Abstract

We experimentally demonstrate the propagation of the conical second harmonic fields generated from a nonlinear crystal with extended defects to investigate their pattern formation. The generated second harmonic waves are found to be the interference of multiple Bessel-like beams that originate from distinct longitudinal layers inside the crystal. To reconstruct the experimental results, we model the individual Bessel-like beam to be the superposition of an ensemble of identical decentered Gaussian waves with random phases. We present that the randomness of the phases leads the Bessel-like beams to show wave profiles with different extent of localization. Moreover, we use the coherent superposition of the developed wave functions with a phase factor to manifest the interference of multiple Bessel-like beams. The relative phases among the Bessel-like beams are shown to be closely related to the near and far-field patterns. With the experimental observations and the theoretical model, the relative phases are decided to successfully reconstruct the propagation characteristics of the multiple Bessel-like beams.

© 2014 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Conical second harmonic generation in KDP crystal assisted by optical elastic scattering

Tianlun Li, Xiaohui Zhao, Yuanlin Zheng, and Xianfeng Chen
Opt. Express 23(18) 23827-23833 (2015)

Nonlinear Raman-Nath second harmonic generation with structured fundamental wave

Haigang Liu, Jun Li, Xiaohui Zhao, Yuanlin Zheng, and Xianfeng Chen
Opt. Express 24(14) 15666-15671 (2016)

Nonlinear Raman-Nath second harmonic generation of hybrid structured fundamental wave

Huaqing Zhou, Haigang Liu, Minghuang Sang, Jun Li, and Xianfeng Chen
Opt. Express 25(4) 3774-3779 (2017)

References

  • View by:
  • |
  • |
  • |

  1. G. S. He and S. H. Liu, Physics of Nonlinear Optics (World Scientific, 1999).
  2. K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical harmonic generation in isotropic materials,” Phys. Rev. Lett. 88(15), 153901 (2002).
    [Crossref] [PubMed]
  3. S. Trillo, C. Conti, P. Di Trapani, O. Jedrkiewicz, J. Trull, G. Valiulis, and G. Bellanca, “Colored conical emission by means of second-harmonic generation,” Opt. Lett. 27(16), 1451–1453 (2002).
    [Crossref] [PubMed]
  4. A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90(24), 243901 (2003).
    [Crossref] [PubMed]
  5. Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
    [Crossref] [PubMed]
  6. H. Zeng, J. Wu, H. Xu, K. Wu, and E. Wu, “Colored conical emission by means of second harmonic generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 92(14), 143903 (2004).
    [Crossref] [PubMed]
  7. G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, “Spatial mode control of radiation generated by frequency difference in periodically poled crystals,” Phys. Rev. Lett. 87(11), 113901 (2001).
    [Crossref] [PubMed]
  8. P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
    [Crossref] [PubMed]
  9. 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]
  10. W. Wang, Y. Sheng, Y. Kong, A. Arie, and W. Krolikowski, “Multiple Čerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure,” Opt. Lett. 35(22), 3790–3792 (2010).
    [Crossref] [PubMed]
  11. J. Chen and X. Chen, “Phase matching in three-dimensional nonlinear photonic crystals,” Phys. Rev. A 80(1), 013801 (2009).
    [Crossref]
  12. L. Mateos, P. Molina, J. Galisteo, C. López, L. E. Bausá, and M. O. Ramírez, “Simultaneous generation of second to fifth harmonic conical beams in a two dimensional nonlinear photonic crystal,” Opt. Express 20(28), 29940–29948 (2012).
    [Crossref] [PubMed]
  13. Y. C. Lin, P. H. Tuan, Y. T. Yu, H. C. Liang, K. W. Su, K. F. Huang, and Y. F. Chen, “Observation of disordered wave functions with conical second-harmonic generation and verification of transition from extended to prelocalized states in weak localization,” Phys. Rev. B 87(4), 045117 (2013).
    [Crossref]
  14. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
    [Crossref]
  15. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt. 43, 1155 (1996).
  16. M. Janssen, Fluctuations and Localization in Mesoscopic Electron Systems (World Scientific, 2001).
  17. K. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University, 1997).
  18. P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Springer, 2006).
  19. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
  20. M. L. Sosnowska, E. Olszyńska, A. Pajączkowska, and A. Kłos, “Extended defects in GdCa4O(BO3)3 crystals,” J. Cryst. Growth 262(1-4), 388–394 (2004).
    [Crossref]
  21. H. Wei and H. Xu, “Hot spots in different metal nanostructures for plasmon-enhanced Raman spectroscopy,” Nanoscale 5(22), 10794–10805 (2013).
    [Crossref] [PubMed]
  22. A. Kudrolli, V. Kidambi, and S. Sridhar, “Experimental studies of chaos and localization in quantum wave functions,” Phys. Rev. Lett. 75(5), 822–825 (1995).
    [Crossref] [PubMed]
  23. M. Moshinsky, “Diffraction in time,” Phys. Rev. 88(3), 625–631 (1952).
    [Crossref]
  24. Č. Brukner and A. Zeilinger, “Diffraction of matter waves in space and in time,” Phys. Rev. A 56(5), 3804–3824 (1997).
    [Crossref]
  25. Th. Hils, J. Felber, R. Gähler, W. Gläser, R. Golub, K. Habicht, and P. Wille, “Matter-wave optics in the time domain: Results of a cold-neutron experiment,” Phys. Rev. A 58(6), 4784–4790 (1998).
    [Crossref]
  26. A. Goussev, “Huygens-Fresnel-Kirchhoff construction for quantum propagators with application to diffraction in space and time,” Phys. Rev. A. 85013626 (2012).
  27. P. O’Connor, J. Gehlen, and E. J. Heller, “Properties of random superpositions of plane waves,” Phys. Rev. Lett. 58(13), 1296–1299 (1987).
    [Crossref] [PubMed]
  28. A. E. Siegman, Lasers (University Science Books, 1986).
  29. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2001).
  30. J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [Crossref] [PubMed]
  31. L. Pitaevskii and S. Stringari, “Interference of bose-einstein condensates in momentum space,” Phys. Rev. Lett. 83(21), 4237–4240 (1999).
    [Crossref]

