Abstract

We describe a spectrogram-based simulated annealing algorithm for designing quasi-phase-matched crystals capable of producing second harmonic generation pulses of any chosen amplitude and phase profile. The approach applies a new and rapid analytic method for calculating the amplitude and phase of the second harmonic generation pulses generated by a quasi-phase-matched crystal containing an arbitrary grating design. The performance of the algorithm is illustrated by examples of femtosecond second harmonic pulses designed according to various target shapes including single, double and triple Gaussian pulses, positive and negative linear chirp and square, triangular and stepped profiles.

© 2005 Optical Society of America

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References

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  1. A. M. Weiner, D. E. Leaird, J.S. Patel, and J.R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28908–920, (1992)
    [Crossref]
  2. E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, H. Kapteyn, and G. Vdovin, “Pulse compression by use of deformable mirrors,” Opt. Lett. 24, 493–495 (1999)
    [Crossref]
  3. A.M. Weiner and A.M. Kanan, “Femtosecond Pulse Shaping for Synthesis, Processing, and Time-to-Space Conversion of Ultrafast Optical Waveforms,” IEEE J. Quantum Electron. QE-4, 317–331 (1998).
  4. F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter pulse compression and shaping,” Opt. Lett. 25575–577 (2000)
    [Crossref]
  5. 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]
  6. M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compressionof ultrashort pulses by use of second-harmonic generationin chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341–1343 (1997).
    [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 KTiOPO 4 with a RbTiOAsO 4 -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 KTiOPO 4,” 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. D. Artigas and D.T. Reid, “Efficient femtosecond optical parametric oscillators based on aperiodically poled nonlinear crystals” Opt. Lett. 27851–853 (2002).
    [Crossref]
  11. 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]
  12. L. Gallmann, G. Steinmeyer, U. Keller, G. Imeshev, M. M. Fejer, and J. P. Meyn, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings” Opt. Lett. 26, 614–616 (2001).
    [Crossref]
  13. L. Gallmann, G. Steinmeyer, G. Imeshev, J. P. Meyn, U. Keller, and M. M. Fejer. “Sub-6-fs blue pulses generated by quas-phase-matcheing second harmonic generation pulse compression” Appl. Phys. B 74S237–S243 (2002).
    [Crossref]
  14. 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]
  15. 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]
  16. G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings” J. Opt. Soc. Am. B 18, 534–539 (2001)
    [Crossref]
  17. S. Helmfrid and G. Arvidsson “Influence of randomly varying domain lengths and nonuniform effective index on second-harmonic generation in quasi-phase-matching waveguides” J. Opt. Soc. Am. B 8797–805 (1991)
    [Crossref]
  18. D. T. Reid, “Engineered quasi-phase-matching for second-garmonic generation” J. Opt. A: Pure Appl. Opt. 5S97–S102 (2003)
    [Crossref]
  19. Y. Zang and B-Y Gu “Optimal design of aperiodically poled lithium niobate crystals for multiple wavelengths parametric amplification” Opt. Comm. 192, 417–425 (2001).
    [Crossref]
  20. W.H. Glenn, “Second harmonic generation by picosecond optical pulses” IEEE J. Quantum Electronics,  QE-5284–290, (1969)
    [Crossref]
  21. E. Sidick, A. Knoesen, and A. Dienes. “Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses” J. Opt. Soc. Am. B 121704–1078 (1995)
    [Crossref]
  22. E. Sidick, A. Knoesen, and A. Dienes “Ultra-short pulse second harmonic generation in quasi-phase matched structures” Pure Appl. Opt. 5709–722 (1996).
    [Crossref]
  23. G. P. Agrawal, “Nonlinear Fiber Optics”, 2nd Edn, (Academic Press)
  24. A. Yariv “Quantum Electronics” 3rd end (New York: Wiley)
  25. R. Buffa. “Transient second-harmonic generation with spatially non-uniform nonlinear coefficients”, Opt. Lett. 27, 1058–1060 (2002)
    [Crossref]
  26. W. H. Press, S. A Teukolsky, W. T. Vetterling, and B. P. Flannery “Numerical Recipes” 2nd end (Cambridge: Cambridge University Press)
  27. D. J. Kane and R. Trebino, “Single-short measurement of the intensity and phase of an arbitrary ultrashort pulses by using frequency-resolved optical gating” Opt. Lett. 18, 823–825 (1993)
    [Crossref] [PubMed]
  28. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum. Electron. 28, 2631 (1992)
    [Crossref]

2003 (1)

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

2002 (4)

2001 (3)

2000 (4)

1999 (3)

1998 (1)

A.M. Weiner and A.M. Kanan, “Femtosecond Pulse Shaping for Synthesis, Processing, and Time-to-Space Conversion of Ultrafast Optical Waveforms,” IEEE J. Quantum Electron. QE-4, 317–331 (1998).

