Abstract

Ultrashort-pulse second-harmonic generation (USP SHG) is examined for fundamental pulses that are not transform limited but still possess coherent spectra. Pulses whose spectra contain various orders of nonlinear phases are considered, and spectral phase compensation (compression) in either the fundamental pulse (FP) or the second harmonic (SH) is investigated with regard to optimizing final SH pulse duration and overall conversion efficiency. For Gaussian shaped or similarly smooth spectra with only quadratic phase it is found that such a phase maps proportionally over to the spectrum of the SH, and the compressed SH pulse shapes and durations are independent of whether the compensation is performed in the FP or in the SH. The optimum compensation configuration and the conversion efficiency depend on the nonlinear interaction length. For a FP with multiple peaked spectra, compensation in the SH gives poor results, accentuating the structure inherent in the pulses. FP’s with higher-order spectral phases were found to give different results for the two different compensation schemes. In particular, it was found that second-plus fourth-order phases are easier to compensate than second- plus third-order. In all the cases examined, the optimum nonlinear interaction length is one to three times the pulse-width-preservation length. These results are useful for both understanding and experimental optimization of USP SHG with state-of-the-art systems and pulses 10 fs or shorter in duration.

© 1995 Optical Society of America

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  1. P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, and J. Schmidt, "Operation of a femtosecond Ti-sapphire solitary laser in the vicinity of zero group-delay dispersion," Opt. Lett. 18, 54 (1993).
    [CrossRef] [PubMed]
  2. M. T. Asaki, C. P. Huang, D. Garvey, J. P. Zhou, H. C. Kapteyn, and M. M. Murnane, "Generation of 11-fs pulses from a self-mode-locked Ti-sapphire laser," Opt. Lett. 18, 977 (1993).
    [CrossRef] [PubMed]
  3. A. Stingl, C. Spielmann, F. Krausz, and R. Szipöcs, "Generation of 11-fs pulses from a Ti:sapphire laser without the use of prisms," Opt. Lett. 19, 204 (1994).
    [CrossRef] [PubMed]
  4. J. P. Zhou, G. Taft, C. P. Huang, M. M. Murnane, and H. C. Kapteyn, "Pulse evolution in a broad-band Ti:sapphire laser," Opt. Lett. 19, 1149 (1994).
    [CrossRef] [PubMed]
  5. S. Backus, M. T. Asaki, C. Shi, H. C. Kapteyn, and M. M. Murname, "Intracavity frequency doubling in a Ti:sapphire laser: generation of 14-fs pulses at 416 nm," Opt. Lett. 19, 399 (1994).
    [PubMed]
  6. J. D. Harvey, J. M. Dudley, P. F. Curley, C. Spielmann, and F. Krausz, "Coherent effects in a self-mode-locked Ti:sapphire laser," Opt. Lett. 19, 972 (1994).
    [CrossRef] [PubMed]
  7. I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. P. Zhou, and C. P. Huang, "Fourth-order dispersion-limited solitary pulses," Opt. Lett. 19, 1465 (1994).
    [CrossRef] [PubMed]
  8. W. H. Knox, M. C. Downer, R. L. Fork, and C. V. Shank, "Amplified femtosecond optical pulses and continuum generation at a 5-kHz repetition rate," Opt. Lett. 9, 552 (1984).
    [CrossRef] [PubMed]
  9. P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, J. P. Heritage, and G. Alphonse, "High-power ultrafast laser diodes," IEEE J. Quantum Electron. 28, 2203 (1992).
    [CrossRef]
  10. E. Sidick, A. Knoesen, and A. Dienes, "Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses," J. Opt. Soc. Am. B 12, 1704 (1995).
    [CrossRef]
  11. W. S. Pelouch, T. Ukachi, E. S. Wachman, and C. L. Tang, "Evaluation of LiB3O5 for second-harmonic generation of femtosecond optical pulses," Appl. Phys. Lett. 57, 111 (1990).
    [CrossRef]
  12. R. J. Ellingson and C. L. Tang, "High-power, high-repetition-rate femtosecond pulses tunable in the visible," Opt. Lett. 18, 438 (1993).
    [CrossRef] [PubMed]
  13. O. E. Martinez, "Achromatic phase matching for second harmonic generation of femtosecond pulses," IEEE J. Quantum Electron. 25, 2464 (1989).
    [CrossRef]
  14. G. Szabó and Z. Bor, "Broadband frequency doubler for femtosecond pulses," Appl. Phys. B 50, 51 (1990).
    [CrossRef]
  15. T. R. Zhang, H. R. Choo, and M. C. Downer, "Phase and group velocity matching for second harmonic generation of femtosecond pulses," Appl. Opt. 29, 3927 (1990).
    [CrossRef] [PubMed]
  16. K. Hayata and M. Koshiba, "Group-velocity-matched second-harmonic generation: an efficient scheme for femtosecond ultraviolet pulse generation in periodically domain-inverted β-BaB2O4," Appl. Phys. Lett. 62, 2188 (1993).
    [CrossRef]
  17. R. Trebino and D. J. Kane, "Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating," J. Opt. Soc. Am. A 10, 1101 (1993).
    [CrossRef]
  18. J. L. A. Chilla and O. E. Martinez, "Direct determination of the amplitude and the phase of femtosecond light pulses," Opt. Lett. 16, 39 (1991).
    [CrossRef] [PubMed]
  19. K. C. Chu, J. P. Heritage, R. S. Grant, K. X. Liu, A. Dienes, W. E. White, and A. Sullivan, "Direct measurement of the spectral phase of femtosecond pulses," Opt. Lett. 20, 904 (1995).
    [CrossRef] [PubMed]

