Abstract

We recently described a general solution to the phase-matching problem that arises when one wishes to perform an arbitrary number of nonlinear optical processes in a single medium [Phys. Rev. Lett. 95, 133901 (2005) ]. Here we outline in detail the implementation of the solution for a one-dimensional photonic quasicrystal, which acts as a simultaneous frequency doubler for three independent optical beams. We confirm this solution experimentally using an electric-field poled KTiOPO4 crystal. In optimizing the device, we find—contrary to common practice—that simple duty cycles of 100% and 0% may yield the highest efficiencies, and we show that our device is more efficient than a comparable device based on periodic quasi-phase matching.

© 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. 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, 2631-2654 (1992).
    [CrossRef]
  3. S.-N. Zhu, Y.-Y. Zhu, and N.-B. Ming, "Quasi-phase-matched third-harmonic generation in a quasiperiodic optical superlattice," Science 278, 843-846 (1997).
    [CrossRef]
  4. Y. S. Kivshar, A. A. Sukhorukov, and S. M. Saltiel, "Two-color multistep cascading and parametric soliton-induced waveguides," Phys. Rev. E 60, R5056-R5059 (1999).
    [CrossRef]
  5. R. C. Pooser and O. Pfister, "Observation of triply coincident nonlinearities in periodically poled KTiOPO4," Opt. Lett. 30, 2635-2637 (2005).
    [CrossRef] [PubMed]
  6. G. I. Stegeman, D. J. Hagan, and L. Torner, "χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons," Opt. Quantum Electron. 28, 1691-1740 (1996).
    [CrossRef]
  7. T. Ellenbogen, A. Arie, and S. M. Saltiel, "Noncollinear double quasi phase matching in one-dimensional poled crystals," Opt. Lett. 32, 262-264 (2007).
    [CrossRef] [PubMed]
  8. V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
    [CrossRef]
  9. N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
    [CrossRef] [PubMed]
  10. A. Arie, N. Habshoosh, and A. Bahabad, "Quasi phase matching in two-dimensional nonlinear photonic crystals," Opt. Quantum Electron. (to be published).
  11. K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, "Multiple nonlinear optical interactions with arbitrary wave vector differences," Phys. Rev. Lett. 88, 023903 (2001).
    [CrossRef]
  12. H. Liu, S. N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, "Multiple-wavelength second-harmonic generation in aperiodic optical superlattices," Appl. Phys. Lett. 81, 3326-3328 (2002).
    [CrossRef]
  13. J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
    [CrossRef]
  14. M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, "Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides," Opt. Lett. 24, 1157-1159 (1999).
    [CrossRef]
  15. R. T. Bratfalean, A. C. Peacock, N. G. R. Broderick, K. Gallo, and R. Lewen, "Harmonic generation in a two-dimensional nonlinear quasicrystal," Opt. Lett. 30, 424-426 (2005).
    [CrossRef] [PubMed]
  16. R. Lifshitz, A. Arie, and A. Bahabad, "Photonic quasicrystals for nonlinear optical frequency conversion," Phys. Rev. Lett. 95, 133901 (2005).
    [CrossRef] [PubMed]
  17. R. Lifshitz, "What is a crystal?" Z. Kristallogr. 222, 313-319 (2007).
    [CrossRef]
  18. N. de Bruijn, "Algebraic theory of Penrose's non-periodic tilings of the plane," Proc. K. Ned. Akad. Wet., Ser. A: Math. Sci. 84, 39-66 (1981).
  19. F. Gahler and J. Rhyner, "Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings," J. Phys. A 19, 267-277 (1986).
    [CrossRef]
  20. D. A. Rabson, T.-L. Ho, and N. D. Mermin, "Aperiodic tilings with nonsymmorphic space groups p2jgm," Acta Crystallogr. 44, 678-688 (1988). http://dx.doi.org/10.1107/S0108767388003733.
  21. D. A. Rabson, T.-L. Ho, and N. D. Mermin, "Space groups of quasicrystallographic tilings," Acta Crystallogr. 45, 538-547 (1989). http://dx.doi.org/10.1107/S0108767389003302.
    [CrossRef]
  22. M. Senechal, Quasicrystals and Geometry (Cambridge U. Press, 1995), pp. 162-166.
  23. K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-916 (1999).
    [CrossRef]
  24. S. Emanueli and A. Arie, "Temperature-dependent dispersion equations for KTiOPO4 and KTiOAsO4," Appl. Opt. 42, 6661-6665 (2003).
    [CrossRef] [PubMed]
  25. A. H. Norton and C. M. de Sterke, "Optimal poling of nonlinear photonic crystals for frequency conversion," Opt. Lett. 28, 188-190 (2003).
    [CrossRef] [PubMed]
  26. R. Lifshitz, "Symmetry of magnetically ordered quasicrystals," Phys. Rev. Lett. 80, 2717-2720 (1998).
    [CrossRef]
  27. R. Lifshitz, "Magnetic quasicrystals: what can we expect to see in their neutron diffraction data?" Mater. Sci. Eng., A 294, 508-511 (2000).
    [CrossRef]
  28. K. Fradkin-Kashi and A. Arie, "Multiple-wavelength quasi-phase-matched nonlinear interactions," IEEE J. Quantum Electron. 35, 1649-1656 (1999).
    [CrossRef]
  29. M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
    [CrossRef]
  30. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  31. G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3596-3639 (1968).
    [CrossRef]

