Abstract

We show that the weak second harmonic light generated from a random distribution of nonlinear domains of transparent Strontium Barium Niobate crystals can display a particularly intense generation in the forward direction. By using a theoretical model able to analyze the optical response of arbitrary distributions of three-dimensional nonlinear volumes of any shape, we found that the physical origin of this observation can be explained in terms of the scattering of light by a single nonlinear domain.

© 2010 Optical Society of America

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  1. R. C. Miller, “Optical harmonic generation in single crystal BaTiO3,” Phys. Rev. 134, A1313–A1319 (1964).
    [CrossRef]
  2. C. F. Dewey, Jr., and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
    [CrossRef]
  3. M. Baudrier-Raybaut, R. Hadar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374–376 (2004).
    [CrossRef] [PubMed]
  4. X. Vidal, and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
    [CrossRef] [PubMed]
  5. S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
    [CrossRef]
  6. R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006).
    [CrossRef]
  7. J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express 15, 15868–15877 (2007).
    [CrossRef] [PubMed]
  8. See for instance, A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997).
  9. P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
    [CrossRef]
  10. J.-X. Cheng, and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002).
    [CrossRef]
  11. E. Yew, and C. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express 14, 1167–1174 (2006).
    [CrossRef] [PubMed]
  12. See for instance, L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge Univ. Press, Cambridge, 2006), pp. 25–30.
  13. R. W. Boyd, Nonlinear Optics, 2nd Ed. (Academic, San Diego, 2003), pp. 37–38.
  14. D. L. Andrews, P. Allcock, and A. A. Demidov, “Theory of second harmonic generation in randomly oriented species,” Chem. Phys. 190, 1–9 (1995).
    [CrossRef]
  15. S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
    [CrossRef] [PubMed]

2007

2006

E. Yew, and C. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express 14, 1167–1174 (2006).
[CrossRef] [PubMed]

X. Vidal, and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[CrossRef] [PubMed]

R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006).
[CrossRef]

2004

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

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef] [PubMed]

2002

1998

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[CrossRef]

1995

D. L. Andrews, P. Allcock, and A. A. Demidov, “Theory of second harmonic generation in randomly oriented species,” Chem. Phys. 190, 1–9 (1995).
[CrossRef]

1975

C. F. Dewey, Jr., and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[CrossRef]

1964

R. C. Miller, “Optical harmonic generation in single crystal BaTiO3,” Phys. Rev. 134, A1313–A1319 (1964).
[CrossRef]

1962

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Allcock, P.

D. L. Andrews, P. Allcock, and A. A. Demidov, “Theory of second harmonic generation in randomly oriented species,” Chem. Phys. 190, 1–9 (1995).
[CrossRef]

Andrews, D. L.

D. L. Andrews, P. Allcock, and A. A. Demidov, “Theory of second harmonic generation in randomly oriented species,” Chem. Phys. 190, 1–9 (1995).
[CrossRef]

Baudrier-Raybaut, M.

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

Botzung-Appert, E.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef] [PubMed]

Brasselet, S.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef] [PubMed]

Cheng, J.-X.

Cojocaru, C.

DeMattei, R. C.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[CrossRef]

Demidov, A. A.

D. L. Andrews, P. Allcock, and A. A. Demidov, “Theory of second harmonic generation in randomly oriented species,” Chem. Phys. 190, 1–9 (1995).
[CrossRef]

Dewey, C. F.

C. F. Dewey, Jr., and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[CrossRef]

Feigelson, R. S.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[CrossRef]

Fischer, R.

Hadar, R.

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

Herrero, R.

Hocker, L. O.

C. F. Dewey, Jr., and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[CrossRef]

Ibanez, A.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef] [PubMed]

Kawai, S.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[CrossRef]

Kivshar, Y. S.

Kivshar, Yu. S.

R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006).
[CrossRef]

Krolikowski, W.

Kupecek, Ph.

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

Le Floch, V.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef] [PubMed]

Lee, H. S.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[CrossRef]

Lemasson, Ph.

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

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Martorell, J.

X. Vidal, and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[CrossRef] [PubMed]

Miller, R. C.

R. C. Miller, “Optical harmonic generation in single crystal BaTiO3,” Phys. Rev. 134, A1313–A1319 (1964).
[CrossRef]

Neshev, D. N.

Nisenhoff, M.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Ogawa, T.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[CrossRef]

Roch, J.-F.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef] [PubMed]

Rosencher, E.

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

Saltiel, S. M.

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Sheppard, C.

Staliunas, K.

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Treussart, F.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef] [PubMed]

Trull, J.

Vidal, X.

X. Vidal, and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[CrossRef] [PubMed]

Vilaseca, R.

Xie, X. S.

Yew, E.

Zyss, J.

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef] [PubMed]

Appl. Phys. Lett.

