Abstract

A technique for numerically simulating second-order nonlinear interactions of light waves modulated spatially and temporally in both amplitude and phase in a uniaxial medium with arbitrary polarization and propagation directions is presented. A three-dimensional grid technique that automatically adapts grid parameters to the evolving sampled amplitudes by means of an analytical fit function is described. By means of spatiotemporal split-step fast-Fourier-transform propagation, diffraction and group-velocity dispersion can be included. We employ this technique for femtosecond noncollinear white-light-seeded parametric amplification in experimental designs presented earlier [Opt. Lett. 22, 1494 (1997); Opt. Lett. 23, 1283 (1998)]. The dependence of phase mismatch on signal wavelength provides a quantitative measure of the achromaticity of phase matching. Our results indicate that the pump pulse characteristics and not phase mismatching limit the amplification and the bandwidth of the parametric amplifier.

© 1999 Optical Society of America

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References

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  1. P. Di Trapani, A. Andreoni, G. P. Banfi, C. Solcia, R. Danielius, A. Piskarskas, P. Foggi, M. Monguzzi, and C. Sozzi, “Group-velocity self-matching of femtosecond pulses in noncollinear parametric generation,” Phys. Rev. A 51, 3164–3168 (1995).
    [CrossRef] [PubMed]
  2. P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, and 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]
  3. P. Matousek, A. W. Parker, P. F. Taday, W. T. Toner, and M. Towrie, “Two independently tunable and synchronised femtosecond pulses generated in the visible at the repetition rate 40 kHz using optical parametric amplifiers,” Opt. Commun. 127, 307–312 (1996).
    [CrossRef]
  4. S. R. Greenfield and M. R. Wasielewski, “Optical parametric amplification of femtosecond pulses tunable from the blue to the infrared with microjoule energies,” Appl. Opt. 34, 2688–2691 (1995).
    [CrossRef] [PubMed]
  5. F. Hache, T. J. Driscoll, M. Cavallari, and G. M. Gale, “Measurement of ultrashort pulse durations by interferometric autocorrelation: influence of various parameters,” Appl. Opt. 35, 3230–3236 (1996).
    [CrossRef] [PubMed]
  6. T. Wilhelm, J. Piel, and E. Riedle, “Sub-20-fs pulses tunable across the visible from a blue-pumped single-pass noncollinear parametric converter,” Opt. Lett. 22, 1494–1496 (1997).
    [CrossRef]
  7. G. Cerullo, M. Nisoli, S. Stagira, and S. De Silvestri, “Sub-8-fs pulses from an ultrabroadband optical parametric amplifier in the visible,” Opt. Lett. 23, 1283–1285 (1998).
    [CrossRef]
  8. G. Cerullo, M. Nisoli, and S. De Silvestri, “Generation of 11 fs pulses tunable across the visible by optical parametric amplification,” Appl. Phys. Lett. 71, 3616–3618 (1997).
    [CrossRef]
  9. A. Shirakawa and T. Kobayashi, “Pulse-front-matched optical parametric amplification for sub-10-fs pulse generation tunable in the visible and near infrared,” Opt. Lett. 23, 1292–1294 (1998).
    [CrossRef]
  10. H. J. Bakker, P. C. M. Planken, and H. G. Muller, “Numeri-cal calculation of optical frequency-conversion processes: a new approach,” J. Opt. Soc. Am. B 6, 1665–1672 (1989).
    [CrossRef]
  11. T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
    [CrossRef]
  12. E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968).
    [CrossRef]
  13. E. Lalor, “The angular spectrum representation of electromagnetic fields in crystals,” J. Math. Phys. 13, 437–449 (1972).
    [CrossRef]
  14. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
  15. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  16. M. Trippenbach and Y. B. Band, “Propagation of light pulses in nonisotropic media,” J. Opt. Soc. Am. B 13, 1403–1411 (1996).
    [CrossRef]
  17. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).
  18. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, 2nd ed. (Springer-Verlag, Berlin, 1997).
  19. R. A. Fisher and W. K. Bischel, “Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses,” J. Appl. Phys. 46, 4921–4934 (1975).
    [CrossRef]
  20. J. Manz and L. Wöste, eds., Femtosecond Chemistry (VCH, Weinheim, Germany, 1995), Vols. 1 and 2.
  21. Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
    [CrossRef]
  22. J. Lindener-Roenneke, “Pump-Tast-Spektroskopie mit durchstimmbaren Femtosekundenpulsen: Realisierung eines experimentellen Aufbas und Untersuchungen zur Dynamik von Cr(CO)6 und Cr(CO)6 ⋅ (CH3OH)nHeteroclustern,” Ph.D. dissertation (Universität zu Köln, Cologne, Germany, 1998).
  23. F. Hache, M. Cavallari, and G. M. Gale, “Ultrafast visible optical parametric oscillators: a route to tunable sub-10-femtosecond pulses?” in Ultrafast Phenomena X, Vol. 62 of Springer Series in Chemical Physics, P. F. Barbara, J. G. Fujimoto, W. H. Knox, and W. Zinth, eds. (Springer-Verlag, Berlin, 1996), pp. 33–35.
  24. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