2007 (2)

2005 (3)

2004 (1)

J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
[CrossRef]

2003 (2)

2002 (1)

H. Liu, S. N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, "Multiple-wavelength second-harmonic generation in aperiodic optical superlattices," Appl. Phys. Lett. 81, 3326-3328 (2002).
[CrossRef]

2001 (1)

K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, "Multiple nonlinear optical interactions with arbitrary wave vector differences," Phys. Rev. Lett. 88, 023903 (2001).
[CrossRef]

2000 (2)

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

R. Lifshitz, "Magnetic quasicrystals: what can we expect to see in their neutron diffraction data?" Mater. Sci. Eng., A 294, 508-511 (2000).
[CrossRef]

1999 (4)

K. Fradkin-Kashi and A. Arie, "Multiple-wavelength quasi-phase-matched nonlinear interactions," IEEE J. Quantum Electron. 35, 1649-1656 (1999).
[CrossRef]

M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, "Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides," Opt. Lett. 24, 1157-1159 (1999).
[CrossRef]

Y. S. Kivshar, A. A. Sukhorukov, and S. M. Saltiel, "Two-color multistep cascading and parametric soliton-induced waveguides," Phys. Rev. E 60, R5056-R5059 (1999).
[CrossRef]

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-916 (1999).
[CrossRef]

1998 (2)

R. Lifshitz, "Symmetry of magnetically ordered quasicrystals," Phys. Rev. Lett. 80, 2717-2720 (1998).
[CrossRef]

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

1997 (1)

S.-N. Zhu, Y.-Y. Zhu, and N.-B. Ming, "Quasi-phase-matched third-harmonic generation in a quasiperiodic optical superlattice," Science 278, 843-846 (1997).
[CrossRef]

1996 (1)

G. I. Stegeman, D. J. Hagan, and L. Torner, "χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons," Opt. Quantum Electron. 28, 1691-1740 (1996).
[CrossRef]

1993 (1)

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

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, 2631-2654 (1992).
[CrossRef]

1989 (1)

D. A. Rabson, T.-L. Ho, and N. D. Mermin, "Space groups of quasicrystallographic tilings," Acta Crystallogr. 45, 538-547 (1989). http://dx.doi.org/10.1107/S0108767389003302.
[CrossRef]

1988 (1)

D. A. Rabson, T.-L. Ho, and N. D. Mermin, "Aperiodic tilings with nonsymmorphic space groups p2jgm," Acta Crystallogr. 44, 678-688 (1988). http://dx.doi.org/10.1107/S0108767388003733.

1986 (1)

F. Gahler and J. Rhyner, "Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings," J. Phys. A 19, 267-277 (1986).
[CrossRef]

1981 (1)

N. de Bruijn, "Algebraic theory of Penrose's non-periodic tilings of the plane," Proc. K. Ned. Akad. Wet., Ser. A: Math. Sci. 84, 39-66 (1981).

1968 (1)

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3596-3639 (1968).
[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]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Arie, A.

T. Ellenbogen, A. Arie, and S. M. Saltiel, "Noncollinear double quasi phase matching in one-dimensional poled crystals," Opt. Lett. 32, 262-264 (2007).
[CrossRef] [PubMed]

R. Lifshitz, A. Arie, and A. Bahabad, "Photonic quasicrystals for nonlinear optical frequency conversion," Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

S. Emanueli and A. Arie, "Temperature-dependent dispersion equations for KTiOPO4 and KTiOAsO4," Appl. Opt. 42, 6661-6665 (2003).
[CrossRef] [PubMed]

K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, "Multiple nonlinear optical interactions with arbitrary wave vector differences," Phys. Rev. Lett. 88, 023903 (2001).
[CrossRef]

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-916 (1999).
[CrossRef]

K. Fradkin-Kashi and A. Arie, "Multiple-wavelength quasi-phase-matched nonlinear interactions," IEEE J. Quantum Electron. 35, 1649-1656 (1999).
[CrossRef]

A. Arie, N. Habshoosh, and A. Bahabad, "Quasi phase matching in two-dimensional nonlinear photonic crystals," Opt. Quantum Electron. (to be published).

