Abstract

We numerically and experimentally investigated the nonlinear Raman-Nath second harmonic generation of hybrid structured fundamental wave whose phase modulation combined periodic and random structure. The second harmonic generation of both one- and two-dimensional hybrid structured fundamental wave were investigated in this paper. The results show that more diffraction spots can be obtained in these hybrid structures than the pure periodic modulation cases. Besides, the total intensity of the second harmonic not only can be dramatically enhanced without altering the diffraction angles, but also is increasing with the degree of randomness of the structure. This study enriches the family of second harmonic generation of structured fundamental wave and has potential application in dynamically controlling second harmonic wave in arbitrary directions.

© 2017 Optical Society of America

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References

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    [Crossref]
  2. S. Das Sarma, A. Kobayashi, and R. E. Prange, “Plasmons in aperiodic structures,” Phys. Rev. B Condens. Matter 34(8), 5309–5314 (1986).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  13. A. M. Vyunishev, A. S. Aleksandrovsky, A. I. Zaitsev, and V. V. Slabko, “Čerenkov nonlinear diffraction in random nonlinear photonic crystal of strontium tetraborate,” Appl. Phys. Lett. 101(21), 211114 (2012).
    [Crossref]
  14. S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett. 34(6), 848–850 (2009).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  17. A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two-dimensional nonlinear beam shaping,” Opt. Lett. 37(11), 2136–2138 (2012).
    [Crossref] [PubMed]
  18. H. Liu, J. Li, X. Zhao, Y. Zheng, and X. Chen, “Nonlinear Raman-Nath second harmonic generation with structured fundamental wave,” Opt. Express 24(14), 15666–15671 (2016).
    [Crossref] [PubMed]
  19. H. Liu, J. Li, X. Fang, X. Zhao, Y. Zheng, and X. Chen, “Scattering-assisted second harmonic generation of structured fundamental wave,” Opt. Express 24(21), 24137–24142 (2016).
    [Crossref] [PubMed]
  20. Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman-Nath diffraction in the frequency doubling process,” J. Phys. B 45(5), 055401 (2012).
    [Crossref]
  21. A. M. Vyunishev, V. V. Slabko, I. S. Baturin, A. R. Akhmatkhanov, and V. Ya. Shur, “Nonlinear Raman-Nath diffraction of femtosecond laser pulses,” Opt. Lett. 39(14), 4231–4234 (2014).
    [Crossref] [PubMed]
  22. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
    [Crossref]

2016 (2)

2014 (1)

2013 (2)

Y. Chen, W. Dang, Y. Zheng, X. Chen, and X. Deng, “Spatial modulation of second-harmonic generation via nonlinear Raman-Nath diffraction in an aperiodically poled lithium tantalite,” Opt. Lett. 38(13), 2298–2300 (2013).
[Crossref] [PubMed]

Y. Sheng, D. L. Ma, and A. Krolikowski, “Randomized nonlinear photonic crystal for broadband optical frequency conversion,” J. Phys. At. Mol. Opt. Phys. 46(21), 215401 (2013).
[Crossref]

2012 (5)

K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Imbrock, C. Denz, and W. Krolikowski, “Enhanced Čerenkov second-harmonic emission in nonlinear photonic structures,” Opt. Lett. 37(11), 1832–1834 (2012).
[Crossref] [PubMed]

A. M. Vyunishev, A. S. Aleksandrovsky, A. I. Zaitsev, and V. V. Slabko, “Čerenkov nonlinear diffraction in random nonlinear photonic crystal of strontium tetraborate,” Appl. Phys. Lett. 101(21), 211114 (2012).
[Crossref]

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[Crossref] [PubMed]

A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two-dimensional nonlinear beam shaping,” Opt. Lett. 37(11), 2136–2138 (2012).
[Crossref] [PubMed]

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman-Nath diffraction in the frequency doubling process,” J. Phys. B 45(5), 055401 (2012).
[Crossref]

2010 (1)

A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photonics Rev. 4(3), 355–373 (2010).
[Crossref]

2009 (1)

