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Spontaneous patterns formation in thin film of two-level centers with a wide-aperture optical feedback
1 April 1995
OPTICS COMMUNICATIONS Optics Communications
115 ( 1995) 559-562
Spontaneous patterns formation in thin film of two-level centers with a wide-aperture optical feedback A.A. Afanas’ev, Yu.A. Logvin, A.M. Samson, B.A. Samson Institute of Physics, Belarusian Academy of Sciences, 70 Kikaryna Ave.. Minsk, 220072, Belarus Received
14 September
1994; revised version received 17 November
1994
Abstract We present a thin film of two-level centers with a feedback mirror as a new object for a spontaneous formation of hexagon patterns, which can be perturbed by lattice penta-hepta defects. A spontaneous switching between two bistable states, initiated by a patterns formation, is detected.
Transverse nonlinear optical effects have attracted the attention of numerious research groups because of their fundamental content and possible applications in optical data processing [ 11. Some of them, namely, regular and turbulent pattern formation in schemes of counterpropagation in Kerr [ 21 or resonant [ 31 media, Kerr slice with feedback mirror [ 41, and in a bistable ring cavity [ 51 demonstrate clear analogy between hydrodynamics and nonlinear optics. The origin of this analogy lies in the loosing of stability of the homogeneous state and subsequent interaction between inhomogeneous modes. As a result regular patterns in a form of rolls or hexagons can appear as fundamental nonlinear modes of a lowest symmetry. In this letter we suggest a thin film of two-level centers with a wide-aperture light feedback, provided by a plane mirror, as a new object for an observation of spontaneous formation of regular patterns, which possess a hexagonal symmetry. Firstly, the film with a thickness much smaller than the light wavelength allows to eliminate the standing-wave effects inside the film. Such a reducing of longitudinal spatial degree of freedom plays the same role as the well-known
0030-4018/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDlOO30-4018(95)00055-O
mean-field approximation for nonlinear interferometers [ 61. The second specific feature of the model is the internal light feedback which is created by the film centers. A detailed consideration (see, e.g. Ref. [ 71) of such a system, starting from the solution of Maxwell equations, shows the presence of the coherent mechanism of light-matter interaction, which results in sophisticated spatio-temporal dynamics. The thin film demonstrates such nonlinear effects as mirrorless optical bistability [ 81, self-pulsations and dynamical chaos [ 9,101, formation of spatio-temporal structures due to switching waves [ 111. Here we show that spontaneous hexagon formation occurs in a control parameter region close to the bistability one. Such a neighbourhood can result in spontaneous switching from one bistable state to another initiated by the development of transverse instability. Our system consists of a thin film of two-level centers illuminated by a light beam propagated in zdirection perpendicularly to the film plane (see Fig. 1). After transmission through the film, light is reflected by a mirror which is set parallel to the film. Due to the small thickness of the film (which is sup-
560
A.A. Afanas’ev et al. /Optics
-I Eo
Communications
115 (1995) 559-562
K
- 1 EC+) T
X
If
4c
12 +
I + 62 + py
4C6
(
)I 2
1 +@+IE/*
’
(3)
Fig. I. The thin film of two-level centres is illuminated by the light beam propagated in z-direction. After transmission through the film, light is reflected by the mirror and goes back.
posed to be much smaller than the wavelength of incident radiation), the Maxwell equations are reduced to simple relations for the field amplitudes [ 71: E(t,rl)
= Ei”(t,rl)
- i2CR(t,ri),
(1)
where E( t, rl) is the dimensionless amplitude of the field acting on two-level centres in the film, Ein( t, rl) is the amplitude of field which consists of the cw homogeneous external one Eo and feedback field 2( EOi2CR( t - 7, rl)) (propagation operator 2 is responsible for the field transformation at transit on the feedback loop, 7 is the feedback delay). The coefficient C is a bistability parameter [7-l 11, proportional to the center density in the film, The value R is the dimensionless polarization obeyed the Bloch equations:
which is identical to the relation for the nonlinear interferometer in the mean field approximation [6]. Bistability in our system takes place at the following condition (4C + 1)2(C - 2)/C
W, = -y(W
+i&R+iWE, + I) + i(E*R - ER*)/2,
(2)
where W is the population difference, the subscript r denotes the time derivative (time t in Eqs. (2) is normalized on a transverse relaxation time T2), y is the ratio of transverse and longitudinal relaxation times, 6 is the normalized detuning of the field from the resonant transition. We should note that because of interference effect between the incident and reflected waves (the phase is implicitly contained in the propagation operator L) the distance between the film and the mirror has to be interferometrically stable. That can be easy realized using a dielectric plate of thickness I between film and mirror. Our description is valid also for the copropagation geometry in a ring cavity, but in this case the cavity length is of great importance and has to be finely adjusted. Below, for the sake of simplicity, we restrict ourselves to the special case of zero bias phase. The steady-states of the system ( 1) ,( 2)) homogeneous on a transverse plane {rl}, are described by the expression for the field intensities:
(4)
When studying the stability of the steady states (3) we suppose that the polarization perturbation 6R, which is proportional to exp( Ar - iklri), takes the factor exp(i0) = exp( ik:l/k) (1 is the distance between the film and the mirror, k is the z-component of the wave vector) after the feedback loop. We admit here that the delay time 7 is much shorter than the relaxation times of the resonant transition. Also, we suppose the reflectivity of the mirror to be equal to unit. Linear stability analysis results in the following equation, which determines the boundaries of the instability area (A = 0): [1+2CW,(l
R,= (-1
> 27S2.