1998 (2)

1997 (3)

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

G. Cerullo, M. Nisoli, and S. De Silvestri, “Generation of 11 fs pulses tunable across the visible by optical parametric amplification,” Appl. Phys. Lett. 71, 3616–3618 (1997).
[CrossRef]

T. Wilhelm, J. Piel, and E. Riedle, “Sub-20-fs pulses tunable across the visible from a blue-pumped single-pass noncollinear parametric converter,” Opt. Lett. 22, 1494–1496 (1997).
[CrossRef]

1996 (3)

F. Hache, T. J. Driscoll, M. Cavallari, and G. M. Gale, “Measurement of ultrashort pulse durations by interferometric autocorrelation: influence of various parameters,” Appl. Opt. 35, 3230–3236 (1996).
[CrossRef] [PubMed]

P. Matousek, A. W. Parker, P. F. Taday, W. T. Toner, and M. Towrie, “Two independently tunable and synchronised femtosecond pulses generated in the visible at the repetition rate 40 kHz using optical parametric amplifiers,” Opt. Commun. 127, 307–312 (1996).
[CrossRef]

M. Trippenbach and Y. B. Band, “Propagation of light pulses in nonisotropic media,” J. Opt. Soc. Am. B 13, 1403–1411 (1996).
[CrossRef]

1995 (4)

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

S. R. Greenfield and M. R. Wasielewski, “Optical parametric amplification of femtosecond pulses tunable from the blue to the infrared with microjoule energies,” Appl. Opt. 34, 2688–2691 (1995).
[CrossRef] [PubMed]

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

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, and 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]

1989 (1)

1975 (1)

R. A. Fisher and W. K. Bischel, “Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses,” J. Appl. Phys. 46, 4921–4934 (1975).
[CrossRef]

1972 (1)

E. Lalor, “The angular spectrum representation of electromagnetic fields in crystals,” J. Math. Phys. 13, 437–449 (1972).
[CrossRef]

1968 (1)

Andreoni, A.

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

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, and 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]

Bakker, H. J.

Band, Y. B.

Banfi, G. P.

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

Bischel, W. K.

R. A. Fisher and W. K. Bischel, “Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses,” J. Appl. Phys. 46, 4921–4934 (1975).
[CrossRef]

Cavallari, M.

Cerullo, G.

G. Cerullo, M. Nisoli, S. Stagira, and S. De Silvestri, “Sub-8-fs pulses from an ultrabroadband optical parametric amplifier in the visible,” Opt. Lett. 23, 1283–1285 (1998).
[CrossRef]

G. Cerullo, M. Nisoli, and S. De Silvestri, “Generation of 11 fs pulses tunable across the visible by optical parametric amplification,” Appl. Phys. Lett. 71, 3616–3618 (1997).
[CrossRef]

Danielius, R.

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

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, and 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]

De Silvestri, S.

G. Cerullo, M. Nisoli, S. Stagira, and S. De Silvestri, “Sub-8-fs pulses from an ultrabroadband optical parametric amplifier in the visible,” Opt. Lett. 23, 1283–1285 (1998).
[CrossRef]

G. Cerullo, M. Nisoli, and S. De Silvestri, “Generation of 11 fs pulses tunable across the visible by optical parametric amplification,” Appl. Phys. Lett. 71, 3616–3618 (1997).
[CrossRef]

Di Trapani, P.

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

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, and 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]

Driscoll, T. J.

Fisher, R. A.

R. A. Fisher and W. K. Bischel, “Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses,” J. Appl. Phys. 46, 4921–4934 (1975).
[CrossRef]

Foggi, P.

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

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, and 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]

Gale, G. M.

Greenfield, S. R.

Hache, F.

