Abstract

We have experimentally measured the distribution of the second-harmonic intensity that is generated inside a highly scattering slab of porous gallium phosphide. Two complementary techniques for determining the distribution are used. First, the spatial distribution of second-harmonic light intensity at the side of a cleaved slab has been recorded. Second, the total second-harmonic radiation at each side of the slab has been measured for several samples at various wavelengths. By combining these measurements with a diffusion model for second-harmonic generation that incorporates extrapolated boundary conditions, we present a consistent picture of the distribution of the second-harmonic intensity inside the slab. We find that the ratio l2ωLc of the mean free path at the second-harmonic frequency to the coherence length, which was suggested by some earlier calculations, cannot describe the second-harmonic yield in our samples. For describing the total second-harmonic yield, our experiments show that the scattering parameter at the fundamental frequency k1ωl1ω is the most relevant parameter in our type of samples.

© 2009 Optical Society of America

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References

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  1. P. J. Campagnola and L. M. Loew, “Second-harmonic imaging microscopy for visualizing biomolecular arrays in cells, tissues and organisms: optical imaging,” Nat. Biotechnol. 21, 1365-1360 (2003).
    [Crossref]
  2. M. Baudrier-Raybaut, R. Haïdar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432, 374-376 (2004).
    [Crossref] [PubMed]
  3. J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second-harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947-3950 (1993).
    [Crossref] [PubMed]
  4. T. Ito and M. Tomita, “Speckle correlation measurement in a disordered medium observed through second-harmonics generation,” Phys. Rev. E 69, 036610 (2004).
    [Crossref]
  5. M. Tomita, “Coherence coupling effect in a space- and time-resolved, nonlinear-correlation measurement in a multiple-scattering medium,” J. Opt. Soc. Am. B 22, 537-546 (2005).
    [Crossref]
  6. V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931-4942 (1991).
    [Crossref]
  7. E. V. Makeev and S. E. Skipetrov, “Second-harmonic generation in suspensions of spherical particles,” Opt. Commun. 224, 139-147 (2003).
    [Crossref]
  8. V. E. Kravtsov and V. M. Agranovich, “Nonlinear backscattering from opaque media,” Phys. Rev. B 43, 13691-13694 (1991).
    [Crossref]
  9. A. Heiderich, R. Maynard, and B. A. van Tiggelen, “Coherent backscattering in nonlinear media,” Opt. Commun. 115, 392-400 (1995).
    [Crossref]
  10. A. G. Mal'shukov and G. D. Mahan, “Nonlinear forward scattering of light in opaque media,” Phys. Rev. B 57, 7701-7704 (1998).
    [Crossref]
  11. T. Wellens, B. Grémaud, D. Delande, and C. Miniatura, “Coherent backscattering of light by nonlinear scatterers,” Phys. Rev. E 71, 055603 (2005).
    [Crossref]
  12. R. W. Boyd, Nonlinear Optics (Academic, 2003).
  13. I. M. Tiginyanu, I. V. Kravetsky, J. Monecke, W. Cordts, G. Marowsky, and H. L. Hartnagel, “Semiconductor sieves as nonlinear optical materials,” Appl. Phys. Lett. 77, 2415-2417 (2000).
    [Crossref]
  14. V. A. Melnikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Yu. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225-228 (2004).
    [Crossref]
  15. L. A. Golovan, V. A. Melnikov, K. P. Bestemyanov, S. V. Zabotnov, V. M. Gordienko, V. Yu. Timoshenko, A. M. Zheltikov, and P. K. Kashkarov, “Disorder-correlated enhancement of second-harmonic generation in strongly photonic porous gallium phosphide,” Appl. Phys. B 81, 353-356 (2005).
    [Crossref]
  16. A mesoscopic property in multiple-scattering of light [after M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313-371 (1999)] refers to those dependencies that are induced by multiple-scattering. In accordance, parameters that are dependent on intrinsic material properties are called microscopic and properties that reflect geometrical specification of the sample including its size are called macroscopic.
    [Crossref]
  17. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  18. E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of peak line shape,” Phys. Rev. Lett. 56, 1471-1474 (1986).
    [Crossref] [PubMed]
  19. D. J. Durian, “Penetration depth for diffusing-wave spectroscopy,” Appl. Opt. 34, 7100-7105 (1995).
    [Crossref] [PubMed]
  20. B. H. Erné, D. Vanmaekelbergh, and J. J. Kelly, “Morphology and strongly enhanced photoresponse of GaP electrodes made porous by anodic etching,” J. Electrochem. Soc. 143, 305-314 (1996).
    [Crossref]
  21. F. J. P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous GaP networks,” Science 284, 141-143 (1999).
    [Crossref] [PubMed]
  22. J. G. Rivas, “Light in strongly scattering semiconductores--diffuse transport and Anderson localization,” Ph.D. dissertation (Universiteit van Amsterdam, 2002).
  23. B. P. J. Bret, “Multiple light scattering in porous gallium phosphide,” Ph.D. dissertation (Universiteit Twente, 2005).
  24. M. U. Vera and D. J. Durian, “Angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215-3224 (1996).
    [Crossref]
  25. D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5to6.0 eV,” Phys. Rev. B 27, 985-1009 (1983).
    [Crossref]
  26. Z. H. Levine, “Optical second harmonic susceptibilities: frequency-dependent formulation with results for GaP and GaAs,” Phys. Rev. B 49, 4532-4538 (1994).
    [Crossref]