2008 (2)

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

A. Tehranchi and R. Kashyap, “Design of novel unapodized and apodized step-chirped quasi-phase matched gratings for broadband frequency converters based on second-harmonic generation,” J. Lightwave Technol. 26(3), 343–349 (2008).
[Crossref]

2004 (1)

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

1998 (1)

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998).
[Crossref]

1995 (2)

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref] [PubMed]

1986 (1)

S. Das Sarma, A. Kobayashi, and R. E. Prange, “Plasmons in aperiodic structures,” Phys. Rev. B Condens. Matter 34(8), 5309–5314 (1986).
[Crossref] [PubMed]

1984 (1)

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
[Crossref]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. Lett. 127(6), 1918 (1962).

Akhmatkhanov, A. R.

Aleksandrovsky, A. S.

A. M. Vyunishev, A. S. Aleksandrovsky, A. I. Zaitsev, and V. V. Slabko, “Čerenkov nonlinear diffraction in random nonlinear photonic crystal of strontium tetraborate,” Appl. Phys. Lett. 101(21), 211114 (2012).
[Crossref]

Arie, A.

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[Crossref] [PubMed]

A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two-dimensional nonlinear beam shaping,” Opt. Lett. 37(11), 2136–2138 (2012).
[Crossref] [PubMed]

A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photonics Rev. 4(3), 355–373 (2010).
[Crossref]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett. 34(6), 848–850 (2009).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. Lett. 127(6), 1918 (1962).

Ayoub, M.

Bang, O.

Baturin, I. S.

Baudrier-Raybaut, M.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

Berger, V.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81(19), 4136–4139 (1998).
[Crossref]

Bloch, N. V.

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[Crossref] [PubMed]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. Lett. 127(6), 1918 (1962).

Bosenberg, W. R.

Byer, R. L.

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref] [PubMed]

Chen, X.

Chen, Y.

Dang, W.

Das Sarma, S.

S. Das Sarma, A. Kobayashi, and R. E. Prange, “Plasmons in aperiodic structures,” Phys. Rev. B Condens. Matter 34(8), 5309–5314 (1986).
[Crossref] [PubMed]

Deng, X.

Denz, C.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. Lett. 127(6), 1918 (1962).

Eckardt, R. C.

Edwards, G. J.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
[Crossref]

Fang, X.

Fejer, M. M.

Fischer, R.

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Haïdar, R.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref] [PubMed]

Hof, A.

A. Hof, “On diffraction by aperiodic structures,” Commun. Math. Phys. 169(1), 25–43 (1995).
[Crossref]

Imbrock, J.

Juwiler, I.

A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two-dimensional nonlinear beam shaping,” Opt. Lett. 37(11), 2136–2138 (2012).
[Crossref] [PubMed]

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[Crossref] [PubMed]

Kalinowski, K.

K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Imbrock, C. Denz, and W. Krolikowski, “Enhanced Čerenkov second-harmonic emission in nonlinear photonic structures,” Opt. Lett. 37(11), 1832–1834 (2012).
[Crossref] [PubMed]

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman-Nath diffraction in the frequency doubling process,” J. Phys. B 45(5), 055401 (2012).
[Crossref]

Kashyap, R.

Kivshar, Y. S.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett. 34(6), 848–850 (2009).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Kobayashi, A.

S. Das Sarma, A. Kobayashi, and R. E. Prange, “Plasmons in aperiodic structures,” Phys. Rev. B Condens. Matter 34(8), 5309–5314 (1986).
[Crossref] [PubMed]

Kong, Q.

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman-Nath diffraction in the frequency doubling process,” J. Phys. B 45(5), 055401 (2012).
[Crossref]

Krolikowski, A.

Y. Sheng, D. L. Ma, and A. Krolikowski, “Randomized nonlinear photonic crystal for broadband optical frequency conversion,” J. Phys. At. Mol. Opt. Phys. 46(21), 215401 (2013).
[Crossref]

Krolikowski, W.

