Файл: Staliunas K., Sanchez-Morcillo V.J. (eds.) Transverse Patterns in Nonlinear Optical Resonators(ST.pdf

References 19

components. In particular, the emergence of Turing-like patterns is predicted to occur in active and passive systems, concrete examples being lasers with a strongly di using population inversion, and degenerate OPOs with a strongly di racting pump wave. In both cases, one field plays the role of activator, and the other the role of inhibitor. It is also shown that the e ect of di u- sion and/or di raction contributes to the stabilization of spatial solitons and allows the existence of complex states resembling molecules of light.

In Chap. 13, we describe the three-dimensional structures of light predicted to occur in resonators described by the three-dimensional Swift– Hohenberg equation. This order parameter equation describes a class of nonlinear optical resonators including the synchronously pumped OPO. Various structures embedded in the envelopes of spatio-temporal light pulses are discussed, in the form of extended patterns (lamellae and tetrahedral patterns), light bubbles (the analogue of the phase solitons in two dimensions) and vortex rings. These structures exist when the OPO resonator length is matched to the length of the pump (mode-locked) laser, which emits a continuous or finite train of picosecond pulses. A three-dimensional modulation can develop on the subharmonic pulses generated, depending on several parameters such as the detuning from the resonance of the OPO cavity, and the mismatch of the resonator lengths for the pump and OPO lasers.

The final chapter, Chap. 14, deals with the influence of noise on spatial structures in nonlinear optics. Noise, which is not considered in the rest of the book, is always present in a real experiment, in the form of vacuum noise (always inevitable) or noise due to technological limitations. It is shown that the noise a ects the pattern formation in several ways. Above the modulation instability threshold, where extended patterns are expected, the noise destroys the long-range order in the pattern. Rolls and other extended structures still exist in the presence of noise, but they may display defects (such as dislocations and disclinations) with a density proportional to the intensity of the noise. Also, below the modulation instability threshold, where no patterns are expected in the ideal (noiseless) case, the noise is amplified and can result in (noisy) patterns. The symmetry of a pattern may show itself even below the pattern formation threshold, thanks to the presence of noise. This can be compared with a single-transverse-mode laser, where the coherence in the radiation develops continuously, and where the spectrum of the luminescence narrows continuously when the generation threshold is approached from below.

References

1.M.C. Cross and P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65, 851 (1993). 1, 3

2.L.A. Lugiato, Spatio-temporal structures part I, Phys. Rep. 219, 293 (1992). 2

201 Introduction

3.C.O. Weiss, Spatio-temporal structures part II, Phys. Rep. 219, 311 (1992). 2

4.L.A. Lugiato, Transverse nonlinear optics: introduction and review, Chaos, Solitons Fractals 4, 1251 (1994). 2

5.L.A. Lugiato, M. Brambilla and A. Gatti, Optical pattern formation, Adv. At. Mol. Opt. Phys. 40, 229 (1998). 2

6.F.T. Arecchi, S. Boccaletti and P.L. Ramazza, Pattern formation and competition in nonlinear optics, Phys. Rep. 318, 83 (1999). 2

7.K. Staliunas, Optical Vortices, Horizons in World Physics, vol. 228, ed. by M. Vasnetsov and K. Staliunas (Nova Science, New York, 1999). 2

8.Y.S. Kivshar, Spatial optical solitons, in Nonlinear Science at the Dawn of the 21st Century, Lecture Notes in Physics, vol. 542, ed. by P.L. Christiansen, M.P. Sørensen and A.C. Scott (Springer, Berlin, Heidelberg, 2000), p.173. 2

9.R. Neubecker and T. Tschudi, Pattern formation in nonlinear optical systems, Chaos Solitons Fractals 10, 615 (1999). 2

10.M.A. Vorontsov and W.B. Miller, Self-organization in nonlinear optics – kaleidoscope of patterns, in Self-Organization in Optical Systems and Applications in Information Technology, ed. by M.A. Vorontsov and W.B. Miller (Springer, Berlin, Heidelberg, 1995) pp. 1-25. 2

11.R. Graham and H. Haken, Laserlight – first example of a second order phase transition far from thermal equilibrium, Z. Phys. 237, 31 (1970). 2, 3

12.S.A. Akhmanov, R.V. Khoklov and A.P. Sukhorukov, in Laser Handbook, ed. by F.T. Arechi and E.O. Schultz-DuBois (North-Holland, Amsterdam, 1972). 2, 11