2005 (3)

T. Wellens, B. Grémaud, D. Delande, and C. Miniatura, “Coherent backscattering of light by nonlinear scatterers,” Phys. Rev. E 71, 055603 (2005).
[Crossref]

L. A. Golovan, V. A. Melnikov, K. P. Bestemyanov, S. V. Zabotnov, V. M. Gordienko, V. Yu. Timoshenko, A. M. Zheltikov, and P. K. Kashkarov, “Disorder-correlated enhancement of second-harmonic generation in strongly photonic porous gallium phosphide,” Appl. Phys. B 81, 353-356 (2005).
[Crossref]

M. Tomita, “Coherence coupling effect in a space- and time-resolved, nonlinear-correlation measurement in a multiple-scattering medium,” J. Opt. Soc. Am. B 22, 537-546 (2005).
[Crossref]

2004 (3)

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

T. Ito and M. Tomita, “Speckle correlation measurement in a disordered medium observed through second-harmonics generation,” Phys. Rev. E 69, 036610 (2004).
[Crossref]

V. A. Melnikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Yu. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225-228 (2004).
[Crossref]

2003 (2)

E. V. Makeev and S. E. Skipetrov, “Second-harmonic generation in suspensions of spherical particles,” Opt. Commun. 224, 139-147 (2003).
[Crossref]

P. J. Campagnola and L. M. Loew, “Second-harmonic imaging microscopy for visualizing biomolecular arrays in cells, tissues and organisms: optical imaging,” Nat. Biotechnol. 21, 1365-1360 (2003).
[Crossref]

2000 (1)

I. M. Tiginyanu, I. V. Kravetsky, J. Monecke, W. Cordts, G. Marowsky, and H. L. Hartnagel, “Semiconductor sieves as nonlinear optical materials,” Appl. Phys. Lett. 77, 2415-2417 (2000).
[Crossref]

1999 (2)

F. J. P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous GaP networks,” Science 284, 141-143 (1999).
[Crossref] [PubMed]

A mesoscopic property in multiple-scattering of light [after M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313-371 (1999)] refers to those dependencies that are induced by multiple-scattering. In accordance, parameters that are dependent on intrinsic material properties are called microscopic and properties that reflect geometrical specification of the sample including its size are called macroscopic.
[Crossref]

1998 (1)

A. G. Mal'shukov and G. D. Mahan, “Nonlinear forward scattering of light in opaque media,” Phys. Rev. B 57, 7701-7704 (1998).
[Crossref]

1996 (2)

M. U. Vera and D. J. Durian, “Angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215-3224 (1996).
[Crossref]

B. H. Erné, D. Vanmaekelbergh, and J. J. Kelly, “Morphology and strongly enhanced photoresponse of GaP electrodes made porous by anodic etching,” J. Electrochem. Soc. 143, 305-314 (1996).
[Crossref]

1995 (2)

A. Heiderich, R. Maynard, and B. A. van Tiggelen, “Coherent backscattering in nonlinear media,” Opt. Commun. 115, 392-400 (1995).
[Crossref]

D. J. Durian, “Penetration depth for diffusing-wave spectroscopy,” Appl. Opt. 34, 7100-7105 (1995).
[Crossref] [PubMed]

1994 (1)

Z. H. Levine, “Optical second harmonic susceptibilities: frequency-dependent formulation with results for GaP and GaAs,” Phys. Rev. B 49, 4532-4538 (1994).
[Crossref]

