A.

K. A. Stankov, V. P. Tzolov, and M. G. Mirkov, “Compensation of group-velocity mismatch in the frequency-doubling modelocker,” Appl. Phys. B 54, 303 (1992).
[Crossref]

K. A. Stankov, V. P. Tzolov, and M. G. Mirkov, “Frequency-doubling mode locker: the influence of group-velocity mismatch,” Opt. Lett. 16, 1119 (1991).
[Crossref] [PubMed]

K. A. Stankov and J. Jethwa, “A new mode-locking technique using a nonlinear mirror,” Opt. Commun. 66, 41 (1988).
[Crossref]

Stegeman, G.

Stegeman, G. I.

Sunderheimer, M. L.

Sutherland, R. L.

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, New York, 1996).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Wetterling, Numerical Recipes (Cambridge University, Cambridge, England, 1989).

Toci, G.

G. Toci, D. McGraw, R. Pini, R. Salimbeni, and M. Vannini, “Time-dependent analysis of a parametric lens detected with a 100-fs Ti:sapphire laser,” Opt. Lett. 20, 1547 (1995).
[Crossref] [PubMed]

G. Toci, R. Pini, R. Salimbeni, S. Siano, and M. Vannini, “Optical modulators based on second order non-linear processes in non-stationary conditions,” Proceedings of the International Conference on Lasers, 1995 (STS, McLean, Va., 1996), p. 761.

G. Toci, R. Pini, R. Salimbeni, and M. Vannini, “Group velocity mismatch effects in ultrafast optical modulators based on cascaded second order nonlinearities,” in Ultrafast Processes in Spectroscopy, O. Svelto, S. De Silvestri, and G. Denardo, eds. (Plenum, New York, 1996).

G. Toci, M. Vannini, and R. Salimbeni, “Temporal dynamic of the non-linear mirror: an analytical description,” Opt. Commun. (to be published).

Torner, L.

Tzolov, V. P.

K. A. Stankov, V. P. Tzolov, and M. G. Mirkov, “Compensation of group-velocity mismatch in the frequency-doubling modelocker,” Appl. Phys. B 54, 303 (1992).
[Crossref]

K. A. Stankov, V. P. Tzolov, and M. G. Mirkov, “Frequency-doubling mode locker: the influence of group-velocity mismatch,” Opt. Lett. 16, 1119 (1991).
[Crossref] [PubMed]

Van Stryland, E. V.

Van Stryland, E. W.

Vannini, M.

G. Toci, D. McGraw, R. Pini, R. Salimbeni, and M. Vannini, “Time-dependent analysis of a parametric lens detected with a 100-fs Ti:sapphire laser,” Opt. Lett. 20, 1547 (1995).
[Crossref] [PubMed]

D. Pierrotet, B. Berman, M. Vannini, and D. McGraw, “Parametric lens,” Opt. Lett. 18, 263 (1993).
[Crossref]

G. Toci, R. Pini, R. Salimbeni, and M. Vannini, “Group velocity mismatch effects in ultrafast optical modulators based on cascaded second order nonlinearities,” in Ultrafast Processes in Spectroscopy, O. Svelto, S. De Silvestri, and G. Denardo, eds. (Plenum, New York, 1996).

G. Toci, R. Pini, R. Salimbeni, S. Siano, and M. Vannini, “Optical modulators based on second order non-linear processes in non-stationary conditions,” Proceedings of the International Conference on Lasers, 1995 (STS, McLean, Va., 1996), p. 761.

G. Toci, M. Vannini, and R. Salimbeni, “Temporal dynamic of the non-linear mirror: an analytical description,” Opt. Commun. (to be published).

Varhenzeele, H.

Velsko, S. P.

S. P. Velsko, M. Webb, L. Davis, and C. Huang, “Phase-matched harmonic generation in lithium triborate,” IEEE J. Quantum Electron. 27, 2182 (1991).
[Crossref]

Wearie, D.

Webb, M.

S. P. Velsko, M. Webb, L. Davis, and C. Huang, “Phase-matched harmonic generation in lithium triborate,” IEEE J. Quantum Electron. 27, 2182 (1991).
[Crossref]

Weber, M. J.

M. J. Weber, CRC Handbook of Laser Science and Technology, Vol. 3 (CRC Press, Boca Raton, Fla., 1986), p. 108.

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

Wetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Wetterling, Numerical Recipes (Cambridge University, Cambridge, England, 1989).

Wherret, B. S.

Zéboulon, A.

