Abstract

Different types of quadratic, radially symmetric, nonlinear photonic crystals are presented. The modulation of the nonlinear coefficient may be a periodic or an aperiodic function of the radial coordinate, whereas the azimuthal symmetry of the crystal may be either continuous or discrete. Nonlinear interactions within these structures are studied in two orientations, transverse and longitudinal, for which the interacting beams propagate either perpendicularly or in the plane of modulation. We show that radially symmetric structures can phase match multiple arbitrary processes in any direction. We study multiple wavelength three-wave mixing interactions and multiple direction interactions and analyze spatially dependent polarization states of the generated harmonics.

© 2008 Optical Society of America

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  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]
  2. 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]
  3. 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]
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    [CrossRef]
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    [CrossRef]
  6. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843-846 (1997).
    [CrossRef]
  7. 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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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London) 432, 374-376 (2004).
    [CrossRef]
  14. 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).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  22. 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]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  26. A. Bahabad, N. Voloch, A. Arie, and R. Lifshitz, “Experimental confirmation of the general solution to the multiple-phase-matching problem,” J. Opt. Soc. Am. B 24, 1916-1921 (2007).
    [CrossRef]
  27. T. Ellenbogen, A. Ganany, and A. Arie, “Nonlinear photonic structures for all-optical deflection,” Opt. Express 16, 3077-3082 (2008).
    [CrossRef] [PubMed]
  28. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics2nd ed. (Wiley, 2007), Chap. 6.

2008 (4)

2007 (4)

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]

A. Bahabad, N. Voloch, A. Arie, and R. Lifshitz, “Experimental confirmation of the general solution to the multiple-phase-matching problem,” J. Opt. Soc. Am. B 24, 1916-1921 (2007).
[CrossRef]

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361-375 (2007).
[CrossRef]

Y. Sheng, J. Dou, J. Li, D. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 101109 (2007).
[CrossRef]

2006 (1)

2005 (3)

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

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, “Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3,” Opt. Commun. 252, 397-401 (2005).
[CrossRef]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasi-crystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

2004 (2)

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London) 432, 374-376 (2004).
[CrossRef]

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 (1)

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

2000 (1)

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]

1999 (2)

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]

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

1998 (1)

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

1997 (2)

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

I. Amidror, “Fourier spectrum of radially periodic images,” J. Opt. Soc. Am. A 14, 816-826 (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]

1968 (1)

I. Freund, “Nonlinear diffraction,” Phys. Rev. Lett. 21, 1404-1406 (1968).
[CrossRef]

Amidror, I.

Arie, A.

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527-529 (2008).
[CrossRef] [PubMed]

A. Bahabad, A. Ganany-Padowicz, and A. Arie, “Engineering two-dimensional nonlinear photonic quasi-crystals,” Opt. Lett. 33, 1386-1388 (2008).
[CrossRef] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Generation of the second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[CrossRef] [PubMed]

T. Ellenbogen, A. Ganany, and A. Arie, “Nonlinear photonic structures for all-optical deflection,” Opt. Express 16, 3077-3082 (2008).
[CrossRef] [PubMed]

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361-375 (2007).
[CrossRef]

A. Bahabad, N. Voloch, A. Arie, and R. Lifshitz, “Experimental confirmation of the general solution to the multiple-phase-matching problem,” J. Opt. Soc. Am. B 24, 1916-1921 (2007).
[CrossRef]

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]

D. Kasimov, A. Arie, E. Winebrand, G. Rosenman, A. Bruner, P. Shaier, and D. Eger, “Annular symmetry nonlinear frequency converters,” Opt. Express 14, 9371-9376 (2006).
[CrossRef] [PubMed]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasi-crystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

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

Bahabad, A.

A. Bahabad, A. Ganany-Padowicz, and A. Arie, “Engineering two-dimensional nonlinear photonic quasi-crystals,” Opt. Lett. 33, 1386-1388 (2008).
[CrossRef] [PubMed]

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361-375 (2007).
[CrossRef]

A. Bahabad, N. Voloch, A. Arie, and R. Lifshitz, “Experimental confirmation of the general solution to the multiple-phase-matching problem,” J. Opt. Soc. Am. B 24, 1916-1921 (2007).
[CrossRef]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasi-crystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

Baudrier-Raybaut, M.

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London) 432, 374-376 (2004).
[CrossRef]

Berger, V.

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

Blau, P.

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 quasi-crystal,” 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]

Bruner, A.

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]

Cheng, B.

Y. Sheng, J. Dou, J. Li, D. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 101109 (2007).
[CrossRef]

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, “Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3,” Opt. Commun. 252, 397-401 (2005).
[CrossRef]

Chou, M. H.

Dou, J.

Y. Sheng, J. Dou, J. Li, D. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 101109 (2007).
[CrossRef]

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]

Eger, D.

Ellenbogen, T.

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,” 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]

Fischer, R.