1993 (1)

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second-harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947-3950 (1993).
[Crossref] [PubMed]

1991 (2)

V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931-4942 (1991).
[Crossref]

V. E. Kravtsov and V. M. Agranovich, “Nonlinear backscattering from opaque media,” Phys. Rev. B 43, 13691-13694 (1991).
[Crossref]

1986 (1)

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of peak line shape,” Phys. Rev. Lett. 56, 1471-1474 (1986).
[Crossref] [PubMed]

1983 (1)

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5to6.0 eV,” Phys. Rev. B 27, 985-1009 (1983).
[Crossref]

Agranovich, V. M.

V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931-4942 (1991).
[Crossref]

V. E. Kravtsov and V. M. Agranovich, “Nonlinear backscattering from opaque media,” Phys. Rev. B 43, 13691-13694 (1991).
[Crossref]

Akkermans, E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of peak line shape,” Phys. Rev. Lett. 56, 1471-1474 (1986).
[Crossref] [PubMed]

Aspnes, D. E.

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5to6.0 eV,” Phys. Rev. B 27, 985-1009 (1983).
[Crossref]

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 432, 374-376 (2004).
[Crossref] [PubMed]

Bestemyanov, K. P.

L. A. Golovan, V. A. Melnikov, K. P. Bestemyanov, S. V. Zabotnov, V. M. Gordienko, V. Yu. Timoshenko, A. M. Zheltikov, and P. K. Kashkarov, “Disorder-correlated enhancement of second-harmonic generation in strongly photonic porous gallium phosphide,” Appl. Phys. B 81, 353-356 (2005).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Bret, B. P. J.

B. P. J. Bret, “Multiple light scattering in porous gallium phosphide,” Ph.D. dissertation (Universiteit Twente, 2005).

Campagnola, P. J.

P. J. Campagnola and L. M. Loew, “Second-harmonic imaging microscopy for visualizing biomolecular arrays in cells, tissues and organisms: optical imaging,” Nat. Biotechnol. 21, 1365-1360 (2003).
[Crossref]

Cordts, W.

I. M. Tiginyanu, I. V. Kravetsky, J. Monecke, W. Cordts, G. Marowsky, and H. L. Hartnagel, “Semiconductor sieves as nonlinear optical materials,” Appl. Phys. Lett. 77, 2415-2417 (2000).
[Crossref]

de Boer, J. F.

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second-harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947-3950 (1993).
[Crossref] [PubMed]

Delande, D.

T. Wellens, B. Grémaud, D. Delande, and C. Miniatura, “Coherent backscattering of light by nonlinear scatterers,” Phys. Rev. E 71, 055603 (2005).
[Crossref]

Durian, D. J.

M. U. Vera and D. J. Durian, “Angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215-3224 (1996).
[Crossref]

D. J. Durian, “Penetration depth for diffusing-wave spectroscopy,” Appl. Opt. 34, 7100-7105 (1995).
[Crossref] [PubMed]

Erné, B. H.

B. H. Erné, D. Vanmaekelbergh, and J. J. Kelly, “Morphology and strongly enhanced photoresponse of GaP electrodes made porous by anodic etching,” J. Electrochem. Soc. 143, 305-314 (1996).
[Crossref]

Fedotov, A. B.

V. A. Melnikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Yu. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225-228 (2004).
[Crossref]

Feng, S.

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second-harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947-3950 (1993).
[Crossref] [PubMed]

Golovan, L. A.

L. A. Golovan, V. A. Melnikov, K. P. Bestemyanov, S. V. Zabotnov, V. M. Gordienko, V. Yu. Timoshenko, A. M. Zheltikov, and P. K. Kashkarov, “Disorder-correlated enhancement of second-harmonic generation in strongly photonic porous gallium phosphide,” Appl. Phys. B 81, 353-356 (2005).
[Crossref]

V. A. Melnikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Yu. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225-228 (2004).
[Crossref]

Gordienko, V. M.

L. A. Golovan, V. A. Melnikov, K. P. Bestemyanov, S. V. Zabotnov, V. M. Gordienko, V. Yu. Timoshenko, A. M. Zheltikov, and P. K. Kashkarov, “Disorder-correlated enhancement of second-harmonic generation in strongly photonic porous gallium phosphide,” Appl. Phys. B 81, 353-356 (2005).
[Crossref]

Grémaud, B.