13.L.A. Lugiato, C. Oldano and L.M. Narducci, Cooperative frequency locking and stationary spatial structures in a laser, J. Opt. Soc. Am. B 5, 879 (1988). 2

14.L.A. Lugiato, G.L. Oppo, J.R. Tredicce, L.M. Narducci and M.A. Pernigo, Instabilities and spatial complexity in a laser, J. Opt. Soc. Am B 7, 1019 (1990). 2, 9

15.P. Coullet, L. Gil and F. Rocca, Optical vortices, Opt. Commun. 73, 403 (1989). 2, 3, 6

16.C.H. Tamm, Frequency locking of two transverse optical modes of a laser, Phys. Rev. A 38, 5960 (1988). 2, 6, 9

17.M. Brambilla, F. Battipede, L.A. Lugiato, V. Penna, F. Prati, C. Tamm, and C.O. Weiss, Transverse laser patterns I: Phase singularity crystals, Phys. Rev. A 43, 5090 (1991). 2, 3, 6

18.M.V. Berry, J.F. Nye and F.P. Wright, The elliptic umbilic di raction catastrophe, Phil. Trans. R. Soc. London 291, 453 (1979). 2, 6

19.N.B. Baranova, B.Ya. Zel’dovich, A.V. Mamaev, N.F. Pilipetskii and V.V. Shkunov, Sov. Phys. JETP Lett. 33, 196 (1981). 2, 6

20.G.-L. Oppo, G.P. D’Alessandro and W.J. Firth, Spatio-temporal instabilities of lasers in models reduced via center manifold techniques, Phys. Rev. A 44, 4712 (1991). 3

21.P.K. Jakobsen, J.V. Moloney, A.C. Newell, R. Indik, Space–time dynamics of wide-gain section lasers, Phys. Rev. A 45, 8129 (1992). 3, 9

22.K. Staliunas, Laser Ginzburg–Landau equation and laser hydrodynamics, Phys. Rev. A 48, 1573 (1993). 3, 6

23.K. Staliunas and C.O. Weiss, Tilted and standing waves and vortex lattices in class-A lasers, Physica D 81, 79 (1995). 3, 6

References 21

24.J. Lega, J.V. Moloney and A.C. Newell, Universal description of laser dynamics near threshold, Physica D 83, 478 (1995). 3

25.K. Wang, N.B. Abraham and L.A. Lugiato, Leading role of optical phase instabilities in the formation of certain laser transverse patterns, Phys. Rev. A 47, 1263 (1993). 3

26.G. D’Alessandro, Spatio-temporal dynamics of a unidirectional ring oscillator with photorefractive gain, Phys. Rev. A 46, 2791 (1992). 4

27.F.T. Arecchi, G. Giacomelli, P.L. Ramazza and S. Residori, Vortices and defect statistics in two dimensional optical chaos, Phys. Rev. Lett. 67, 3749 (1991). 4, 6

28.D. Hennequin, L. Dambly, D. Dangoise and P. Glorieux, Basic transverse dynamics of a photorefractive oscillator, J. Opt. Soc. Am. B 11, 676 (1994). 4, 6

29.K. Staliunas, M.F.H. Tarroja, G. Slekys, C.O. Weiss and L. Dambly, Analogy between photorefractive oscillators and class-A lasers, Phys. Rev. A 51, 4140 (1995). 4

30.P. Mandel, M. Georgiou and T. Erneux, Transverse e ects in coherently driven nonlinear cavities, Phys. Rev. A 47, 4277 (1993). 4

31.M. Tlidi, M. Georgiou and P. Mandel, Transverse patterns in nascent optical bistability, Phys. Rev. A 48, 4605 (1993). 4, 9

32.L.A. Lugiato and R. Lefever, Spatial dissipative structures in passive optical systems, Phys. Rev. Lett. 58, 2209 (1987). 4

33.L.A. Lugiato and C. Oldano, Stationary spatial patterns in passive optical systems: two-level atoms, Phys. Rev. A 37, 3896 (1988). 4

34.G.L. Oppo, M. Brambilla and L.A. Lugiato, Formation and evolution of roll patterns in optical parametric oscillators, Phys. Rev. A 49, 2028 (1994). 4, 9

35.S. Longhi, Traveling-wave states and secondary instabilities in optical parametric oscillators, Phys. Rev. A 53, 4488 (1996). 4, 9

36.K. Staliunas, Transverse pattern formation in optical parametric oscillators, J. Mod. Opt. 42, 1261 (1995). 4

