Abstract

Pulse compression through noncollinear sum-frequency generation in β-barium borate crystals is investigated. With dispersion to all orders taken into account, numerical calculations, carried out in the frequency domain, were performed for input pulse durations ranging from 10 to 50 fs. Fundamental pulses of a 10-fs duration at 800 nm were compressed down to 2.5 fs while a 2.5-fs second-harmonic signal was generated.

© 2001 Optical Society of America

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References

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  1. Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity-dispersion,” Phys. Rev. A 41, 5645–5649 (1990).
    [CrossRef] [PubMed]
  2. A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991).
    [CrossRef]
  3. Y. Wang and B. L. Davies, “Frequency-doubling pulse compressor for picosecond high power neodymium laser pulses,” Opt. Lett. 17, 1459–1461 (1992).
    [CrossRef]
  4. P. Heinz, A. Laubereau, A. Dubietis, and A. Piskarskas, “Fiberless two-step parametric compression of sub-picosecond laser pulses,” Lith. Phys. Rev. 33, 314–317 (1993).
  5. A. Umbrasas, J. C. Diels, G. Valiulis, J. Jacob, and A. Piskarskas, “Generation of femtosecond pulses through second harmonic compression of the output of a Nd:YAG laser,” Opt. Lett. 20, 2228–2230 (1995).
    [CrossRef]
  6. J. Biegert, V. Kubecek, and J.-C. Diels, “Pulse compression: Type II second harmonic pulse compression,” in Encyclopedia of Electrical and Electronics Engineering (#17), J. G. Webster, ed., (IEEE, New York, 1998).
  7. M. Nisoli, S. D. Silvestri, G. Valiulius, and A. Varanavicius, “Fivefold femtosecond pulse compression by sum frequency generation,” Appl. Phys. Lett. 68, 3540–3542 (1996).
    [CrossRef]
  8. O. E. Martinez, “Matrix formalism for dispersive cavities,” IEEE J. Quantum Electron. QE-25, 296–300 (1989).
    [CrossRef]
  9. A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. 22, 1071–1073 (1997).
    [CrossRef] [PubMed]
  10. R. Danielius, A. Piskarskas, P. DiTrapani, A. Andreoni, C. Solcia, and 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]
  11. T. R. Zhang, H. R. Choo, and M. C. Downer, “Phase and group-velocity matching for SHG of femtosecond pulses,” Appl. Opt. 29, 3927–3933 (1990).
    [CrossRef] [PubMed]
  12. T. J. Driscoll, G. M. Gale, and F. Hache, “Ti:sapphire 2nd-harmonic-pumped visible range femtosecond optical parametric oscillator,” Opt. Commun. 110, 638–644 (1994).
    [CrossRef]
  13. G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. 20, 1562–1564 (1995).
    [CrossRef] [PubMed]
  14. G. Cerullo, M. Nisoli, and S. DeSilvestri, “Generation of 11 fs pulses tunable across the visible by optical parametric amplification,” Appl. Phys. Lett. 71, 3616–3618 (1997).
    [CrossRef]
  15. A. Shirakawa, I. Sakane, 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]
  16. G. Cerullo, M. Nisoli, S. Stagira, and S. DeSilvestri, “Sub-8-fs pulses from an ultrabroadband optical parametric amplifier in the visible,” Opt. Lett. 23, 1283–1285 (1998).
    [CrossRef]
  17. A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268–2270 (1999).
    [CrossRef]
  18. R. Danielius, A. Dubietis, and A. Piskarskas, “Femtosecond high-contrast pulses from a parametric generator pumped by the self-compressed second harmonic of a Nd:glass laser,” Opt. Lett. 20, 2225–2227 (1995).
    [CrossRef] [PubMed]
  19. D. A. Guk and V. G. Dmitirev, “Some characteristics of second harmonic generation under conditions of strong energy exchange between interacting waves,” Kvant. Elektron. (Moscow) 18, 106–110 (1990).
  20. J. Biegert, V. Kubecek, and J.-C. Diels, “Second harmonic pulse compression,” in Ultrafast Phenomena XI (Springer-Verlag, New York, 1998), pp. 84–86.
  21. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, Boston, 1995).
  22. H. Tan, G. P. Banfi, and A. Tomaselli, “Optical frequency mixing through cascaded second-order processes in β-barium borate,” Appl. Phys. Lett. 63, 2472–2474 (1993).
    [CrossRef]
  23. A. Umbrasas, J. C. Diels, J. Jacob, and A. Piskarskas, “Parametric oscillation and compression in KTP crystals,” Opt. Lett. 19, 1753–1755 (1994).
    [CrossRef] [PubMed]