Appl. Opt. (1)

Appl. Phys. B (1)

K. A. Stankov, V. P. Tzolov, and M. G. Mirkov, “Compensation of group-velocity mismatch in the frequency-doubling modelocker,” Appl. Phys. B 54, 303 (1992).
[Crossref]

IEEE J. Quantum Electron. (6)

R. C. Eckardt and J. Reintjes, “Phase-matching limitation of high efficiency second harmonic generation,” IEEE J. Quantum Electron. 20, 1178 (1984).
[Crossref]

K. Kato, “Second harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron. QE-22, 1013 (1986).
[Crossref]

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P,BaB2O4,LiIO3,MgO:LiNbO3 and KTP measured by phase-matched second harmonic generation,” IEEE J. Quantum Electron. QE-26, 922 (1990).
[Crossref]

S. P. Velsko, M. Webb, L. Davis, and C. Huang, “Phase-matched harmonic generation in lithium triborate,” IEEE J. Quantum Electron. 27, 2182 (1991).
[Crossref]

G. C. Ghosh and G. C. Bhar, “Temperature dispersion in ADP, KDP, KD*P for nonlinear devices,” IEEE J. Quantum Electron. QE-18, 143 (1982).
[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref] [PubMed]

G. Cerullo, S. De Silvestri, A. Monguzzi, D. Segala, and V. Magni, “Self-starting mode locking of a cw Nd:YAG laser using cascaded second-order nonlinearities,” Opt. Lett. 20, 746 (1995).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref]

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[Crossref]

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Figures (9)

Fig. 1
Fig. 1

Phase-modulation coefficient ϕ2 [Eq. (10b)] for the single-pass configuration at the output of the nonlinear medium (ξ=1) as a function of the local time u1=t-L/v1 [see Eq. (2)] varying the steadiness parameter σ for the specified values of the phase-mismatch parameter δ: (a), δ=2π; (b), δ=4π; (c), δ =6π. Mesh spacing is Δσ=0.2, Δu1/τ=0.1. Gray-scale levels in the three-dimensional mesh surfaces and in the uppermost contour maps correspond to the vertical-axis main-division values.

Fig. 2
Fig. 2

Amplitude-modulation coefficient α2 [Eq. (10a)] for the single-pass configuration at the output of the nonlinear medium (ξ=1) as a function of the local time u1=t-L/v1 [see Eq. (2)], varying the steadiness parameter σ for the specified values of the phase-mismatch parameter δ: (a), δ=2π; (b), δ=4π; (c), δ=6π. Mesh spacing is Δσ=0.2, Δu1/τ=0.1. Gray-scale levels in the three-dimensional mesh surfaces and in the uppermost contour maps correspond to the vertical-axis main-division values.

Fig. 3
Fig. 3

Fourth-order coefficient ϕ4 of the phase modulation [Eq. (10b)] for the single-pass configuration at the output of the nonlinear medium (ξ=1) as a function of the local time u1=t -L/v1 [see Eq. (2)], varying the steadiness parameter σ for the specified values of the phase-mismatch parameter δ: (a), δ =2π; (b), δ=4π; (c), δ=6π. Mesh spacing is Δσ=0.2, Δu1/τ=0.1. Gray-scale levels in the three-dimensional mesh surfaces and in the uppermost contour maps correspond to the vertical-axis main-division values.

Fig. 4
Fig. 4

Fourth-order coefficient α4 of the phase modulation [Eq. (10a)] for the single-pass configuration at the output of the nonlinear medium (ξ=1) as a function of the local time u1=t -L/v1 [see Eq. (2)], varying the steadiness parameter σ for the specified values of the phase-mismatch parameter δ: (a), δ =2π; (b), δ=4π; (c), δ=6π. Mesh spacing is Δσ=0.2, Δu1/τ=0.1. Gray-scale levels in the three-dimensional mesh surfaces and in the uppermost contour maps correspond to the vertical-axis main-division values.

Fig. 5
Fig. 5

Ideal experimental layout describing the second-order cascaded effect in the double-pass configuration. Open and solid black shapes represent the ω1 and ω2 pulse envelopes, respectively; Δt is the additional time delay between the F and SH pulses before the second pass.

Fig. 6
Fig. 6

Phase-modulation coefficient ϕ2 [Eqs. (19b) and (20b)] for the double-pass configuration at the output of the nonlinear medium (ξ=1) as a function of the local time u1=t-L/v1 [see Eq. (2)], varying the steadiness parameter σ for the specified values of the phase-mismatch parameter δ: (a), δ=2π; (b), δ =4π; (c), δ=6π, with r1=r2=1, Θ=0, Δt=(1/v2 -1/v1)L. Mesh spacing is Δσ=0.2, Δu1/τ=0.1. Gray-scale levels in the three-dimensional mesh surfaces and in the uppermost contour maps correspond to the vertical-axis main-division values.