T. Wellens, B. Grémaud, D. Delande, and C. Miniatura, “Coherent backscattering of light by nonlinear scatterers,” Phys. Rev. E 71, 055603 (2005).
[Crossref]

Grigorishin, K. I.

V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931-4942 (1991).
[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 432, 374-376 (2004).
[Crossref] [PubMed]

Hartnagel, H. L.

I. M. Tiginyanu, I. V. Kravetsky, J. Monecke, W. Cordts, G. Marowsky, and H. L. Hartnagel, “Semiconductor sieves as nonlinear optical materials,” Appl. Phys. Lett. 77, 2415-2417 (2000).
[Crossref]

Heiderich, A.

A. Heiderich, R. Maynard, and B. A. van Tiggelen, “Coherent backscattering in nonlinear media,” Opt. Commun. 115, 392-400 (1995).
[Crossref]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Ito, T.

T. Ito and M. Tomita, “Speckle correlation measurement in a disordered medium observed through second-harmonics generation,” Phys. Rev. E 69, 036610 (2004).
[Crossref]

Kashkarov, P. K.

L. A. Golovan, V. A. Melnikov, K. P. Bestemyanov, S. V. Zabotnov, V. M. Gordienko, V. Yu. Timoshenko, A. M. Zheltikov, and P. K. Kashkarov, “Disorder-correlated enhancement of second-harmonic generation in strongly photonic porous gallium phosphide,” Appl. Phys. B 81, 353-356 (2005).
[Crossref]

V. A. Melnikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Yu. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225-228 (2004).
[Crossref]

Kelly, J. J.

B. H. Erné, D. Vanmaekelbergh, and J. J. Kelly, “Morphology and strongly enhanced photoresponse of GaP electrodes made porous by anodic etching,” J. Electrochem. Soc. 143, 305-314 (1996).
[Crossref]

Konorov, S. O.

V. A. Melnikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Yu. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225-228 (2004).
[Crossref]

Kravetsky, I. V.

I. M. Tiginyanu, I. V. Kravetsky, J. Monecke, W. Cordts, G. Marowsky, and H. L. Hartnagel, “Semiconductor sieves as nonlinear optical materials,” Appl. Phys. Lett. 77, 2415-2417 (2000).
[Crossref]

Kravtsov, V. E.

V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931-4942 (1991).
[Crossref]

V. E. Kravtsov and V. M. Agranovich, “Nonlinear backscattering from opaque media,” Phys. Rev. B 43, 13691-13694 (1991).
[Crossref]

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 432, 374-376 (2004).
[Crossref] [PubMed]

Lagendijk, A.

F. J. P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous GaP networks,” Science 284, 141-143 (1999).
[Crossref] [PubMed]

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second-harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947-3950 (1993).
[Crossref] [PubMed]

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 432, 374-376 (2004).
[Crossref] [PubMed]

Levine, Z. H.

Z. H. Levine, “Optical second harmonic susceptibilities: frequency-dependent formulation with results for GaP and GaAs,” Phys. Rev. B 49, 4532-4538 (1994).
[Crossref]

Loew, L. M.

P. J. Campagnola and L. M. Loew, “Second-harmonic imaging microscopy for visualizing biomolecular arrays in cells, tissues and organisms: optical imaging,” Nat. Biotechnol. 21, 1365-1360 (2003).
[Crossref]

Mahan, G. D.

A. G. Mal'shukov and G. D. Mahan, “Nonlinear forward scattering of light in opaque media,” Phys. Rev. B 57, 7701-7704 (1998).
[Crossref]

Makeev, E. V.

E. V. Makeev and S. E. Skipetrov, “Second-harmonic generation in suspensions of spherical particles,” Opt. Commun. 224, 139-147 (2003).
[Crossref]

Mal'shukov, A. G.

A. G. Mal'shukov and G. D. Mahan, “Nonlinear forward scattering of light in opaque media,” Phys. Rev. B 57, 7701-7704 (1998).
[Crossref]

Marowsky, G.

I. M. Tiginyanu, I. V. Kravetsky, J. Monecke, W. Cordts, G. Marowsky, and H. L. Hartnagel, “Semiconductor sieves as nonlinear optical materials,” Appl. Phys. Lett. 77, 2415-2417 (2000).
[Crossref]

Maynard, R.

