Abstract

The effect of a spatially filtered negative feedback on a laser that supports many transverse modes is studied. The Lyapunov theorem is used to find an analytical expression of the parameter domain in which the laser can be stabilized in the plane-wave state by the feedback. The prediction of the Lyapunov theorem is compared with that of the Routh–Hurwitz criterion and is verified by the results of numerical simulation. The numerical studies also show that the spatially filtered feedback can direct the laser to the plane-wave state from a distant initial state.

© 2000 Optical Society of America

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  1. E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
    [CrossRef] [PubMed]
  2. W. L. Ditto, S. N. Rauseo, and M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
    [CrossRef] [PubMed]
  3. E. R. Hunt, “Stabilizing high-period orbits in a chaotic system: the diode resonator,” Phys. Rev. Lett. 67, 1953–1955 (1991).
    [CrossRef] [PubMed]
  4. K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170, 421–428 (1992).
    [CrossRef]
  5. R. Roy, T. W. Murphy, Jr., T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1262 (1992).
    [CrossRef] [PubMed]
  6. S. Bielawski, D. Derozier, and P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
    [CrossRef] [PubMed]
  7. C. Reyl, L. Flepp, R. Badii, and E. Brun, “Control of NMR-laser chaos in high-dimensional embedding space,” Phys. Rev. E 47, 267–272 (1993).
    [CrossRef]
  8. D. J. Gauthier, D. W. Sukow, H. M. Concannon, and J. E. S. Socolar, “Stabilizing unstable periodic orbits in a fast diode resonator,” Phys. Rev. E 50, 2343–2346 (1994).
    [CrossRef]
  9. J. M. Perez, J. Steinshnider, R. E. Stallcup, and A. F. Aviles, “Control of chaos in a CO2 laser,” Appl. Phys. Lett. 65, 1216–1218 (1994).
    [CrossRef]
  10. W. Lu and R. Harrison, “Controlling chaos using continuous interference feedback: proposal for all optical devices,” Opt. Commun. 109, 457–461 (1994).
    [CrossRef]
  11. J.-H. Dai, H.-W. Yin, and H.-J. Zhang, “Controlling chaos in a hybrid optical bistable system,” Opt. Commun. 120, 85–90 (1995).
    [CrossRef]
  12. C. Simmendinger and O. Hess, “Controlling delay-induced chaotic behavior of a semiconductor laser with optical feedback,” Phys. Lett. A 216, 97–105 (1996).
    [CrossRef]
  13. D. J. Gauthier, “Controlling lasers by use of extended time-delay autosynchronization,” Opt. Lett. 23, 703–705 (1998).
    [CrossRef]
  14. G. Hu and K. He, “Controlling chaos in systems described by partial differential equations,” Phys. Rev. Lett. 71, 3794–3797 (1993).
    [CrossRef]
  15. G. Hu and Z. Qu, “Controlling spatiotemporal chaos in coupled map lattice systems,” Phys. Rev. Lett. 72, 68–71 (1994).
    [CrossRef]
  16. F. Qin, E. E. Wolf, and H.-C. Chang, “Controlling spatiotemporal patterns on a catalytic wafer,” Phys. Rev. Lett. 72, 1459–1462 (1994).
    [CrossRef] [PubMed]
  17. C. Lourenco, M. Hougardy, and A. Babloyantz, “Control of low-dimensional spatiotemporal chaos in Fourier space,” Phys. Rev. E 52, 1528–1532 (1995).
    [CrossRef]
  18. W. Lu, D. Yu, and R. G. Harrison, “Control of patterns in spatiotemporal chaos in optics,” Phys. Rev. Lett. 76, 3316–3319 (1996); “Tracking periodic patterns into spatiotemporal chaotic regimes,” Phys. Rev. Lett. 78, 4375–4378 (1997).
    [CrossRef] [PubMed]
  19. M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar, “Controlling extended systems with spatially filtered, time-delayed feedback,” Phys. Rev. E 55, 2119–2126 (1997).
    [CrossRef]
  20. 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. 77, 4007–4010 (1996); 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. Express 1, 39–43 (1997), http://epubs.osa.org/opticsexpress.
    [CrossRef] [PubMed]
  21. G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A 58, 2577–2585 (1998).
    [CrossRef]
  22. A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80, 3499–3502 (1998); S. J. Jensen, M. Schwab, and C. Denz, “Manipulation, stabilization, and control of pattern formation using Fourier space filtering,” Phys. Rev. Lett. 81, 1614–1617 (1998).
    [CrossRef]
  23. R. O. Grigoriev, M. C. Cross, and H. G. Schuster, “Pinning control of spatiotemporal chaos,” Phys. Rev. Lett. 79, 2795–2798 (1997).
    [CrossRef]
  24. L. A. Lugiato, C. Oldano, and L. M. Narducci, “Cooperative frequency locking and stationary spatial structures in lasers,” J. Opt. Soc. Am. B 5, 879–888 (1988).
    [CrossRef]
  25. H. Lin and N. B. Abraham, “Mode formation and beating in the transverse pattern dynamics in a laser,” Opt. Commun. 79, 476–488 (1990).
    [CrossRef]
  26. H. Lin and N. B. Abraham, “Transverse pattern variations in a laser with a parabolic excitation profile,” J. Opt. Soc. Am. B 8, 2429–2436 (1991).
    [CrossRef]
  27. H. Lin, “Numerical and experimental study of transverse dynamics in a laser with several transverse modes,” Ph.D. dissertation (Bryn Mawr College, Bryn Mawr, Penn., 1991).
  28. L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, “Instabilities in passive and active optical systems with a Gaussian transverse intensity profile,” Phys. Rev. A 30, 1366–1376 (1984).
    [CrossRef]
  29. C.-T. Chen, Linear System Theory and Design (Holt, Rinehart & Winston, New York, 1984), pp. 412–417.