1997 (2)

1996 (1)

E. Sidick, A. Knoesen, and A. Dienes “Ultra-short pulse second harmonic generation in quasi-phase matched structures” Pure Appl. Opt. 5709–722 (1996).
[Crossref]

1995 (1)

1993 (1)

1992 (2)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum. Electron. 28, 2631 (1992)
[Crossref]

A. M. Weiner, D. E. Leaird, J.S. Patel, and J.R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28908–920, (1992)
[Crossref]

1991 (1)

1969 (1)

W.H. Glenn, “Second harmonic generation by picosecond optical pulses” IEEE J. Quantum Electronics,  QE-5284–290, (1969)
[Crossref]

Agrawal, G. P.

G. P. Agrawal, “Nonlinear Fiber Optics”, 2nd Edn, (Academic Press)

Arbore, M. A.

Artigas, D.

Arvidsson, G.

Backus, S.

Beddard, T.

Buffa, R.

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum. Electron. 28, 2631 (1992)
[Crossref]

Cheng, Z.

Chou, M. H.

Dienes, A.

E. Sidick, A. Knoesen, and A. Dienes “Ultra-short pulse second harmonic generation in quasi-phase matched structures” Pure Appl. Opt. 5709–722 (1996).
[Crossref]

E. Sidick, A. Knoesen, and A. Dienes. “Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses” J. Opt. Soc. Am. B 121704–1078 (1995)
[Crossref]

Ebrahimzadeh, M.

Faller, P.

Fejer, M. M.

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]

L. Gallmann, G. Steinmeyer, G. Imeshev, J. P. Meyn, U. Keller, and M. M. Fejer. “Sub-6-fs blue pulses generated by quas-phase-matcheing second harmonic generation pulse compression” Appl. Phys. B 74S237–S243 (2002).
[Crossref]

G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings” J. Opt. Soc. Am. B 18, 534–539 (2001)
[Crossref]

L. Gallmann, G. Steinmeyer, U. Keller, G. Imeshev, M. M. Fejer, and J. P. Meyn, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings” Opt. Lett. 26, 614–616 (2001).
[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]

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]

M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compressionof ultrashort pulses by use of second-harmonic generationin chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341–1343 (1997).
[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. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum. Electron. 28, 2631 (1992)
[Crossref]

Fermann, M.

Flannery, B. P.

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

Gallmann, L.

L. Gallmann, G. Steinmeyer, G. Imeshev, J. P. Meyn, U. Keller, and M. M. Fejer. “Sub-6-fs blue pulses generated by quas-phase-matcheing second harmonic generation pulse compression” Appl. Phys. B 74S237–S243 (2002).
[Crossref]

L. Gallmann, G. Steinmeyer, U. Keller, G. Imeshev, M. M. Fejer, and J. P. Meyn, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings” Opt. Lett. 26, 614–616 (2001).
[Crossref]

Galvanauskas, A.

Glenn, W.H.

W.H. Glenn, “Second harmonic generation by picosecond optical pulses” IEEE J. Quantum Electronics,  QE-5284–290, (1969)
[Crossref]

Gu, B-Y

Y. Zang and B-Y Gu “Optimal design of aperiodically poled lithium niobate crystals for multiple wavelengths parametric amplification” Opt. Comm. 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, IEEE J. Quantum. Electron. 28, 2631 (1992)
[Crossref]

Kanan, A.M.

A.M. Weiner and A.M. Kanan, “Femtosecond Pulse Shaping for Synthesis, Processing, and Time-to-Space Conversion of Ultrafast Optical Waveforms,” IEEE J. Quantum Electron. QE-4, 317–331 (1998).

Kane, D. J.

Kapteyn, H.

Karlsson, H.

Kasriel, S.

Keller, U.

L. Gallmann, G. Steinmeyer, G. Imeshev, J. P. Meyn, U. Keller, and M. M. Fejer. “Sub-6-fs blue pulses generated by quas-phase-matcheing second harmonic generation pulse compression” Appl. Phys. B 74S237–S243 (2002).
[Crossref]

L. Gallmann, G. Steinmeyer, U. Keller, G. Imeshev, M. M. Fejer, and J. P. Meyn, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings” Opt. Lett. 26, 614–616 (2001).
[Crossref]

Knoesen, A.

E. Sidick, A. Knoesen, and A. Dienes “Ultra-short pulse second harmonic generation in quasi-phase matched structures” Pure Appl. Opt. 5709–722 (1996).
[Crossref]

E. Sidick, A. Knoesen, and A. Dienes. “Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses” J. Opt. Soc. Am. B 121704–1078 (1995)
[Crossref]

Laude, V.