37.G.J. de Valc´arcel, K. Staliunas, E. Rold´an and V.J. S´anchez-Morcillo, Transverse patterns in degenerate optical parametric oscillation and degenerate four-wave mixing, Phys. Rev. A 54, 1609 (1996). 4, 6, 9

38.S. Longhi and A. Geraci, Swift–Hohenberg equation for optical parametric oscillators, Phys. Rev. A 54, 4581 (1996). 4

39.G.J. de Valc´arcel, Order-parameter equations for transverse pattern formation in nonlinear optical systems with nonplanar resonators, Phys. Rev. A 56, 1542 (1997). 4

40.Lord Rayleigh, On the dynamics of revolving fluids, Proc. R. Soc. London Ser. A 93, 148 (1916). 5

41.R.C. DiPrima and H.L. Swinney, in Hydrodynamical Instabilities and the Transition to Turbulence, ed. by H.L. Swinney and J.P. Gollub (Springer, New York, 1981). 5

42.A.M. Turing, The Chemical Basis of Morphogenesis, Phil. Trans. R. Soc. London B 237, 37 (1952). 5

43.H. Meinhardt, Models of Biological Pattern Formation (Academic Press, London, 1982). 5

44.H. Haken, Synergetics, an Introduction (Springer, Berlin, 1977). 5

45.I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems, J. Chem. Phys. 48, 1695 (1968). 5

221 Introduction

46.G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (Willey, New York, 1977). 5

47.M. Brambilla, L.A. Lugiato, V. Penna, F. Prati, C. Tamm and C.O.Weiss, Transverse laser patterns II: variational principle for pattern selection, spatial multistability, and laser hydrodynamics, Phys. Rev. A 43, 5114 (1991). 6

48.K. Staliunas, Dynamics of optical vortices in a laser beam, Opt. Commun. 90, 123 (1992). 6

49.F.T. Arecchi, Space–time complexity in nonlinear optics, Physica D 51, 450 (1991). 6

50.P. Coullet, L. Gil and J. Lega, Defect-mediated turbulence, Phys. Rev. Lett. 62, 1619 (1989). 6

51.L. Gil, K. Emilsson and G.L. Oppo, Dynamics of spiral waves in a spatially inhomogeneous Hopf bifurcation, Phys. Rev. A 45, R567 (1992). 6

52.G. Huyet, M.C. Martinoni, J.R. Tredicce and S. Rica, Spatiotemporal dynamics of lasers with large Fresnel number, Phys. Rev. Lett. 75, 4027 (1995). 6

53.G. Indebetow and S.R. Liu, Defect-mediated spatial complexity and chaos in a phase-conjugate resonator, Opt. Commun. 91, 321 (1992). 6

54.R. Neubecker, M. Kreuzer and T. Tschudi, Phase defects in a nonlinear Fabry– Perot resonator, Opt. Commun. 96, 117 (1993). 6

55.G.L. Lippi, T. Ackemann, L.M. Ho er, A. Gahl and W. Lange, Interplay of linear and nonlinear e ects in the formation of optical vortices in a nonlinear resonator, Phys. Rev. A 48, R4043 (1993). 6

56.G. Balzer, C. Denz, O. Knaup and T. Tschudi, Circling vortices and pattern dynamics in a unidirectional photorefractive ring oscillator, Chaos, Solitons Fractals 10, 725 (1998). 6

57.C.O. Weiss, H.R. Telle, K. Staliunas and M. Brambilla, Restless optical vortices, Phys. Rev. A 47, R1616 (1993). 6

58.K. Staliunas and C.O. Weiss, Nonstationary vortex lattices in large aperture class-B lasers, J. Opt. Soc. Am. B 12, 1142 (1995). 6

59.Y.F. Chen and Y.P. Lan, Transverse pattern formation of optical vortices in a microchip laser with a large Fresnel number, Phys. Rev. A 65, 013802 (2002). 6

60.K.A. Gorshkov, L.N. Korzinov, M.I. Rabinowich and L.S. Tsimring, Random pinning of localized states at the birth of deterministic disorder within gradient models, J. Stat. Phys. 74, 1033 (1994). 6

61.L.S. Tsimring and I. Aranson, Cellular and localized structures in a vibrated granular layer, Phys. Rev. Lett. 79, 213 (1997). 6

62.J. Dewel, P. Borkmans, A. De Wit, B. Rudovics, J.J. Perraud, E. Dulos, J. Boissonade and P. De Kepper, Pattern selection and localized structures in reaction–di usion systems, Physica A 213, 181 (1995). 6