2013 (2)

Y. C. Lin, P. H. Tuan, Y. T. Yu, H. C. Liang, K. W. Su, K. F. Huang, and Y. F. Chen, “Observation of disordered wave functions with conical second-harmonic generation and verification of transition from extended to prelocalized states in weak localization,” Phys. Rev. B 87(4), 045117 (2013).
[Crossref]

H. Wei and H. Xu, “Hot spots in different metal nanostructures for plasmon-enhanced Raman spectroscopy,” Nanoscale 5(22), 10794–10805 (2013).
[Crossref] [PubMed]

2012 (2)

2010 (1)

2009 (1)

J. Chen and X. Chen, “Phase matching in three-dimensional nonlinear photonic crystals,” Phys. Rev. A 80(1), 013801 (2009).
[Crossref]

2008 (1)

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]

2006 (1)

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

2004 (3)

H. Zeng, J. Wu, H. Xu, K. Wu, and E. Wu, “Colored conical emission by means of second harmonic generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 92(14), 143903 (2004).
[Crossref] [PubMed]

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

M. L. Sosnowska, E. Olszyńska, A. Pajączkowska, and A. Kłos, “Extended defects in GdCa4O(BO3)3 crystals,” J. Cryst. Growth 262(1-4), 388–394 (2004).
[Crossref]

2003 (1)

A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90(24), 243901 (2003).
[Crossref] [PubMed]

2002 (2)

2001 (1)

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, “Spatial mode control of radiation generated by frequency difference in periodically poled crystals,” Phys. Rev. Lett. 87(11), 113901 (2001).
[Crossref] [PubMed]

1999 (1)

L. Pitaevskii and S. Stringari, “Interference of bose-einstein condensates in momentum space,” Phys. Rev. Lett. 83(21), 4237–4240 (1999).
[Crossref]

1998 (1)

Th. Hils, J. Felber, R. Gähler, W. Gläser, R. Golub, K. Habicht, and P. Wille, “Matter-wave optics in the time domain: Results of a cold-neutron experiment,” Phys. Rev. A 58(6), 4784–4790 (1998).
[Crossref]

1997 (1)

Č. Brukner and A. Zeilinger, “Diffraction of matter waves in space and in time,” Phys. Rev. A 56(5), 3804–3824 (1997).
[Crossref]

1996 (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt. 43, 1155 (1996).

