Abstract

We investigate the temporal dynamics of transverse optical patterns spontaneously formed in a photorefractive single-feedback system with a virtual feedback mirror. The linear stability analysis for the system is reviewed and extended to the region of larger propagation lengths. The stationary patterns obtained experimentally are classified as a function of feedback reflectivity and feedback mirror position. Inserting masks into the feedback path permits pattern selection and control by Fourier filtering. When an asymmetry that is due to noncollinear pump beams is introduced, the otherwise stationary hexagons show several complex but periodic rotationlike motions. Furthermore, the competition of hexagonal and square patterns can be observed by the appropriate choice of feedback mirror position and coupling strength. The origin of this behavior is discussed. The temporal evolution of the patterns is illustrated by a method based on unfolding the angular distribution of the spots in the far field.

[Optical Society of America ]

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  1. J. Pender and L. Hesselink , Degenerate conical emissions in atomic-sodium vapor , J. Opt. Soc. Am. B JOBPDE 7 , 1361 ( 1990
    [CrossRef]
  2. A. Petrossian , M. Pinard , A. Ma tre , J. Y. Courtois , and G. Grynberg , Transverse pattern formation for counterpropagating beams in rubidium vapor , Europhys. Lett. EULEEJ 18 , 689 ( 1992
    [CrossRef]
  3. R. Macdonald and H. J. Eichler , Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror , Opt. Commun. OPCOB8 89 , 289 ( 1992
    [CrossRef]
  4. M. Tamburrini , M. Bonavita , S. Wabnitz , and E. Santamato , Hexagonally patterned beam filamentation in a thin liquid-crystal film with single feedback mirror , Opt. Lett. OPLEDP 18 , 855 ( 1993
    [CrossRef]
  5. B. Thu ring , R. Neubecker , and T. Tschudi , Transverse pattern formation in an LCLV feedback system , Opt. Commun. OPCOB8 102 , 111 ( 1993
    [CrossRef]
  6. T. Honda , Hexagonal pattern formation due to counterpropagation in KNbO 3 , Opt. Lett. OPLEDP 18 , 598 ( 1993
    [CrossRef]
  7. J. Glu ckstad and M. Saffman , Spontaneous pattern formation in a thin film of bacteriorhodopsin with mixed absorptive dispersive nonlinearity , Opt. Lett. OPLEDP 20 , 551 ( 1995
    [CrossRef]
  8. M. A. Vorontsov and W. J. Firth , Pattern formation and competition in nonlinear optical systems with two-dimensional feedback , Phys. Rev. A PLRAAN 49 , 2891 ( 1994
    [CrossRef] [PubMed]
  9. T. Honda , H. Matsumoto , M. Sedlatschek , C. Denz , and T. Tschudi , Spontaneous formation of hexagons, squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror , Opt. Commun. OPCOB8 133 , 293 ( 1997
    [CrossRef]
  10. T. Honda , Flow and controlled rotation of spontaneous optical hexagon in KNbO 3 , Opt. Lett. OPLEDP 20 , 851 ( 1995
    [CrossRef] [PubMed]
  11. A. V. Mamaev and M. Saffman , Modulational instability and pattern formation in the field of noncollinear pump beams , Opt. Lett. OPLEDP 22 , 283 ( 1997
    [CrossRef] [PubMed]
  12. T. Honda and P. P. Banerjee , Threshold for spontaneous pattern formation in reflection-grating-dominated photorefractive media with mirror feedback , Opt. Lett. OPLEDP 21 , 779 ( 1996
    [CrossRef] [PubMed]
  13. M. Saffman , A. A. Zozulya , and D. Z. Anderson , Transverse instability of energy-exchanging counterpropagating waves in photorefractive media , J. Opt. Soc. Am. B JOBPDE 11 , 1409 ( 1994
    [CrossRef]
  14. N. V. Kukhtarev , T. Kukhtareva , H. J. Caulfield , P. P. Banerjee , H. L. Yu , and L. Hesselink , Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO 3 :Fe , Opt. Eng. OPEGAR 34 , 2261 ( 1995
    [CrossRef]
  15. A. I. Chernykh , B. I. Sturman , M. Aguilar , and F. Agullo -Lo pez , Threshold for pattern formation in a medium with a local photorefractive response , J. Opt. Soc. Am. B JOBPDE 14 , 1754 ( 1997
    [CrossRef]
  16. E. V. Degtiarev and M. A. Vorontsov , Spatial filtering in nonlinear two-dimensional feedback systems: phase-distortion suppression , J. Opt. Soc. Am. B JOBPDE 12 , 1238 ( 1995
    [CrossRef]
  17. R. Martin , A. J. Scroggie , G.-L. Oppo , and W. J. Firth , Stabilization, selection, and tracking of unstable patterns by Fourier space techniques , Phys. Rev. Lett. PRLTAO 77 , 4007 ( 1996
    [CrossRef] [PubMed]
  18. R. Martin , G.-L. Oppo , G. K. Harkness , A. J. Scroggie , and W. J. Firth , Controlling pattern formation and spatio-temporal disorder in nonlinear optics , Opt. Expr. ZZZZZZ 1 , 39 ( 1997
    [CrossRef]
  19. B. Thu ring , A. Schreiber , M. Kreuzer , and T. Tschudi , Spatio-temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system , Physica D PDNPDT 96 , 282 ( 1996
    [CrossRef]
  20. A. Petrossian , L. Dambly , and G. Grynberg , Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror , Europhys. Lett. EULEEJ 29 , 209 ( 1995
    [CrossRef]
  21. O. Sandfuchs , J. Leonardy , F. Kaiser , and M. R. Belic , Transverse instabilities in photorefractive counterpropagating two-wave mixing , Opt. Lett. OPLEDP 22 , 498 ( 1997
    [CrossRef] [PubMed]
  22. O. Sandfuchs , F. Kaiser , and M. R. Belic , Spatiotemporal pattern formation in counterpropagating two-wave mixing with an externally applied field , J. Opt. Soc. Am. B JOBPDE 15 , 2070 ( 1998
    [CrossRef]