+-
+cosf3)12+
[S+2CWSsin812
lEoI* 1 + a2 - 8(CW,)*( 4Y
1 + cos0)
= o,
(5)
( I + 4CWS)2 + s2
where W, is the steady-state solution for the population difference. An analysis of Eq. (5) shows that at S = 0 the homogeneous perturbation with kl = 0 has the lowest threshold, which corresponds to an instability of a middle branch of S-shaped bistable characteristics. At S # 0 (regardless of the sign of 6) inhomogeneous perturbations with nonzero vector kl dominate. The most unstable wavevector value depends not only on the internal system parameters like in Ref. [ 41, but also on the intensity and frequency of the external pump Eo. The regions of instability of nonzero spatial harmonics, as well as the bistability domain, on the plane of control parameters (2C, 10 = I&l*) are drawn in Fig. 2. It is seen from Fig. 2 that spatially inhomogeneous perturbations develop below the bistability threshold C < Cl. Increasing of parameter C > Cl leads to an appearence of the bistability region (bounded by the dashed line in Fig. 2), which splits the domain of spatial instability on two subzones. One of them lies on the lower branch of the bistable curve
A.A. Afanas ‘ev et al. /Optics Communications 115 (1995) 559-562
561
26.00
IO 26.00
24.00
16.00
14.00 5550 2ci
Fig. 2. Domains 6 = 2.y
=
of instability
1. The
instability
of the plane-wave
2c solutions for
1519.5419 56
19.6 1_.1
19.56
19.62
of unhomogeneous perturbations external mtensity
occurs inside the bands U, L and the closed loop bounded by the solid line. The region of the bistability is bounded by the dashed
Fig. 3. Bistahle characteristic jEI’( lE#)
line. The region of the switching induced by patterns formation
ity regions (crosses). The arrow illustrates the switching induced
near the switch-up edge of the band L is shaded.
by patterns formation.
(band L in Fig. 2)) the other one on the upper branch (band U) .It should be noted that the width of the region of the spatial instability on the upper branch is quite narrow, and therefore is undistinguished in Fig. 2. The relative disposition of the regions of instability of spatially inhomogeneous perturbations on the bistable curve is illustrated in Fig. 3. Both these regions adjoin the switch-up and switch-down points. The numerical simulation of the Eqs. ( 1) ,(2) was carried out on a Cartesian grid 128 x 128 using the second-order Runge-Kutt method for the Bloch equations and the fast Fourier transform for the light propagation to the mirror and back. We started from a ground state ( W( t = 0) = - 1) of two-level centers, perturbated by a small additive transverse noise and we monitored the transverse distribution of output light intensity and its spectrum of transverse spatial frequencies. In order to eliminate a development of high spatial harmonics we suppressed them with a filter. Physically it can be ensured by a diaphragm or just by a finite size of the feedback mirror. The typical transverse light distribution after a longtime (hundreds relaxation times) evolution is presented in Fig. 4. The first stage of development of this hexagonal pattern from a nonexcited state was the establishement of a quasy-homogeneous solution cor-
with transverse instabil-
Fig. 4. Contour plot of the output transverse intensity distribution at S = 2.7 = I, C = 3.7, I,, = 19.6. The location of the defect is marked by the circle.