A. Heiderich, R. Maynard, and B. A. van Tiggelen, “Coherent backscattering in nonlinear media,” Opt. Commun. 115, 392-400 (1995).
[Crossref]

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of peak line shape,” Phys. Rev. Lett. 56, 1471-1474 (1986).
[Crossref] [PubMed]

Melnikov, V. A.

L. A. Golovan, V. A. Melnikov, K. P. Bestemyanov, S. V. Zabotnov, V. M. Gordienko, V. Yu. Timoshenko, A. M. Zheltikov, and P. K. Kashkarov, “Disorder-correlated enhancement of second-harmonic generation in strongly photonic porous gallium phosphide,” Appl. Phys. B 81, 353-356 (2005).
[Crossref]

V. A. Melnikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Yu. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225-228 (2004).
[Crossref]

Miniatura, C.

T. Wellens, B. Grémaud, D. Delande, and C. Miniatura, “Coherent backscattering of light by nonlinear scatterers,” Phys. Rev. E 71, 055603 (2005).
[Crossref]

Monecke, J.

I. M. Tiginyanu, I. V. Kravetsky, J. Monecke, W. Cordts, G. Marowsky, and H. L. Hartnagel, “Semiconductor sieves as nonlinear optical materials,” Appl. Phys. Lett. 77, 2415-2417 (2000).
[Crossref]

Muzychenko, D. A.

V. A. Melnikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Yu. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225-228 (2004).
[Crossref]

Nieuwenhuizen, Th. M.

A mesoscopic property in multiple-scattering of light [after M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313-371 (1999)] refers to those dependencies that are induced by multiple-scattering. In accordance, parameters that are dependent on intrinsic material properties are called microscopic and properties that reflect geometrical specification of the sample including its size are called macroscopic.
[Crossref]

Rivas, J. G.

J. G. Rivas, “Light in strongly scattering semiconductores--diffuse transport and Anderson localization,” Ph.D. dissertation (Universiteit van Amsterdam, 2002).

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 432, 374-376 (2004).
[Crossref] [PubMed]

Schuurmans, F. J. P.

F. J. P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous GaP networks,” Science 284, 141-143 (1999).
[Crossref] [PubMed]

Skipetrov, S. E.

E. V. Makeev and S. E. Skipetrov, “Second-harmonic generation in suspensions of spherical particles,” Opt. Commun. 224, 139-147 (2003).
[Crossref]

Sprik, R.

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second-harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947-3950 (1993).
[Crossref] [PubMed]

Studna, A. A.

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5to6.0 eV,” Phys. Rev. B 27, 985-1009 (1983).
[Crossref]

Tiginyanu, I. M.

I. M. Tiginyanu, I. V. Kravetsky, J. Monecke, W. Cordts, G. Marowsky, and H. L. Hartnagel, “Semiconductor sieves as nonlinear optical materials,” Appl. Phys. Lett. 77, 2415-2417 (2000).
[Crossref]

Timoshenko, V. Yu.

L. A. Golovan, V. A. Melnikov, K. P. Bestemyanov, S. V. Zabotnov, V. M. Gordienko, V. Yu. Timoshenko, A. M. Zheltikov, and P. K. Kashkarov, “Disorder-correlated enhancement of second-harmonic generation in strongly photonic porous gallium phosphide,” Appl. Phys. B 81, 353-356 (2005).
[Crossref]

Tomita, M.

M. Tomita, “Coherence coupling effect in a space- and time-resolved, nonlinear-correlation measurement in a multiple-scattering medium,” J. Opt. Soc. Am. B 22, 537-546 (2005).
[Crossref]

T. Ito and M. Tomita, “Speckle correlation measurement in a disordered medium observed through second-harmonics generation,” Phys. Rev. E 69, 036610 (2004).
[Crossref]

van de Lagemaat, J.

F. J. P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous GaP networks,” Science 284, 141-143 (1999).
[Crossref] [PubMed]

van Rossum, M. C. W.

A mesoscopic property in multiple-scattering of light [after M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313-371 (1999)] refers to those dependencies that are induced by multiple-scattering. In accordance, parameters that are dependent on intrinsic material properties are called microscopic and properties that reflect geometrical specification of the sample including its size are called macroscopic.
[Crossref]

van Tiggelen, B. A.

A. Heiderich, R. Maynard, and B. A. van Tiggelen, “Coherent backscattering in nonlinear media,” Opt. Commun. 115, 392-400 (1995).
[Crossref]

Vanmaekelbergh, D.