Laurell, F.

Leaird, D. E.

A. M. Weiner, D. E. Leaird, J.S. Patel, and J.R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28908–920, (1992)
[Crossref]

Loza-Alvarez, P.

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum. Electron. 28, 2631 (1992)
[Crossref]

Maginnis, K.

Marco, O.

Meyn, J. P.

L. Gallmann, G. Steinmeyer, G. Imeshev, J. P. Meyn, U. Keller, and M. M. Fejer. “Sub-6-fs blue pulses generated by quas-phase-matcheing second harmonic generation pulse compression” Appl. Phys. B 74S237–S243 (2002).
[Crossref]

L. Gallmann, G. Steinmeyer, U. Keller, G. Imeshev, M. M. Fejer, and J. P. Meyn, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings” Opt. Lett. 26, 614–616 (2001).
[Crossref]

Mourou, G.

Murnane, M.

Patel, J.S.

A. M. Weiner, D. E. Leaird, J.S. Patel, and J.R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28908–920, (1992)
[Crossref]

Press, W. H.

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

Reid, D. T.

Reid, D.T.

Russek, U.

Sibbett, W.

Sidick, E.

E. Sidick, A. Knoesen, and A. Dienes “Ultra-short pulse second harmonic generation in quasi-phase matched structures” Pure Appl. Opt. 5709–722 (1996).
[Crossref]

E. Sidick, A. Knoesen, and A. Dienes. “Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses” J. Opt. Soc. Am. B 121704–1078 (1995)
[Crossref]

Spielmann, Ch.

Steinmeyer, G.

L. Gallmann, G. Steinmeyer, G. Imeshev, J. P. Meyn, U. Keller, and M. M. Fejer. “Sub-6-fs blue pulses generated by quas-phase-matcheing second harmonic generation pulse compression” Appl. Phys. B 74S237–S243 (2002).
[Crossref]

L. Gallmann, G. Steinmeyer, U. Keller, G. Imeshev, M. M. Fejer, and J. P. Meyn, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings” Opt. Lett. 26, 614–616 (2001).
[Crossref]

Teukolsky, S. A

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

Torner, L.

Tournois, P.

Trebino, R.

Vdovin, G.

Verluise, F.

Vetterling, W. T.

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

Weiner, A. M.

A. M. Weiner, D. E. Leaird, J.S. Patel, and J.R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28908–920, (1992)
[Crossref]

Weiner, A.M.

A.M. Weiner and A.M. Kanan, “Femtosecond Pulse Shaping for Synthesis, Processing, and Time-to-Space Conversion of Ultrafast Optical Waveforms,” IEEE J. Quantum Electron. QE-4, 317–331 (1998).

Wullert, J.R.

A. M. Weiner, D. E. Leaird, J.S. Patel, and J.R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28908–920, (1992)
[Crossref]

Yariv, A.

A. Yariv “Quantum Electronics” 3rd end (New York: Wiley)

Zang, Y.

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

Zeek, E.

Appl. Phys. B (1)

L. Gallmann, G. Steinmeyer, G. Imeshev, J. P. Meyn, U. Keller, and M. M. Fejer. “Sub-6-fs blue pulses generated by quas-phase-matcheing second harmonic generation pulse compression” Appl. Phys. B 74S237–S243 (2002).
[Crossref]

IEEE J. Quantum Electron. (2)

A. M. Weiner, D. E. Leaird, J.S. Patel, and J.R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28908–920, (1992)
[Crossref]

A.M. Weiner and A.M. Kanan, “Femtosecond Pulse Shaping for Synthesis, Processing, and Time-to-Space Conversion of Ultrafast Optical Waveforms,” IEEE J. Quantum Electron. QE-4, 317–331 (1998).

IEEE J. Quantum Electronics (1)

W.H. Glenn, “Second harmonic generation by picosecond optical pulses” IEEE J. Quantum Electronics,  QE-5284–290, (1969)
[Crossref]

IEEE J. Quantum. Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, IEEE J. Quantum. Electron. 28, 2631 (1992)
[Crossref]

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

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

J. Opt. Soc. Am. B (6)

Opt. Comm. (1)

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

Opt. Lett. (11)

E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, H. Kapteyn, and G. Vdovin, “Pulse compression by use of deformable mirrors,” Opt. Lett. 24, 493–495 (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 KTiOPO 4,” 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]

D. Artigas and D.T. Reid, “Efficient femtosecond optical parametric oscillators based on aperiodically poled nonlinear crystals” Opt. Lett. 27851–853 (2002).
[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]