C. F. Dewey, Jr., and L. O. Hocker, “Enhanced nonlinear optical effects in rotationally twinned crystals,” Appl. Phys. Lett. 26, 442–444 (1975).
[CrossRef]

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, “Second-harmonic generation from needlelike ferroelectric domains in Sr0.6Ba0.4Nd2O6 single crystals,” Appl. Phys. Lett. 73, 768–770 (1998).
[CrossRef]

R. Fischer, S. M. Saltiel, D. N. Neshev, W. Krolikowski, and Yu. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89, 191105 (2006).
[CrossRef]

Chem. Phys.

D. L. Andrews, P. Allcock, and A. A. Demidov, “Theory of second harmonic generation in randomly oriented species,” Chem. Phys. 190, 1–9 (1995).
[CrossRef]

J. Opt. Soc. Am. B

Nature

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

Opt. Express

Phys. Rev.

R. C. Miller, “Optical harmonic generation in single crystal BaTiO3,” Phys. Rev. 134, A1313–A1319 (1964).
[CrossRef]

Phys. Rev. Lett.

X. Vidal, and J. Martorell, “Generation of light in media with a random distribution of nonlinear domains,” Phys. Rev. Lett. 97, 013902 (2006).
[CrossRef] [PubMed]

S. Brasselet, V. Le Floch, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
[CrossRef] [PubMed]

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Other

See for instance, L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge Univ. Press, Cambridge, 2006), pp. 25–30.

R. W. Boyd, Nonlinear Optics, 2nd Ed. (Academic, San Diego, 2003), pp. 37–38.

See for instance, A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, New York, 1997).

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Figures (3)

Fig. 1.
Fig. 1.

(a) SEM picture of one of the SBN crystals used in the experiments. Main panel displays one of the surfaces parallel to the c-axis. The picture was taken after etching was carried out to make the nonlinear domains visible. The inset corresponds to an optical microscope picture taken from a surface perpendicular to the c-axis. (b) Picture of the second-harmonic light ring generated by the considered SBN crystal when the fundamental beam is propagating along the crystallographic c-axis. The spot in the center corresponds to a third harmonic generation in the direction of the incoming pump beam. Inset shows the experimental second harmonic intensity measured along one diameter of the generated light ring displayed in the main panel. The white arrows in the inset mark the angular locations of the secondary ring observed in the main figure. (c) Second-harmonic intensity scattered on a plane perpendicular to the c-axis of the SBN crystal as a function of the in-plane observation angle ϕobs . Left inset shows a picture taken from the light generation in this configuration. Right inset displays the second-harmonic peak power as a function of the thickness of the SBN crystal (blue circles). Red line in right inset corresponds to a linear fit through the data points that intersects with the origin.

Fig. 2.
Fig. 2.

(a),(b): Schematics of the two different configurations considered in the theoretical analysis. (c),(d): Computed far-field second-harmonic light distributions from a single nonlinear three-dimensional cylindrical volume (diameter dc =13µm and height h=26µm) obtained for the case in which the fundamental beam is propagating in the direction perpendicular and parallel to the c-axis, respectively. (e),(f): As in (c) and (d), respectively, but for a nonlinear volume consisting of a three-dimensional parallelepiped with rounded edges.

Fig. 3.
Fig. 3.

(a) Simulated average far-field second-harmonic emission pattern for a random distribution of domains (see text for details on the distribution) computed for the configuration in which the fundamental beam propagates along the c-axis. Right inset: second harmonic intensity measured along one diameter of the generated light ring displayed in the main panel. Black arrows in this inset mark the locations of the inner right observed in the corresponding main figure. (b) Simulated SHG from the random distribution of nonlinear domains considered in (a), but now assuming that the fundamental beam propagates normally to the c-axis. Top inset: same as black line in main figure, but plotted now in linear scale. Bottom inset: calculated growth of second harmonic versus the number of cylinders. (c) Simulated second-harmonic intensity computed for random structures formed by a combination of nonlinear cylinders and rectangular parallelepipeds with rounded corners. Results for structures in which the percentage of nonlinear volumes that are parallelepipeds ranges from 0 (all the nonlinear volumes in the random structure are cylinders, see black line) to 75% (i.e., 25% of volumes in the random structure are cylinders, see cyan line) are shown.

Equations (3)

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E ( 2 ω ) ( r ) = ( 2 ω ) 2 c 2 V NL d r G ̂ ( r , r ) P ( 2 ω ) ( r )
E ( 2 ω ) ( r ) = ( 2 ω ) 2 c 2 [ E 0 ( ω ) ] 2 exp ( i k ( 2 ω ) r obs ) 4 π r obs I ̂ Δ k ( ϕ obs , θ obs ) p ( 2 ω )
I ( 2 ω ) ( r ) = ( 2 ω 2 k ( 2 ω ) 2 ε 0 c 3 n 2 n 1 2 ) ( d c 4 h 2 r obs 2 ) d eff 2 [ I ( ω ) ] 2 sinc 2 [ Δ k z ( ϕ obs , θ obs ) h 2 ] { J 1 [ Δ k ( ϕ obs , θ obs ) r c ] Δ k ( ϕ obs , θ obs ) r c } 2

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