1998 (2)

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A 58, 2577–2585 (1998).
[CrossRef]

D. J. Gauthier, “Controlling lasers by use of extended time-delay autosynchronization,” Opt. Lett. 23, 703–705 (1998).
[CrossRef]

1997 (2)

R. O. Grigoriev, M. C. Cross, and H. G. Schuster, “Pinning control of spatiotemporal chaos,” Phys. Rev. Lett. 79, 2795–2798 (1997).
[CrossRef]

M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar, “Controlling extended systems with spatially filtered, time-delayed feedback,” Phys. Rev. E 55, 2119–2126 (1997).
[CrossRef]

1996 (1)

C. Simmendinger and O. Hess, “Controlling delay-induced chaotic behavior of a semiconductor laser with optical feedback,” Phys. Lett. A 216, 97–105 (1996).
[CrossRef]

1995 (2)

J.-H. Dai, H.-W. Yin, and H.-J. Zhang, “Controlling chaos in a hybrid optical bistable system,” Opt. Commun. 120, 85–90 (1995).
[CrossRef]

C. Lourenco, M. Hougardy, and A. Babloyantz, “Control of low-dimensional spatiotemporal chaos in Fourier space,” Phys. Rev. E 52, 1528–1532 (1995).
[CrossRef]

1994 (5)

G. Hu and Z. Qu, “Controlling spatiotemporal chaos in coupled map lattice systems,” Phys. Rev. Lett. 72, 68–71 (1994).
[CrossRef]

F. Qin, E. E. Wolf, and H.-C. Chang, “Controlling spatiotemporal patterns on a catalytic wafer,” Phys. Rev. Lett. 72, 1459–1462 (1994).
[CrossRef] [PubMed]

D. J. Gauthier, D. W. Sukow, H. M. Concannon, and J. E. S. Socolar, “Stabilizing unstable periodic orbits in a fast diode resonator,” Phys. Rev. E 50, 2343–2346 (1994).
[CrossRef]

J. M. Perez, J. Steinshnider, R. E. Stallcup, and A. F. Aviles, “Control of chaos in a CO2 laser,” Appl. Phys. Lett. 65, 1216–1218 (1994).
[CrossRef]

W. Lu and R. Harrison, “Controlling chaos using continuous interference feedback: proposal for all optical devices,” Opt. Commun. 109, 457–461 (1994).
[CrossRef]

1993 (3)