L. Gallmann, G. Steinmeyer, U. Keller, G. Imeshev, M. M. Fejer, and J. P. Meyn, “Generation of sub-6-fs blue pulses by frequency doubling with quasi-phase-matching gratings” Opt. Lett. 26, 614–616 (2001).
[Crossref]

F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter pulse compression and shaping,” Opt. Lett. 25575–577 (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 compressionof ultrashort pulses by use of second-harmonic generationin chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341–1343 (1997).
[Crossref]

D. J. Kane and R. Trebino, “Single-short measurement of the intensity and phase of an arbitrary ultrashort pulses by using frequency-resolved optical gating” Opt. Lett. 18, 823–825 (1993)
[Crossref] [PubMed]

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

Pure Appl. Opt. (1)

E. Sidick, A. Knoesen, and A. Dienes “Ultra-short pulse second harmonic generation in quasi-phase matched structures” Pure Appl. Opt. 5709–722 (1996).
[Crossref]

Other (3)

G. P. Agrawal, “Nonlinear Fiber Optics”, 2nd Edn, (Academic Press)

A. Yariv “Quantum Electronics” 3rd end (New York: Wiley)

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

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Figures (8)

Fig. 1.
Fig. 1.

Schematic of the arbitrary quasi-phasematched crystal design used in the calculations. The crystal is comprised of n alternately inverted domains (directions represented by arrows) of individual lengths qm (1≤mn). At the entry face the first domain has a length of q 1 and at the exit face the last domain has a length of qn .

Fig. 2.
Fig. 2.

Results showing the second harmonic pulses obtained using different QPM crystal designs and calculated using the analytic model described here (solid lines) and a conventional pulse propagation model (symbols). See text for full details.

Fig. 3.
Fig. 3.

Flowchart of the simulated annealing algorithm for designing arbitrary QPM gratings. R is a random number between 0 and 1.

Fig. 4.
Fig. 4.

Target pulses chosen in testing the simulated annealing algorithm for designing arbitrary QPM gratings. See text for full details.

Fig. 5.
Fig. 5.

Left to right for target pulses of (a) a 150fs Gaussian pulse and (b) a 200fs Gaussian pulse: PG-FROG spectrograms of the target and calculated SHG pulses; the SHG power evolution through the crystal calculated by numerical code (symbols) and the new analytic method (solid curve), and; the distribution of domain sizes throughout the crystal.

Fig. 6.
Fig. 6.

Left to right for target pulses of (a) a 150fs Gaussian double-pulse and (b) a 150fs Gaussian triple-pulse: PG-FROG spectrograms of the target and calculated SHG pulses; the SHG power evolution through the crystal calculated by numerical code (symbols) and the new analytic method (solid curve), and; the distribution of domain sizes throughout the crystal.

Fig. 7.
Fig. 7.

Left to right for target pulses of (a) a positively chirped 300fs Gaussian pulse and (b) a negatively chirped 300fs Gaussian pulse: PG-FROG spectrograms of the target and calculated SHG pulses; the SHG power evolution through the crystal calculated by numerical code (symbols) and the new analytic method (solid curve), and; the distribution of domain sizes throughout the crystal.

Fig. 8.
Fig. 8.

Left to right for target pulses of (a) a 400fs square pulse, (b) a 200fs triangular pulse, and (c) a 400fs stepped square pulse: PG-FROG spectrograms of the target and calculated SHG pulses; the SHG power evolution through the crystal calculated by numerical code (symbols) and the new analytic method (solid curve), and; the distribution of domain sizes throughout the crystal.

Equations (8)

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E crys ( Ω ) = κ d ijk Δ k ( Ω ) [ 1 ( 1 ) n exp ( i Δ k ( Ω ) Q n ) + m = 1 n 1 2 ( 1 ) m exp ( i Δ k ( Ω ) Q m ) ]
E 2 ( Ω ) = F { E 1 ( t ) 2 } E crys ( Ω )
E 2 ( t ) = F 1 { E 2 ( Ω ) }
i A 1 ( z , t ) z + i k 1 A 1 ( z , t ) t 1 2 k 1 2 A 1 ( z , t ) t 2 + σ ( z ) Γ 1 A 1 * ( z , t ) A 2 ( z , t ) exp ( i Δ k o z ) = 0
i A 2 ( z , t ) z + i k 2 A 2 ( z , t ) t 1 2 k 2 2 A 2 ( z , t ) t 2 + σ ( z ) Γ 2 A 1 2 ( z , t ) exp ( i Δ k o z ) = 0
I FROG ( Ω , τ ) = E sig ( t , τ ) exp ( i Ω t ) dt 2
E sig ( t , τ ) = E ( t ) E ( t τ ) 2
e k = τ Ω I ̂ FROG SHG I ̂ FROG target 2

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