1999 (1)

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268–2270 (1999).
[CrossRef]

1998 (2)

1997 (2)

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

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. 22, 1071–1073 (1997).
[CrossRef] [PubMed]

1996 (2)

R. Danielius, A. Piskarskas, P. DiTrapani, A. Andreoni, C. Solcia, and 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]

M. Nisoli, S. D. Silvestri, G. Valiulius, and A. Varanavicius, “Fivefold femtosecond pulse compression by sum frequency generation,” Appl. Phys. Lett. 68, 3540–3542 (1996).
[CrossRef]

1995 (3)

1994 (2)

T. J. Driscoll, G. M. Gale, and F. Hache, “Ti:sapphire 2nd-harmonic-pumped visible range femtosecond optical parametric oscillator,” Opt. Commun. 110, 638–644 (1994).
[CrossRef]

A. Umbrasas, J. C. Diels, J. Jacob, and A. Piskarskas, “Parametric oscillation and compression in KTP crystals,” Opt. Lett. 19, 1753–1755 (1994).
[CrossRef] [PubMed]

1993 (2)

H. Tan, G. P. Banfi, and A. Tomaselli, “Optical frequency mixing through cascaded second-order processes in β-barium borate,” Appl. Phys. Lett. 63, 2472–2474 (1993).
[CrossRef]

P. Heinz, A. Laubereau, A. Dubietis, and A. Piskarskas, “Fiberless two-step parametric compression of sub-picosecond laser pulses,” Lith. Phys. Rev. 33, 314–317 (1993).

1992 (1)

1991 (1)

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991).
[CrossRef]

1990 (3)

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity-dispersion,” Phys. Rev. A 41, 5645–5649 (1990).
[CrossRef] [PubMed]

T. R. Zhang, H. R. Choo, and M. C. Downer, “Phase and group-velocity matching for SHG of femtosecond pulses,” Appl. Opt. 29, 3927–3933 (1990).
[CrossRef] [PubMed]

D. A. Guk and V. G. Dmitirev, “Some characteristics of second harmonic generation under conditions of strong energy exchange between interacting waves,” Kvant. Elektron. (Moscow) 18, 106–110 (1990).

1989 (1)

O. E. Martinez, “Matrix formalism for dispersive cavities,” IEEE J. Quantum Electron. QE-25, 296–300 (1989).
[CrossRef]

Andreoni, A.

Banfi, G. P.

H. Tan, G. P. Banfi, and A. Tomaselli, “Optical frequency mixing through cascaded second-order processes in β-barium borate,” Appl. Phys. Lett. 63, 2472–2474 (1993).
[CrossRef]

Cavallari, M.

Cerullo, G.

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

Choo, H. R.

Danielius, R.

Davies, B. L.

DeSilvestri, S.

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

Diels, J. C.

DiTrapani, P.

Dmitirev, V. G.

D. A. Guk and V. G. Dmitirev, “Some characteristics of second harmonic generation under conditions of strong energy exchange between interacting waves,” Kvant. Elektron. (Moscow) 18, 106–110 (1990).

Downer, M. C.

Dragila, R.

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity-dispersion,” Phys. Rev. A 41, 5645–5649 (1990).
[CrossRef] [PubMed]

Driscoll, T. J.

G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. 20, 1562–1564 (1995).
[CrossRef] [PubMed]

T. J. Driscoll, G. M. Gale, and F. Hache, “Ti:sapphire 2nd-harmonic-pumped visible range femtosecond optical parametric oscillator,” Opt. Commun. 110, 638–644 (1994).
[CrossRef]

Dubietis, A.

