Abstract

An optimal control technique is applied to design a pair of input laser pulses that control the temporal shape of an optical pulse generated in a process of transient sum-frequency mixing. Specific calculations performed for a model crystal in the case of fifth-harmonic generation are discussed as an example. The efficiency of the generation process is enhanced by the use of temporally tailored laser pulses.

© 2002 Optical Society of America

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References

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  1. C. R. Vidal, in Tunable Lasers, L. F. Mollenauer and J. C. White, eds. (Springer-Verlag, Berlin, 1987), pp. 57–113.
  2. J. H. Glownia, M. Kaschke, and P. P. Sorokin, “Amplification of 193-nm femtosecond seed pulses generated by third-order, nonresonant, difference-frequency mixing in xenon,” Opt. Lett. 17, 337–339 (1992).
    [CrossRef] [PubMed]
  3. A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
    [CrossRef] [PubMed]
  4. J. Ringling, O. Kittelmann, F. Noack, G. Korn, and J. Squier, “Tunable femtosecond pulses in the near vacuum ultraviolet generated by frequency conversion of amplified Ti: sapphire laser pulses,” Opt. Lett. 18, 2035–2037 (1993).
    [CrossRef]
  5. F. Seifert, J. Ringling, F. Noack, V. Petrov, and O. Kittelmann, “Generation of femtosecond pulses to as low as 172.7 nm by sum-frequency mixing in lithium triborate,” Opt. Lett. 19, 1538–1540 (1994).
    [CrossRef] [PubMed]
  6. V. Petrov, F. Rotermund, and F. Noack, “Generation of femtosecond pulses down to 166 nm by sum-frequency mixing in KB5O84H2O,” Electron. Lett. 34, 1748–1750 (1998).
    [CrossRef]
  7. R. Danielius, A. Dubietis, A. Piskarskas, G. Valiulis, and A. Varanavicius, “Generation of compressed 600–720-nm tunable femtosecond pulses by transient frequency mixing in a β-barium borate crystal,” Opt. Lett. 21, 216–218 (1996).
    [CrossRef] [PubMed]
  8. A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–305 (1991).
    [CrossRef]
  9. A. Dubietis, G. Tamosauskas, A. Varanavicius, G. Valiulis, and R. Danielius, “Highly efficient subpicosecond pulse generation at 211 nm,” J. Opt. Soc. Am. B 17, 48–52 (2000).
    [CrossRef]
  10. A. Dubietis, G. Tamosauskas, A. Varanavicius, G. Valiulis, and R. Danielius, “Generation of femtosecond radiation at 211 nm by femtosecond pulse upconversion in the field of a picosecond pulse,” Opt. Lett. 25, 1116–1118 (2000).
    [CrossRef]
  11. N. Wang and H. Rabitz, “Optimal control of optical pulse propagation in a medium of three-level systems,” Phys. Rev. A 52, R17–R20 (1995).
    [CrossRef] [PubMed]
  12. N. Wang and H. Rabitz, “Optimal control of population transfer in an optically dense medium,” J. Chem. Phys. 104, 1173–1178 (1996).
    [CrossRef]
  13. N. Wang and H. Rabitz, “Optimal control of pulse amplification without inversion,” Phys. Rev. A 53, 1879–1885 (1996).
    [CrossRef] [PubMed]
  14. R. Buffa, “Optimal control of population transfer through the continuum,” Opt. Commun. 153, 240–244 (1998).
    [CrossRef]
  15. R. Buffa and S. Cavalieri, “Optimal control of type I second-harmonic generation with ultrashort laser pulses,” J. Opt. Soc. Am. B 17, 1901–1905 (2000).
    [CrossRef]
  16. R. Buffa, “Transient sum-frequency mixing with temporally tailored laser pulses,” Opt. Lett. 26, 722–724 (2001).
    [CrossRef]
  17. W. H. Press, B. P. Flannery, S. A. Teukolski, and T. Wetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1989).
  18. G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B 17, 304–318 (2000).
    [CrossRef]
  19. D. Yelin, D. Meshulach, and Y. Silberberg, “Adaptive femtosecond pulse compression,” Opt. Lett. 22, 1793–1795 (1997).
    [CrossRef]
  20. D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive real-time femtosecond pulse shaping,” J. Opt. Soc. Am. B 15, 1615–1619 (1998).
    [CrossRef]

2001 (1)

2000 (4)

1998 (3)

V. Petrov, F. Rotermund, and F. Noack, “Generation of femtosecond pulses down to 166 nm by sum-frequency mixing in KB5O84H2O,” Electron. Lett. 34, 1748–1750 (1998).
[CrossRef]

R. Buffa, “Optimal control of population transfer through the continuum,” Opt. Commun. 153, 240–244 (1998).
[CrossRef]

D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive real-time femtosecond pulse shaping,” J. Opt. Soc. Am. B 15, 1615–1619 (1998).
[CrossRef]

1997 (1)

1996 (3)

N. Wang and H. Rabitz, “Optimal control of population transfer in an optically dense medium,” J. Chem. Phys. 104, 1173–1178 (1996).
[CrossRef]