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]

Bahabad, A.

R. Lifshitz, A. Arie, and A. Bahabad, "Photonic quasicrystals for nonlinear optical frequency conversion," Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

A. Arie, N. Habshoosh, and A. Bahabad, "Quasi phase matching in two-dimensional nonlinear photonic crystals," Opt. Quantum Electron. (to be published).

Berger, V.

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
[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]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3596-3639 (1968).
[CrossRef]

Bratfalean, R. T.

Brener, I.

Broderick, N. G. R.

R. T. Bratfalean, A. C. Peacock, N. G. R. Broderick, K. Gallo, and R. Lewen, "Harmonic generation in a two-dimensional nonlinear quasicrystal," Opt. Lett. 30, 424-426 (2005).
[CrossRef] [PubMed]

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

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, 2631-2654 (1992).
[CrossRef]

Chou, M. H.

de Bruijn, N.

N. de Bruijn, "Algebraic theory of Penrose's non-periodic tilings of the plane," Proc. K. Ned. Akad. Wet., Ser. A: Math. Sci. 84, 39-66 (1981).

de Sterke, C. M.

Du, J.

J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
[CrossRef]

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]

Ellenbogen, T.

Emanueli, S.

Fejer, M. M.

M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, "Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides," Opt. Lett. 24, 1157-1159 (1999).
[CrossRef]

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, 2631-2654 (1992).
[CrossRef]

Fradkin, K.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-916 (1999).
[CrossRef]

Fradkin-Kashi, K.

K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, "Multiple nonlinear optical interactions with arbitrary wave vector differences," Phys. Rev. Lett. 88, 023903 (2001).
[CrossRef]

K. Fradkin-Kashi and A. Arie, "Multiple-wavelength quasi-phase-matched nonlinear interactions," IEEE J. Quantum Electron. 35, 1649-1656 (1999).
[CrossRef]

Gahler, F.

F. Gahler and J. Rhyner, "Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings," J. Phys. A 19, 267-277 (1986).
[CrossRef]

Gallo, K.

Habshoosh, N.

A. Arie, N. Habshoosh, and A. Bahabad, "Quasi phase matching in two-dimensional nonlinear photonic crystals," Opt. Quantum Electron. (to be published).

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, "χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons," Opt. Quantum Electron. 28, 1691-1740 (1996).
[CrossRef]

Hanna, D. C.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

He, J. L.

J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
[CrossRef]

Ho, T.-L.

D. A. Rabson, T.-L. Ho, and N. D. Mermin, "Space groups of quasicrystallographic tilings," Acta Crystallogr. 45, 538-547 (1989). http://dx.doi.org/10.1107/S0108767389003302.
[CrossRef]

D. A. Rabson, T.-L. Ho, and N. D. Mermin, "Aperiodic tilings with nonsymmorphic space groups p2jgm," Acta Crystallogr. 44, 678-688 (1988). http://dx.doi.org/10.1107/S0108767388003733.

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, 2631-2654 (1992).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar, A. A. Sukhorukov, and S. M. Saltiel, "Two-color multistep cascading and parametric soliton-induced waveguides," Phys. Rev. E 60, R5056-R5059 (1999).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3596-3639 (1968).
[CrossRef]

Lewen, R.

Liao, J.

J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
[CrossRef]

Lifshitz, R.

R. Lifshitz, "What is a crystal?" Z. Kristallogr. 222, 313-319 (2007).
[CrossRef]

R. Lifshitz, A. Arie, and A. Bahabad, "Photonic quasicrystals for nonlinear optical frequency conversion," Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

R. Lifshitz, "Magnetic quasicrystals: what can we expect to see in their neutron diffraction data?" Mater. Sci. Eng., A 294, 508-511 (2000).
[CrossRef]

R. Lifshitz, "Symmetry of magnetically ordered quasicrystals," Phys. Rev. Lett. 80, 2717-2720 (1998).
[CrossRef]

Lin, X. C.

H. Liu, S. N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, "Multiple-wavelength second-harmonic generation in aperiodic optical superlattices," Appl. Phys. Lett. 81, 3326-3328 (2002).
[CrossRef]

Ling, W. J.