1995 (2)

1994 (5)

1993 (5)

1992 (1)

P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, J. P. Heritage, and G. Alphonse, "High-power ultrafast laser diodes," IEEE J. Quantum Electron. 28, 2203 (1992).
[CrossRef]

1991 (1)

1990 (3)

G. Szabó and Z. Bor, "Broadband frequency doubler for femtosecond pulses," Appl. Phys. B 50, 51 (1990).
[CrossRef]

T. R. Zhang, H. R. Choo, and M. C. Downer, "Phase and group velocity matching for second harmonic generation of femtosecond pulses," Appl. Opt. 29, 3927 (1990).
[CrossRef] [PubMed]

W. S. Pelouch, T. Ukachi, E. S. Wachman, and C. L. Tang, "Evaluation of LiB3O5 for second-harmonic generation of femtosecond optical pulses," Appl. Phys. Lett. 57, 111 (1990).
[CrossRef]

1989 (1)

O. E. Martinez, "Achromatic phase matching for second harmonic generation of femtosecond pulses," IEEE J. Quantum Electron. 25, 2464 (1989).
[CrossRef]

1984 (1)

Alphonse, G.

P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, J. P. Heritage, and G. Alphonse, "High-power ultrafast laser diodes," IEEE J. Quantum Electron. 28, 2203 (1992).
[CrossRef]

Andreadakis, N. C.

P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, J. P. Heritage, and G. Alphonse, "High-power ultrafast laser diodes," IEEE J. Quantum Electron. 28, 2203 (1992).
[CrossRef]

Asaki, M. T.

Backus, S.

Bor, Z.

G. Szabó and Z. Bor, "Broadband frequency doubler for femtosecond pulses," Appl. Phys. B 50, 51 (1990).
[CrossRef]

Brabec, T.

Chilla, J. L. A.

Choo, H. R.

Christov, I. P.

Chu, K. C.

Curley, P. F.

Delfyett, P. J.

P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, J. P. Heritage, and G. Alphonse, "High-power ultrafast laser diodes," IEEE J. Quantum Electron. 28, 2203 (1992).
[CrossRef]

Dienes, A.

Downer, M. C.

Dudley, J. M.

Ellingson, R. J.

Florez, L. T.

P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, J. P. Heritage, and G. Alphonse, "High-power ultrafast laser diodes," IEEE J. Quantum Electron. 28, 2203 (1992).
[CrossRef]

Fork, R. L.

Garvey, D.

Gmitter, T.

P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, J. P. Heritage, and G. Alphonse, "High-power ultrafast laser diodes," IEEE J. Quantum Electron. 28, 2203 (1992).
[CrossRef]

Grant, R. S.

Harvey, J. D.

Hayata, K.

K. Hayata and M. Koshiba, "Group-velocity-matched second-harmonic generation: an efficient scheme for femtosecond ultraviolet pulse generation in periodically domain-inverted β-BaB2O4," Appl. Phys. Lett. 62, 2188 (1993).
[CrossRef]

Heritage, J. P.

K. C. Chu, J. P. Heritage, R. S. Grant, K. X. Liu, A. Dienes, W. E. White, and A. Sullivan, "Direct measurement of the spectral phase of femtosecond pulses," Opt. Lett. 20, 904 (1995).
[CrossRef] [PubMed]

P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, J. P. Heritage, and G. Alphonse, "High-power ultrafast laser diodes," IEEE J. Quantum Electron. 28, 2203 (1992).
[CrossRef]

Huang, C. P.

Kane, D. J.

Kapteyn, H. C.