F. J. P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous GaP networks,” Science 284, 141-143 (1999).
[Crossref] [PubMed]

B. H. Erné, D. Vanmaekelbergh, and J. J. Kelly, “Morphology and strongly enhanced photoresponse of GaP electrodes made porous by anodic etching,” J. Electrochem. Soc. 143, 305-314 (1996).
[Crossref]

Vera, M. U.

M. U. Vera and D. J. Durian, “Angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215-3224 (1996).
[Crossref]

Wellens, T.

T. Wellens, B. Grémaud, D. Delande, and C. Miniatura, “Coherent backscattering of light by nonlinear scatterers,” Phys. Rev. E 71, 055603 (2005).
[Crossref]

Wolf, P. E.

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of peak line shape,” Phys. Rev. Lett. 56, 1471-1474 (1986).
[Crossref] [PubMed]

Yu. Timoshenko, V.

V. A. Melnikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Yu. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225-228 (2004).
[Crossref]

Zabotnov, S. V.

L. A. Golovan, V. A. Melnikov, K. P. Bestemyanov, S. V. Zabotnov, V. M. Gordienko, V. Yu. Timoshenko, A. M. Zheltikov, and P. K. Kashkarov, “Disorder-correlated enhancement of second-harmonic generation in strongly photonic porous gallium phosphide,” Appl. Phys. B 81, 353-356 (2005).
[Crossref]

Zheltikov, A. M.

L. A. Golovan, V. A. Melnikov, K. P. Bestemyanov, S. V. Zabotnov, V. M. Gordienko, V. Yu. Timoshenko, A. M. Zheltikov, and P. K. Kashkarov, “Disorder-correlated enhancement of second-harmonic generation in strongly photonic porous gallium phosphide,” Appl. Phys. B 81, 353-356 (2005).
[Crossref]

V. A. Melnikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Yu. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225-228 (2004).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (2)

V. A. Melnikov, L. A. Golovan, S. O. Konorov, D. A. Muzychenko, A. B. Fedotov, A. M. Zheltikov, V. Yu. Timoshenko, and P. K. Kashkarov, “Second-harmonic generation in strongly scattering porous gallium phosphide,” Appl. Phys. B 79, 225-228 (2004).
[Crossref]

L. A. Golovan, V. A. Melnikov, K. P. Bestemyanov, S. V. Zabotnov, V. M. Gordienko, V. Yu. Timoshenko, A. M. Zheltikov, and P. K. Kashkarov, “Disorder-correlated enhancement of second-harmonic generation in strongly photonic porous gallium phosphide,” Appl. Phys. B 81, 353-356 (2005).
[Crossref]

Appl. Phys. Lett. (1)

I. M. Tiginyanu, I. V. Kravetsky, J. Monecke, W. Cordts, G. Marowsky, and H. L. Hartnagel, “Semiconductor sieves as nonlinear optical materials,” Appl. Phys. Lett. 77, 2415-2417 (2000).
[Crossref]

J. Electrochem. Soc. (1)

B. H. Erné, D. Vanmaekelbergh, and J. J. Kelly, “Morphology and strongly enhanced photoresponse of GaP electrodes made porous by anodic etching,” J. Electrochem. Soc. 143, 305-314 (1996).
[Crossref]

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

Nat. Biotechnol. (1)

P. J. Campagnola and L. M. Loew, “Second-harmonic imaging microscopy for visualizing biomolecular arrays in cells, tissues and organisms: optical imaging,” Nat. Biotechnol. 21, 1365-1360 (2003).
[Crossref]

Nature (1)

M. Baudrier-Raybaut, R. Haïdar, 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. Commun. (2)

A. Heiderich, R. Maynard, and B. A. van Tiggelen, “Coherent backscattering in nonlinear media,” Opt. Commun. 115, 392-400 (1995).
[Crossref]

E. V. Makeev and S. E. Skipetrov, “Second-harmonic generation in suspensions of spherical particles,” Opt. Commun. 224, 139-147 (2003).
[Crossref]

Phys. Rev. B (5)

V. E. Kravtsov and V. M. Agranovich, “Nonlinear backscattering from opaque media,” Phys. Rev. B 43, 13691-13694 (1991).
[Crossref]

A. G. Mal'shukov and G. D. Mahan, “Nonlinear forward scattering of light in opaque media,” Phys. Rev. B 57, 7701-7704 (1998).
[Crossref]

V. E. Kravtsov, V. M. Agranovich, and K. I. Grigorishin, “Theory of second harmonic generation in strongly scattering media,” Phys. Rev. B 44, 4931-4942 (1991).
[Crossref]

