Abstract

We report on the experimental observation of speckle formation from a transparent crystal formed by a random distribution of nonlinear domains. The angular distribution of second-harmonic light generated by a transparent strontium barium niobate crystal is measured for different diameters of the fundamental beam and crystal thicknesses. Distinct manifestations of speckle pattern formation are found in these experiments. By using a theoretical Green’s function formalism, we explain the reported observations as a result of the linear interference among the second-harmonic waves generated in all directions by each of the nonlinear domains forming the nonlinear crystal.

© 2011 Optical Society of America

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2010

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X. Vidal and J. Martorell, Phys. Rev. Lett. 97, 013902 (2006).
[CrossRef] [PubMed]

2005

T. Volk, D. Isakov, N. Ivanov, L. Ivleva, K. Betzler, A. Tunyagi, and M. Wöhlecke, J. Appl. Phys. 97, 074102 (2005).
[CrossRef]

2004

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, Nature 432, 374 (2004).
[CrossRef] [PubMed]

2003

A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef] [PubMed]

1998

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, Appl. Phys. Lett. 73, 768 (1998).
[CrossRef]

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N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, Nature 368, 436 (1994).
[CrossRef]

1985

M. P. Van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef] [PubMed]

P. E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985).
[CrossRef] [PubMed]

1976

Balachandran, R. M.

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, Nature 368, 436 (1994).
[CrossRef]

Baudrier-Raybaut, M.

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, Nature 432, 374 (2004).
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Betzler, K.

T. Volk, D. Isakov, N. Ivanov, L. Ivleva, K. Betzler, A. Tunyagi, and M. Wöhlecke, J. Appl. Phys. 97, 074102 (2005).
[CrossRef]

A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef] [PubMed]

Bravo-Abad, J.

DeMattei, R. C.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, Appl. Phys. Lett. 73, 768 (1998).
[CrossRef]

Domínguez-Juárez, J. L.

Feigelson, R. S.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, Appl. Phys. Lett. 73, 768 (1998).
[CrossRef]

Gomes, A. S. L.

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, Nature 368, 436 (1994).
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Goodman, J. W.

Haidar, R.

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, Nature 432, 374 (2004).
[CrossRef] [PubMed]

Isakov, D.

T. Volk, D. Isakov, N. Ivanov, L. Ivleva, K. Betzler, A. Tunyagi, and M. Wöhlecke, J. Appl. Phys. 97, 074102 (2005).
[CrossRef]

Ivanov, N.

T. Volk, D. Isakov, N. Ivanov, L. Ivleva, K. Betzler, A. Tunyagi, and M. Wöhlecke, J. Appl. Phys. 97, 074102 (2005).
[CrossRef]

Ivleva, L.

T. Volk, D. Isakov, N. Ivanov, L. Ivleva, K. Betzler, A. Tunyagi, and M. Wöhlecke, J. Appl. Phys. 97, 074102 (2005).
[CrossRef]

Kawai, S.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, Appl. Phys. Lett. 73, 768 (1998).
[CrossRef]

Kupecek, Ph.

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, Nature 432, 374 (2004).
[CrossRef] [PubMed]

Lagendijk, A.

M. P. Van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef] [PubMed]

Lawandy, N. M.

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, Nature 368, 436 (1994).
[CrossRef]

Lee, H. S.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, Appl. Phys. Lett. 73, 768 (1998).
[CrossRef]

Lemasson, Ph.

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, Nature 432, 374 (2004).
[CrossRef] [PubMed]

Maret, G.

P. E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985).
[CrossRef] [PubMed]

Martorell, J.

Ogawa, T.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, Appl. Phys. Lett. 73, 768 (1998).
[CrossRef]

Rosencher, E.

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, Nature 432, 374 (2004).
[CrossRef] [PubMed]

Sauvain, E.

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, Nature 368, 436 (1994).
[CrossRef]

Tunyagi, A.

T. Volk, D. Isakov, N. Ivanov, L. Ivleva, K. Betzler, A. Tunyagi, and M. Wöhlecke, J. Appl. Phys. 97, 074102 (2005).
[CrossRef]

Tunyagi, A. R.

