Abstract

We report on the experimental observation of a dynamic instability in the interaction of counterpropagating self-trapped beams in a photorefractive strontium barium niobate crystal. While the interaction of copropagating spatial optical solitons exhibits only transient dynamics, resulting in a final steady state, the counterpropagating geometry supports a dynamic instability mediated by intrinsic feedback. Experimental observations are compared with and found to be in qualitative agreement with numerical simulations.

© 2005 Optical Society of America

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  1. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
    [CrossRef] [PubMed]
  2. G. I. Stegman and M. Segev, Science 286, 1518 (1999).
    [CrossRef]
  3. O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
    [CrossRef]
  4. O. Cohen, S. Lan, T. Carmon, J. A. Giormaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
    [CrossRef]
  5. D. Kip, Ch. Herden, and M. Wesner, Ferroelectrics 274, 135 (2002).
    [CrossRef]
  6. M. Beli?, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, Phys. Rev. E 68, R025601 (2003).
  7. C. Rotschild, O. Cohen, O. Manela, T. Carmon, and M. Segev, J. Opt. Soc. Am. B 21, 1354 (2004).
    [CrossRef]
  8. R. W. Boyd, M. A. Raymer, and L. M. Narducci, Optical Instabilities (Cambridge U. Press, Cambridge, England, 1986).
  9. Y. Silberberg and I. Bar, Phys. Rev. Lett. 48, 1541 (1982).
    [CrossRef]
  10. W. J. Firth and C. Pare, Opt. Lett. 13, 1096 (1988).
    [CrossRef] [PubMed]
  11. F. T. Arecchi, S. Boccaletti, and P. Ramazza, Phys. Rep. 318, 1 (1999).
    [CrossRef]
  12. K. Motzek, Ph. Jander, A. Desyatnikov, M. Beli?, C. Denz, and F. Kaiser, Phys. Rev. E 68, 066611 (2003).
    [CrossRef]
  13. M. Beli?, M. Petrovi?, D. Jovi?, D. Arsenovi?, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, Opt. Express 12, 708 (2004), http://www.opticsexpress.org.
    [CrossRef]
  14. It should be noted that the threshold parameter is actually the coupling strength, which is the interaction length times the PR coupling constant. Holding the latter at a fixed value, we restricted the investigation to the interaction length threshold.

2004 (2)

2003 (2)

K. Motzek, Ph. Jander, A. Desyatnikov, M. Beli?, C. Denz, and F. Kaiser, Phys. Rev. E 68, 066611 (2003).
[CrossRef]

M. Beli?, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, Phys. Rev. E 68, R025601 (2003).

2002 (3)

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

O. Cohen, S. Lan, T. Carmon, J. A. Giormaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
[CrossRef]

D. Kip, Ch. Herden, and M. Wesner, Ferroelectrics 274, 135 (2002).
[CrossRef]

1999 (2)

G. I. Stegman and M. Segev, Science 286, 1518 (1999).
[CrossRef]

F. T. Arecchi, S. Boccaletti, and P. Ramazza, Phys. Rep. 318, 1 (1999).
[CrossRef]

1994 (1)

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

1988 (1)

1982 (1)

Y. Silberberg and I. Bar, Phys. Rev. Lett. 48, 1541 (1982).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, Phys. Rep. 318, 1 (1999).
[CrossRef]

Arsenovic, D.

Bar, I.

Y. Silberberg and I. Bar, Phys. Rev. Lett. 48, 1541 (1982).
[CrossRef]

Belic, M.

M. Beli?, M. Petrovi?, D. Jovi?, D. Arsenovi?, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, Opt. Express 12, 708 (2004), http://www.opticsexpress.org.
[CrossRef]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Beli?, C. Denz, and F. Kaiser, Phys. Rev. E 68, 066611 (2003).
[CrossRef]

M. Beli?, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, Phys. Rev. E 68, R025601 (2003).

Boccaletti, S.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, Phys. Rep. 318, 1 (1999).
[CrossRef]

Boyd, R. W.

R. W. Boyd, M. A. Raymer, and L. M. Narducci, Optical Instabilities (Cambridge U. Press, Cambridge, England, 1986).

Carmon, T.

Cohen, O.

Crosignani, B.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Denz, C.

M. Beli?, M. Petrovi?, D. Jovi?, D. Arsenovi?, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, Opt. Express 12, 708 (2004), http://www.opticsexpress.org.
[CrossRef]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Beli?, C. Denz, and F. Kaiser, Phys. Rev. E 68, 066611 (2003).
[CrossRef]

M. Beli?, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, Phys. Rev. E 68, R025601 (2003).

Desyatnikov, A.

K. Motzek, Ph. Jander, A. Desyatnikov, M. Beli?, C. Denz, and F. Kaiser, Phys. Rev. E 68, 066611 (2003).
[CrossRef]

M. Beli?, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, Phys. Rev. E 68, R025601 (2003).

DiPorto, P.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Firth, W. J.

Fleischer, J. W.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

Giormaine, J. A.

Herden, Ch.

D. Kip, Ch. Herden, and M. Wesner, Ferroelectrics 274, 135 (2002).
[CrossRef]

Jander, Ph.