responding to a choosen set of control parameters in a plane-wave approximation. Then this solution was broken up because of excitation of off-axis perturbations with the most unstable wavevector kl,, resulting in a ring-like spatial spectrum, or a disordered set of spots in a near-field intensity distribution. The last stage was the ordering of spots into a hexagonal lattice
562
A.A. Afanas’ev et al. /Optics
due to an interplay between the off-axis components. The process for the hexagonalization is quite common and can be clarified carrying out a nonlinear stability analysis which keeps higher orders of perturbations. In Fig. 4 can clearly be seen the presence of pentahepta defects of the regular hexagonal structure. The total number and positions of defects are completely determined by the initial noise. Defects of a such kind were earlier observed in Ref. [ 51 in a simulation of pattern formation in a bistable ring cavity. Also, we have observed a spontaneous switching from one bistable state to another initiated by a pattern formation. This happens near the upper switchup edge of the bistability domain (shaded in Fig. 2) and is caused by a competition between homogeneous and inhomogeneous components. Temporarily established homogeneous solution, corresponding to a lower branch of the bistability curve (see Fig. 3)) is destroyed due to a development of off-axis perturbations. If a transverse modulation depth in a resulting pattern is sufficient to “hook” the upper branch, the homogeneous solution corresponding to a second bistable state, appears and begins to predominate. As a result, the pattern dies, have performed its part as a catalyst for the spontaneous switching. We detected only unidirectional spontaneous switching from lower branch to upper, which means the upper branch is globally more stable. A similar effect at the transmission of a gaussian beam through a ring cavity with a Kerr slice was described in Ref. [ 121. As a special feature of pattern formation in a thin film of two-level centers, one can point out the following. In contrast with the resonators schemes [5,13 1, here we have more simple relations for the incident, reflected and transmitted fields, and therefore, our threshold condition for pattern formation is independent of the sign of the detuning between the field and the two-state transition. It is not the case for all previously considered pattern formation schemes [ 25,12,13], where distinctions occur for focusing and defocusing media. This particularity of the thin film model will determine the special requirements for corresponding experiment. Because of the importance of the phase relations, change of the distance between the film and mirror in a part of the light wavelength may
Communications
115 (1995) 559-562
lead to the dissappearance of bistability and/or pattern formation (in contrast with a diffusive Kerr slice). Also special attention has to be given to the coherence length of the light source. In conclusion, we have shown that a thin film of twolevel centers with a wide-aperture feedback is capable to produce a nonlinear spatio-temporal complexity resulting in a formation of regular hexagon patterns, which can be perturbed by lattice defects. An interplay between formation of inhomogeneous structures and bistability effect lies in the basis of a new kind of switching, namely, spontaneous switching initiated by patterns formation. This work was partially supported by the Soros Grants awarded through the American Physical Society and the International Science Foundation.
References I I] N.V. Abraham and W.J. Firth, J. Opt. Sot. Am. B 7 ( 1990) 95 1. 21 R. Chang, W.J. Firth, R. Indik, J.V. Moloney and E.M. Wright, Optics Comm. 88 ( 1992) 167. 31 G. Grynberg, E. LeBihan, P Verkerk. P Simoneau, J.J.R. Leite, D. Bloch. S. Le Boiteaux and M. Ducloy, Optics Comm. 66 (1988) 321. 4) G. D’Alessandro and W.3. Firth, Phys. Rev. Len. 66 ( 1991) 2597. 51 W.J. Firth, A.J. Scroggie, L.A. Lugiato and G.S. McDonald, Phys. Rev. A 46 ( 1992) R3609. 61 L.A. Lugiato, in: Progress in Optics, Vol. 21, ed. E. Wolf (North-Holland, Amsterdam, 1984) p. 71. 71 M. Benedict and E.D. Trifonov, Phys. Rev. A 38 (1988) 2854. 181 SM. Zakharov and E.A. Manykin. Poverkhnost’. No. 2 (1988) 137. [9] A.M. Basharov, Sov. Phys. JETP 67 (1988) 1741. 1lo] Yu. A. Logvin and A.M. Samson Sov. Phys. JETP 75 ( 1992) 250. [ I1 ] Yu. A. Logvin. A.M. Samson and V.M. Volkov, to appear in Solitons, Chaos and Fractals, 1994. [ 121 W.J. Firth, Complexity and defects in passive nonlinear optical systems, in: Nonlinear Dynamics and Spatial Complexity in Optical Systems, eds. R.G. Harrison and J.S. Uppal (SUSSP & Inst. Phys. Publ., Bristol and Philadelphia. 1993). 1131 P Mandel, M. Georgiou and T. Emeux, Phys. Rev. A 47 (1993) 4277.