S. Bielawski, D. Derozier, and P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

C. Reyl, L. Flepp, R. Badii, and E. Brun, “Control of NMR-laser chaos in high-dimensional embedding space,” Phys. Rev. E 47, 267–272 (1993).
[CrossRef]

G. Hu and K. He, “Controlling chaos in systems described by partial differential equations,” Phys. Rev. Lett. 71, 3794–3797 (1993).
[CrossRef]

1992 (2)

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170, 421–428 (1992).
[CrossRef]

R. Roy, T. W. Murphy, Jr., T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1262 (1992).
[CrossRef] [PubMed]

1991 (2)

E. R. Hunt, “Stabilizing high-period orbits in a chaotic system: the diode resonator,” Phys. Rev. Lett. 67, 1953–1955 (1991).
[CrossRef] [PubMed]

H. Lin and N. B. Abraham, “Transverse pattern variations in a laser with a parabolic excitation profile,” J. Opt. Soc. Am. B 8, 2429–2436 (1991).
[CrossRef]

1990 (3)

H. Lin and N. B. Abraham, “Mode formation and beating in the transverse pattern dynamics in a laser,” Opt. Commun. 79, 476–488 (1990).
[CrossRef]

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

W. L. Ditto, S. N. Rauseo, and M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
[CrossRef] [PubMed]

1988 (1)

1984 (1)

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, “Instabilities in passive and active optical systems with a Gaussian transverse intensity profile,” Phys. Rev. A 30, 1366–1376 (1984).
[CrossRef]

Abraham, N. B.

H. Lin and N. B. Abraham, “Transverse pattern variations in a laser with a parabolic excitation profile,” J. Opt. Soc. Am. B 8, 2429–2436 (1991).
[CrossRef]

H. Lin and N. B. Abraham, “Mode formation and beating in the transverse pattern dynamics in a laser,” Opt. Commun. 79, 476–488 (1990).
[CrossRef]

Aviles, A. F.

J. M. Perez, J. Steinshnider, R. E. Stallcup, and A. F. Aviles, “Control of chaos in a CO2 laser,” Appl. Phys. Lett. 65, 1216–1218 (1994).
[CrossRef]

Babloyantz, A.

C. Lourenco, M. Hougardy, and A. Babloyantz, “Control of low-dimensional spatiotemporal chaos in Fourier space,” Phys. Rev. E 52, 1528–1532 (1995).
[CrossRef]

Badii, R.

C. Reyl, L. Flepp, R. Badii, and E. Brun, “Control of NMR-laser chaos in high-dimensional embedding space,” Phys. Rev. E 47, 267–272 (1993).
[CrossRef]

Bielawski, S.

S. Bielawski, D. Derozier, and P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

Bleich, M. E.

M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar, “Controlling extended systems with spatially filtered, time-delayed feedback,” Phys. Rev. E 55, 2119–2126 (1997).
[CrossRef]

Brun, E.

C. Reyl, L. Flepp, R. Badii, and E. Brun, “Control of NMR-laser chaos in high-dimensional embedding space,” Phys. Rev. E 47, 267–272 (1993).
[CrossRef]

Chang, H.-C.

F. Qin, E. E. Wolf, and H.-C. Chang, “Controlling spatiotemporal patterns on a catalytic wafer,” Phys. Rev. Lett. 72, 1459–1462 (1994).
[CrossRef] [PubMed]

Concannon, H. M.

D. J. Gauthier, D. W. Sukow, H. M. Concannon, and J. E. S. Socolar, “Stabilizing unstable periodic orbits in a fast diode resonator,” Phys. Rev. E 50, 2343–2346 (1994).
[CrossRef]

Cross, M. C.

R. O. Grigoriev, M. C. Cross, and H. G. Schuster, “Pinning control of spatiotemporal chaos,” Phys. Rev. Lett. 79, 2795–2798 (1997).
[CrossRef]

Dai, J.-H.

J.-H. Dai, H.-W. Yin, and H.-J. Zhang, “Controlling chaos in a hybrid optical bistable system,” Opt. Commun. 120, 85–90 (1995).
[CrossRef]

Derozier, D.

S. Bielawski, D. Derozier, and P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

Ditto, W. L.