Knoesen, A.

Knox, W. H.

Koshiba, M.

K. Hayata and M. Koshiba, "Group-velocity-matched second-harmonic generation: an efficient scheme for femtosecond ultraviolet pulse generation in periodically domain-inverted β-BaB2O4," Appl. Phys. Lett. 62, 2188 (1993).
[CrossRef]

Krausz, F.

Liu, K. X.

Martinez, O. E.

J. L. A. Chilla and O. E. Martinez, "Direct determination of the amplitude and the phase of femtosecond light pulses," Opt. Lett. 16, 39 (1991).
[CrossRef] [PubMed]

O. E. Martinez, "Achromatic phase matching for second harmonic generation of femtosecond pulses," IEEE J. Quantum Electron. 25, 2464 (1989).
[CrossRef]

Murname, M. M.

Murnane, M. M.

Pelouch, W. S.

W. S. Pelouch, T. Ukachi, E. S. Wachman, and C. L. Tang, "Evaluation of LiB3O5 for second-harmonic generation of femtosecond optical pulses," Appl. Phys. Lett. 57, 111 (1990).
[CrossRef]

Schmidt, J.

Shank, C. V.

Shi, C.

Sidick, E.

Silberberg, Y.

P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, J. P. Heritage, and G. Alphonse, "High-power ultrafast laser diodes," IEEE J. Quantum Electron. 28, 2203 (1992).
[CrossRef]

Spielmann, C.

Spielmann, Ch.

Stingl, A.

Stoffel, N.

P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, J. P. Heritage, and G. Alphonse, "High-power ultrafast laser diodes," IEEE J. Quantum Electron. 28, 2203 (1992).
[CrossRef]

Sullivan, A.

Szabó, G.

G. Szabó and Z. Bor, "Broadband frequency doubler for femtosecond pulses," Appl. Phys. B 50, 51 (1990).
[CrossRef]

Szipöcs, R.

Taft, G.

Tang, C. L.

R. J. Ellingson and C. L. Tang, "High-power, high-repetition-rate femtosecond pulses tunable in the visible," Opt. Lett. 18, 438 (1993).
[CrossRef] [PubMed]

W. S. Pelouch, T. Ukachi, E. S. Wachman, and C. L. Tang, "Evaluation of LiB3O5 for second-harmonic generation of femtosecond optical pulses," Appl. Phys. Lett. 57, 111 (1990).
[CrossRef]

Trebino, R.

Ukachi, T.

W. S. Pelouch, T. Ukachi, E. S. Wachman, and C. L. Tang, "Evaluation of LiB3O5 for second-harmonic generation of femtosecond optical pulses," Appl. Phys. Lett. 57, 111 (1990).
[CrossRef]

Wachman, E. S.

W. S. Pelouch, T. Ukachi, E. S. Wachman, and C. L. Tang, "Evaluation of LiB3O5 for second-harmonic generation of femtosecond optical pulses," Appl. Phys. Lett. 57, 111 (1990).
[CrossRef]

White, W. E.

Wintner, E.

Zhang, T. R.

Zhou, J. P.

Appl. Opt. (1)

Appl. Phys. B (1)

G. Szabó and Z. Bor, "Broadband frequency doubler for femtosecond pulses," Appl. Phys. B 50, 51 (1990).
[CrossRef]

Appl. Phys. Lett. (2)

K. Hayata and M. Koshiba, "Group-velocity-matched second-harmonic generation: an efficient scheme for femtosecond ultraviolet pulse generation in periodically domain-inverted β-BaB2O4," Appl. Phys. Lett. 62, 2188 (1993).
[CrossRef]

W. S. Pelouch, T. Ukachi, E. S. Wachman, and C. L. Tang, "Evaluation of LiB3O5 for second-harmonic generation of femtosecond optical pulses," Appl. Phys. Lett. 57, 111 (1990).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. J. Delfyett, L. T. Florez, N. Stoffel, T. Gmitter, N. C. Andreadakis, Y. Silberberg, J. P. Heritage, and G. Alphonse, "High-power ultrafast laser diodes," IEEE J. Quantum Electron. 28, 2203 (1992).
[CrossRef]

O. E. Martinez, "Achromatic phase matching for second harmonic generation of femtosecond pulses," IEEE J. Quantum Electron. 25, 2464 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Lett. (11)

R. J. Ellingson and C. L. Tang, "High-power, high-repetition-rate femtosecond pulses tunable in the visible," Opt. Lett. 18, 438 (1993).
[CrossRef] [PubMed]