H. Liu, S. N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, "Multiple-wavelength second-harmonic generation in aperiodic optical superlattices," Appl. Phys. Lett. 81, 3326-3328 (2002).
[CrossRef]

Liu, H.

J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
[CrossRef]

H. Liu, S. N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, "Multiple-wavelength second-harmonic generation in aperiodic optical superlattices," Appl. Phys. Lett. 81, 3326-3328 (2002).
[CrossRef]

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, 2631-2654 (1992).
[CrossRef]

Mermin, N. D.

D. A. Rabson, T.-L. Ho, and N. D. Mermin, "Space groups of quasicrystallographic tilings," Acta Crystallogr. 45, 538-547 (1989). http://dx.doi.org/10.1107/S0108767389003302.
[CrossRef]

D. A. Rabson, T.-L. Ho, and N. D. Mermin, "Aperiodic tilings with nonsymmorphic space groups p2jgm," Acta Crystallogr. 44, 678-688 (1988). http://dx.doi.org/10.1107/S0108767388003733.

Ming, N. B.

J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
[CrossRef]

H. Liu, S. N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, "Multiple-wavelength second-harmonic generation in aperiodic optical superlattices," Appl. Phys. Lett. 81, 3326-3328 (2002).
[CrossRef]

Ming, N.-B.

S.-N. Zhu, Y.-Y. Zhu, and N.-B. Ming, "Quasi-phase-matched third-harmonic generation in a quasiperiodic optical superlattice," Science 278, 843-846 (1997).
[CrossRef]

Nada, N.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

Norton, A. H.

Offerhaus, H. L.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

Parameswaran, K. R.

Peacock, A. C.

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]

Pfister, O.

Pooser, R. C.

Rabson, D. A.

D. A. Rabson, T.-L. Ho, and N. D. Mermin, "Space groups of quasicrystallographic tilings," Acta Crystallogr. 45, 538-547 (1989). http://dx.doi.org/10.1107/S0108767389003302.
[CrossRef]

D. A. Rabson, T.-L. Ho, and N. D. Mermin, "Aperiodic tilings with nonsymmorphic space groups p2jgm," Acta Crystallogr. 44, 678-688 (1988). http://dx.doi.org/10.1107/S0108767388003733.

Rhyner, J.

F. Gahler and J. Rhyner, "Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings," J. Phys. A 19, 267-277 (1986).
[CrossRef]

Richardson, D. J.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

Rosenman, G.

K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, "Multiple nonlinear optical interactions with arbitrary wave vector differences," Phys. Rev. Lett. 88, 023903 (2001).
[CrossRef]

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-916 (1999).
[CrossRef]

Ross, G. W.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

Saitoh, M.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

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[CrossRef]

Senechal, M.

M. Senechal, Quasicrystals and Geometry (Cambridge U. Press, 1995), pp. 162-166.

Skliar, A.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-916 (1999).
[CrossRef]

Stegeman, G. I.

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Sukhorukov, A. A.

Y. S. Kivshar, A. A. Sukhorukov, and S. M. Saltiel, "Two-color multistep cascading and parametric soliton-induced waveguides," Phys. Rev. E 60, R5056-R5059 (1999).
[CrossRef]

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, "χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons," Opt. Quantum Electron. 28, 1691-1740 (1996).
[CrossRef]

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K. Fradkin-Kashi, A. Arie, P. Urenski, and G. Rosenman, "Multiple nonlinear optical interactions with arbitrary wave vector differences," Phys. Rev. Lett. 88, 023903 (2001).
[CrossRef]

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J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
[CrossRef]

Watanabe, K.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

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J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
[CrossRef]

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H. Liu, S. N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, "Multiple-wavelength second-harmonic generation in aperiodic optical superlattices," Appl. Phys. Lett. 81, 3326-3328 (2002).
[CrossRef]

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M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

Yao, A. Y.

H. Liu, S. N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, "Multiple-wavelength second-harmonic generation in aperiodic optical superlattices," Appl. Phys. Lett. 81, 3326-3328 (2002).
[CrossRef]

Zhu, S. N.

J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
[CrossRef]

H. Liu, S. N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, "Multiple-wavelength second-harmonic generation in aperiodic optical superlattices," Appl. Phys. Lett. 81, 3326-3328 (2002).
[CrossRef]

Zhu, S.-N.

S.-N. Zhu, Y.-Y. Zhu, and N.-B. Ming, "Quasi-phase-matched third-harmonic generation in a quasiperiodic optical superlattice," Science 278, 843-846 (1997).
[CrossRef]

Zhu, Y. Y.

