Abstract

We apply the optimal control technique to the topic of type I second-harmonic generation with ultrashort laser pulses. We design temporally tailored fundamental-frequency laser pulses that generate, in the regime of pump depletion, temporally symmetric second-harmonic optical pulses.

© 2000 Optical Society of America

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References

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  1. V. Krylov, A. Rebane, A. G. Kalintsev, H. Schwoerer, and U. P. Wild, “Second-harmonic generation of amplified femtosecond Ti:sapphire laser pulses,” Opt. Lett. 20, 198–200 (1995).
    [CrossRef] [PubMed]
  2. E. Sidick, A. Knoesen, and A. Dienes, “Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses,” J. Opt. Soc. Am. B 12, 1704–1712 (1995); E. Sidick, A. Knoesen, and A. Dienes, “Ultrashort-pulse second-harmonic generation. II. Non-transform-limited fundamental pulses,” J. Opt. Soc. Am. B 12, 1713–1722 (1995).
    [CrossRef]
  3. J. Zhang, J. Y. Huang, H. Wang, K. S. Wong, and G. K. Wong, “Second-harmonic generation from regeneratively amplified femtosecond laser pulses in BBO and LBO crystals,” J. Opt. Soc. Am. B 15, 200–209 (1998).
    [CrossRef]
  4. 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); N. Wang and H. Rabitz, “Optimal control of population transfer in an optically dense medium,” J. Chem. Phys. 104, 1173–1178 (1996); N. Wang and H. Rabitz, “Optimal control of pulse amplification without inversion,” Phys. Rev. A PLRAAN 53, 1879–1885 (1996).
    [CrossRef] [PubMed]
  5. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995).
    [CrossRef]
  6. M. M. Wefers and K. A. Nelson, “Analysis of programmable ultrashort waveform generation using liquid-crystal spatial light modulators,” J. Opt. Soc. Am. B 12, 1343–1362 (1995).
    [CrossRef]
  7. M. A. Dugan, J. X. Tull, and W. S. Warren, “High-resolution acousto-optic shaping of unamplified and amplified femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 2348–2358 (1997).
    [CrossRef]
  8. G. Imeshev, A. Galvanauskas, D. Harter, M. A. Arbore, M. Proctor, and M. M. Fejer, “Engineerable femtosecond pulse shaping by second-harmonic generation with Fourier synthetic quasi-phase-matching gratings,” Opt. Lett. 23, 864–866 (1998); 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]
  9. D. Yelin, D. Meshulach, and Y. Silberberg, “Adaptive femtosecond pulse compression,” Opt. Lett. 22, 1793–1795 (1997); D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive real-time femtosecond pulse shaping,” J. Opt. Soc. Am. B 15, 1615–1619 (1998).
    [CrossRef]
  10. J. Comly and E. Garmire, “Second harmonic generation from short pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
    [CrossRef]
  11. W. H. Press, B. P. Flannery, S. A. Teukolski, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge University, London, 1989).

1998

1997

1995

1968

J. Comly and E. Garmire, “Second harmonic generation from short pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
[CrossRef]

Comly, J.

J. Comly and E. Garmire, “Second harmonic generation from short pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
[CrossRef]

Dugan, M. A.

Garmire, E.

J. Comly and E. Garmire, “Second harmonic generation from short pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
[CrossRef]

Huang, J. Y.

Kalintsev, A. G.

Krylov, V.

Nelson, K. A.

Rebane, A.

Schwoerer, H.

Tull, J. X.

Wang, H.

Warren, W. S.

Wefers, M. M.

Weiner, A. M.

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995).
[CrossRef]

Wild, U. P.

Wong, G. K.

Wong, K. S.

Zhang, J.

Appl. Phys. Lett.

J. Comly and E. Garmire, “Second harmonic generation from short pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Prog. Quantum Electron.