W. L. Ditto, S. N. Rauseo, and M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
[CrossRef] [PubMed]

Firth, W. J.

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A 58, 2577–2585 (1998).
[CrossRef]

Flepp, L.

C. Reyl, L. Flepp, R. Badii, and E. Brun, “Control of NMR-laser chaos in high-dimensional embedding space,” Phys. Rev. E 47, 267–272 (1993).
[CrossRef]

Gauthier, D. J.

D. J. Gauthier, “Controlling lasers by use of extended time-delay autosynchronization,” Opt. Lett. 23, 703–705 (1998).
[CrossRef]

D. J. Gauthier, D. W. Sukow, H. M. Concannon, and J. E. S. Socolar, “Stabilizing unstable periodic orbits in a fast diode resonator,” Phys. Rev. E 50, 2343–2346 (1994).
[CrossRef]

Gills, Z.

R. Roy, T. W. Murphy, Jr., T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1262 (1992).
[CrossRef] [PubMed]

Glorieux, P.

S. Bielawski, D. Derozier, and P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

Grebogi, C.

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

Grigoriev, R. O.

R. O. Grigoriev, M. C. Cross, and H. G. Schuster, “Pinning control of spatiotemporal chaos,” Phys. Rev. Lett. 79, 2795–2798 (1997).
[CrossRef]

Harkness, G. K.

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A 58, 2577–2585 (1998).
[CrossRef]

Harrison, R.

W. Lu and R. Harrison, “Controlling chaos using continuous interference feedback: proposal for all optical devices,” Opt. Commun. 109, 457–461 (1994).
[CrossRef]

He, K.

G. Hu and K. He, “Controlling chaos in systems described by partial differential equations,” Phys. Rev. Lett. 71, 3794–3797 (1993).
[CrossRef]

Hess, O.

C. Simmendinger and O. Hess, “Controlling delay-induced chaotic behavior of a semiconductor laser with optical feedback,” Phys. Lett. A 216, 97–105 (1996).
[CrossRef]

Hochheiser, D.

M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar, “Controlling extended systems with spatially filtered, time-delayed feedback,” Phys. Rev. E 55, 2119–2126 (1997).
[CrossRef]

Horowicz, R. J.

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, “Instabilities in passive and active optical systems with a Gaussian transverse intensity profile,” Phys. Rev. A 30, 1366–1376 (1984).
[CrossRef]

Hougardy, M.

C. Lourenco, M. Hougardy, and A. Babloyantz, “Control of low-dimensional spatiotemporal chaos in Fourier space,” Phys. Rev. E 52, 1528–1532 (1995).
[CrossRef]

Hu, G.

G. Hu and Z. Qu, “Controlling spatiotemporal chaos in coupled map lattice systems,” Phys. Rev. Lett. 72, 68–71 (1994).
[CrossRef]

G. Hu and K. He, “Controlling chaos in systems described by partial differential equations,” Phys. Rev. Lett. 71, 3794–3797 (1993).
[CrossRef]

Hunt, E. R.

R. Roy, T. W. Murphy, Jr., T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1262 (1992).
[CrossRef] [PubMed]

E. R. Hunt, “Stabilizing high-period orbits in a chaotic system: the diode resonator,” Phys. Rev. Lett. 67, 1953–1955 (1991).
[CrossRef] [PubMed]

Lin, H.

H. Lin and N. B. Abraham, “Transverse pattern variations in a laser with a parabolic excitation profile,” J. Opt. Soc. Am. B 8, 2429–2436 (1991).
[CrossRef]

H. Lin and N. B. Abraham, “Mode formation and beating in the transverse pattern dynamics in a laser,” Opt. Commun. 79, 476–488 (1990).
[CrossRef]

Lourenco, C.

C. Lourenco, M. Hougardy, and A. Babloyantz, “Control of low-dimensional spatiotemporal chaos in Fourier space,” Phys. Rev. E 52, 1528–1532 (1995).
[CrossRef]

Lu, W.

