Abstract

We numerically design quasi-phase matched crystals with domains of arbitrary sizes for second harmonic generation by femtosecond pulses, taking into account both group velocity mismatch and dispersion. An efficient simulated-annealing algorithm is developed to design quasi-phase matching gratings which can yield the desired amplitude and phase of second-harmonic pulses in the presence of pump depletion. The method is illustrated with reference to single, double-hump and chirped fs Gaussian pulses in a lithium niobate crystal.

© 2007 Optical Society of America

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [Crossref]
  2. M. A. Arbore, O. Marco, and M. M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett. 22, 865–867 (1997).
    [Crossref] [PubMed]
  3. M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultra-short pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341–1343 (1997).
    [Crossref]
  4. D. Artigas, D. T. Reid, M. M. Fejer, and L. Torner, “Pulse compression and gain enhancement in a degenerate optical parametric amplifier based on aperiodically poled crystals,” Opt. Lett. 27, 442–444 (2002)
    [Crossref]
  5. D. Artigas and D. T. Reid, “Efficient femtosecond optical parametric oscillators based on aperiodically poled nonlinear crystals,” Opt. Lett. 27, 851–853 (2002).
    [Crossref]
  6. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation - Tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2653 (1992).
    [Crossref]
  7. P. Loza-Alvarez, D. T. Reid, P. Faller, M. Ebrahimzadeh, and W. Sibbett, “Simultaneous second-harmonic generation and femtosecond-pulse compression in aperiodically poled KTiOPO4 with a RbTiOAsO4 -based optical parametric oscillator,” J. Opt. Soc. Am. B 16, 1553–1560 (1999).
    [Crossref]
  8. P. Loza-Alvarez, D. T. Reid, P. Faller, M. Ebrahimzadeh, W. Sibbett, H. Karlsson, and F. Laurell, “Simultaneous femtosecond-pulse compression and second-harmonic generation in aperiodically poled KTiOPO4,” Opt. Lett. 24, 1071–1073 (1999).
    [Crossref]
  9. T. Beddard, M. Ebrahimzadeh, D. T. Reid, and W. Sibbett, “Five-optical-cycle pulse generation in the mid infrared from an optical parametric oscillator based on aperiodically poled lithium niobate,” Opt. Lett. 25, 1052–1054 (2000).
    [Crossref]
  10. 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]
  11. G. Imeshev, M. A. Arbore, S. Kasriel, and M. M. Fejer, “Pulse shaping and compression by second-harmonic generation with quasi-phase-matching gratings in the presence of arbitrary dispersion,” J. Opt. Soc. Am. B 17, 1420–1437 (2000).
    [Crossref]
  12. R. Buffa, “Transient second-harmonic generation with spatially non-uniform nonlinear coefficients,” Opt. Lett. 27, 1058–1060 (2002).
    [Crossref]
  13. 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]
  14. M. Conforti, F. Baronio, and C. De Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high efficiency conversion,” Opt. Lett. (to appear).
    [PubMed]
  15. S. Helmfrid and G. Arvidsson, “Influence of randomly varying domain lengths and non-uniform effective index on second-harmonic generation in quasi-phase-matching waveguides,” J. Opt. Soc. Am. B 8, 797–805 (1991).
    [Crossref]
  16. D. T. Reid, “Engineered quasi-phase-matching for second-harmonic generation,” J. Opt. A: Pure Appl. Opt. 5, S97–S102 (2003).
    [Crossref]
  17. U. Sapaev and D. T. Reid, “General second-harmonic pulse shaping in grating-engineered quasi-phase-matched nonlinear crystals,” Opt. Express 13, 3264–3276 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-9-3264
    [Crossref] [PubMed]
  18. U. Sapaev, “Optimum shaping of a spectral response of second harmonic generation process in the aperiodic quasi-phase matched nonlinear crystal,” Opt. Spectrosc. 102, 1023–1027 (2007).
  19. E. A. Morozov, A. A. Kaminski, A. S. Chirkin, and D. B. Yusupov, “Second optical harmonic generation in nonlinear crystals with a disordered domain structure,” JETP Lett. 73, 647–650 (2001).
    [Crossref]
  20. W. H. Press, S. A Teukolsky, W. T. Vetterling, and B. P. Flannery “Numerical Recipes”, 2nd end (Cambridge University Press).
  21. 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]
  22. 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).
    [Crossref]
  23. N. C. Kothari and X. Carlotti “Transient second-harmonic generation: influence of effective group-velocity dispersion,” J. Opt. Soc. Am. B 5, 756–764 (1988).
    [Crossref]
  24. Y. Zang and B-Y. Gu, “Optimal design of aperiodically poled lithium niobate crystals for multiple wavelengths parametric amplification,” Opt. Commun. 192, 417–425 (2001).
    [Crossref]
  25. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, Springer, Berlin (1999).