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5to6.0 eV,” Phys. Rev. B 27, 985-1009 (1983).
[Crossref]

Z. H. Levine, “Optical second harmonic susceptibilities: frequency-dependent formulation with results for GaP and GaAs,” Phys. Rev. B 49, 4532-4538 (1994).
[Crossref]

Phys. Rev. E (3)

M. U. Vera and D. J. Durian, “Angular distribution of diffusely transmitted light,” Phys. Rev. E 53, 3215-3224 (1996).
[Crossref]

T. Wellens, B. Grémaud, D. Delande, and C. Miniatura, “Coherent backscattering of light by nonlinear scatterers,” Phys. Rev. E 71, 055603 (2005).
[Crossref]

T. Ito and M. Tomita, “Speckle correlation measurement in a disordered medium observed through second-harmonics generation,” Phys. Rev. E 69, 036610 (2004).
[Crossref]

Phys. Rev. Lett. (2)

J. F. de Boer, A. Lagendijk, R. Sprik, and S. Feng, “Transmission and reflection correlations of second-harmonic waves in nonlinear random media,” Phys. Rev. Lett. 71, 3947-3950 (1993).
[Crossref] [PubMed]

E. Akkermans, P. E. Wolf, and R. Maynard, “Coherent backscattering of light by disordered media: analysis of peak line shape,” Phys. Rev. Lett. 56, 1471-1474 (1986).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

A mesoscopic property in multiple-scattering of light [after M. C. W. van Rossum and Th. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy, and diffusion,” Rev. Mod. Phys. 71, 313-371 (1999)] refers to those dependencies that are induced by multiple-scattering. In accordance, parameters that are dependent on intrinsic material properties are called microscopic and properties that reflect geometrical specification of the sample including its size are called macroscopic.
[Crossref]

Science (1)

F. J. P. Schuurmans, D. Vanmaekelbergh, J. van de Lagemaat, and A. Lagendijk, “Strongly photonic macroporous GaP networks,” Science 284, 141-143 (1999).
[Crossref] [PubMed]

Other (4)

J. G. Rivas, “Light in strongly scattering semiconductores--diffuse transport and Anderson localization,” Ph.D. dissertation (Universiteit van Amsterdam, 2002).

B. P. J. Bret, “Multiple light scattering in porous gallium phosphide,” Ph.D. dissertation (Universiteit Twente, 2005).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

R. W. Boyd, Nonlinear Optics (Academic, 2003).

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Figures (7)

Fig. 1
Fig. 1 Mean free path of a porous-GaP sample is wavelength-dependent. As a representative, we plot the mean free path as a function of vacuum wavelength for sample 2. Similar curves are also obtained for all samples using total-transmission measurements.
Fig. 2
Fig. 2 Experimental setup for measuring the total second-harmonic radiation in forward and backward directions. The porous-GaP slab is illuminated with a parallel beam of 150 fs infrared pulses. The integrating sphere collects the fundamental and the second-harmonic light radiated in all directions from one side of the sample. The sample position is either on the top of the sphere for transmission measurements or attached beneath the sphere for reflection measurements. A silicon photodetector (PD-Si) behind a cold glass filter measures the second-harmonic signal in the visible range. In parallel, a germanium photodetector (PD-Ge) measures the transmitted fundamental light at the infrared range.
Fig. 3
Fig. 3 Experimental setup for effusion microscopy measurements. The sample consists of a thin porous and highly scattering layer laying on top of a transparent substrate. The CCD camera images the second-harmonic signal that is radiated from the narrow cross section of the porous part of the sample while it is illuminated by a parallel beam of infrared pulses.
Fig. 4
Fig. 4 Backward-radiated second-harmonic signal V 2 , detected by the silicon detector, is scaled by the sample thickness and plotted for three of the samples versus the signal detected by a germanium detector V 0 in bilogarithmic scales. The second-harmonic power is proportional to V 2 and shows a power-law dependence ( exponent = 1.87 ± 0.03 ) on the incident power, which is proportional to V 0 . Numbers in the legend indicate sample tags as introduced in Table 1.
Fig. 5
Fig. 5 Inset, micrograph of the second-harmonic effusion intensity from a section of the porous-GaP slab while it is illuminated with a parallel beam of infrared laser pulses from the left. Brighter regions indicate higher effusion of second-harmonic light. Outset, the measured second-harmonic intensity is averaged parallel to the substrate and its peak is normalized to 1. The result of the experiment (symbols) is plotted versus the position inside the sample and is compared with the prediction of the stationary diffusion model (solid curve), found from the numeric calculation with no adjustable fitting parameters. Our theoretical value for distribution of the second-harmonic intensity in the bulk, which is presented in Subsection 2C, is shown by the dotted curve. The theoretical fundamental-frequency intensity distribution inside the slab (dashed curve) is plotted for comparison.
Fig. 6
Fig. 6 Ratio η between total second-harmonic light measured in the backward and the forward direction is plotted versus the optical thickness L l 1 ω for various wavelengths and samples. The stationary diffusion prediction from Subsection 2C is plotted as a solid curve. We find a good agreement between theory and measurements. Numbers 1–8 in the legend correspond to the sample numbers introduced in Table 1.
Fig. 7
Fig. 7 (a) Second-harmonic normalized yield as defined in Eq. (17) is plotted versus the ratio of the mean free path at the second-harmonic frequency to the coherence length l 2 ω L c for various wavelengths and samples. The normalized yield is defined as the total second-harmonic intensity generated in the backward direction divided by the square of incident intensity and the thickness of the slab and normalized for frequency dependent material properties of GaP. The dotted curve shows the value calculated from the theoretical model of Kravtsov et al. [6] plotted for comparison. No agreement has been found between their theory and our measurements. Numbers 1–8 in the legend correspond to the sample numbers introduced in Table 1. (b) Same data of (a) is plotted versus the scattering strength at the fundamental frequency. The overall trend can be described by a power-law relation, γ ( k 1 ω l 1 ω ) β , β = 2.0 ± 0.3 , which is shown by the dashed curve.

