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

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]

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]

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]

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]

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 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]

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]

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

Torner, L.

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)

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]

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]

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]

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]

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]

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]

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]

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)

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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