N. Wang and H. Rabitz, “Optimal control of pulse amplification without inversion,” Phys. Rev. A 53, 1879–1885 (1996).
[CrossRef] [PubMed]

R. Danielius, A. Dubietis, A. Piskarskas, G. Valiulis, and A. Varanavicius, “Generation of compressed 600–720-nm tunable femtosecond pulses by transient frequency mixing in a β-barium borate crystal,” Opt. Lett. 21, 216–218 (1996).
[CrossRef] [PubMed]

1995 (1)

N. Wang and H. Rabitz, “Optimal control of optical pulse propagation in a medium of three-level systems,” Phys. Rev. A 52, R17–R20 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

1992 (2)

J. H. Glownia, M. Kaschke, and P. P. Sorokin, “Amplification of 193-nm femtosecond seed pulses generated by third-order, nonresonant, difference-frequency mixing in xenon,” Opt. Lett. 17, 337–339 (1992).
[CrossRef] [PubMed]

A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
[CrossRef] [PubMed]

1991 (1)

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–305 (1991).
[CrossRef]

Arbore, M. A.

Buffa, R.

Cavalieri, S.

Danielius, R.

Dubietis, A.

Fejer, M. M.

Fermann, M.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolski, and T. Wetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1989).

Galvanauskas, A.

Glownia, J. H.

Harter, D.

Ibragimov, E. A.

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–305 (1991).
[CrossRef]

Imeshev, G.

Kaschke, M.

Kittelmann, O.

Korn, G.

Meshulach, D.

Mossavi, K.

A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
[CrossRef] [PubMed]

Noack, F.

Petrov, V.

V. Petrov, F. Rotermund, and F. Noack, “Generation of femtosecond pulses down to 166 nm by sum-frequency mixing in KB5O84H2O,” Electron. Lett. 34, 1748–1750 (1998).
[CrossRef]

F. Seifert, J. Ringling, F. Noack, V. Petrov, and O. Kittelmann, “Generation of femtosecond pulses to as low as 172.7 nm by sum-frequency mixing in lithium triborate,” Opt. Lett. 19, 1538–1540 (1994).
[CrossRef] [PubMed]

Piskarskas, A.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolski, and T. Wetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1989).

Rabitz, H.

N. Wang and H. Rabitz, “Optimal control of pulse amplification without inversion,” Phys. Rev. A 53, 1879–1885 (1996).
[CrossRef] [PubMed]

N. Wang and H. Rabitz, “Optimal control of population transfer in an optically dense medium,” J. Chem. Phys. 104, 1173–1178 (1996).
[CrossRef]

N. Wang and H. Rabitz, “Optimal control of optical pulse propagation in a medium of three-level systems,” Phys. Rev. A 52, R17–R20 (1995).
[CrossRef] [PubMed]

Ringling, J.

Rotermund, F.

V. Petrov, F. Rotermund, and F. Noack, “Generation of femtosecond pulses down to 166 nm by sum-frequency mixing in KB5O84H2O,” Electron. Lett. 34, 1748–1750 (1998).
[CrossRef]

Seifert, F.

Silberberg, Y.

Sorokin, P. P.

Squier, J.

Stabinis, A.

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–305 (1991).
[CrossRef]

Tamosauskas, G.

Teukolski, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolski, and T. Wetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1989).

Tünnermann, A.

A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
[CrossRef] [PubMed]

Valiulis, G.

Varanavicius, A.

Vidal, C. R.

C. R. Vidal, in Tunable Lasers, L. F. Mollenauer and J. C. White, eds. (Springer-Verlag, Berlin, 1987), pp. 57–113.

Wang, N.

N. Wang and H. Rabitz, “Optimal control of population transfer in an optically dense medium,” J. Chem. Phys. 104, 1173–1178 (1996).
[CrossRef]

N. Wang and H. Rabitz, “Optimal control of pulse amplification without inversion,” Phys. Rev. A 53, 1879–1885 (1996).
[CrossRef] [PubMed]

N. Wang and H. Rabitz, “Optimal control of optical pulse propagation in a medium of three-level systems,” Phys. Rev. A 52, R17–R20 (1995).
[CrossRef] [PubMed]

Wellegehausen, B.

A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
[CrossRef] [PubMed]

Wetterling, T.

W. H. Press, B. P. Flannery, S. A. Teukolski, and T. Wetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1989).

Yelin, D.