Fig. 7
Fig. 7

Amplitude-modulation coefficient α2 [Eqs. (19a) and (20a)] for the double-pass configuration at the output of the nonlinear medium (ξ=1) as a function of the local time u1=t -L/v1 [see Eqs. (2)], varying the steadiness parameter σ for the specified values of the phase-mismatch parameter δ: (a), δ =2π; (b), δ=4π; (c), δ=6π, with r1=r2=1, Θ=0, Δt =(1/v2-1/v1)L. Mesh spacing is Δσ=0.2, Δu1/τ=0.1. Gray-scale levels in the three-dimensional mesh surfaces and in the uppermost contour maps correspond to the vertical-axis main-division values.

Fig. 8
Fig. 8

Time dependence of the phase-modulation coefficient ϕ2 and normalized phase modulation ϕnum (calculated from the numerical integration) evaluated at σ=0.6, (a) δ=2π, (b) δ =4π, (c) δ=6π, Θ=0, Δt=2στ, r1=r2=1 for increasing values of η0. Lower frame: residuals (ϕnum-ϕ2).

Fig. 9
Fig. 9

Time dependence of the phase-modulation coefficient α2 and normalized phase modulation αnum (calculated from the numerical integration) evaluated at σ=0.6, (a) δ=2π, (b) δ =4π, (c) δ=6π, Θ=0, Δt=2στ, r1=r2=1 for increasing values of η0. Lower frame: residuals (αnum-α2).

Tables (1)

Tables Icon

Table 1 Optical Parameters and Figure of Merit M of Several Nonlinear Materials for the Type I Second-Order Cascaded Process at 1064 nm

Equations (53)