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527-529 (2008).
[CrossRef] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Generation of the second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[CrossRef] [PubMed]

Fradkin, K.

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

Freund, I.

I. Freund, “Nonlinear diffraction,” Phys. Rev. Lett. 21, 1404-1406 (1968).
[CrossRef]

Gallo, K.

Ganany, A.

Ganany-Padowicz, A.

Habshoosh, N.

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361-375 (2007).
[CrossRef]

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]

Haïdar, R.

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London) 432, 374-376 (2004).
[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]

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]

Kasimov, D.

Katz, M.

Kivshar, Y. S.

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Generation of the second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[CrossRef] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527-529 (2008).
[CrossRef] [PubMed]

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]

Krolikowski1, W.

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527-529 (2008).
[CrossRef] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Generation of the second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[CrossRef] [PubMed]

Kupecek, Ph.

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London) 432, 374-376 (2004).
[CrossRef]

Lemasson, Ph.

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London) 432, 374-376 (2004).
[CrossRef]

Lewen, R.

Li, J.

Y. Sheng, J. Dou, J. Li, D. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 101109 (2007).
[CrossRef]

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.

A. Bahabad, N. Voloch, A. Arie, and R. Lifshitz, “Experimental confirmation of the general solution to the multiple-phase-matching problem,” J. Opt. Soc. Am. B 24, 1916-1921 (2007).
[CrossRef]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasi-crystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

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]

Ma, B.

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, “Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3,” Opt. Commun. 252, 397-401 (2005).
[CrossRef]

Ma, D.

Y. Sheng, J. Dou, J. Li, D. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 101109 (2007).
[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]

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]

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic 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]

Neshev, D. N.

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527-529 (2008).
[CrossRef] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Generation of the second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[CrossRef] [PubMed]

Ni, P.

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, “Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3,” Opt. Commun. 252, 397-401 (2005).
[CrossRef]

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]

Oron, M. B.

Parameswaran, K. R.

Peacock, A. C.

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]

Rosencher, E.

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London) 432, 374-376 (2004).
[CrossRef]

Rosenman, G.

D. Kasimov, A. Arie, E. Winebrand, G. Rosenman, A. Bruner, P. Shaier, and D. Eger, “Annular symmetry nonlinear frequency converters,” Opt. Express 14, 9371-9376 (2006).
[CrossRef] [PubMed]

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

Ruschin, S.

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]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics2nd ed. (Wiley, 2007), Chap. 6.

Saltiel, S. M.

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Generation of the second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[CrossRef] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Spatiotemporal toroidal waves from the transverse second-harmonic generation,” Opt. Lett. 33, 527-529 (2008).
[CrossRef] [PubMed]

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]

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]

Shaier, P.

Sheng, Y.

Y. Sheng, J. Dou, J. Li, D. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 101109 (2007).
[CrossRef]

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, “Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3,” Opt. Commun. 252, 397-401 (2005).
[CrossRef]

Stegeman, G. I.

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]

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]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics2nd ed. (Wiley, 2007), Chap. 6.

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]

Urenski, P.

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

Voloch, N.

Wang, H. T.

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]

Wang, T.

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, “Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3,” Opt. Commun. 252, 397-401 (2005).
[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]

Winebrand, E.

Xu, F.

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]

Xu, Z. 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]

Yamada, 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]

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]

Zhang, D.

Y. Sheng, J. Dou, J. Li, D. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 101109 (2007).
[CrossRef]

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, “Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3,” Opt. Commun. 252, 397-401 (2005).
[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]

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic 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]

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

Appl. Phys. B (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]

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]

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]

Y. Sheng, J. Dou, J. Li, D. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 101109 (2007).
[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, 2631-2654 (1992).
[CrossRef]

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

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

Nature (London) (1)

M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature (London) 432, 374-376 (2004).
[CrossRef]

Opt. Commun. (1)

T. Wang, B. Ma, Y. Sheng, P. Ni, B. Cheng, and D. Zhang, “Large-angle acceptance of quasi-phase-matched second-harmonic generation in homocentrically poled LiNbO3,” Opt. Commun. 252, 397-401 (2005).
[CrossRef]

Opt. Express (2)

Opt. Lett. (6)

Opt. Quantum Electron. (2)

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]

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361-375 (2007).
[CrossRef]

Phys. Rev. E (1)

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]

Phys. Rev. Lett. (6)

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski1, A. Arie, and Y. S. Kivshar, “Generation of the second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[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]

I. Freund, “Nonlinear diffraction,” Phys. Rev. Lett. 21, 1404-1406 (1968).
[CrossRef]

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

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

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasi-crystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[CrossRef] [PubMed]

Science (1)

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

Other (1)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics2nd ed. (Wiley, 2007), Chap. 6.