J. L. A. Chilla and O. E. Martinez, "Direct determination of the amplitude and the phase of femtosecond light pulses," Opt. Lett. 16, 39 (1991).
[CrossRef] [PubMed]

K. C. Chu, J. P. Heritage, R. S. Grant, K. X. Liu, A. Dienes, W. E. White, and A. Sullivan, "Direct measurement of the spectral phase of femtosecond pulses," Opt. Lett. 20, 904 (1995).
[CrossRef] [PubMed]

P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, and J. Schmidt, "Operation of a femtosecond Ti-sapphire solitary laser in the vicinity of zero group-delay dispersion," Opt. Lett. 18, 54 (1993).
[CrossRef] [PubMed]

M. T. Asaki, C. P. Huang, D. Garvey, J. P. Zhou, H. C. Kapteyn, and M. M. Murnane, "Generation of 11-fs pulses from a self-mode-locked Ti-sapphire laser," Opt. Lett. 18, 977 (1993).
[CrossRef] [PubMed]

A. Stingl, C. Spielmann, F. Krausz, and R. Szipöcs, "Generation of 11-fs pulses from a Ti:sapphire laser without the use of prisms," Opt. Lett. 19, 204 (1994).
[CrossRef] [PubMed]

J. P. Zhou, G. Taft, C. P. Huang, M. M. Murnane, and H. C. Kapteyn, "Pulse evolution in a broad-band Ti:sapphire laser," Opt. Lett. 19, 1149 (1994).
[CrossRef] [PubMed]

S. Backus, M. T. Asaki, C. Shi, H. C. Kapteyn, and M. M. Murname, "Intracavity frequency doubling in a Ti:sapphire laser: generation of 14-fs pulses at 416 nm," Opt. Lett. 19, 399 (1994).
[PubMed]

J. D. Harvey, J. M. Dudley, P. F. Curley, C. Spielmann, and F. Krausz, "Coherent effects in a self-mode-locked Ti:sapphire laser," Opt. Lett. 19, 972 (1994).
[CrossRef] [PubMed]

I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. P. Zhou, and C. P. Huang, "Fourth-order dispersion-limited solitary pulses," Opt. Lett. 19, 1465 (1994).
[CrossRef] [PubMed]

W. H. Knox, M. C. Downer, R. L. Fork, and C. V. Shank, "Amplified femtosecond optical pulses and continuum generation at a 5-kHz repetition rate," Opt. Lett. 9, 552 (1984).
[CrossRef] [PubMed]

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Figures (11)

Fig. 1
Fig. 1

Block diagrams of dispersion-compensation schemes commonly used in association with short-pulse SHG. The φa(ω) and φb(ω) represent the spectral phases provided by compensators A and B, respectively.

Fig. 2
Fig. 2

Pulse-width ratio τp2/τp1 versus the compensator phase coefficient b2 for four different values of L. The input pulse has (a) a Gaussian pulse spectrum containing a quadratic phase with r2 = 1 and (b) a sech pulse spectrum containing a quadratic phase with r2 = 2.71.

Fig. 3
Fig. 3

Temporal profiles of a FP having a double-peaked spectral amplitude |Ã(0, ω)| = exp[−(Ω1 + 2)2] + 0.7 exp [−(Ω1 − 2)2] and those of the corresponding SH obtained for L = Lτ. (a) r2 = 1, (b) r2 = 0 (precompensated FP). In both parts of the figure the SH intensities are normalized by the same factor. In (a) the dotted–dashed curve is the profile of the SH compensated after its generation with b2 = −0.5. For this FP, τp0 = 0.728τ, and T/0.728 is the time normalized by τp0.

Fig. 4
Fig. 4

Normalized spectral intensities of the pulses shown in Fig. 3. Ωi = Ω1 for the FP and Ωi = Ω2 for the SH.

Fig. 5
Fig. 5

Temporal profiles of (a) a FP having a Gaussian spectrum and only cubic spectral phase with r3 = 0.5 and the corresponding SH obtained for (b) L = Lτ and (c) L = 3Lτ. The three different curves in (b) and (c) show the uncompensated SH (solid curves), the cubic phase-compensated SH (dashed curves; b3 = −0.09 when L = Lτ and b3 = −0.305 when L = 3Lτ), and the completely compensated (entire phase removed) SH (dotted-dashed curves).

Fig. 6
Fig. 6

Spectral intensity and phase of the SH obtained for L = Lτ from a Gaussian spectral shaped FP containing quadratic phase spectral phases with r2 = 2 and r3 = 0.5 versus Ω2. The dashed curve is the spectral intensity of the FP.