A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef] [PubMed]

Ulex, M.

A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef] [PubMed]

Van Albada, M. P.

M. P. Van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef] [PubMed]

Vidal, X.

Volk, T.

T. Volk, D. Isakov, N. Ivanov, L. Ivleva, K. Betzler, A. Tunyagi, and M. Wöhlecke, J. Appl. Phys. 97, 074102 (2005).
[CrossRef]

Wöhlecke, M.

T. Volk, D. Isakov, N. Ivanov, L. Ivleva, K. Betzler, A. Tunyagi, and M. Wöhlecke, J. Appl. Phys. 97, 074102 (2005).
[CrossRef]

Wolf, P. E.

P. E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985).
[CrossRef] [PubMed]

Appl. Phys. Lett.

S. Kawai, T. Ogawa, H. S. Lee, R. C. DeMattei, and R. S. Feigelson, Appl. Phys. Lett. 73, 768 (1998).
[CrossRef]

J. Appl. Phys.

T. Volk, D. Isakov, N. Ivanov, L. Ivleva, K. Betzler, A. Tunyagi, and M. Wöhlecke, J. Appl. Phys. 97, 074102 (2005).
[CrossRef]

J. Opt. Soc. Am.

Nature

M. Baudrier-Raybaut, R. Haidar, Ph. Kupecek, Ph. Lemasson, and E. Rosencher, Nature 432, 374 (2004).
[CrossRef] [PubMed]

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, and E. Sauvain, Nature 368, 436 (1994).
[CrossRef]

Opt. Express

Phys. Rev. Lett.

A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef] [PubMed]

X. Vidal and J. Martorell, Phys. Rev. Lett. 97, 013902 (2006).
[CrossRef] [PubMed]

M. P. Van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985).
[CrossRef] [PubMed]

P. E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Speckle from the 532 nm wavelength SHG from a poled crystal detected with a CCD camera located at the focal plane of a 150 mm focal lens. Images taken using fundamental beam diameters of (a)  2 mm and (b)  4 mm . The inset on (b) shows the SHG speckle from a crystal with smaller domains. (c) Sequence when the crystal was rotated from 0 ° to 5 ° with respect to the incident beam. A 50 mm focal lens and 1 mm fundamental beam diameter were used in this case. (d) Intensity as a function of the angle of emission with respect to the fundamental beam. Inset: experimental scheme. (e) Intensity of the averaged speckle as a function of the crystal length obtained by translating the fundamental beam across a wedge-shaped crystal.

Fig. 2
Fig. 2

Simulated SHG intensity from 303 μm × 303 μm structures consisting of (a)  3 μm × 3 μm square domains when 2% of them are in a given polarization and (b)  1 μm × 1 μm square domains, 50% in each polarization. Examples of the structures are shown in the insets on the right, using white or black squares depending on the polarization. Left inset on (a): intensity fluctuations when the width of the structure is reduced to 75 μm . Left inset on (b): intensity at 5 ° from a similar composition of domains in which the length of the structure is changed. The result is averaged over 30 different random structures.

Fig. 3
Fig. 3

SHG intensity as a function of the angle of emission with respect to the fundamental beam in the direction (a) perpendicular and (b) parallel to the c axis. The thin lines correspond to the experimental data and the thick lines to the fittings with the theoretical model for cylindrical domains of 6 μm diameter and 580 μm length.

Equations (2)

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E i ( 2 ω ) = ( 2 ω ) 2 e i | k ( 2 ω ) | r c 2 4 π r × V ( δ i , j k ^ i ( 2 ω ) k ^ j ( 2 ω ) ) ( χ j , k , l ( 2 ) E k ( ω ) E l ( ω ) ) e i Δ k · r d r ,
I ( 2 ω ) = 2 ω 4 n ( 2 ω ) [ χ z , z , z ( 2 ) ] 2 c 5 π 2 r 2 n ( ω ) 2 ε 0 [ I ( ω ) ] 2 h 2 L 4 × sinc 2 [ Δ k x L 2 ] sinc 2 [ Δ k y L 2 ] | i = 1 N e i ( Δ k x x i + Δ k y y i ) | 2 ,

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