Foggi, P.

Gale, G. M.

G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. 20, 1562–1564 (1995).
[CrossRef] [PubMed]

T. J. Driscoll, G. M. Gale, and F. Hache, “Ti:sapphire 2nd-harmonic-pumped visible range femtosecond optical parametric oscillator,” Opt. Commun. 110, 638–644 (1994).
[CrossRef]

Guk, D. A.

D. A. Guk and V. G. Dmitirev, “Some characteristics of second harmonic generation under conditions of strong energy exchange between interacting waves,” Kvant. Elektron. (Moscow) 18, 106–110 (1990).

Hache, F.

G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. 20, 1562–1564 (1995).
[CrossRef] [PubMed]

T. J. Driscoll, G. M. Gale, and F. Hache, “Ti:sapphire 2nd-harmonic-pumped visible range femtosecond optical parametric oscillator,” Opt. Commun. 110, 638–644 (1994).
[CrossRef]

Heinz, P.

P. Heinz, A. Laubereau, A. Dubietis, and A. Piskarskas, “Fiberless two-step parametric compression of sub-picosecond laser pulses,” Lith. Phys. Rev. 33, 314–317 (1993).

Ibragimov, E. A.

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991).
[CrossRef]

Jacob, J.

Kobayashi, T.

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268–2270 (1999).
[CrossRef]

A. Shirakawa, I. Sakane, 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]

Laubereau, A.

P. Heinz, A. Laubereau, A. Dubietis, and A. Piskarskas, “Fiberless two-step parametric compression of sub-picosecond laser pulses,” Lith. Phys. Rev. 33, 314–317 (1993).

Martinez, O. E.

O. E. Martinez, “Matrix formalism for dispersive cavities,” IEEE J. Quantum Electron. QE-25, 296–300 (1989).
[CrossRef]

Nisoli, M.

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

M. Nisoli, S. D. Silvestri, G. Valiulius, and A. Varanavicius, “Fivefold femtosecond pulse compression by sum frequency generation,” Appl. Phys. Lett. 68, 3540–3542 (1996).
[CrossRef]

Piskarskas, A.

Sakane, I.

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268–2270 (1999).
[CrossRef]

A. Shirakawa, I. Sakane, 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]

Shirakawa, A.

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268–2270 (1999).
[CrossRef]

A. Shirakawa, I. Sakane, 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]

Silvestri, S. D.

M. Nisoli, S. D. Silvestri, G. Valiulius, and A. Varanavicius, “Fivefold femtosecond pulse compression by sum frequency generation,” Appl. Phys. Lett. 68, 3540–3542 (1996).
[CrossRef]

Solcia, C.

Stabinis, A.

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991).
[CrossRef]

Stagira, S.

Takasaka, M.

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268–2270 (1999).
[CrossRef]

Tamosauskas, G.

Tan, H.

H. Tan, G. P. Banfi, and A. Tomaselli, “Optical frequency mixing through cascaded second-order processes in β-barium borate,” Appl. Phys. Lett. 63, 2472–2474 (1993).
[CrossRef]

Tomaselli, A.

H. Tan, G. P. Banfi, and A. Tomaselli, “Optical frequency mixing through cascaded second-order processes in β-barium borate,” Appl. Phys. Lett. 63, 2472–2474 (1993).
[CrossRef]

Umbrasas, A.

Valiulis, G.

Valiulius, G.

M. Nisoli, S. D. Silvestri, G. Valiulius, and A. Varanavicius, “Fivefold femtosecond pulse compression by sum frequency generation,” Appl. Phys. Lett. 68, 3540–3542 (1996).
[CrossRef]

Varanavicius, A.

M. Nisoli, S. D. Silvestri, G. Valiulius, and A. Varanavicius, “Fivefold femtosecond pulse compression by sum frequency generation,” Appl. Phys. Lett. 68, 3540–3542 (1996).
[CrossRef]

Wang, Y.