1995 (1)

A. Kudrolli, V. Kidambi, and S. Sridhar, “Experimental studies of chaos and localization in quantum wave functions,” Phys. Rev. Lett. 75(5), 822–825 (1995).
[Crossref] [PubMed]

1987 (3)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

P. O’Connor, J. Gehlen, and E. J. Heller, “Properties of random superpositions of plane waves,” Phys. Rev. Lett. 58(13), 1296–1299 (1987).
[Crossref] [PubMed]

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

1952 (1)

M. Moshinsky, “Diffraction in time,” Phys. Rev. 88(3), 625–631 (1952).
[Crossref]

Arie, A.

W. Wang, Y. Sheng, Y. Kong, A. Arie, and W. Krolikowski, “Multiple Čerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure,” Opt. Lett. 35(22), 3790–3792 (2010).
[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]

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt. 43, 1155 (1996).

Bausá, L. E.

Bellanca, G.

Betzler, K.

A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90(24), 243901 (2003).
[Crossref] [PubMed]

Boyd, R. W.

K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical harmonic generation in isotropic materials,” Phys. Rev. Lett. 88(15), 153901 (2002).
[Crossref] [PubMed]

Brukner, C.

Č. Brukner and A. Zeilinger, “Diffraction of matter waves in space and in time,” Phys. Rev. A 56(5), 3804–3824 (1997).
[Crossref]

Cancio, P.

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, “Spatial mode control of radiation generated by frequency difference in periodically poled crystals,” Phys. Rev. Lett. 87(11), 113901 (2001).
[Crossref] [PubMed]

Chen, J.

J. Chen and X. Chen, “Phase matching in three-dimensional nonlinear photonic crystals,” Phys. Rev. A 80(1), 013801 (2009).
[Crossref]

Chen, X.

J. Chen and X. Chen, “Phase matching in three-dimensional nonlinear photonic crystals,” Phys. Rev. A 80(1), 013801 (2009).
[Crossref]

Chen, Y. F.

Y. C. Lin, P. H. Tuan, Y. T. Yu, H. C. Liang, K. W. Su, K. F. Huang, and Y. F. Chen, “Observation of disordered wave functions with conical second-harmonic generation and verification of transition from extended to prelocalized states in weak localization,” Phys. Rev. B 87(4), 045117 (2013).
[Crossref]

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

Conti, C.

De Natale, P.

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, “Spatial mode control of radiation generated by frequency difference in periodically poled crystals,” Phys. Rev. Lett. 87(11), 113901 (2001).
[Crossref] [PubMed]

Di Trapani, P.

Durnin, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Felber, J.

Th. Hils, J. Felber, R. Gähler, W. Gläser, R. Golub, K. Habicht, and P. Wille, “Matter-wave optics in the time domain: Results of a cold-neutron experiment,” Phys. Rev. A 58(6), 4784–4790 (1998).
[Crossref]

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]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt. 43, 1155 (1996).

Gaeta, A. L.

K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical harmonic generation in isotropic materials,” Phys. Rev. Lett. 88(15), 153901 (2002).
[Crossref] [PubMed]

Gähler, R.

Th. Hils, J. Felber, R. Gähler, W. Gläser, R. Golub, K. Habicht, and P. Wille, “Matter-wave optics in the time domain: Results of a cold-neutron experiment,” Phys. Rev. A 58(6), 4784–4790 (1998).
[Crossref]

Galisteo, J.

Gehlen, J.

P. O’Connor, J. Gehlen, and E. J. Heller, “Properties of random superpositions of plane waves,” Phys. Rev. Lett. 58(13), 1296–1299 (1987).
[Crossref] [PubMed]

Giusfredi, G.

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, “Spatial mode control of radiation generated by frequency difference in periodically poled crystals,” Phys. Rev. Lett. 87(11), 113901 (2001).
[Crossref] [PubMed]

Gläser, W.

Th. Hils, J. Felber, R. Gähler, W. Gläser, R. Golub, K. Habicht, and P. Wille, “Matter-wave optics in the time domain: Results of a cold-neutron experiment,” Phys. Rev. A 58(6), 4784–4790 (1998).
[Crossref]

Golub, R.

Th. Hils, J. Felber, R. Gähler, W. Gläser, R. Golub, K. Habicht, and P. Wille, “Matter-wave optics in the time domain: Results of a cold-neutron experiment,” Phys. Rev. A 58(6), 4784–4790 (1998).
[Crossref]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Goussev, A.