Kobayashi, T.

Lalor, E.

E. Lalor, “The angular spectrum representation of electromagnetic fields in crystals,” J. Math. Phys. 13, 437–449 (1972).
[CrossRef]

E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968).
[CrossRef]

Lin, Q.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Matousek, P.

P. Matousek, A. W. Parker, P. F. Taday, W. T. Toner, and M. Towrie, “Two independently tunable and synchronised femtosecond pulses generated in the visible at the repetition rate 40 kHz using optical parametric amplifiers,” Opt. Commun. 127, 307–312 (1996).
[CrossRef]

Monguzzi, M.

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

Muller, H. G.

Nishikawa, T.

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

Nisoli, M.

G. Cerullo, M. Nisoli, S. Stagira, and S. De Silvestri, “Sub-8-fs pulses from an ultrabroadband optical parametric amplifier in the visible,” Opt. Lett. 23, 1283–1285 (1998).
[CrossRef]

G. Cerullo, M. Nisoli, and S. De Silvestri, “Generation of 11 fs pulses tunable across the visible by optical parametric amplification,” Appl. Phys. Lett. 71, 3616–3618 (1997).
[CrossRef]

Parker, A. W.

P. Matousek, A. W. Parker, P. F. Taday, W. T. Toner, and M. Towrie, “Two independently tunable and synchronised femtosecond pulses generated in the visible at the repetition rate 40 kHz using optical parametric amplifiers,” Opt. Commun. 127, 307–312 (1996).
[CrossRef]

Piel, J.

Piskarskas, A.

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

Planken, P. C. M.

Riedle, E.

Shirakawa, A.

Solcia, C.

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

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, and 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]

Sozzi, C.

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

Stagira, S.

Taday, P. F.

P. Matousek, A. W. Parker, P. F. Taday, W. T. Toner, and M. Towrie, “Two independently tunable and synchronised femtosecond pulses generated in the visible at the repetition rate 40 kHz using optical parametric amplifiers,” Opt. Commun. 127, 307–312 (1996).
[CrossRef]

Toner, W. T.

P. Matousek, A. W. Parker, P. F. Taday, W. T. Toner, and M. Towrie, “Two independently tunable and synchronised femtosecond pulses generated in the visible at the repetition rate 40 kHz using optical parametric amplifiers,” Opt. Commun. 127, 307–312 (1996).
[CrossRef]

Towrie, M.

P. Matousek, A. W. Parker, P. F. Taday, W. T. Toner, and M. Towrie, “Two independently tunable and synchronised femtosecond pulses generated in the visible at the repetition rate 40 kHz using optical parametric amplifiers,” Opt. Commun. 127, 307–312 (1996).
[CrossRef]

Trippenbach, M.

Uesugi, N.

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

Wang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Wasielewski, M. R.

Wilhelm, T.

Xu, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Zhang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

G. Cerullo, M. Nisoli, and S. De Silvestri, “Generation of 11 fs pulses tunable across the visible by optical parametric amplification,” Appl. Phys. Lett. 71, 3616–3618 (1997).
[CrossRef]

IEEE J. Quantum Electron. (1)

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

J. Appl. Phys. (2)

R. A. Fisher and W. K. Bischel, “Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses,” J. Appl. Phys. 46, 4921–4934 (1975).
[CrossRef]

T. Nishikawa and N. Uesugi, “Effects of walk-off and group velocity difference on the optical parametric generation in KTiOPO4 crystals,” J. Appl. Phys. 77, 4941–4947 (1995).
[CrossRef]

J. Math. Phys. (1)

E. Lalor, “The angular spectrum representation of electromagnetic fields in crystals,” J. Math. Phys. 13, 437–449 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

P. Di Trapani, A. Andreoni, P. Foggi, C. Solcia, R. Danielius, and 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. Matousek, A. W. Parker, P. F. Taday, W. T. Toner, and M. Towrie, “Two independently tunable and synchronised femtosecond pulses generated in the visible at the repetition rate 40 kHz using optical parametric amplifiers,” Opt. Commun. 127, 307–312 (1996).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

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

Other (8)

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, 2nd ed. (Springer-Verlag, Berlin, 1997).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

J. Manz and L. Wöste, eds., Femtosecond Chemistry (VCH, Weinheim, Germany, 1995), Vols. 1 and 2.

J. Lindener-Roenneke, “Pump-Tast-Spektroskopie mit durchstimmbaren Femtosekundenpulsen: Realisierung eines experimentellen Aufbas und Untersuchungen zur Dynamik von Cr(CO)6 und Cr(CO)6 ⋅ (CH3OH)nHeteroclustern,” Ph.D. dissertation (Universität zu Köln, Cologne, Germany, 1998).