2007 (1)

U. Sapaev, “Optimum shaping of a spectral response of second harmonic generation process in the aperiodic quasi-phase matched nonlinear crystal,” Opt. Spectrosc. 102, 1023–1027 (2007).

2005 (1)

2003 (1)

D. T. Reid, “Engineered quasi-phase-matching for second-harmonic generation,” J. Opt. A: Pure Appl. Opt. 5, S97–S102 (2003).
[Crossref]

2002 (3)

2001 (2)

Y. Zang and B-Y. Gu, “Optimal design of aperiodically poled lithium niobate crystals for multiple wavelengths parametric amplification,” Opt. Commun. 192, 417–425 (2001).
[Crossref]

E. A. Morozov, A. A. Kaminski, A. S. Chirkin, and D. B. Yusupov, “Second optical harmonic generation in nonlinear crystals with a disordered domain structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

2000 (4)

1999 (2)

1997 (2)

1995 (1)

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation - Tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2653 (1992).
[Crossref]

1991 (1)

1988 (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]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Arbore, M. A.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Artigas, D.

Arvidsson, G.

Baronio, F.

M. Conforti, F. Baronio, and C. De Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high efficiency conversion,” Opt. Lett. (to appear).
[PubMed]

Beddard, T.

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]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Buffa, R.

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation - Tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2653 (1992).
[Crossref]

Carlotti, X.

Cavalieri, S.

Chirkin, A. S.

E. A. Morozov, A. A. Kaminski, A. S. Chirkin, and D. B. Yusupov, “Second optical harmonic generation in nonlinear crystals with a disordered domain structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

Chou, M. H.

Conforti, M.

M. Conforti, F. Baronio, and C. De Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high efficiency conversion,” Opt. Lett. (to appear).
[PubMed]

De Angelis, C.

M. Conforti, F. Baronio, and C. De Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high efficiency conversion,” Opt. Lett. (to appear).
[PubMed]

Dienes, A

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, Springer, Berlin (1999).

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Ebrahimzadeh, M.

Faller, P.

Fejer, M. M.

Fermann, M.

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]

Flannery, B. P.

W. H. Press, S. A Teukolsky, W. T. Vetterling, and B. P. Flannery “Numerical Recipes”, 2nd end (Cambridge University Press).

Galvanauskas, A.

Gu, B-Y.

Y. Zang and B-Y. Gu, “Optimal design of aperiodically poled lithium niobate crystals for multiple wavelengths parametric amplification,” Opt. Commun. 192, 417–425 (2001).
[Crossref]

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, Springer, Berlin (1999).

Harter, D.

Helmfrid, S.

Imeshev, G.

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation - Tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2653 (1992).
[Crossref]

Kaminski, A. A.

E. A. Morozov, A. A. Kaminski, A. S. Chirkin, and D. B. Yusupov, “Second optical harmonic generation in nonlinear crystals with a disordered domain structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

Karlsson, H.

Kasriel, S.

Knoesen, A

Kothari, N. C.

Laurell, F.

Loza-Alvarez, P.

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation - Tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2653 (1992).
[Crossref]

Marco, O.

Morozov, E. A.