Tables (1)

Tables Icon

Table 1 Summary of Specifications of the Samples that are Analyzed in this Paper

Equations (17)

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Γ = A ε 0 ω n 2 ω n 1 ω 2 [ χ ( 2 ) ] 2 arccot [ π l 2 ω ( 1 2 + l 2 ω l 1 ω ) L c ] ,
Γ = B ρ υ 2 ω Σ 2 ω ,
S 2 ω ( r , t ) = Γ U 1 ω 2 ( r , t ) .
D 1 ω d 2 U 1 ω ( z ) d z 2 = S 1 ω ( z ) ,
D 2 ω d 2 U 2 ω ( z ) d z 2 = Γ U 1 ω 2 ( z ) ,
U 1 ω ( z ) ζ 1 ω d U 1 ω ( z ) d z = 0 at z = 0 ,
U 1 ω ( z ) + ζ 1 ω d U 1 ω ( z ) d z = 0 at z = L ,
U 2 ω ( z ) ζ 2 ω d U 2 ω ( z ) d z = 0 at z = 0 ,
U 2 ω ( z ) + ζ 2 ω d U 2 ω ( z ) d z = 0 at z = L ,
U 1 ω ( z ) = 3 I 0 [ ( l 1 ω + ζ 1 ω ) ( L + ζ 1 ω z ) l 1 ω ( L + ζ 1 ω + ζ 1 ω ) 2 3 exp ( z l 1 ω ) ] ,
R 2 ω = + D 2 ω υ 2 ω d U 2 ω ( z ) d z z = 0 ,
T 2 ω = D 2 ω υ 2 ω d U 2 ω ( z ) d z z = L .
U 2 ω ( z ) = ( ζ 2 ω + z ) R 2 ω Γ 0 z 0 z 1 U 1 ω 2 ( z 2 ) d z 2 d z 1 .
R 2 ω = 9 Γ I 0 2 L 4 ( 1 + ζ 1 ω l 1 ω ) 2 [ 1 + 4 ( 2 ζ 2 ω + ζ 2 ω ) 3 L 8 l 1 ω 2 ( 5 l 1 ω + 6 ζ 1 ω ) 9 L ( l 1 ω + ζ 1 ω ) 2 ] + O ( l 1 ω L ) ,
T 2 ω = 3 Γ I 0 2 L 4 ( 1 + ζ 1 ω l 1 ω ) 2 ( 1 + 4 ( 2 ζ 2 ω + ζ 1 ω ) L ) + O ( l 1 ω L ) .
η R 2 ω T 2 ω .
γ R 2 ω ω n 2 ω n 1 ω 2 [ χ ( 2 ) ] 2 L I 0 2 ,

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