F. Hache, M. Cavallari, and G. M. Gale, “Ultrafast visible optical parametric oscillators: a route to tunable sub-10-femtosecond pulses?” in Ultrafast Phenomena X, Vol. 62 of Springer Series in Chemical Physics, P. F. Barbara, J. G. Fujimoto, W. H. Knox, and W. Zinth, eds. (Springer-Verlag, Berlin, 1996), pp. 33–35.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

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Figures (10)

Fig. 1
Fig. 1

Model of a time-dependent field: The wave vector moves along a curve parameterized by ω. The vector g(/ω)k|Ω describes the dispersion of k(ω) to linear order near the central frequency Ω. The curve is drawn with vanishing curvature at k(Ω), which corresponds to h=0 (see text).

Fig. 2
Fig. 2

Relationship of coordinate frames: The cube with edges along the x, y, and z directions (laboratory frame) represents the BBO slab; axes, a, b, and c are the crystal axes (c is the optical axis). The transformation between these two frames is described in terms of Eulerian angles: Rotation of the crystal frame about c through angle ϕ takes a into intermediate axis x and b into y. Rotation of the xyc frame about y through angle θ (in the example shown θ is negative) takes x into x and c into z.

Fig. 3
Fig. 3

Modeled white-light continuum. Time-dependent complex envelope (real and imaginary parts) and intensity.

Fig. 4
Fig. 4

Modeled white-light continuum. Instantaneous frequency, ω¯(η).

Fig. 5
Fig. 5

Kinematic interpretation of the achromatic phase-matching condition g1=g2: It requires the fronts of the signal and the idler fields (depicted as shaded and white bars, respectively) to be parallel and their group-velocity components that are normal to their common front to be equal. In general, the components v1 and v2 parallel to the front will not be equal. Signal and idler pulses subsequently move away from each other and lose their initial overlap.

Fig. 6
Fig. 6

Illustration of (a) wave and (b) beam directions and pulse fronts: In (a) the axis tick marks are given in inverse micrometers, whereas in (b) they are given in micrometers per femtosecond.

Fig. 7
Fig. 7

(a) ΔkLeff and (b) expected relative parametric gain Gsinc2ΔkLeff versus signal frequency ω1. The calculation of phase mismatch Δk assumed that a monochromatic wave of frequency ω1 was parametrically amplified. Leff equals the crystal thickness of 1 mm. The signal center frequency Ω1 on which the first-order phase matching is based enters the calculation as a parameter. The calculation was carried out for five values of Ω1. The corresponding wavelengths are indicated.

Fig. 8
Fig. 8

(a) Signal and (b) idler spectra during parametric amplification. Each of the 40 curves corresponds to one simulation step. The scale is logarithmic.

Fig. 9
Fig. 9

Complex envelope of signal field (a) before and (b) after compression with an element of negative linear GVD. Intensities are shown in (c). The pulse in (b) contains residual phase structure that derives from higher-order dispersion terms.

Fig. 10
Fig. 10

Influence of pump pulse duration and shape on bandwidth and amplification.

Tables (6)

Tables Icon

Table 1 External and Internal Angles of Incidence and Refractive Indices

Tables Icon

Table 2 Grid Extensions and Number of Grid Points for Each Coordinatea

Tables Icon

Table 3 Investigation of the Effects of Various Factors on the Parametric Amplification by Means of Simulationa

Tables Icon

Table 4 GVM between Signal and Pump Field, Effective Interaction Length Leff, Group Delay β, Chirp Parameter b, and Coupling Coefficient χeff(2) for Various Wavelengthsa

Tables Icon

Table 5 Signal Bandwidths Estimated (Δω˜) and Obtained from Simulation (ΔωFWHM) for Various Signal Wavelengths

Tables Icon

Table 6 Simulation Results: Amplification Factors and Autocorrelation Widths Before (τac0) and After (τacmin) Optimal Compression with a Simulated Element of Linear Negative GVD βopt for Various Signal Wavelengthsa

Equations (49)