W. Lu and R. Harrison, “Controlling chaos using continuous interference feedback: proposal for all optical devices,” Opt. Commun. 109, 457–461 (1994).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, C. Oldano, and L. M. Narducci, “Cooperative frequency locking and stationary spatial structures in lasers,” J. Opt. Soc. Am. B 5, 879–888 (1988).
[CrossRef]

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, “Instabilities in passive and active optical systems with a Gaussian transverse intensity profile,” Phys. Rev. A 30, 1366–1376 (1984).
[CrossRef]

Maier, T. D.

R. Roy, T. W. Murphy, Jr., T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1262 (1992).
[CrossRef] [PubMed]

Martin, R.

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A 58, 2577–2585 (1998).
[CrossRef]

Moloney, J. V.

M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar, “Controlling extended systems with spatially filtered, time-delayed feedback,” Phys. Rev. E 55, 2119–2126 (1997).
[CrossRef]

Murphy Jr., T. W.

R. Roy, T. W. Murphy, Jr., T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1262 (1992).
[CrossRef] [PubMed]

Narducci, L. M.

L. A. Lugiato, C. Oldano, and L. M. Narducci, “Cooperative frequency locking and stationary spatial structures in lasers,” J. Opt. Soc. Am. B 5, 879–888 (1988).
[CrossRef]

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, “Instabilities in passive and active optical systems with a Gaussian transverse intensity profile,” Phys. Rev. A 30, 1366–1376 (1984).
[CrossRef]

Oldano, C.

Oppo, G.-L.

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A 58, 2577–2585 (1998).
[CrossRef]

Ott, E.

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

Perez, J. M.

J. M. Perez, J. Steinshnider, R. E. Stallcup, and A. F. Aviles, “Control of chaos in a CO2 laser,” Appl. Phys. Lett. 65, 1216–1218 (1994).
[CrossRef]

Pyragas, K.

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170, 421–428 (1992).
[CrossRef]

Qin, F.

F. Qin, E. E. Wolf, and H.-C. Chang, “Controlling spatiotemporal patterns on a catalytic wafer,” Phys. Rev. Lett. 72, 1459–1462 (1994).
[CrossRef] [PubMed]

Qu, Z.

G. Hu and Z. Qu, “Controlling spatiotemporal chaos in coupled map lattice systems,” Phys. Rev. Lett. 72, 68–71 (1994).
[CrossRef]

Rauseo, S. N.

W. L. Ditto, S. N. Rauseo, and M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
[CrossRef] [PubMed]

Reyl, C.

C. Reyl, L. Flepp, R. Badii, and E. Brun, “Control of NMR-laser chaos in high-dimensional embedding space,” Phys. Rev. E 47, 267–272 (1993).
[CrossRef]

Roy, R.

R. Roy, T. W. Murphy, Jr., T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1262 (1992).
[CrossRef] [PubMed]

Schuster, H. G.

R. O. Grigoriev, M. C. Cross, and H. G. Schuster, “Pinning control of spatiotemporal chaos,” Phys. Rev. Lett. 79, 2795–2798 (1997).
[CrossRef]

Scroggie, A. J.

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A 58, 2577–2585 (1998).
[CrossRef]

Simmendinger, C.

C. Simmendinger and O. Hess, “Controlling delay-induced chaotic behavior of a semiconductor laser with optical feedback,” Phys. Lett. A 216, 97–105 (1996).
[CrossRef]

Socolar, J. E. S.

M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar, “Controlling extended systems with spatially filtered, time-delayed feedback,” Phys. Rev. E 55, 2119–2126 (1997).
[CrossRef]

D. J. Gauthier, D. W. Sukow, H. M. Concannon, and J. E. S. Socolar, “Stabilizing unstable periodic orbits in a fast diode resonator,” Phys. Rev. E 50, 2343–2346 (1994).
[CrossRef]

Spano, M. L.

W. L. Ditto, S. N. Rauseo, and M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
[CrossRef] [PubMed]

Stallcup, R. E.

J. M. Perez, J. Steinshnider, R. E. Stallcup, and A. F. Aviles, “Control of chaos in a CO2 laser,” Appl. Phys. Lett. 65, 1216–1218 (1994).
[CrossRef]

Steinshnider, J.

J. M. Perez, J. Steinshnider, R. E. Stallcup, and A. F. Aviles, “Control of chaos in a CO2 laser,” Appl. Phys. Lett. 65, 1216–1218 (1994).
[CrossRef]

Strini, G.