Kreuzer, M

B. Thu ring , A. Schreiber , M. Kreuzer , and T. Tschudi , Spatio-temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system , Physica D PDNPDT 96 , 282 ( 1996
[CrossRef]

Matre, A

A. Petrossian , M. Pinard , A. Ma tre , J. Y. Courtois , and G. Grynberg , Transverse pattern formation for counterpropagating beams in rubidium vapor , Europhys. Lett. EULEEJ 18 , 689 ( 1992
[CrossRef]

Neubecker, R

B. Thu ring , R. Neubecker , and T. Tschudi , Transverse pattern formation in an LCLV feedback system , Opt. Commun. OPCOB8 102 , 111 ( 1993
[CrossRef]

Pinard, M

A. Petrossian , M. Pinard , A. Ma tre , J. Y. Courtois , and G. Grynberg , Transverse pattern formation for counterpropagating beams in rubidium vapor , Europhys. Lett. EULEEJ 18 , 689 ( 1992
[CrossRef]

Sandfuchs, O

Schreiber, A

B. Thu ring , A. Schreiber , M. Kreuzer , and T. Tschudi , Spatio-temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system , Physica D PDNPDT 96 , 282 ( 1996
[CrossRef]

Thuring, B

B. Thu ring , R. Neubecker , and T. Tschudi , Transverse pattern formation in an LCLV feedback system , Opt. Commun. OPCOB8 102 , 111 ( 1993
[CrossRef]

Yu, H. L

N. V. Kukhtarev , T. Kukhtareva , H. J. Caulfield , P. P. Banerjee , H. L. Yu , and L. Hesselink , Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO 3 :Fe , Opt. Eng. OPEGAR 34 , 2261 ( 1995
[CrossRef]

Other (22)

J. Pender and L. Hesselink , Degenerate conical emissions in atomic-sodium vapor , J. Opt. Soc. Am. B JOBPDE 7 , 1361 ( 1990
[CrossRef]

A. Petrossian , M. Pinard , A. Ma tre , J. Y. Courtois , and G. Grynberg , Transverse pattern formation for counterpropagating beams in rubidium vapor , Europhys. Lett. EULEEJ 18 , 689 ( 1992
[CrossRef]