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Figures (8)

Fig. 1
Fig. 1

Family of radial photonic structures. The black and white areas denote negative and positive signs of nonlinear coefficients, respectively. (a) Periodic annular photonic structure characterized by period Λ. (b) Aperiodic continuous radial photonic crystal characterized by several periods Λ n . (c) Discrete periodic radial photonic crystal characterized by azimuthal angle φ. (d) Discrete periodic radial photonic crystal characterized by azimuthal angle φ and radial period Λ.

Fig. 2
Fig. 2

THG with continuous radial symmetry. (a) Longitudinal QPM; the pump beam propagates in the modulation plane. (b) Wave vector diagram for THG. (c) Transverse QPM; the pump beam is orthogonal to the modulation plane. (d) Wave vector diagram for simultaneous conical SHG and THG.

Fig. 3
Fig. 3

Density shaping of noncollinear processes by phase-reversed structures. Phase matching is obtained when the (black) circle (Ewald sphere) intersects with the (blue) circles (Fourier spectrum circles). (a) Periodic annular structure. The noncollinear processes are usually with large walk-off and they are hard to detect experimentally. (b) Phase-reversed structure Λ p h a s e = 3 Λ QPM . (c) Phase reversed structure Λ QPM Λ p hase .

Fig. 4
Fig. 4

Phase matching with continuous angular symmetry. (a) Fourier spectrum of a photonic structure that supports phase matching for all directions. The Ewald construction (red) circle coincides with a circle (blue) upon the Fourier space. (b) Annular periodic photonic structure multiplied with 1D periodic structure. (c) Far-field simulation of SH beam at the crystal output. (d) Far-field simulation of SH beam at the crystal output in a 1D periodic structure. Note that the far-field image of the radial structure is wider than the far-field output of the 1D structure. (e) Scaled intensity of SH accumulated along the propagation path of the beams in z direction. (f) Scaled intensity of SH accumulated along the propagation path of the beams in z direction in a 1D periodic structure.

Fig. 5
Fig. 5

Phase matching arbitrary multiple processes in any direction by radial nonlinear photonic quasi-crystal. (a) Quasi-periodic radial nonlinear photonic crystal (b) Numerical simulation results of SH efficiency of radial photonic structure (shown in the inset) that supports three nonlinear process of 1.53, 1.55, and 1.57 μ m simultaneously. This phase matching applies for each input angle of pump beam, i.e., with radial symmetry. (inset) The Fourier coefficients of the source 1D quasi-periodic structure. (c) Nonlinear radial quasi-periodic structure spanned from 1D quasi-periodic structure (d) that was constructed by DGM. (e) Scheme of phase matching arbitrary processes in any direction.

Fig. 6
Fig. 6

Applications of radial photonic crystals by transverse QPM. (a) Multicolored rings scheme. Each colored ring results from a different diffraction condition. (b) Simulation results of the efficiency of phase-reversed structure that supports transverse phase matching of 1500 nm and 1600 nm . (c) Simulated far-field output of each process–two SH rings of different colors. (d) Discrete radial structure for azimuthal intensity shaping. (e) Azimuthal intensity shaping of SH rings.

Fig. 7
Fig. 7

Azimuthal transparency window for [o-oo] interactions. (a) Polarization and propagation directions of interacting waves in [o-oo] interactions. The polarization of SH generated wave is P local = sin ( 2 γ ) x ̂ + cos ( 2 γ ) y ̂ . The projection of the generated polarization onto the direction of the propagating wave is sin ( 3 γ + ρ ) . (b) Transparency window for longitudinal THG with continuous radial symmetry using [o-oo] interactions for 3 m symmetry point group crystals.

Fig. 8
Fig. 8

Special polarization states by transverse QPM in radial structures. (a) and (b) Radial and azimuthal polarization states of pump beams, respectively. (c) Generated π 2 2 φ polarization state of the SH for both pump polarization states (radial or azimuthal) shown in (a) and (b). In case of THG the TH wave will maintain the polarization state of the pump.

Equations (11)

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g ( r , φ ) = Π m n [ s i g n ( cos ( q n r ) ) s i g n ( cos ( m φ ) ) ] ,
G T 1 = k 2 ω 2 2 k ω 2 ,
G T 2 = k 3 ω 2 3 k ω 2 G T 1 .
g ( r ) = m , n = C m n exp ( i G m n r ) ,
C m n jinc ( m π 2 ) jinc ( n π 2 ) ,
G m n = 2 π m Λ QPM + 2 π n Λ phase ,
cos ( ρ m n ) = ( 2 k ω ) 2 + ( k 2 ω ) 2 G m n 2 4 k ω k 2 ω .
d ( x , y ) = sign ( cos ( x k s hift ) ) sign ( cos ( r k R ) ) .
D ( k x , k y ) = [ δ ( k x k s hift ) + δ ( k x + k s hift ) ] F ( k R ) .
Γ SH ( Γ p ump = γ ) = π 2 2 γ ,
Γ TH ( Γ p ump = γ , Γ SH = π 2 2 γ ) = π 2 γ ( π 2 2 γ ) = γ .

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