63.N.N. Rosanov, Di ractive autosolitons in nonlinear interferometers, J. Opt. Soc. Am. B 7, 1057 (1990). 6

64.N.N. Rosanov, Transverse Patterns in Wide-Aperture Nonlinear Optical Systems, Progress in Optics, vol. 35, ed. by E. Wolf (North-Holland, Amsterdam, 1996). 6

65.S. Fauve and O. Thual, Solitary waves generated by subcritical instabilities in dissipative systems, Phys. Rev. Lett. 64, 282 (1990). 6

66.K. Staliunas and V.J. S´anchez-Morcillo, Localized structures in degenerate optical parametric oscillators, Opt. Commun. 139, 306 (1997). 6

References 23

67.S. Longhi, Localized structures in degenerate optical parametric oscillators, Phys. Scr. 56, 611 (1997). 6

68.S. Trillo and M. Haelterman, Excitation and bistability of self-trapped signal beams in optical parametric oscillators, Opt. Lett. 23, 1514 (1998). 6

69.S. Longhi, Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation, Opt. Commun. 149, 335 (1998). 6

70.G.J. de Valc´arcel, E. Rold´an and K. Staliunas, Cavity solitons in nondegenerate optical parametric oscillators, Opt. Commun. 181, 207 (2000). 6

71.C. Etrich, U. Peschel and F. Lederer, Solitary waves in quadratically nonlinear resonators, Phys. Rev. Lett. 79, 2454 (1997). 6

72.S. Longhi, Spatial solitary waves and patterns in type II second-harmonic generation, Opt. Lett. 23, 346 (1998). 6

73.U. Peschel, D. Michaelis, C. Etrich and F. Lederer, Formation, motion, and decay of vectorial cavity solitons, Phys. Rev. E 58, 2745 (1998). 6, 7

74.D.V. Skryabin and W.J. Firth, Interaction of cavity solitons in degenerate optical parametric oscillators, Opt. Lett. 24, 1056 (1999). 7

75.D.V. Skryabin, Instabilities of cavity solitons in optical parametric oscillators, Phys. Rev. E 60, R3508 (1999). 7

76.W.J. Firth and A. Lord, Two dimensional solitons in a Kerr cavity, J. Mod. Opt. 43, 1071 (1996). 7

77.V.J. S´anchez-Morcillo, I. P´erez-Arjona, F. Silva, G.J. de Valc´arcel and E. Rold´an, Vectorial Kerr-cavity solitons, Opt. Lett. 25, 957 (2000). 7

78.M. Sa man, D. Montgomery and D.Z. Anderson, Collapse of a transversemode continuum in a self-imaging photorefractively pumped ring resonator, Opt. Lett. 19, 5180 (1994). 7, 10

79.G. Slekys, K. Staliunas and C.O. Weiss, Spatial solitons in optical photorefractive oscillators with saturable absorber, Opt. Commun. 149, 113 (1998). 7

80.W. Kr´olikowski, M. Sa man, B. Luther-Davies and C. Denz, Anomalous interaction of spatial solitons in photorefractive media, Phys. Rev. Lett. 80, 3240 (1998). 7

81.V.B. Taranenko, K. Staliunas and C.O. Weiss, Spatial soliton laser: localized structures in a laser with a saturable absorber in a self-imaging resonator, Phys. Rev. A 56, 1582 (1997). 7

82.K. Staliunas, V.B. Taranenko, G. Slekys, R. Viselga and C.O. Weiss, Dynamical spatial localized structures in lasers with intracavity saturable absorbers, Phys. Rev. A 57, 599 (1998). 7

83.A.M. Dunlop, E.M. Wright and W.J. Firth, Spatial soliton laser, Opt. Commun. 147, 393 (1998). 7

84.M. Brambilla, L.A. Lugiato, F. Prati, L. Spinelli and W.J. Firth, Spatial soliton pixels in semiconductor devices, Phys. Rev. Lett. 79, 2042 (1997). 7

85.D. Michaelis, U. Peschel and F. Lederer, Multistable localized structures and superlattices in semiconductor optical resonators, Phys. Rev. A 56, R3366 (1997). 7

86.L. Spinelli, G. Tissoni, M. Brambilla, F. Prati and L.A. Lugiato, Spatial solitons in semiconductor microcavities, Phys. Rev. A 58, 2542 (1998). 7

87.G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini and L.A. Lugiato, Cavity solitons in passive bulk semiconductor microcavities. I. Microscopic model and modulational instabilities, J. Opt. Soc. Am. B 16, 2083