Fig. 7
Fig. 7

(a) Temporal profile of the Gaussian spectrum FP when r2 = 2 and r3 = 0.5 and that of the corresponding SH obtained for L = Lτ. The pulse widths are τp1/τp0 = 1.93 and τp2/τp0 = 1.65, respectively. (b) Temporal profiles of the SH pulses obtained by compensating the SH in (a) with different amounts of spectral phase.

Fig. 8
Fig. 8

(a) Spectral intensity and phase and (b) temporal profile of the SH pulse obtained from the FP shown in Fig. 7(a) for L = 3Lτ.

Fig. 9
Fig. 9

Temporal profile of the Gaussian spectrum FP when r2 = 2 and r4 = 0.5 and those of the corresponding SH pulses obtained for L = Lτ. The dotted curve is obtained when the SH (dashed curve) is shaped by use of a quadratic phase compensator with b2 = −1.5.

Fig. 10
Fig. 10

Normalized FWHM width τac/τp0 of the SH intensity autocorrelation trace versus the quadratic spectral phase coefficient b2. The fundamental pulse is the same as in Fig. 2(a).

Fig. 11
Fig. 11

Intensity autocorrelation trace of the uncompensated SH pulse shown in Fig. 7(a). The dashed curve is the fitted sech pulse autocorrelation trace and has the same FWHM width as the solid curve.

Tables (1)

Tables Icon

Table 1 Coefficients bi of the Fitted SH Spectral Phase φ2(ω) (Even-Order Coefficients All Zero) versus the Nonlinear Medium Thickness La

Equations (17)

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A ˜ a ( ω ) = η a A ˜ ( ω ) exp [ - i φ a ( ω ) ] = η a A ˜ ( ω ) exp [ - i ( a 2 Ω 1 2 + a 3 Ω 1 3 + a 4 Ω 1 4 + ) ] ,
B ˜ b ( ω ) = η b B ˜ ( ω ) exp [ - i φ b ( ω ) ] = η b B ˜ ( ω ) exp [ - i ( b 2 Ω 2 2 + b 3 Ω 2 3 + b 4 Ω 2 4 + ) ] ,
R I = η a 2 I 2 a η b I 2 b .
L τ = C τ Δ f B 1 γ 2 - γ 1 ,
φ 1 ( ω ) = r 2 Ω 1 2 + r 3 Ω 1 3 + r 4 Ω 1 4 + ,
φ 2 ( ω ) = s 2 Ω 2 2 + s 3 Ω 2 3 + s 4 Ω 2 4 + ,
A ˜ ( 0 , ω ) = τ A 0 [ π ( 1 + i 4 r 2 ) ] 1 / 2 exp ( - Ω 1 2 / 4 ) exp ( - i r 2 Ω 1 2 ) ,
A ( z , T ) = A 0 exp [ - ( T + η z / 2 ) 2 1 + i 4 r 2 ] ,
τ p 1 = τ p 0 [ 1 + ( 4 r 2 ) 2 ] 1 / 2 ,
τ p 1 Δ f B 1 = 2 ln 2 π [ 1 + ( 4 r 2 ) 2 ] 1 / 2 .
B ( L , T ) = - i ρ 2 L A 0 2 0 1 d ζ exp { - 2 [ T + η L ( ζ - 1 / 2 ) ] 2 1 + i 4 r 2 } ,
B ˜ ( L , ω ) = - i ρ 2 L A 0 2 τ [ π ( 1 + i 4 r 2 ) 2 ] 1 / 2 × exp [ - ( 1 + i 4 r 2 ) Ω 2 2 8 ] 0 1 d ζ exp ( i η Ω 2 L ζ ) ,
B b ( L , T ) = - i ρ 2 L A 0 2 [ 1 + i 4 r 2 1 + i 4 ( r 2 + 2 b 2 ) ] 1 / 2 × 0 1 d ζ exp { - 2 [ T + η L ( ζ - 1 / 2 ) ] 2 1 + i 4 ( r 2 + 2 b 2 ) } .
B a ( L , T ) = - i ρ 2 L A 0 2 1 + i 4 r 2 1 + i 4 ( r 2 + a 2 ) 0 1 d ζ × exp { - 2 [ T + η L ( ζ - 1 / 2 ) ] 2 1 + i 4 ( r 2 + a 2 ) } .
R I = η a 2 η b { 1 + ( 4 r 2 ) 2 1 + [ 4 ( r 2 + a 2 ) ] 2 } 1 / 2 .
A ( z , T ) = A 0 sech [ t - γ 1 z ) / τ ] = A 0 sech ( T + η z / 2 ) ,
A ˜ ( 0 , ω ) = A 0 π τ sech ( π Ω 1 / 2 ) .

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