A. Goussev, “Huygens-Fresnel-Kirchhoff construction for quantum propagators with application to diffraction in space and time,” Phys. Rev. A. 85013626 (2012).

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Habicht, K.

Th. Hils, J. Felber, R. Gähler, W. Gläser, R. Golub, K. Habicht, and P. Wille, “Matter-wave optics in the time domain: Results of a cold-neutron experiment,” Phys. Rev. A 58(6), 4784–4790 (1998).
[Crossref]

He, J. L.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Heller, E. J.

P. O’Connor, J. Gehlen, and E. J. Heller, “Properties of random superpositions of plane waves,” Phys. Rev. Lett. 58(13), 1296–1299 (1987).
[Crossref] [PubMed]

Hils, Th.

Th. Hils, J. Felber, R. Gähler, W. Gläser, R. Golub, K. Habicht, and P. Wille, “Matter-wave optics in the time domain: Results of a cold-neutron experiment,” Phys. Rev. A 58(6), 4784–4790 (1998).
[Crossref]

Homoelle, D.

K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical harmonic generation in isotropic materials,” Phys. Rev. Lett. 88(15), 153901 (2002).
[Crossref] [PubMed]

Huang, K. F.

Y. C. Lin, P. H. Tuan, Y. T. Yu, H. C. Liang, K. W. Su, K. F. Huang, and Y. F. Chen, “Observation of disordered wave functions with conical second-harmonic generation and verification of transition from extended to prelocalized states in weak localization,” Phys. Rev. B 87(4), 045117 (2013).
[Crossref]

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

Jedrkiewicz, O.

Ji, S. H.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Kidambi, V.

A. Kudrolli, V. Kidambi, and S. Sridhar, “Experimental studies of chaos and localization in quantum wave functions,” Phys. Rev. Lett. 75(5), 822–825 (1995).
[Crossref] [PubMed]

Kivshar, Y. S.

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]

Klos, A.

M. L. Sosnowska, E. Olszyńska, A. Pajączkowska, and A. Kłos, “Extended defects in GdCa4O(BO3)3 crystals,” J. Cryst. Growth 262(1-4), 388–394 (2004).
[Crossref]

Kong, Y.

Krolikowski, W.

W. Wang, Y. Sheng, Y. Kong, A. Arie, and W. Krolikowski, “Multiple Čerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure,” Opt. Lett. 35(22), 3790–3792 (2010).
[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]

Kudrolli, A.

A. Kudrolli, V. Kidambi, and S. Sridhar, “Experimental studies of chaos and localization in quantum wave functions,” Phys. Rev. Lett. 75(5), 822–825 (1995).
[Crossref] [PubMed]

Liang, H. C.

Y. C. Lin, P. H. Tuan, Y. T. Yu, H. C. Liang, K. W. Su, K. F. Huang, and Y. F. Chen, “Observation of disordered wave functions with conical second-harmonic generation and verification of transition from extended to prelocalized states in weak localization,” Phys. Rev. B 87(4), 045117 (2013).
[Crossref]

Lin, Y. C.

Y. C. Lin, P. H. Tuan, Y. T. Yu, H. C. Liang, K. W. Su, K. F. Huang, and Y. F. Chen, “Observation of disordered wave functions with conical second-harmonic generation and verification of transition from extended to prelocalized states in weak localization,” Phys. Rev. B 87(4), 045117 (2013).
[Crossref]

López, C.

Lu, T. H.

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

Mateos, L.

Mazzotti, D.

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, “Spatial mode control of radiation generated by frequency difference in periodically poled crystals,” Phys. Rev. Lett. 87(11), 113901 (2001).
[Crossref] [PubMed]

Miceli, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Ming, N. B.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Molina, P.

Moll, K. D.

K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical harmonic generation in isotropic materials,” Phys. Rev. Lett. 88(15), 153901 (2002).
[Crossref] [PubMed]

Moshinsky, M.

M. Moshinsky, “Diffraction in time,” Phys. Rev. 88(3), 625–631 (1952).
[Crossref]

Neshev, D. N.

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]

O’Connor, P.

P. O’Connor, J. Gehlen, and E. J. Heller, “Properties of random superpositions of plane waves,” Phys. Rev. Lett. 58(13), 1296–1299 (1987).
[Crossref] [PubMed]

Olszynska, E.