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, “Instabilities in passive and active optical systems with a Gaussian transverse intensity profile,” Phys. Rev. A 30, 1366–1376 (1984).
[CrossRef]

Sukow, D. W.

D. J. Gauthier, D. W. Sukow, H. M. Concannon, and J. E. S. Socolar, “Stabilizing unstable periodic orbits in a fast diode resonator,” Phys. Rev. E 50, 2343–2346 (1994).
[CrossRef]

Wolf, E. E.

F. Qin, E. E. Wolf, and H.-C. Chang, “Controlling spatiotemporal patterns on a catalytic wafer,” Phys. Rev. Lett. 72, 1459–1462 (1994).
[CrossRef] [PubMed]

Yin, H.-W.

J.-H. Dai, H.-W. Yin, and H.-J. Zhang, “Controlling chaos in a hybrid optical bistable system,” Opt. Commun. 120, 85–90 (1995).
[CrossRef]

Yorke, J. A.

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

Zhang, H.-J.

J.-H. Dai, H.-W. Yin, and H.-J. Zhang, “Controlling chaos in a hybrid optical bistable system,” Opt. Commun. 120, 85–90 (1995).
[CrossRef]

Appl. Phys. Lett. (1)

J. M. Perez, J. Steinshnider, R. E. Stallcup, and A. F. Aviles, “Control of chaos in a CO2 laser,” Appl. Phys. Lett. 65, 1216–1218 (1994).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (3)

H. Lin and N. B. Abraham, “Mode formation and beating in the transverse pattern dynamics in a laser,” Opt. Commun. 79, 476–488 (1990).
[CrossRef]

W. Lu and R. Harrison, “Controlling chaos using continuous interference feedback: proposal for all optical devices,” Opt. Commun. 109, 457–461 (1994).
[CrossRef]

J.-H. Dai, H.-W. Yin, and H.-J. Zhang, “Controlling chaos in a hybrid optical bistable system,” Opt. Commun. 120, 85–90 (1995).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (2)

C. Simmendinger and O. Hess, “Controlling delay-induced chaotic behavior of a semiconductor laser with optical feedback,” Phys. Lett. A 216, 97–105 (1996).
[CrossRef]

K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170, 421–428 (1992).
[CrossRef]

Phys. Rev. A (3)

S. Bielawski, D. Derozier, and P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, and W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A 58, 2577–2585 (1998).
[CrossRef]

L. A. Lugiato, R. J. Horowicz, G. Strini, and L. M. Narducci, “Instabilities in passive and active optical systems with a Gaussian transverse intensity profile,” Phys. Rev. A 30, 1366–1376 (1984).
[CrossRef]

Phys. Rev. E (4)

M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar, “Controlling extended systems with spatially filtered, time-delayed feedback,” Phys. Rev. E 55, 2119–2126 (1997).
[CrossRef]

C. Reyl, L. Flepp, R. Badii, and E. Brun, “Control of NMR-laser chaos in high-dimensional embedding space,” Phys. Rev. E 47, 267–272 (1993).
[CrossRef]

D. J. Gauthier, D. W. Sukow, H. M. Concannon, and J. E. S. Socolar, “Stabilizing unstable periodic orbits in a fast diode resonator,” Phys. Rev. E 50, 2343–2346 (1994).
[CrossRef]

C. Lourenco, M. Hougardy, and A. Babloyantz, “Control of low-dimensional spatiotemporal chaos in Fourier space,” Phys. Rev. E 52, 1528–1532 (1995).
[CrossRef]

Phys. Rev. Lett. (8)

R. Roy, T. W. Murphy, Jr., T. D. Maier, Z. Gills, and E. R. Hunt, “Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system,” Phys. Rev. Lett. 68, 1259–1262 (1992).
[CrossRef] [PubMed]

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64, 1196–1199 (1990).
[CrossRef] [PubMed]

W. L. Ditto, S. N. Rauseo, and M. L. Spano, “Experimental control of chaos,” Phys. Rev. Lett. 65, 3211–3214 (1990).
[CrossRef] [PubMed]