J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
[CrossRef]

H. Liu, S. N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, "Multiple-wavelength second-harmonic generation in aperiodic optical superlattices," Appl. Phys. Lett. 81, 3326-3328 (2002).
[CrossRef]

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S.-N. Zhu, Y.-Y. Zhu, and N.-B. Ming, "Quasi-phase-matched third-harmonic generation in a quasiperiodic optical superlattice," Science 278, 843-846 (1997).
[CrossRef]

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Appl. Opt. (1)

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J. Liao, J. L. He, H. Liu, J. Du, F. Xu, H. T. Wang, S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Red, yellow, green and blue--four-color light from a single, aperiodically poled LiTaO3 crystal," Appl. Phys. B 78, 265-267 (2004).
[CrossRef]

Appl. Phys. Lett. (3)

H. Liu, S. N. Zhu, Y. Y. Zhu, N. B. Ming, X. C. Lin, W. J. Ling, A. Y. Yao, and Z. Y. Xu, "Multiple-wavelength second-harmonic generation in aperiodic optical superlattices," Appl. Phys. Lett. 81, 3326-3328 (2002).
[CrossRef]

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-916 (1999).
[CrossRef]

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435-436 (1993).
[CrossRef]

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K. Fradkin-Kashi and A. Arie, "Multiple-wavelength quasi-phase-matched nonlinear interactions," IEEE J. Quantum Electron. 35, 1649-1656 (1999).
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[CrossRef]

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Y. S. Kivshar, A. A. Sukhorukov, and S. M. Saltiel, "Two-color multistep cascading and parametric soliton-induced waveguides," Phys. Rev. E 60, R5056-R5059 (1999).
[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

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S.-N. Zhu, Y.-Y. Zhu, and N.-B. Ming, "Quasi-phase-matched third-harmonic generation in a quasiperiodic optical superlattice," Science 278, 843-846 (1997).
[CrossRef]

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R. Lifshitz, "What is a crystal?" Z. Kristallogr. 222, 313-319 (2007).
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A. Arie, N. Habshoosh, and A. Bahabad, "Quasi phase matching in two-dimensional nonlinear photonic crystals," Opt. Quantum Electron. (to be published).

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Figures (4)

Fig. 1
Fig. 1

Illustration of the LAB solution for designing a one-dimensional NPC for multiple collinear optical processes, using the dual-grid method. (a) Required mismatch vectors. (b) Dual grid, in which each family of lines is shown with a different gray level (different color online). (c) Tiling of the real-space line according to the order in which lines of different families appear in the dual grid. (d) Associating a given duty cycle with each tiling vector. Positively poled segments are shown in dark gray (blue online), and negatively poled segments are shown in white.

Fig. 2
Fig. 2

Optical microscope image of the demonstrated NPC. The prominent elements correspond to the a ( 1 ) tiling vector, poled with a 100% duty cycle. Their width is 8.5 μ m . The distances between these elements are quasiperiodically ordered along with the a ( 2 ) and a ( 3 ) tiling vectors, whose widths are 8.1 μ m and 7.9 μ m , respectively, and which are poled with a 0% duty cycle.

Fig. 3
Fig. 3

Spectral shaping. Each panel shows the magnitude of the Fourier transform for a 1 cm long NPC made to phase match the three collinear processes, described in the text. In each panel, one of the tiling vectors is given a duty cycle of 100%, denoted as 1, and the remaining two a duty cycle of 0%, or 0. Each panel also shows a piece of the corresponding real-space representation of the NPC, where the smallest element size is 8 μ m .

Fig. 4
Fig. 4

Normalized pump wavelength response. Experiment and simulation results of second-harmonic generation as a function of pump wavelength. This figure corresponds with panel (a) of Fig. 3.

Tables (1)

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Table 1 Conversion Efficiencies a

Equations (5)

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E 3 ( Δ k ) = Γ A g ( r ) exp ( i Δ k r ) d 2 r ,
( K ( 1 ) K ( 2 ) K ( 3 ) ) [ Δ k ( 1 ) q 2 ( 1 ) q 3 ( 1 ) Δ k ( 2 ) q 2 ( 2 ) q 3 ( 2 ) Δ k ( 3 ) q 2 ( 3 ) q 3 ( 3 ) ] ,
( A ( 1 ) A ( 2 ) A ( 3 ) ) [ a ( 1 ) b 2 ( 1 ) b 3 ( 1 ) a ( 2 ) b 2 ( 2 ) b 3 ( 2 ) a ( 3 ) b 2 ( 3 ) b 3 ( 3 ) ] ,
A ( i ) K ( j ) = 2 π δ i j .
( 1 3 2 π ) 2 = 0.045 .

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