M. L. Sosnowska, E. Olszyńska, A. Pajączkowska, and A. Kłos, “Extended defects in GdCa4O(BO3)3 crystals,” J. Cryst. Growth 262(1-4), 388–394 (2004).
[Crossref]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Pajaczkowska, A.

M. L. Sosnowska, E. Olszyńska, A. Pajączkowska, and A. Kłos, “Extended defects in GdCa4O(BO3)3 crystals,” J. Cryst. Growth 262(1-4), 388–394 (2004).
[Crossref]

Pitaevskii, L.

L. Pitaevskii and S. Stringari, “Interference of bose-einstein condensates in momentum space,” Phys. Rev. Lett. 83(21), 4237–4240 (1999).
[Crossref]

Ramírez, M. O.

Saltiel, S. M.

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]

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt. 43, 1155 (1996).

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt. 43, 1155 (1996).

Sheng, Y.

Sosnowska, M. L.

M. L. Sosnowska, E. Olszyńska, A. Pajączkowska, and A. Kłos, “Extended defects in GdCa4O(BO3)3 crystals,” J. Cryst. Growth 262(1-4), 388–394 (2004).
[Crossref]

Spagnolo, G. S.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt. 43, 1155 (1996).

Sridhar, S.

A. Kudrolli, V. Kidambi, and S. Sridhar, “Experimental studies of chaos and localization in quantum wave functions,” Phys. Rev. Lett. 75(5), 822–825 (1995).
[Crossref] [PubMed]

Stringari, S.

L. Pitaevskii and S. Stringari, “Interference of bose-einstein condensates in momentum space,” Phys. Rev. Lett. 83(21), 4237–4240 (1999).
[Crossref]

Su, K. W.

Y. C. Lin, P. H. Tuan, Y. T. Yu, H. C. Liang, K. W. Su, K. F. Huang, and Y. F. Chen, “Observation of disordered wave functions with conical second-harmonic generation and verification of transition from extended to prelocalized states in weak localization,” Phys. Rev. B 87(4), 045117 (2013).
[Crossref]

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

Sun, J.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Trillo, S.

Trull, J.

Tuan, P. H.

Y. C. Lin, P. H. Tuan, Y. T. Yu, H. C. Liang, K. W. Su, K. F. Huang, and Y. F. Chen, “Observation of disordered wave functions with conical second-harmonic generation and verification of transition from extended to prelocalized states in weak localization,” Phys. Rev. B 87(4), 045117 (2013).
[Crossref]

Tunyagi, A. R.

A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90(24), 243901 (2003).
[Crossref] [PubMed]

Ulex, M.

A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90(24), 243901 (2003).
[Crossref] [PubMed]

Valiulis, G.

Wang, H. T.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Wang, W.

Wei, H.

H. Wei and H. Xu, “Hot spots in different metal nanostructures for plasmon-enhanced Raman spectroscopy,” Nanoscale 5(22), 10794–10805 (2013).
[Crossref] [PubMed]

Wille, P.

Th. Hils, J. Felber, R. Gähler, W. Gläser, R. Golub, K. Habicht, and P. Wille, “Matter-wave optics in the time domain: Results of a cold-neutron experiment,” Phys. Rev. A 58(6), 4784–4790 (1998).
[Crossref]

Wu, E.

H. Zeng, J. Wu, H. Xu, K. Wu, and E. Wu, “Colored conical emission by means of second harmonic generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 92(14), 143903 (2004).
[Crossref] [PubMed]

Wu, J.

H. Zeng, J. Wu, H. Xu, K. Wu, and E. Wu, “Colored conical emission by means of second harmonic generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 92(14), 143903 (2004).
[Crossref] [PubMed]

Wu, K.

H. Zeng, J. Wu, H. Xu, K. Wu, and E. Wu, “Colored conical emission by means of second harmonic generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 92(14), 143903 (2004).
[Crossref] [PubMed]

Xu, H.

H. Wei and H. Xu, “Hot spots in different metal nanostructures for plasmon-enhanced Raman spectroscopy,” Nanoscale 5(22), 10794–10805 (2013).
[Crossref] [PubMed]

H. Zeng, J. Wu, H. Xu, K. Wu, and E. Wu, “Colored conical emission by means of second harmonic generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 92(14), 143903 (2004).
[Crossref] [PubMed]

Xu, P.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Yu, X. Q.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Yu, Y. T.