E. A. Morozov, A. A. Kaminski, A. S. Chirkin, and D. B. Yusupov, “Second optical harmonic generation in nonlinear crystals with a disordered domain structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, Springer, Berlin (1999).

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Press, W. H.

W. H. Press, S. A Teukolsky, W. T. Vetterling, and B. P. Flannery “Numerical Recipes”, 2nd end (Cambridge University Press).

Reid, D. T.

U. Sapaev and D. T. Reid, “General second-harmonic pulse shaping in grating-engineered quasi-phase-matched nonlinear crystals,” Opt. Express 13, 3264–3276 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-9-3264
[Crossref] [PubMed]

D. T. Reid, “Engineered quasi-phase-matching for second-harmonic generation,” J. Opt. A: Pure Appl. Opt. 5, S97–S102 (2003).
[Crossref]

D. Artigas, D. T. Reid, M. M. Fejer, and L. Torner, “Pulse compression and gain enhancement in a degenerate optical parametric amplifier based on aperiodically poled crystals,” Opt. Lett. 27, 442–444 (2002)
[Crossref]

D. Artigas and D. T. Reid, “Efficient femtosecond optical parametric oscillators based on aperiodically poled nonlinear crystals,” Opt. Lett. 27, 851–853 (2002).
[Crossref]

T. Beddard, M. Ebrahimzadeh, D. T. Reid, and W. Sibbett, “Five-optical-cycle pulse generation in the mid infrared from an optical parametric oscillator based on aperiodically poled lithium niobate,” Opt. Lett. 25, 1052–1054 (2000).
[Crossref]

P. Loza-Alvarez, D. T. Reid, P. Faller, M. Ebrahimzadeh, W. Sibbett, H. Karlsson, and F. Laurell, “Simultaneous femtosecond-pulse compression and second-harmonic generation in aperiodically poled KTiOPO4,” Opt. Lett. 24, 1071–1073 (1999).
[Crossref]

P. Loza-Alvarez, D. T. Reid, P. Faller, M. Ebrahimzadeh, and W. Sibbett, “Simultaneous second-harmonic generation and femtosecond-pulse compression in aperiodically poled KTiOPO4 with a RbTiOAsO4 -based optical parametric oscillator,” J. Opt. Soc. Am. B 16, 1553–1560 (1999).
[Crossref]

Sapaev, U.

U. Sapaev, “Optimum shaping of a spectral response of second harmonic generation process in the aperiodic quasi-phase matched nonlinear crystal,” Opt. Spectrosc. 102, 1023–1027 (2007).

U. Sapaev and D. T. Reid, “General second-harmonic pulse shaping in grating-engineered quasi-phase-matched nonlinear crystals,” Opt. Express 13, 3264–3276 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-9-3264
[Crossref] [PubMed]

Sibbett, W.

Sidick, E

Teukolsky, S. A

W. H. Press, S. A Teukolsky, W. T. Vetterling, and B. P. Flannery “Numerical Recipes”, 2nd end (Cambridge University Press).

Torner, L.

Vetterling, W. T.

W. H. Press, S. A Teukolsky, W. T. Vetterling, and B. P. Flannery “Numerical Recipes”, 2nd end (Cambridge University Press).

Yusupov, D. B.

E. A. Morozov, A. A. Kaminski, A. S. Chirkin, and D. B. Yusupov, “Second optical harmonic generation in nonlinear crystals with a disordered domain structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

Zang, Y.

Y. Zang and B-Y. Gu, “Optimal design of aperiodically poled lithium niobate crystals for multiple wavelengths parametric amplification,” Opt. Commun. 192, 417–425 (2001).
[Crossref]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation - Tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2653 (1992).
[Crossref]

J. Appl. Phys. (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]

J. Opt. A: Pure Appl. Opt. (1)

D. T. Reid, “Engineered quasi-phase-matching for second-harmonic generation,” J. Opt. A: Pure Appl. Opt. 5, S97–S102 (2003).
[Crossref]