Y. Wang and B. L. Davies, “Frequency-doubling pulse compressor for picosecond high power neodymium laser pulses,” Opt. Lett. 17, 1459–1461 (1992).
[CrossRef]

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity-dispersion,” Phys. Rev. A 41, 5645–5649 (1990).
[CrossRef] [PubMed]

Zhang, T. R.

Appl. Opt. (1)

Appl. Phys. Lett. (4)

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

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268–2270 (1999).
[CrossRef]

M. Nisoli, S. D. Silvestri, G. Valiulius, and A. Varanavicius, “Fivefold femtosecond pulse compression by sum frequency generation,” Appl. Phys. Lett. 68, 3540–3542 (1996).
[CrossRef]

H. Tan, G. P. Banfi, and A. Tomaselli, “Optical frequency mixing through cascaded second-order processes in β-barium borate,” Appl. Phys. Lett. 63, 2472–2474 (1993).
[CrossRef]

IEEE J. Quantum Electron. (1)

O. E. Martinez, “Matrix formalism for dispersive cavities,” IEEE J. Quantum Electron. QE-25, 296–300 (1989).
[CrossRef]

Kvant. Elektron. (Moscow) (1)

D. A. Guk and V. G. Dmitirev, “Some characteristics of second harmonic generation under conditions of strong energy exchange between interacting waves,” Kvant. Elektron. (Moscow) 18, 106–110 (1990).

Lith. Phys. Rev. (1)

P. Heinz, A. Laubereau, A. Dubietis, and A. Piskarskas, “Fiberless two-step parametric compression of sub-picosecond laser pulses,” Lith. Phys. Rev. 33, 314–317 (1993).

Opt. Commun. (2)

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991).
[CrossRef]

T. J. Driscoll, G. M. Gale, and F. Hache, “Ti:sapphire 2nd-harmonic-pumped visible range femtosecond optical parametric oscillator,” Opt. Commun. 110, 638–644 (1994).
[CrossRef]

Opt. Lett. (9)

G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. 20, 1562–1564 (1995).
[CrossRef] [PubMed]

A. Shirakawa, I. Sakane, 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]

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

R. Danielius, A. Dubietis, and A. Piskarskas, “Femtosecond high-contrast pulses from a parametric generator pumped by the self-compressed second harmonic of a Nd:glass laser,” Opt. Lett. 20, 2225–2227 (1995).
[CrossRef] [PubMed]

Y. Wang and B. L. Davies, “Frequency-doubling pulse compressor for picosecond high power neodymium laser pulses,” Opt. Lett. 17, 1459–1461 (1992).
[CrossRef]

A. Umbrasas, J. C. Diels, G. Valiulis, J. Jacob, and A. Piskarskas, “Generation of femtosecond pulses through second harmonic compression of the output of a Nd:YAG laser,” Opt. Lett. 20, 2228–2230 (1995).
[CrossRef]

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. 22, 1071–1073 (1997).
[CrossRef] [PubMed]

R. Danielius, A. Piskarskas, P. DiTrapani, A. Andreoni, C. Solcia, and 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]

A. Umbrasas, J. C. Diels, J. Jacob, and A. Piskarskas, “Parametric oscillation and compression in KTP crystals,” Opt. Lett. 19, 1753–1755 (1994).
[CrossRef] [PubMed]

Phys. Rev. A (1)

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity-dispersion,” Phys. Rev. A 41, 5645–5649 (1990).
[CrossRef] [PubMed]

Other (3)

J. Biegert, V. Kubecek, and J.-C. Diels, “Pulse compression: Type II second harmonic pulse compression,” in Encyclopedia of Electrical and Electronics Engineering (#17), J. G. Webster, ed., (IEEE, New York, 1998).

J. Biegert, V. Kubecek, and J.-C. Diels, “Second harmonic pulse compression,” in Ultrafast Phenomena XI (Springer-Verlag, New York, 1998), pp. 84–86.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, Boston, 1995).

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Figures (8)

Fig. 1
Fig. 1

Three pulses represented in a temporal frame of reference moving with the group velocity of the second harmonic. Initially, at z=0, the fundamental e is delayed with respect to the fundamental o wave, as shown in (a) and (c). For low-input energies (a), the second harmonic will be broadened as the two fundamental pulses move into each other (b). For sufficiently high-input energies (c), the overlap between the two fundamentals remains small because they are depleted by the upconversion process, and the second-harmonic pulse remains short (d).