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

2E-(·E)-1c22t21+χ¯¯(1)E=10c22t2PNL,
Ej(r, t)=1/2jE(r, t)exp(iKj·r-iΩjt)+c.c.,
kz(o)=kz(o)(kx, ky, ω)=o2(ω) ω2c2-kx2-ky21/2
kz(e)
=kz(e)(kx, ky, ω)=-sin θ cos θ E-oo sin2 θ+E cos2 θkx+[o(Ek02-ky2)(o sin2 θ+E cos2 θ)-oEkx2]1/2o sin2 θ+E cos2 θ
E(kx-Kx, ky-Ky, ω-Ω)=exp[ikz(kx, ky, ω)z]E(kx-Kx, ky-Ky, ω-Ω)
E(r, t)=116π3dkxdkydω×exp(ikxx+ikyy-iωt)×E(kx-Kx, ky-Ky, ω-Ω)+c.c.
k(ω)=K+(ω-Ω)g+(ω-Ω)22h+O[(ω-Ω)3],
Kk(Ω),
gωkΩ,
h2ω2kΩ,
E(r, t)=1/2j exp(iK·r-iΩt)dω×exp[-i(ω-Ω)(t-g·r)]E(ω-Ωj)+c.c.
[kx(ω), ky(ω)]=[Kx+(ω-Ω)gx, Ky+(ω-Ω)gy].
(x, y, η)jklT=[xmin, ymin, ηmin(xmin, ymin)]T+jTη+kTx+lTy,
Tηδη(0, 0, 1)T,
Txδx(1, 0, ux)T,
Tyδy(0, 1, uy)T,
v=cΩ(/ω)n|Ω+n(Ω).
l1·[E1(r, t-g1·r)+g1E˙1(r, t-g1·r)]
=i Ω124c2χeff(2)E2*(r, t-g2·r)E3(r, t-g3·r)×exp(iΔk·r),
l2·[E2(r, t-g2·r)+g2E˙2(r, t-g2·r)]
=i Ω224c2χeff(2)E1*(r, t-g1·r)E3(r, t-g3·r)×exp(iΔk·r),
l3·[E3(r, t-g3·r)+g3E˙3(r, t-g3·r)]
=i Ω324c2χeff(2)E1(r, t-g1·r)E2(r, t-g2·r)×exp(-iΔk·r),
γ(z)r0+1llzll(z-z0),
ζ(z)η0+gl[γ(z)-r0].
ddzE1[γ (z), ζ(z)]|z=z0
=i Ω124l1zc2χeff(2)E2*[γ (z), ζ(z)-g21·γ (z)]E3[γ (z), ζ(z)-g31·γ (z)]exp[iΔk·γ (z)],
ddzE2[γ (z), ζ(z)]|z=z0
=i Ω224l2zc2χeff(2)E1*[γ (z), ζ(z)-g12·γ (z)]E3[γ (z), ζ(z)-g32·γ (z)]exp[iΔk·γ (z)],
ddzE3[γ (z), ζ(z)]|z=z0
=i Ω324l3zc2χeff(2)E1[γ (z), ζ(z)-g13·γ (z)]E2[γ (z), ζ(z)-g23·γ (z)]exp[-iΔk·γ (z)],
Efit(Δx, Δy, Δη)=E0fit exp[(Δx, Δy, Δη)·Γ¯¯(Δx, Δy, Δη)T+ϱ·(Δx, Δy, Δη)T],
ux=-Γηx/Γηη,uy=-Γηy/Γηη.
x0=X-(Nx-1)/2δx,
y0=Y-(Ny-1)/2δy,
η0(0, 0)=H-(Nη-1)/2δη-uxX-uyY.
Econt(η)=12π-Δω/2Δω/2dω expi β2ω2-iηω.
ϖinst(η)=-Im E˙(η)E(η)
bβ7.687Δω-4+β2,
K1+K2=K3.
k1(ω1)=K1+(ω1-Ω1)g1+O[(ω1-Ω1)2],
k2(ω2)=K2+(ω2-Ω2)g2+O[(ω-Ω2)2].
k2(ω1)=K3-K1-(ω1-Ω1)g2+O[(ω-Ω2)2].
0=ω1(k1+k2)Ω1=ω1[K3+(ω1-Ω1)g12]Ω1=g1-g2,
g1=g2.
v1=cos Ψ12v2,
Δk(ω1)n(Ω1+Ω2-ω1) Ω1+Ω2-ω1c-|K3-K1-(ω1-Ω1)g1|0.
Δω˜=b(Δη+τ3).

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