J. Opt. Soc. Am. B (7)

S. Helmfrid and G. Arvidsson, “Influence of randomly varying domain lengths and non-uniform effective index on second-harmonic generation in quasi-phase-matching waveguides,” J. Opt. Soc. Am. B 8, 797–805 (1991).
[Crossref]

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]

G. Imeshev, M. A. Arbore, S. Kasriel, and M. M. Fejer, “Pulse shaping and compression by second-harmonic generation with quasi-phase-matching gratings in the presence of arbitrary dispersion,” J. Opt. Soc. Am. B 17, 1420–1437 (2000).
[Crossref]

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]

P. Loza-Alvarez, D. T. Reid, P. Faller, M. Ebrahimzadeh, and W. Sibbett, “Simultaneous second-harmonic generation and femtosecond-pulse compression in aperiodically poled KTiOPO4 with a RbTiOAsO4 -based optical parametric oscillator,” J. Opt. Soc. Am. B 16, 1553–1560 (1999).
[Crossref]

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).
[Crossref]

N. C. Kothari and X. Carlotti “Transient second-harmonic generation: influence of effective group-velocity dispersion,” J. Opt. Soc. Am. B 5, 756–764 (1988).
[Crossref]

JETP Lett. (1)

E. A. Morozov, A. A. Kaminski, A. S. Chirkin, and D. B. Yusupov, “Second optical harmonic generation in nonlinear crystals with a disordered domain structure,” JETP Lett. 73, 647–650 (2001).
[Crossref]

Opt. Commun. (1)

Y. Zang and B-Y. Gu, “Optimal design of aperiodically poled lithium niobate crystals for multiple wavelengths parametric amplification,” Opt. Commun. 192, 417–425 (2001).
[Crossref]

Opt. Express (1)

Opt. Lett. (7)

R. Buffa, “Transient second-harmonic generation with spatially non-uniform nonlinear coefficients,” Opt. Lett. 27, 1058–1060 (2002).
[Crossref]

P. Loza-Alvarez, D. T. Reid, P. Faller, M. Ebrahimzadeh, W. Sibbett, H. Karlsson, and F. Laurell, “Simultaneous femtosecond-pulse compression and second-harmonic generation in aperiodically poled KTiOPO4,” Opt. Lett. 24, 1071–1073 (1999).
[Crossref]

T. Beddard, M. Ebrahimzadeh, D. T. Reid, and W. Sibbett, “Five-optical-cycle pulse generation in the mid infrared from an optical parametric oscillator based on aperiodically poled lithium niobate,” Opt. Lett. 25, 1052–1054 (2000).
[Crossref]

M. A. Arbore, O. Marco, and M. M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett. 22, 865–867 (1997).
[Crossref] [PubMed]

M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultra-short pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341–1343 (1997).
[Crossref]

D. Artigas, D. T. Reid, M. M. Fejer, and L. Torner, “Pulse compression and gain enhancement in a degenerate optical parametric amplifier based on aperiodically poled crystals,” Opt. Lett. 27, 442–444 (2002)
[Crossref]

D. Artigas and D. T. Reid, “Efficient femtosecond optical parametric oscillators based on aperiodically poled nonlinear crystals,” Opt. Lett. 27, 851–853 (2002).
[Crossref]

Opt. Spectrosc. (1)

U. Sapaev, “Optimum shaping of a spectral response of second harmonic generation process in the aperiodic quasi-phase matched nonlinear crystal,” Opt. Spectrosc. 102, 1023–1027 (2007).

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Other (3)

M. Conforti, F. Baronio, and C. De Angelis, “From femtosecond infrared to picosecond visible pulses: temporal shaping with high efficiency conversion,” Opt. Lett. (to appear).
[PubMed]

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, Springer, Berlin (1999).

W. H. Press, S. A Teukolsky, W. T. Vetterling, and B. P. Flannery “Numerical Recipes”, 2nd end (Cambridge University Press).

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Figures (9)

Fig. 1.
Fig. 1.

Scheme of the arbitrary QPM grating, with δ(z) the dimensionless sign-changing aperiodic function of amplitude ∣δ(z)∣ = 1. The grating is comprised of N inverted domains with individual lengths qm(1 ≤ m ≤ N).