Fig. 2
Fig. 2

Experimental setup precompensating for the pulse-front tilt that occurs from propagation through an interface under an angle 2β=3.22°. For the chosen type II second-harmonic generation of 800-nm pulses, θ=2°, which is the angle between the incoming fundamental beams for noncollinear interaction. This requires a pulse-front tilt of γ=0.62° to precompensate for the propagation effect at the air-to-crystal interface. We also show that a very thin prism (SF 10) can achieve γ with a ratio a/b=0.2 for prism base length to beam diameter.

Fig. 3
Fig. 3

Contour map showing the pulse duration of the emerging second-harmonic signal as function of the o,e input pulse duration and the predelay between the o and e fundamental pulses. Results are calculated for the type II second-harmonic generation of 800-nm pulses with single-pulse energies of 400 µJ and for an interaction length of 250 µm inside a BBO crystal. The angle between the noncollinearly interacting fundamental waves inside the crystal is 2°.

Fig. 4
Fig. 4

(a) e fundamental and (b) second-harmonic (SH) pulse durations, plotted as a function of propagation inside the BBO crystal for an input pulse duration of 10 fs and single-pulse energies of 400 µJ. Results for three different input predelays between the o and e fundamental pulses are given: 20 fs (light), 25 fs (medium), and 30 fs (dark).

Fig. 5
Fig. 5

(a) Fundamental and (b) second-harmonic waves as they propagate and interact inside a 250-µm-long BBO crystal; note the different scales for (a) and (b). Group-velocity mismatch leads to a compressed second-harmonic pulse with a FWHM of 2.5 fs (b). Furthermore, considerable pulse reshaping can be seen for the extraordinary fundamental in (a), leading to a shortened FWHM from 10 fs to 2.5 fs with a shoulder. Pulse energies were 400 µJ for each fundamental pulse, at a FWHM of 10 fs with a predelay of 20 fs.

Fig. 6
Fig. 6

Contour map showing the pulse duration of the emerging second-harmonic signal as function of the o,e input pulse duration and the predelay between the o and e fundamental pulses. Results are calculated for the type II second-harmonic generation of 800-nm pulses with single-pulse energies of 200 µJ and for an interaction length of 250 µm inside a BBO crystal. The angle between the noncollinearly interacting fundamental waves inside the crystal is 2°.

Fig. 7
Fig. 7

(a) e-fundamental and (b) second-harmonic (SH) pulse durations, plotted as a function of propagation inside the BBO crystal for an input pulse duration of 10 fs and single-pulse energies of 200 µJ. Results for three different input predelays between the o and e fundamental pulses are given: 20 fs (light), 25 fs (medium), and 30 fs (dark).

Fig. 8
Fig. 8

(a) o and e Fundamental (a) and (b) second-harmonic waves as they propagate and interact inside a 250-µm-long BBO crystal; note the different scales for (a) and (b). Group-velocity mismatch leads to a compressed second-harmonic pulse with a FWHM of 3.5 fs (b). Furthermore, considerable pulse reshaping can be seen for the extraordinary fundamental in (a), leading to a shortened FWHM from 10 fs to 5.3 fs. Pulse energies were 200 µJ for each fundamental pulse, at a FWHM of 10 fs with a predelay of 20 fs.

Tables (1)

Tables Icon

Table 1 Group Velocities for the o, e, and Second-Harmonic e Wavesa

Equations (29)