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z+1ν1 tE1=i 4πω12k1c2 cos2 β1 χeff(2)(ω1)E1*E2 ×exp[-i(2k1-k2)z],
z+1ν2 tE2=i 8πω12k2c2 cos2 β2 χeff(2)(ω2)E12 ×exp[i(2k1-k2)z],
ρ1=E1/|E1|pk,
ρ2=[(k2 cos2 β2)/(2k1 cos2 β1)]1/2[E2/|E1|pk],fieldamplitudes,
η0=[(4πω12)/(c2 cos β1 cos β2)]×[2/k1k2]1/2χeff(2)L|E1|pk,couplingcoefficient,
ξ=z/L,normalizedcoordinate,
δ=(2k1-k2)L=ΔkL,phase-mismatchparameter,
u1,2=t-ξL/ν1,2,localtime,
ξ ρ1(ξ, u1)=iη0ρ1*(ξ, u1)ρ2(ξ, u1)exp[-iδξ],
ξ ρ2(ξ, u2)=iη0ρ12(ξ, u2)exp[iδξ],
ρi(ξ, t)=n=0η0nρi,n(ξ, t)(i=1, 2),
ρ1,n(ξ, u1)=i0ξ i+j=n-1ρ1i*(x, u1)ρ2,j(x, u1)×exp(-iδx)dx(i, j0),
ρ2,n(ξ, u2)=i0ξ i+j=n-1ρ1i(x, u2)ρ1,j(x, u2)×exp(iδx)dx,(i, j0),
|ρ2(ξ)|=|ρ2,1(ξ)η0+ρ2,3(ξ)η03+ρ2,5(ξ)η05|,
δ=nπ1-4η0nπ2+4η0nπ4+352η0nπ6-16(11n2π2-362)η0nπ8-8(133n2π2-3728)η0nπ10,
σ=1v2-1v1 L2τ,
ρ1(u1)=ρ1,0(u1)exp[-A(u1)+iΦ(u1)].
A(u1)=α2(u1)η02+α4(u1)η04=-η02 Re[ρ1,2(u1)]ρ1,0(u1)+η04{Im[ρ1,2(u1)]2-Re[ρ1,2(u1)]2}2ρ1,02(u1)+Re[ρ1,4(u1)]ρ1,0(u1),
Φ(u1)=ϕ2(u1)η02+ϕ4(u1)η04=η02 Im[ρ1,2(u1)]ρ1,0(u1)+η04-{Re[ρ1,2(u1)]Im[ρ1,2(u1)]}ρ1,02(u1)+Im[ρ1,4(u1)]ρ1,0(u1)
ρ1,n(-δ)=ρ1,n*(δ),
ρ2,n(-δ)=-ρ2,n*(δ).
ρ1,2(u1)/ρ1,0(u1)=-0ξρ1,0*(u1)0xρ1,02(u1+2στx)exp(iδx)dx×exp(-iδx)dx/ρ1,0(u1).
γ1(u1, σ, δ)=-0ξρ2,1(u1-2στx)exp(-iδx)dx,
ε1,4(η0)=η04|ρ1,4(u1, 1)|pk/η02|ρ1,2(u1, 1)|pk
ε1,4<0.24η02(δ=2π),
ε1,4<0.093η02(δ=4π),
ε1,4<0.056η02(δ=6π).
ρ1,in(ξ=0, t)=r1 exp(iθ1)[ρ1,0(ξ=1, t)+η02ρ1,2(ξ=1, t)+],
ρ2,in(ξ=0, t)=r2 exp[i(δ+θ2)][η0ρ2,1(ξ=1, t+Δt)+η03ρ2,3(ξ=1, t+Δt)+],
ρ1(u1, ξ)=r1 exp(iθ1)ρ1,0(u1)+iη02r1r2×exp[i(θ2+δ-θ1)]0ξρ1,0*(u1, x)×ρ2,1(u1-2στx+Δt, 1)exp(-iδx)dx+iη02r13 exp(iθ1)0ξρ1,0*(u1, x)×ρ2,1(u1-2στx, x)exp(-iδx)dx+η02r1 exp(iθ1)ρ1,2(u1, 1).
ρ1(u1, ξ=1)=r1 exp(iθ1)ρ1,0(u1, ξ=0)×exp[-A(u1)+iΦ(u1)]=r1 exp(iθ1)×ρ1,0(u1, ξ=0)×exp{[-α2(u1)+iϕ2(u1)]η02},
α2(u1)=-1η02 Reρ1(u1, ξ=1)r1 exp(iθ1)ρ1,0(u1, ξ=0)-1,
ϕ2(u1)=1η02 Imρ1(u1, ξ=1)r1 exp(iθ1)ρ1,0(u1, ξ=0),
α2(u1)=-Re{r2 exp[i(θ2+δ-2θ1)]×γ2(u1+Δt, σ, δ)+(1+r12)γ1(u1, σ, δ)},
ϕ2(u1)=Im{r2 exp[i(θ2+δ-2θ1)]γ2(u1+Δt, σ, δ)+(1+r12)γ1(u1, σ, δ)},
γ2(u1+Δt, σ, δ)=i0ξρ2,1(u1-2στx+Δt, 1)×exp(-iδx)dx,
ρ2(u2, ξ)=r2 exp[i(δ+θ2)]η0ρ2,1(u2+Δt, 1)+ir12 exp(i2θ1)η00ξρ1,02×(u1+2στx)exp(iδx)dx,
σpk(δ)0.5|δ|/π,
Φpk0.7η02π/|δ|.
|Δk|=π(1/v2-1/v1)/τ.
Φpk=0.74ω12c2 cos2 β1 cos2 β2×(χeff(2))2n1n2(1/v2-1/v1)|E1|pk2τL.
M=(χeff(2))2n1n2(1/v2-1/v1),
z+1v1 tE1b=i 4πω12k1c2 cos2 β1 χeff(2)(ω1)E1b*E2b ×exp[i(2k1-k2)L]×exp[-i(2k1-k2)z],
z+1v2 tE2b=i 8πω12k2c2 cos2 β2 χeff(2)(ω2)×E1b2 exp[-i(2k1-k2)L]×exp[i(2k1-k2)z].
ρ1,in(t)=sech(t/τ),
ρ1,0(u1)=sech(u1/τ),
ρ1,2(u1, ξ)=-sechu1τξ2σ tanhu1τ+14σ2 lncosh(u1/τ-2σξ)cosh(u1/τ),
ρ2,1(u2, ξ)=i2σ tanhu2τ+2σξ-tanhu2τ.
ρ1(u1, ξ=1)
=r1 exp(iθ1)sechu1τ+r2 exp(iΘ) η024σ2 sechu1τln×cosh(u1+Δt)τ-2σ2cosh(u1+Δt)τcosh(u1+Δt)τ-4σ+(1+r12)η02ρ12(u1, ξ=1).
ρ2(u2,ξ=1)=r2 exp[i(δ+θ2)]η0ρ2,1(u2+Δt, ξ=1)+ir12 exp(i2θ1)η0ρ2,1(u2,ξ=1),
ϕnum=arcsin[Im(ρ1)/|ρ1|]/η02,
αnum=-ln(|ρ1|/|ρ10|)/η02.

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