R. Macdonald and H. J. Eichler , Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror , Opt. Commun. OPCOB8 89 , 289 ( 1992
[CrossRef]

M. Tamburrini , M. Bonavita , S. Wabnitz , and E. Santamato , Hexagonally patterned beam filamentation in a thin liquid-crystal film with single feedback mirror , Opt. Lett. OPLEDP 18 , 855 ( 1993
[CrossRef]

B. Thu ring , R. Neubecker , and T. Tschudi , Transverse pattern formation in an LCLV feedback system , Opt. Commun. OPCOB8 102 , 111 ( 1993
[CrossRef]

T. Honda , Hexagonal pattern formation due to counterpropagation in KNbO 3 , Opt. Lett. OPLEDP 18 , 598 ( 1993
[CrossRef]

J. Glu ckstad and M. Saffman , Spontaneous pattern formation in a thin film of bacteriorhodopsin with mixed absorptive dispersive nonlinearity , Opt. Lett. OPLEDP 20 , 551 ( 1995
[CrossRef]

M. A. Vorontsov and W. J. Firth , Pattern formation and competition in nonlinear optical systems with two-dimensional feedback , Phys. Rev. A PLRAAN 49 , 2891 ( 1994
[CrossRef] [PubMed]

T. Honda , H. Matsumoto , M. Sedlatschek , C. Denz , and T. Tschudi , Spontaneous formation of hexagons, squares and squeezed hexagons in a photorefractive phase conjugator with virtually internal feedback mirror , Opt. Commun. OPCOB8 133 , 293 ( 1997
[CrossRef]

T. Honda , Flow and controlled rotation of spontaneous optical hexagon in KNbO 3 , Opt. Lett. OPLEDP 20 , 851 ( 1995
[CrossRef] [PubMed]

A. V. Mamaev and M. Saffman , Modulational instability and pattern formation in the field of noncollinear pump beams , Opt. Lett. OPLEDP 22 , 283 ( 1997
[CrossRef] [PubMed]

T. Honda and P. P. Banerjee , Threshold for spontaneous pattern formation in reflection-grating-dominated photorefractive media with mirror feedback , Opt. Lett. OPLEDP 21 , 779 ( 1996
[CrossRef] [PubMed]

M. Saffman , A. A. Zozulya , and D. Z. Anderson , Transverse instability of energy-exchanging counterpropagating waves in photorefractive media , J. Opt. Soc. Am. B JOBPDE 11 , 1409 ( 1994
[CrossRef]

N. V. Kukhtarev , T. Kukhtareva , H. J. Caulfield , P. P. Banerjee , H. L. Yu , and L. Hesselink , Broadband dynamic, holographically self-recorded, and static hexagonal scattering patterns in photorefractive KNbO 3 :Fe , Opt. Eng. OPEGAR 34 , 2261 ( 1995
[CrossRef]

A. I. Chernykh , B. I. Sturman , M. Aguilar , and F. Agullo -Lo pez , Threshold for pattern formation in a medium with a local photorefractive response , J. Opt. Soc. Am. B JOBPDE 14 , 1754 ( 1997
[CrossRef]

E. V. Degtiarev and M. A. Vorontsov , Spatial filtering in nonlinear two-dimensional feedback systems: phase-distortion suppression , J. Opt. Soc. Am. B JOBPDE 12 , 1238 ( 1995
[CrossRef]

R. Martin , A. J. Scroggie , G.-L. Oppo , and W. J. Firth , Stabilization, selection, and tracking of unstable patterns by Fourier space techniques , Phys. Rev. Lett. PRLTAO 77 , 4007 ( 1996
[CrossRef] [PubMed]

R. Martin , G.-L. Oppo , G. K. Harkness , A. J. Scroggie , and W. J. Firth , Controlling pattern formation and spatio-temporal disorder in nonlinear optics , Opt. Expr. ZZZZZZ 1 , 39 ( 1997
[CrossRef]

B. Thu ring , A. Schreiber , M. Kreuzer , and T. Tschudi , Spatio-temporal dynamics due to competing spatial instabilities in a coupled LCLV feedback system , Physica D PDNPDT 96 , 282 ( 1996
[CrossRef]