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman-Nath diffraction in the frequency doubling process,” J. Phys. B 45(5), 055401 (2012).
[Crossref]

K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Imbrock, C. Denz, and W. Krolikowski, “Enhanced Čerenkov second-harmonic emission in nonlinear photonic structures,” Opt. Lett. 37(11), 1832–1834 (2012).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett. 34(6), 848–850 (2009).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Kupecek, P.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

Lawrence, M.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984).
[Crossref]

Lemasson, P.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

Li, J.

Liu, H.

Ma, D. L.

Y. Sheng, D. L. Ma, and A. Krolikowski, “Randomized nonlinear photonic crystal for broadband optical frequency conversion,” J. Phys. At. Mol. Opt. Phys. 46(21), 215401 (2013).
[Crossref]

Miller, G. D.

Myers, L. E.

Neshev, D. N.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett. 34(6), 848–850 (2009).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. Lett. 127(6), 1918 (1962).

Prange, R. E.

S. Das Sarma, A. Kobayashi, and R. E. Prange, “Plasmons in aperiodic structures,” Phys. Rev. B Condens. Matter 34(8), 5309–5314 (1986).
[Crossref] [PubMed]

Roedig, P.

Rosencher, E.

M. Baudrier-Raybaut, R. Haïdar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004).
[Crossref] [PubMed]

Saltiel, S. M.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett. 34(6), 848–850 (2009).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100(10), 103902 (2008).
[Crossref] [PubMed]

Shapira, A.

A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two-dimensional nonlinear beam shaping,” Opt. Lett. 37(11), 2136–2138 (2012).
[Crossref] [PubMed]

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[Crossref] [PubMed]

Shemer, K.

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[Crossref] [PubMed]

Sheng, Y.

Y. Sheng, D. L. Ma, and A. Krolikowski, “Randomized nonlinear photonic crystal for broadband optical frequency conversion,” J. Phys. At. Mol. Opt. Phys. 46(21), 215401 (2013).
[Crossref]

K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Imbrock, C. Denz, and W. Krolikowski, “Enhanced Čerenkov second-harmonic emission in nonlinear photonic structures,” Opt. Lett. 37(11), 1832–1834 (2012).
[Crossref] [PubMed]

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman-Nath diffraction in the frequency doubling process,” J. Phys. B 45(5), 055401 (2012).
[Crossref]

Shiloh, R.

A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Two-dimensional nonlinear beam shaping,” Opt. Lett. 37(11), 2136–2138 (2012).
[Crossref] [PubMed]

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[Crossref] [PubMed]

Shur, V. Ya.

Slabko, V. V.

A. M. Vyunishev, V. V. Slabko, I. S. Baturin, A. R. Akhmatkhanov, and V. Ya. Shur, “Nonlinear Raman-Nath diffraction of femtosecond laser pulses,” Opt. Lett. 39(14), 4231–4234 (2014).
[Crossref] [PubMed]

A. M. Vyunishev, A. S. Aleksandrovsky, A. I. Zaitsev, and V. V. Slabko, “Čerenkov nonlinear diffraction in random nonlinear photonic crystal of strontium tetraborate,” Appl. Phys. Lett. 101(21), 211114 (2012).
[Crossref]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65(25), 3152–3155 (1990).
[Crossref] [PubMed]

Tehranchi, A.

Voloch, N.

A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photonics Rev. 4(3), 355–373 (2010).
[Crossref]

Vyunishev, A. M.

A. M. Vyunishev, V. V. Slabko, I. S. Baturin, A. R. Akhmatkhanov, and V. Ya. Shur, “Nonlinear Raman-Nath diffraction of femtosecond laser pulses,” Opt. Lett. 39(14), 4231–4234 (2014).
[Crossref] [PubMed]

A. M. Vyunishev, A. S. Aleksandrovsky, A. I. Zaitsev, and V. V. Slabko, “Čerenkov nonlinear diffraction in random nonlinear photonic crystal of strontium tetraborate,” Appl. Phys. Lett. 101(21), 211114 (2012).
[Crossref]

Wang, W.