Y. C. Lin, P. H. Tuan, Y. T. Yu, H. C. Liang, K. W. Su, K. F. Huang, and Y. F. Chen, “Observation of disordered wave functions with conical second-harmonic generation and verification of transition from extended to prelocalized states in weak localization,” Phys. Rev. B 87(4), 045117 (2013).
[Crossref]

Zeilinger, A.

Č. Brukner and A. Zeilinger, “Diffraction of matter waves in space and in time,” Phys. Rev. A 56(5), 3804–3824 (1997).
[Crossref]

Zeng, H.

H. Zeng, J. Wu, H. Xu, K. Wu, and E. Wu, “Colored conical emission by means of second harmonic generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 92(14), 143903 (2004).
[Crossref] [PubMed]

Zhu, S. N.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

Zhu, Y. Y.

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[Crossref] [PubMed]

J. Cryst. Growth (1)

M. L. Sosnowska, E. Olszyńska, A. Pajączkowska, and A. Kłos, “Extended defects in GdCa4O(BO3)3 crystals,” J. Cryst. Growth 262(1-4), 388–394 (2004).
[Crossref]

J. Mod. Opt. (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt. 43, 1155 (1996).

Nanoscale (1)

H. Wei and H. Xu, “Hot spots in different metal nanostructures for plasmon-enhanced Raman spectroscopy,” Nanoscale 5(22), 10794–10805 (2013).
[Crossref] [PubMed]

Opt. Commun. (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. (1)

M. Moshinsky, “Diffraction in time,” Phys. Rev. 88(3), 625–631 (1952).
[Crossref]

Phys. Rev. A (3)

Č. Brukner and A. Zeilinger, “Diffraction of matter waves in space and in time,” Phys. Rev. A 56(5), 3804–3824 (1997).
[Crossref]

Th. Hils, J. Felber, R. Gähler, W. Gläser, R. Golub, K. Habicht, and P. Wille, “Matter-wave optics in the time domain: Results of a cold-neutron experiment,” Phys. Rev. A 58(6), 4784–4790 (1998).
[Crossref]

J. Chen and X. Chen, “Phase matching in three-dimensional nonlinear photonic crystals,” Phys. Rev. A 80(1), 013801 (2009).
[Crossref]

Phys. Rev. A. (1)

A. Goussev, “Huygens-Fresnel-Kirchhoff construction for quantum propagators with application to diffraction in space and time,” Phys. Rev. A. 85013626 (2012).

Phys. Rev. B (1)

Y. C. Lin, P. H. Tuan, Y. T. Yu, H. C. Liang, K. W. Su, K. F. Huang, and Y. F. Chen, “Observation of disordered wave functions with conical second-harmonic generation and verification of transition from extended to prelocalized states in weak localization,” Phys. Rev. B 87(4), 045117 (2013).
[Crossref]

Phys. Rev. Lett. (11)

A. Kudrolli, V. Kidambi, and S. Sridhar, “Experimental studies of chaos and localization in quantum wave functions,” Phys. Rev. Lett. 75(5), 822–825 (1995).
[Crossref] [PubMed]

K. D. Moll, D. Homoelle, A. L. Gaeta, and R. W. Boyd, “Conical harmonic generation in isotropic materials,” Phys. Rev. Lett. 88(15), 153901 (2002).
[Crossref] [PubMed]

A. R. Tunyagi, M. Ulex, and K. Betzler, “Noncollinear optical frequency doubling in strontium barium niobate,” Phys. Rev. Lett. 90(24), 243901 (2003).
[Crossref] [PubMed]

Y. F. Chen, K. W. Su, T. H. Lu, and K. F. Huang, “Manifestation of weak localization and long-range correlation in disordered wave functions from conical second harmonic generation,” Phys. Rev. Lett. 96(3), 033905 (2006).
[Crossref] [PubMed]

H. Zeng, J. Wu, H. Xu, K. Wu, and E. Wu, “Colored conical emission by means of second harmonic generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 92(14), 143903 (2004).
[Crossref] [PubMed]

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, “Spatial mode control of radiation generated by frequency difference in periodically poled crystals,” Phys. Rev. Lett. 87(11), 113901 (2001).
[Crossref] [PubMed]