M. Beli?, M. Petrovi?, D. Jovi?, D. Arsenovi?, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, Opt. Express 12, 708 (2004), http://www.opticsexpress.org.
[CrossRef]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Beli?, C. Denz, and F. Kaiser, Phys. Rev. E 68, 066611 (2003).
[CrossRef]

M. Beli?, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, Phys. Rev. E 68, R025601 (2003).

Jovic, D.

Kaiser, F.

Kip, D.

D. Kip, Ch. Herden, and M. Wesner, Ferroelectrics 274, 135 (2002).
[CrossRef]

Lan, S.

Mandel, P.

Manela, O.

Motzek, K.

Narducci, L. M.

R. W. Boyd, M. A. Raymer, and L. M. Narducci, Optical Instabilities (Cambridge U. Press, Cambridge, England, 1986).

Odoulov, S.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

Pare, C.

Petrovic, M.

Ramazza, P.

F. T. Arecchi, S. Boccaletti, and P. Ramazza, Phys. Rep. 318, 1 (1999).
[CrossRef]

Raymer, M. A.

R. W. Boyd, M. A. Raymer, and L. M. Narducci, Optical Instabilities (Cambridge U. Press, Cambridge, England, 1986).

Rotschild, C.

Segev, M.

C. Rotschild, O. Cohen, O. Manela, T. Carmon, and M. Segev, J. Opt. Soc. Am. B 21, 1354 (2004).
[CrossRef]

O. Cohen, S. Lan, T. Carmon, J. A. Giormaine, and M. Segev, Opt. Lett. 27, 2013 (2002).
[CrossRef]

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

G. I. Stegman and M. Segev, Science 286, 1518 (1999).
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Silberberg, Y.

Y. Silberberg and I. Bar, Phys. Rev. Lett. 48, 1541 (1982).
[CrossRef]

Stegman, G. I.

G. I. Stegman and M. Segev, Science 286, 1518 (1999).
[CrossRef]

Strinic, A.

M. Beli?, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, Phys. Rev. E 68, R025601 (2003).

Tlidi, M.

Uzdin, R.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

Valley, G. C.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Wesner, M.

D. Kip, Ch. Herden, and M. Wesner, Ferroelectrics 274, 135 (2002).
[CrossRef]

Yariv, A.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Ferroelectrics (1)

D. Kip, Ch. Herden, and M. Wesner, Ferroelectrics 274, 135 (2002).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (2)

Phys. Rep. (1)

F. T. Arecchi, S. Boccaletti, and P. Ramazza, Phys. Rep. 318, 1 (1999).
[CrossRef]

Phys. Rev. E (2)

K. Motzek, Ph. Jander, A. Desyatnikov, M. Beli?, C. Denz, and F. Kaiser, Phys. Rev. E 68, 066611 (2003).
[CrossRef]

M. Beli?, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, Phys. Rev. E 68, R025601 (2003).

Phys. Rev. Lett. (3)

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Y. Silberberg and I. Bar, Phys. Rev. Lett. 48, 1541 (1982).
[CrossRef]

Science (1)

G. I. Stegman and M. Segev, Science 286, 1518 (1999).
[CrossRef]

Other (2)

R. W. Boyd, M. A. Raymer, and L. M. Narducci, Optical Instabilities (Cambridge U. Press, Cambridge, England, 1986).

It should be noted that the threshold parameter is actually the coupling strength, which is the interaction length times the PR coupling constant. Holding the latter at a fixed value, we restricted the investigation to the interaction length threshold.

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Figures (3)

Fig. 1
Fig. 1

Experimental setup. Two beams are rendered mutually incoherent with an oscillating piezo-mounted mirror (PM) and focused on opposite faces of a PR Ce:SBN60 crystal. Both crystal faces are imaged onto a CCD camera, allowing for synchronous observation of reflections of both exit and input beams (Ms, mirrors; Ls, lenses; PH, pinhole; PBS, polarizing beam splitter; BS, beam splitter). Inset, CP soliton interaction in the numerical model (no beam bending, only attractive forces). Two beams individually self-focus in the PR medium (solid curves). Propagating in close proximity, incoherent beams attract, as the combined intensity of both beams creates a common lens (dashed curves) HV, high voltage.

Fig. 2
Fig. 2

Images of one exit face. (a) Separated beams: The beam leaving the crystal is visible as the bright spot. The second beam entering the crystal at this plane is visible in reflection (faint spot). (b) Strong interaction and the splitting of beams: Although most of the output beam overlaps with the input beam, a fraction is split off into a second channel. Images (a) and (b) correspond to t = 9 s and t = 417 s of the time series displayed as Fig. 3(b).

Fig. 3
Fig. 3

Temporal plot of system dynamics. (a) Below threshold ( L 1 = 5 mm ) , the resulting stable and stationary state consists of two symmetrically overlapping solitons. (b) Above threshold ( L 2 = 23 mm ) , irregular dynamics are observed. (c) Close up of later development of a similar experiment starting at t = 15 min . (d) Numerical simulation qualitatively corresponding to experimental parameters for (c). Fast oscillations for which the beams overlap result from confinement to one transverse dimension and can be considered a numerical artifact. In its place, experimental observations feature a splitting of the beams into two parts (compare Fig. 2).

Equations (3)

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i z F + x 2 F = Γ E 0 F ,
i z B + x 2 B = Γ E 0 B ,
τ t E 0 + E 0 = F 2 + B 2 1 + F 2 + B 2 ,

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