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

2x2+2y2+2z2-1c2 2t2E(x, y, z, t)
=μ0 2t2 P(x, y, z, t),
P=PL+PNL.
2z2-1c2 2t2E(x, t)=μ0 2t2[PL+PNL].
z+i Ωc (Ω)z-i Ωc (Ω)E˜(z, Ω)
=μ0Ω2P˜NL(Ω, z),
(Ω)=[1+χ(Ω)],
E˜(Ω, z)=E˜2(Ω, z)+E˜o(Ω, z)+E˜e(Ω, z)
=½ E˜2[(Ω-ω2), z]×exp[-ik2(Ω-ω2)z]1ˆe+½ E˜o[(Ω-ω), z]×exp[-iko(Ω-ω)z]1ˆo+½ E˜e[(Ω-ω), z]×exp[-ike(Ω-ω)z]1ˆe+c.c.,
k2(Ω)=Ωc 2(Ω)=Ωcn2(Ω)
ko(Ω)=Ωc o(Ω)=Ωcno(Ω)
ke(Ω)=Ωc e(Ω)=Ωcne(Ω).
E˜2(ΔΩ, z)=E˜2(Ω-2ω, z)
E˜o(ΔΩ, z)=E˜o(Ω-ω, z)
E˜e(ΔΩ, z)=E˜e(Ω-ω, z).
2z2 E˜2(ΔΩ, z)-2ik2 z E˜2(ΔΩ, z)
=2μ0(ω+ΔΩ)2P˜NL(ΔΩ, z)exp[ik2(ΔΩ)z],
z E˜2(ΔΩ, z)=i μ0k2(ω+ΔΩ)2P˜NL(ΔΩ, z)×exp[ik2(ΔΩ)z]-i2k2 2z2 E˜2(ΔΩ, z).
P˜NL(t, z)=0χ(2)[Eo(t, z)Ee(t, z)1ˆe+E2(t, z)Eo*(t, z)1ˆe+E2(t, z)Ee*(t, z)1ˆo].
E˜2(ΔΩ, z)z=-i ω2χ(2)4n2c -E˜o(ΔΩ, z)E˜e(ΔΩ-ΔΩ, z)×exp{-i[ko(ΔΩ)+ke(ΔΩ-ΔΩ)-k2(ΔΩ)]z}dΔΩ+i2k2 2E˜2(ΔΩ, z)z2,
E˜o(ΔΩ, z)z=-i ωχ(2)4noc -E˜2(ΔΩ, z)E˜e*(ΔΩ-ΔΩ, z)×exp{i[ko(ΔΩ)+ke(ΔΩ-ΔΩ)-k2(ΔΩ)]z}dΔΩ+i2ko 2E˜o(ΔΩ, z)z2,
E˜e(ΔΩ, z)z=-i ωχ(2)4nec -E˜2(ΔΩ, z)E˜o*(ΔΩ-ΔΩ, z)×exp{i[ke(ΔΩ)+ko(ΔΩ-ΔΩ)-k2(ΔΩ)]z}dΔΩ+i2ke 2E˜e(ΔΩ, z)z2.
E˜2(t, z)=-E˜2(ΔΩ, z)exp-ik2(ΔΩ)-k2-ΔΩvg2z×exp(-iΔΩ)dΔΩ,
E˜o(t, z)=-E˜o(ΔΩ, z)exp-iko(ΔΩ)-k2-ΔΩvg2z×exp(-iΔΩ)dΔΩ,
E˜e(t, z)=-E˜e(ΔΩ, z)exp-ike(ΔΩ)-k2-ΔΩvg2z×exp(-iΔΩ)dΔΩ,
1vg2=k2|2ω,
E˜2(ΔΩ, z)z=-i ω2χ(2)4n2c -E˜o(ΔΩ, z)E˜e(ΔΩ-ΔΩ, z)×exp{-i[ko(ΔΩ)+ke(ΔΩ-ΔΩ)-k2(ΔΩ)]z}dΔΩ,
E˜o(ΔΩ, z)z=-i ωχ(2)4noc -E˜2(ΔΩ, z)E˜e*(ΔΩ-ΔΩ, z)×exp{i[ko(ΔΩ)+ke(ΔΩ-ΔΩ)-k2(ΔΩ)]z}dΔΩ,
E˜e(ΔΩ, z)z=-i ωχ(2)4nec -E˜2(ΔΩ, z)E˜o*(ΔΩ-ΔΩ, z)×exp{i[ke(ΔΩ)+ko(ΔΩ-ΔΩ)-k2(ΔΩ)]z}dΔΩ.

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