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman-Nath diffraction in the frequency doubling process,” J. Phys. B 45(5), 055401 (2012).
[Crossref]

Zaitsev, A. I.

A. M. Vyunishev, A. S. Aleksandrovsky, A. I. Zaitsev, and V. V. Slabko, “Čerenkov nonlinear diffraction in random nonlinear photonic crystal of strontium tetraborate,” Appl. Phys. Lett. 101(21), 211114 (2012).
[Crossref]

Zhao, X.

Zheng, Y.

Appl. Phys. Lett. (1)

A. M. Vyunishev, A. S. Aleksandrovsky, A. I. Zaitsev, and V. V. Slabko, “Čerenkov nonlinear diffraction in random nonlinear photonic crystal of strontium tetraborate,” Appl. Phys. Lett. 101(21), 211114 (2012).
[Crossref]

Commun. Math. Phys. (1)

A. Hof, “On diffraction by aperiodic structures,” Commun. Math. Phys. 169(1), 25–43 (1995).
[Crossref]

J. Lightwave Technol. (1)

J. Phys. At. Mol. Opt. Phys. (1)

Y. Sheng, D. L. Ma, and A. Krolikowski, “Randomized nonlinear photonic crystal for broadband optical frequency conversion,” J. Phys. At. Mol. Opt. Phys. 46(21), 215401 (2013).
[Crossref]

J. Phys. B (1)

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman-Nath diffraction in the frequency doubling process,” J. Phys. B 45(5), 055401 (2012).
[Crossref]

Laser Photonics Rev. (1)

A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photonics Rev. 4(3), 355–373 (2010).
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Figures (6)

Fig. 1
Fig. 1 Schematic of the hybrid structured FW whose phase modulation combine periodic and random structure.
Fig. 2
Fig. 2 (a)(b) The calculated spectral density of the SH field S 2 ( k x , y ) as the function of the external emission angle corresponding to the pure periodic modulation cases of ϕ = π / 2 and ϕ = π , respectively.
Fig. 3
Fig. 3 (a) The hybrid structured FW with ϕ = π / 2 . (b) The calculated spectral density of the SH field S 2 ( k x , y ) as the function of the external emission angle corresponding to (a). (c)(d) The experimental results of the Raman-Nath SH signals of the pure periodic and hybrid structure of the FW, respectively.
Fig. 4
Fig. 4 (a) The hybrid structured FW with ϕ = π . (b) The calculated spectral density of the SH field S 2 ( k x , y ) as the function of the external emission angle corresponding to (a). (c)(d) The experimental results of the Raman-Nath SH signals of the pure periodic and hybrid structure of FW, respectively.
Fig. 5
Fig. 5 (a) The ring shaped hybrid structured FW with ϕ = π / 2 . (b) The calculated spectral density of the SH field S 2 ( k x , k z , y ) as the function of the external emission angle corresponding to (a). (c) The experimental results of the Raman-Nath SH signals of the hybrid modulation.
Fig. 6
Fig. 6 (a) The experimental results of different randomness of the structure. (b) The intensity of the SH as a function of the number of random bar.

Equations (4)

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E 1 = A 1 ( x ) exp [ i ( k 1 y ω t ) ] m = + C m exp ( i m G 0 x ) ,
( y + i 2 k 2 2 x 2 ) A 2 ( x , y ) = i β 2 A 1 2 ( x ) exp ( i Δ k y ) [ m = + C m exp ( i m G 0 x ) ] 2 ,
A 2 ( k x , y ) = a π 2 A 1 2 β 2 y exp ( i k x 2 2 k 2 y ) sin c [ y ( Δ k k x 2 2 k 2 ) / 2 ] q = + b q exp [ a 2 ( q G 0 + k x ) 2 / 8 ] ,
S 2 ( k x , y ) = 1 2 π a 2 β 2 2 y 2 A 1 4 { sin c [ y ( Δ k k x 2 2 k 2 ) / 2 ] } 2 { q = + b q exp [ a 2 ( q G 0 + k x ) 2 / 8 ] } 2 .

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