P. Xu, S. H. Ji, S. N. Zhu, X. Q. Yu, J. Sun, H. T. Wang, J. L. He, Y. Y. Zhu, and N. B. Ming, “Conical second harmonic generation in a two-dimensional χ(2) photonic crystal: a hexagonally poled LiTaO3 crystal,” Phys. Rev. Lett. 93(13), 133904 (2004).
[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]

P. O’Connor, J. Gehlen, and E. J. Heller, “Properties of random superpositions of plane waves,” Phys. Rev. Lett. 58(13), 1296–1299 (1987).
[Crossref] [PubMed]

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

L. Pitaevskii and S. Stringari, “Interference of bose-einstein condensates in momentum space,” Phys. Rev. Lett. 83(21), 4237–4240 (1999).
[Crossref]

Other (7)

A. E. Siegman, Lasers (University Science Books, 1986).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2001).

G. S. He and S. H. Liu, Physics of Nonlinear Optics (World Scientific, 1999).

M. Janssen, Fluctuations and Localization in Mesoscopic Electron Systems (World Scientific, 2001).

K. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University, 1997).

P. Sheng, Introduction to Wave Scattering, Localization and Mesoscopic Phenomena (Springer, 2006).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 Experimental setup for the generation of conical SH waves in a diode-pumped Q-switched Nd:YAG laser.
Fig. 2
Fig. 2 (a) Schematic diagram for the phase-matching condition of the conical SHG process, where k 1 and k 1 are the wave vectors for the axial and scattered off-axis fundamental fields, and k 2 is the wave vector for the phase-matched decentered SH field. (b)-(f) Experimental results for the propagation of the generated SH wave.
Fig. 3
Fig. 3 (a) Schematic diagram for the fundamental input beam passing through scatters to form two hot-spots located on different layers, where the dashed arrows and the solid arrows signify the wave vectors of the fundamental and the SH fields. (b)-(p) Propagation of various experimental SH waves.
Fig. 4
Fig. 4 Propagation of the experimental extended state.
Fig. 5
Fig. 5 (a)-(e) Theoretical simulations of U o ( r , ϕ ; Δ ) with Δ varying from 0 to 2 π .
Fig. 6
Fig. 6 (a)-(e) Reconstructed patterns U ( r , ϕ , z ; Δ ) for the propagation of the experimental results in Figs. 2(b)-2(f).
Fig. 7
Fig. 7 (a)-(e) Theoretical simulations of V o ( r , ϕ ) for two Bessel-like beams that originate from different layers. (f)-(j) Corresponding far-field simulations of the distributions V ( r , ϕ , z ) .
Fig. 8
Fig. 8 Theoretical simulations for the experimental results in Fig. 3.
Fig. 9
Fig. 9 (a) Schematic diagram for the hot-spots that randomly spread in the 3D nonlinear medium. (b)-(f) Corresponding theoretical simulations of the extended state in Fig. 4.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

u o ( r , ϕ ; α ) = exp [ ( r ω o ) 2 ] exp [ i κ r cos ( α ϕ ) ] ,
U B G ( r , ϕ ) = 0 2 π u o ( r , ϕ ; α ) d α = 2 π exp ( r 2 ω 0 2 ) J 0 ( κ r ) .
U o ( r , ϕ ; Δ ) = q = 0 M 1 exp ( i θ q ) u o ( r , ϕ ; α q ) 2 π M ,
U ( r , ϕ , z ; Δ ) = k 2 exp ( i k 2 z ) i 2 π z 0 0 2 π U o ( r , ϕ ; Δ ) exp ( i k 2 | r r | 2 2 z ) r d ϕ d r .
U ( r , ϕ , z ; Δ ) = q = 0 M 1 exp ( i θ q ) u ( r , ϕ , z ; α q ) 2 π M ,
u ( r , ϕ , z ; α q ) = ω o ω ( z ) exp { i [ k 2 z κ 2 z 2 k 2 Φ ( z ) ] } × exp { F ( z ) [ r 2 + ( κ z k 2 ) 2 2 κ z r k 2 cos ( α q ϕ ) ] } , × exp [ i κ r cos ( α q ϕ ) ]
V o ( r , ϕ ) = n = 0 N 1 exp ( i δ n ) U o n ( | r r n | , ϕ ; Δ n ) ,
V ( r , ϕ , z ) = n = 0 N 1 exp ( i δ n ) U n ( | r r n | , ϕ , z ; Δ n ) .
S = I exp ( x , y ) I s i m ( x , y ) d x d y ,

Metrics