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995).
[CrossRef]

Other

G. Imeshev, A. Galvanauskas, D. Harter, M. A. Arbore, M. Proctor, and M. M. Fejer, “Engineerable femtosecond pulse shaping by second-harmonic generation with Fourier synthetic quasi-phase-matching gratings,” Opt. Lett. 23, 864–866 (1998); 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]

D. Yelin, D. Meshulach, and Y. Silberberg, “Adaptive femtosecond pulse compression,” Opt. Lett. 22, 1793–1795 (1997); D. Meshulach, D. Yelin, and Y. Silberberg, “Adaptive real-time femtosecond pulse shaping,” J. Opt. Soc. Am. B 15, 1615–1619 (1998).
[CrossRef]

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

E. Sidick, A. Knoesen, and A. Dienes, “Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses,” J. Opt. Soc. Am. B 12, 1704–1712 (1995); E. Sidick, A. Knoesen, and A. Dienes, “Ultrashort-pulse second-harmonic generation. II. Non-transform-limited fundamental pulses,” J. Opt. Soc. Am. B 12, 1713–1722 (1995).
[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); N. Wang and H. Rabitz, “Optimal control of population transfer in an optically dense medium,” J. Chem. Phys. 104, 1173–1178 (1996); N. Wang and H. Rabitz, “Optimal control of pulse amplification without inversion,” Phys. Rev. A PLRAAN 53, 1879–1885 (1996).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Temporal shape of the FF[χ1L(τ)]2 (left) and SH[χ2L(τ)]2 (right) pulses generated by Gaussian FF laser pulses [χ10(τ)]2 of peak intensities equal to (a) 4, (b) 9, and (c) 25. The corresponding conversion efficiencies are 29% (a), 49% (b), and 74% (c).

Fig. 2
Fig. 2

(a) Temporal shape of the FF pulse [χ10(τ)]2 generating (b) the FF (left) and SH (right) pulses [χiL(τ)]2. The SH pulse reproduces a Gaussian shape (circles) of width equal to 1 (FWHM) with a precision better than 3×10-4. The corresponding conversion efficiency is 31%.

Fig. 3
Fig. 3

(a) Temporal shape of the FF pulse [χ10(τ)]2 generating (b) the FF (left) and SH (right) pulses [χiL(τ)]2. The SH pulse reproduces a Gaussian shape (circles) of width equal to 1 (FWHM) with a precision better than 5×10-4. The corresponding conversion efficiency is 54%.

Fig. 4
Fig. 4

(a) Temporal shape of the FF pulse [χ10(τ)]2 generating (b) the FF (left) and SH (right) pulses [χiL(τ)]2. The SH pulse reproduces a Gaussian shape (circles) of width equal to 1 (FWHM) with a precision better than 1.5×10-3. The corresponding conversion efficiency is 80%.

Fig. 5
Fig. 5

Normalized objective functional φ versus number of iterations of the solving algorithm obtained in the case of Figs. 2 (diamonds), 3 (circles), and 4 (dots) when one chooses as the initial FF electric field amplitude χ10(τ) the one generating the SH pulses shown in Fig. 1.

Equations (19)

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2Ez2-μ0ε0 2Et2+2PLt2=μ0 2PNLt2.
E=12[E1 exp[j(ω1t-k1z)]+E2 exp[j(2ω1t-k2z)]+c.c.],
z+1u1 t-j g12 2t2E1=-jγ1E1*E2 exp(jΔkz),
z+1u2 t-j g22 2t2E2=-jγ2E12 exp(-jΔkz),
LˆX+f(X)=0,
X(η, τ)=X1(η, τ)X2(η, τ),
Lˆ=η+1ν1 τ00η+1ν2 τ,
f(X)=Γ1X1X2-Γ2X12,
X(η=0, τ)=X10(τ)0,
X(η=1, τ)=X1L(τ)X2L(τ).
Φ=12 -+[X2L(τ)-Xr(τ)]2dτ,
λ(η, τ)=λ1(η, τ)λ2(η, τ),
J=Φ+01dη-+dτλT[LˆX+f(X)],
Lˆλ+g(X, λ)=0,
g(X, λ)=2Γ2λ2X1-Γ1λ1X2-Γ1λ1X1.
λ(η=1, τ)=0X2L(τ)-Xr(τ).
δJδX10(τ)=λ1(η=0, τ).
X10(τ)X10(τ)+αλ1(η=0, τ),
φ=-+[X2L(τ)-Xr(τ)]2dτ-+[Xr(τ)]2dτ

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