A. Petrossian , L. Dambly , and G. Grynberg , Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror , Europhys. Lett. EULEEJ 29 , 209 ( 1995
[CrossRef]

O. Sandfuchs , J. Leonardy , F. Kaiser , and M. R. Belic , Transverse instabilities in photorefractive counterpropagating two-wave mixing , Opt. Lett. OPLEDP 22 , 498 ( 1997
[CrossRef] [PubMed]

O. Sandfuchs , F. Kaiser , and M. R. Belic , Spatiotemporal pattern formation in counterpropagating two-wave mixing with an externally applied field , J. Opt. Soc. Am. B JOBPDE 15 , 2070 ( 1998
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Principle of the interaction geometry: SB’s, spatial sidebands; M, mirror; v.M., virtual mirror; L, propagation length; l, crystal length; θ’s, sideband angles; p.r., photorefractive.

Fig. 2
Fig. 2

Sideband angle θ as a function of normalized virtual mirror position n0L/l according to the linear stability analysis.

Fig. 3
Fig. 3

Threshold curve for n0L/l=1.4, indicating the leap in the sideband angle. The minimum of the threshold curve is passed over from the first to the second branch, resulting in a leap of Δθ of the sideband angle.

Fig. 4
Fig. 4

Sideband angle θ as a function of virtual mirror position for the case of a virtual mirror position inside the crystal. Filled circles, experimental points; solid curve, theoretical curve. Inset, experimentally obtained hexagonal pattern in the far field, including second- and third-order spots.

Fig. 5
Fig. 5

Minima of threshold curves as a function of virtual mirror position for the parameter region -1n0L/l1. Selected parameter values are n0L/l=-1, 0 (point A), n0L/l=-0.7, -0.3 (point B), n0L/l=-0.5 (point C), and n0L/l=1 (point D). The largest observable sideband angle θmax is indicated, corresponding to a coupling strength of γl5.5.

Fig. 6
Fig. 6

Experimental setup: OD, optical diode; L’s, lenses; M’s, mirrors; BS’s, beam splitters; MLS, microscope lens system.

Fig. 7
Fig. 7

Dependence of pattern type on feedback reflectivity and mirror position. I, no observable pattern; II, weak hexagonal pattern; III, pattern without geometric symmetry; IV, washed-out hexagonal pattern with emphasis on two spots opposite each other; V, static hexagonal pattern.

Fig. 8
Fig. 8

Method for illustrating pattern dynamics based on the unfolding of the angular spectrum of the far-field pattern.

Fig. 9
Fig. 9

Rock ’n’ roll motion. Two spots, opposite each other, are nearly stable; the other four spots exhibit a rotation and a fast leap back to the initial position.

Fig. 10
Fig. 10

Frequency of the rock ’n’ roll motion as a function of the tilt angle (full angle between the incoming and the feedback beam outside the crystal).

Fig. 11
Fig. 11

Rocking motion with two time scales. All six spots show a rotation and a fast leap back on a long time scale, whereas on a shorter time scale the spots oscillate periodically.  

Fig. 12
Fig. 12

Experimentally observed far-field pattern of (left) hexagonal and (right) square symmetry. For strong coupling of γl11.5 and an appropriate choice of the virtual mirror position (e.g., n0L/l-0.25), the patterns alternate in time.

Fig. 13
Fig. 13

Angular distribution of spots during competition of hexagons and squares. Top, radius k=k1; bottom, radius k=2k1.

Equations (9)

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

Fz-i2k0n0 2F=iγ |B|2|F|2+|B|2 F,
Bz+i2k0n0 2B=-iγ* |F|2|F|2+|B|2 B,
F(r)=F0(z)[1+F+1(z)exp(ikr)+F-1(z)exp(-ikr)],
B(r)=B0(z)[1+B+1(z)exp(ikr)+B-1(z)exp(-ikr)],
F±1(0)=0,
B±1(l)=exp(2ikdn0L)F±1(l),
cos wl cos kdl+γI2w sin wl cos kd(l+2n0L)
+γR+2kd2w sin wl sin kdl
-γR2w sin wl sin kd(l+2n0L)=0,

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