Fig. 2.
Fig. 2.

Integration scheme. Here the number of steps in each domain is j=4.

Fig. 3.
Fig. 3.

(Left) Normalized SHG conversion and (Right) domain size variation versus normalized crystal length (units of LNL ), for various QPM conditions: black lines, uniform grating; blue lines, positively chirped; red lines, negatively chirped; green and magenta lines, randomly varied. Here ε=100*(qm-lo )/lo .

Fig. 4.
Fig. 4.

Illustration of the synthesis task. A fundamental frequency Gaussian shaped pulse (red profile) excites with duration τ=100 fs and peak intensity I0=5 GW/cm2 the QPM structure to be determined (blue box) in order to generate the desired SH target pulses (green profiles on the right).

Fig. 5.
Fig. 5.

Left: FF intensity distributions (green (ξ=0), magenta (ξ=ξo)), desired target profile (∙) and obtained SH profile (blue). Center: PG-FROG spectrograms of target (center left) and final (center right) SH pulses. Right: FF (red) and SH (blue) power versus propagation length. (a) case (i) with a flat phase Gaussian pulse of 100 fs; (b) case (ii) with a 150 fs pulse; (c) case (iii) with 200 fs duration.

Fig. 6.
Fig. 6.

Left: FF intensity distributions (green (ξ=0), magenta (ξ=ξo)), desired target profile (∙) and obtained SH profile (blue). Center: PG-FROG spectrograms of target (center left) and final (center right) SH pulses. Right: FF (red) and SH (blue) power versus propagation length. (a) Case (iv) with two equal width pulses; (b) case (v) with two unequal width pulses of 100 and 200 fs, respectively.

Fig. 7.
Fig. 7.

Left: FF intensity distributions (green (ξ=0), magenta (ξ=ξo)), desired target profile (∙) and obtained SH profile (blue). Center: PG-FROG spectrograms of target (center left) and final (center right) SH pulses. Right: FF (red) and SH (blue) power versus propagation length. (a) Case (vi) with positive chirp ∼150 fs2 on a 100 fs pulse; (b) case (vii) with negative chirp ∼-150 fs2.

Fig. 8.
Fig. 8.

Domain size distribution along the crystal and corresponding number of domains N for the seven target profiles: (i) 100 fs Gaussian, N=37; (ii) 150 fs Gaussian, N=75; (iii) 200 fs Gaussian, N=120; (iv) 100 fs twin-pulses with a separation of 300 fs, N=135; (v) 100 and 200fs pulses with a separation of 400 fs, N=185; (vi) a positively chirped 100 fs Gaussian pulse (φ″ ∼ 150 fs2), N=54; (vii) a negatively chirped 100 fs Gaussian pulse (φ″ ∼ −150 fs2), N=66.

Fig. 9.
Fig. 9.

Calculated SH pulse shape with different resolution in domain size: 100 nm (blue line), 500 nm (green line), 1 μm (red line).

Equations (8)

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

A 1 z + 1 V 1 A 1 t i α 1 2 2 A 1 t 2 = 1 δ ( z ) ( A 1 ) * A 2 exp ( i Δ kz )
A 2 z + 1 V 2 A 2 t i α 2 2 2 A 2 t 2 = 2 δ ( z ) ( A 1 ) 2 exp ( i Δ kz )
A 1 ( z , t ) z = 0 = A 0 exp ( 2 ln 2 ( 1 τ ) 2 + i φ 1 )
A 2 ( z , t ) z = 0 = 0
a 1 ξ i β 1 2 a 1 μ 2 = i δ ( ξ ) ( a 1 ) * a 2 exp ( i Δ S ξ )
a 2 ξ + ρ a 2 μ i β 2 2 a 2 μ 2 = i δ ( ξ ) ( a 1 ) 2 exp ( i Δ S ξ )
a 1 ( ξ , μ ) ξ = 0 = exp ( 2 ln 2 μ 2 + 1 )
a 2 ( ξ , μ ) ξ = 0 = 0

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