Electron. Lett. (1)

V. Petrov, F. Rotermund, and F. Noack, “Generation of femtosecond pulses down to 166 nm by sum-frequency mixing in KB5O84H2O,” Electron. Lett. 34, 1748–1750 (1998).
[CrossRef]

J. Chem. Phys. (1)

N. Wang and H. Rabitz, “Optimal control of population transfer in an optically dense medium,” J. Chem. Phys. 104, 1173–1178 (1996).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (2)

R. Buffa, “Optimal control of population transfer through the continuum,” Opt. Commun. 153, 240–244 (1998).
[CrossRef]

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–305 (1991).
[CrossRef]

Opt. Lett. (7)

Phys. Rev. A (3)

N. Wang and H. Rabitz, “Optimal control of optical pulse propagation in a medium of three-level systems,” Phys. Rev. A 52, R17–R20 (1995).
[CrossRef] [PubMed]

N. Wang and H. Rabitz, “Optimal control of pulse amplification without inversion,” Phys. Rev. A 53, 1879–1885 (1996).
[CrossRef] [PubMed]

A. Tünnermann, K. Mossavi, and B. Wellegehausen, “Nonlinear-optical processes in the near-resonant two-photon excitation of xenon by femtosecond KrF-excimer-laser pulses,” Phys. Rev. A 46, 2707–2717 (1992).
[CrossRef] [PubMed]

Other (2)

C. R. Vidal, in Tunable Lasers, L. F. Mollenauer and J. C. White, eds. (Springer-Verlag, Berlin, 1987), pp. 57–113.

W. H. Press, B. P. Flannery, S. A. Teukolski, and T. Wetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1989).

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Figures (9)

Fig. 1
Fig. 1

Energy conversion efficiency along the crystal in the case of Gaussian input pulses Xi0(t) of parameters (2.8). (a) X10=1.5, X20=2.6; (b) X10=1.5, X20=3.5; (c) X10=1.6, X20=5.2.

Fig. 2
Fig. 2

Temporal shape of the SF pulses X3L(τ) generated by the Gaussian input pulses Xi0(τ) of Fig. 1.

Fig. 3
Fig. 3

φ parameter (top panel) and SF energy conversion efficiency (bottom panel) versus the number of iterations of the solving algorithm in the case of a Gaussian target pulse Xr(τ) with peak value Xr0=2.5 and FWHM=1. (a) β=0; (b) β=0.01; (c) β=0.1.

Fig. 4
Fig. 4

Optimal input pulses that generate a Gaussian SF pulse X3L(τ) of peak value equal to 2.5 and FWHM equal to 1. The best Gaussian fits, shown as dashed curves reproduce pulse shapes with peaks equal to 1.74 (a) and 3.04 (b), and FWHM equal to 0.804 (a) and 0.616 (b).

Fig. 5
Fig. 5

Energy conversion efficiency of the SF pulse X3(η, τ) generated along the crystal by the input pulses of Fig. 4.

Fig. 6
Fig. 6

SF pulses X3L(τ) generated by the input pulses of Fig. 4. The best Gaussian fit of the dashed pulse reproduces a shape with peak value of 2.36 and FWHM of 1.04.

Fig. 7
Fig. 7

Optimal input pulses that generate a Gaussian SF pulse X3L(τ) with a peak value of 3.5 and FWHM of 1. The best Gaussian fits, shown as dashed curves, reproduce shapes with peak values of 2.41 (a) and 4.10 (b), and FWHM of 0.640 (a) and 0.632 (b).

Fig. 8
Fig. 8

Energy conversion efficiency of the SF pulse X3(η, τ) generated along the crystal by the input pulses of Fig. 7.

Fig. 9
Fig. 9

SF pulses X3L(τ) generated by the input pulses of Fig. 7. The best Gaussian fit of the dashed curve reproduces a shape with peak value of 2.90 and FWHM of 1.13.

Equations (23)

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z+1u1t-jg122t2E1=-jγ1E2*E3 exp(jΔkz),
z+1u2t-jg222t2E2=-jγ2E1*E3 exp(jΔkz),
z+1u3t-jg322t2E3=-jγ3E1E2 exp(-jΔkz).
LˆX+f(X)=0,
X(η, τ)=X1(η, τ)X2(η, τ)X3(η, τ),
Lˆ=η+L0L13τ000η+L0L23τ000η,
f(X)=Γ1X2X3Γ2X1X3-Γ3X1X2,
X0(τ)=X(η=0, τ)=X10(τ)X20(τ)0,
XL(τ)=X(η=L/L0, τ)=X1L(τ)X2L(τ)X3L(τ).
X10(t)=X10 exp{-ln(2)[2(t-T0)/T0]2},
X20(t)=X20 exp{-ln(2)[2t/T0]2},
Xr(t)=Xr0 exp{-ln(2)[2(t-Δt)/T0]2},
L=L0=L23=3L13,
Γ1=0.1,Γ2=0.4,Γ3=0.5.
Φ=12-+[X3L(τ)-Xr(τ)]2dτ+β-+{[X10(τ)]2+[X20(τ)]2}dτ.
λ(η, τ)=λ1(η, τ)λ2(η, τ)λ3(η, τ),
J=Φ+0L/L0dη-+dτλT[LˆX+f(X)],
Lˆλ+g(X, λ)=0,
g(X, λ)=Γ3X2λ3-Γ2X3λ2Γ3X1λ3-Γ1X3λ1-Γ1X2λ1-Γ2X1λ2.
λ(η=L/L0, τ)=00X3L(τ)-Xr(τ).
δJδX0(τ)=λ1(η=0, τ)λ2(η=0, τ)0+βX0(τ).
X0(τ)X0(τ)+αˆδJδX0(τ),
φ=-+[X3L(τ)-Xr(τ)]2dτ-+[Xr(τ)]2dτ

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