E. R. Hunt, “Stabilizing high-period orbits in a chaotic system: the diode resonator,” Phys. Rev. Lett. 67, 1953–1955 (1991).
[CrossRef] [PubMed]

G. Hu and K. He, “Controlling chaos in systems described by partial differential equations,” Phys. Rev. Lett. 71, 3794–3797 (1993).
[CrossRef]

G. Hu and Z. Qu, “Controlling spatiotemporal chaos in coupled map lattice systems,” Phys. Rev. Lett. 72, 68–71 (1994).
[CrossRef]

F. Qin, E. E. Wolf, and H.-C. Chang, “Controlling spatiotemporal patterns on a catalytic wafer,” Phys. Rev. Lett. 72, 1459–1462 (1994).
[CrossRef] [PubMed]

R. O. Grigoriev, M. C. Cross, and H. G. Schuster, “Pinning control of spatiotemporal chaos,” Phys. Rev. Lett. 79, 2795–2798 (1997).
[CrossRef]

Other (5)

C.-T. Chen, Linear System Theory and Design (Holt, Rinehart & Winston, New York, 1984), pp. 412–417.

H. Lin, “Numerical and experimental study of transverse dynamics in a laser with several transverse modes,” Ph.D. dissertation (Bryn Mawr College, Bryn Mawr, Penn., 1991).

W. Lu, D. Yu, and R. G. Harrison, “Control of patterns in spatiotemporal chaos in optics,” Phys. Rev. Lett. 76, 3316–3319 (1996); “Tracking periodic patterns into spatiotemporal chaotic regimes,” Phys. Rev. Lett. 78, 4375–4378 (1997).
[CrossRef] [PubMed]

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. 77, 4007–4010 (1996); 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. Express 1, 39–43 (1997), http://epubs.osa.org/opticsexpress.
[CrossRef] [PubMed]

A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80, 3499–3502 (1998); S. J. Jensen, M. Schwab, and C. Denz, “Manipulation, stabilization, and control of pattern formation using Fourier space filtering,” Phys. Rev. Lett. 81, 1614–1617 (1998).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Stable domain of the plane-wave state obtained from the Lyapunov theorem in the parameter plane (Δ, s), where the two boundaries—the dashed and solid curves—correspond to 2C=2.0 and 2C=3.5, respectively. The other parameters are κ=0.1 and γ=0.5. The boundary of s is higher for smaller values of 2C.

Fig. 2
Fig. 2

(a) Stable domains of the plane-wave state numerically obtained from the Routh–Hurwitz criterion in the parameter plane (Δ, s), where the three boundaries—the solid, dotted, and dashed curves—correspond to a1=0.1, a=2.0, and a=1.2, respectively. The other parameters are κ=0.1, γ=0.5, and 2C=3.5. The stable domain varies significantly with a1. (b) Stable domains obtained from the Routh–Hurwitz criterion for 2C=2.0 (dashed curve) and 2C=3.5 (solid curve), respectively. The other parameters are a1=0.1, κ=0.1, and γ=0.5.

Fig. 3
Fig. 3

(a) Inhomogeneous, stationary transverse intensity pattern (CFL state), and (b) temporal evolution of the first three dominant modes from the initial plane-wave state to the CFL state for κ=0.1, γ=0.5, Δ=0.4, 2C=3.5, a1=0.4, and s=0. (c) Behavior of the first three modes for s=0.23. The higher-order modes are successfully suppressed.

Fig. 4
Fig. 4

(a) Transverse intensity pattern at two different times, t1 and t2, within one modulation period of the moduli, and (b) evolution of the first three modes from the initial plane-wave state to the periodically pulsating state for κ=0.1, γ=0.5, Δ=0.4, 2C=3.5, a1=0.01, and s=0. (c) Behavior of the first three modes for s=0.23.

Fig. 5
Fig. 5

(a) Average total intensity, and (b) phase diagram at x=0 for κ=0.1, γ=1.0, Δ=0.2, 2C=3.5, a1=0.0088, and s=0. (c) The irregular pulsation in 〈I〉 is suppressed when the feedback is turned on at t=20000, where s=0.02.

Equations (35)

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Et=-κ[(1-iΔ)E-2CP-ia 2Ex2+Econ],
Pt=-γ[(1+iΔ)P-ED],
Dt=-γ[D-1+(1/2)(E*P+EP*)],
ωn,c=ω0,c+κaπ2n2,
|E|s2=2C-1-Δ2,
Ps=(1-iΔ)Es2C,
Ds=(1+Δ2)2C.
E(x, t)=n=0fn(t)cos nπx,
P(x, t)=n=0pn(t)cos nπx,
D(x, t)=n=0dn(t)cos nπx,
|fns|2=(2C-1-Δ2)δn,0,
pns=1-iΔ2Cfns,
dns=1+Δ22Cδn,0,
dfndt=-κ[(1-iΔ)fn-2Cpn+ianfn+s(fn-fnδn,0)],
dpndt=-γ(1+iΔ)pn-12(1+δn,0)ml0*fmdl,
ddndt=-γdn-δn,0+14(1+δn,0)ml0*(fmpl*+fm*pl),
dundt=-κ{[1+s(1-δn,0)]un+(Δ-an)vn-2Cgn},
dvndt=-κ{[1+s(1-δn,0)]vn-(Δ-an)un-2Chn},
dgndt=-γgn-Δhn-f0swn-1+Δ22Cun,
dhndt=-γhn+Δgn-1+Δ22Cvn,
dwndt=-γwn+f0sgn+f0s2C(un-Δvn),
dRdt=MR,
λ5+C1(n)λ4+C2(n)λ3+C3(n)λ2+C4(n)λ+C5(n)=0,
an2Δf0s22C,
N=2κ[1+s(1δn,0)]02Cκ+γ1+Δ22C0γf0s2C02κ[1+s(1δn,0)]02Cκ+γ1+Δ22Cγf0sΔ2C(2Cκ+γ1+Δ22C)02γ0(γγ)f0s02Cκ+γ1+Δ22C02γ0γf0s2Cγf0sΔ2C(γγ)f0s02γ.
4γγ-(γ-γ)2f0s2>0,
s>2Cκ-γ 1+Δ22C2+4κγΔ24κγ+γγ2κf0s2Δ2C2 14γγ-(γ-γ)2f0s2,
s2+bs+c>0,
b=14κγ[4γγ-(γ-γ)2f0s2]×8κγ[4γγ-(γ-γ)2f0s2]+4γγ(γ-γ) f0s22C2Cκ+γ 1+Δ22C-γγ(1+Δ2)f0s2C2-2Cκ+γ 1+Δ22C2[8γγ-(γ-γ)2f0s2],
c=1+14κ2γ[4γγ-(γ-γ)2f0s2]×4κγγ(γ-γ) f0s22C2Cκ+γ 1+Δ22C-γγ(1+Δ2)f0s2C2+2Cκ+γ 1+Δ22C2γγ2(1+Δ2)f0s2C2-κ[8γγ-(γ-γ)2f0s2]-γ(γ-γ) f0s22C2Cκ+γ 1+Δ22C+γ2Cκ+γ 1+Δ22C2.
C1(n)=γ+2γ+2κ(1+s),
C2(n)=2γγ+γγf0s2+2κ(2γ+γ)+(γ-κ)2(1+Δ2)+κ2an(an-2Δ)+κ2(2+s)s+2κ(2γ+γ)s,
C3(n)=γ(1+Δ2)(γ-κ)2+γγ(γ+κ)f0s2+2κγγ(2+f0s2)+κ2an(an-2Δ)(2γ+γ)+κ2(2γ+γ)s2+2κ[γ(γ-κ)(1+Δ2)+γγ(2+f0s2)+2κγ+κγ]s,
C4(n)=2κγγ(κ+γ)f0s2+κ2γan[an(γ+γΔ2+2γ+γf0s2)-γΔ(4+f0s2)]+κ2[γ2(1+Δ2)+2γγ+γγf0s2]s2+κγγ[2(γ-κ)(1+Δ2)+(2γ+3κ)f0s2+4κ]s,
C5(n)=κ2γ2γ{an[an(1+Δ2+f0s2)-2Δf0s2]+2 f0s2s+(1+Δ2+f0s2)s2},

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