OPTICS COMMUNICATIONS Optics Communications
115 ( 1995) 559-562
Spontaneous patterns formation in thin film of two-level centers with a wide-aperture optical feedback A.A. Afanas’ev, Yu.A. Logvin, A.M. Samson, B.A. Samson Institute of Physics, Belarusian Academy of Sciences, 70 Kikaryna Ave.. Minsk, 220072, Belarus Received
14 September
1994; revised version received 17 November
1994
Abstract We present a thin film of two-level centers with a feedback mirror as a new object for a spontaneous formation of hexagon patterns, which can be perturbed by lattice penta-hepta defects. A spontaneous switching between two bistable states, initiated by a patterns formation, is detected.
Transverse nonlinear optical effects have attracted the attention of numerious research groups because of their fundamental content and possible applications in optical data processing [ 11. Some of them, namely, regular and turbulent pattern formation in schemes of counterpropagation in Kerr [ 21 or resonant [ 31 media, Kerr slice with feedback mirror [ 41, and in a bistable ring cavity [ 51 demonstrate clear analogy between hydrodynamics and nonlinear optics. The origin of this analogy lies in the loosing of stability of the homogeneous state and subsequent interaction between inhomogeneous modes. As a result regular patterns in a form of rolls or hexagons can appear as fundamental nonlinear modes of a lowest symmetry. In this letter we suggest a thin film of two-level centers with a wide-aperture light feedback, provided by a plane mirror, as a new object for an observation of spontaneous formation of regular patterns, which possess a hexagonal symmetry. Firstly, the film with a thickness much smaller than the light wavelength allows to eliminate the standing-wave effects inside the film. Such a reducing of longitudinal spatial degree of freedom plays the same role as the well-known
0030-4018/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDlOO30-4018(95)00055-O
mean-field approximation for nonlinear interferometers [ 61. The second specific feature of the model is the internal light feedback which is created by the film centers. A detailed consideration (see, e.g. Ref. [ 71) of such a system, starting from the solution of Maxwell equations, shows the presence of the coherent mechanism of light-matter interaction, which results in sophisticated spatio-temporal dynamics. The thin film demonstrates such nonlinear effects as mirrorless optical bistability [ 81, self-pulsations and dynamical chaos [ 9,101, formation of spatio-temporal structures due to switching waves [ 111. Here we show that spontaneous hexagon formation occurs in a control parameter region close to the bistability one. Such a neighbourhood can result in spontaneous switching from one bistable state to another initiated by the development of transverse instability. Our system consists of a thin film of two-level centers illuminated by a light beam propagated in zdirection perpendicularly to the film plane (see Fig. 1). After transmission through the film, light is reflected by a mirror which is set parallel to the film. Due to the small thickness of the film (which is sup-
560
A.A. Afanas’ev et al. /Optics
-I Eo
Communications
115 (1995) 559-562
K
- 1 EC+) T
X
If
4c
12 +
I + 62 + py
4C6
(
)I 2
1 +@+IE/*
’
(3)
Fig. I. The thin film of two-level centres is illuminated by the light beam propagated in z-direction. After transmission through the film, light is reflected by the mirror and goes back.
posed to be much smaller than the wavelength of incident radiation), the Maxwell equations are reduced to simple relations for the field amplitudes [ 71: E(t,rl)
= Ei”(t,rl)
- i2CR(t,ri),
(1)
where E( t, rl) is the dimensionless amplitude of the field acting on two-level centres in the film, Ein( t, rl) is the amplitude of field which consists of the cw homogeneous external one Eo and feedback field 2( EOi2CR( t - 7, rl)) (propagation operator 2 is responsible for the field transformation at transit on the feedback loop, 7 is the feedback delay). The coefficient C is a bistability parameter [7-l 11, proportional to the center density in the film, The value R is the dimensionless polarization obeyed the Bloch equations:
which is identical to the relation for the nonlinear interferometer in the mean field approximation [6]. Bistability in our system takes place at the following condition (4C + 1)2(C - 2)/C
W, = -y(W
+i&R+iWE, + I) + i(E*R - ER*)/2,
(2)
where W is the population difference, the subscript r denotes the time derivative (time t in Eqs. (2) is normalized on a transverse relaxation time T2), y is the ratio of transverse and longitudinal relaxation times, 6 is the normalized detuning of the field from the resonant transition. We should note that because of interference effect between the incident and reflected waves (the phase is implicitly contained in the propagation operator L) the distance between the film and the mirror has to be interferometrically stable. That can be easy realized using a dielectric plate of thickness I between film and mirror. Our description is valid also for the copropagation geometry in a ring cavity, but in this case the cavity length is of great importance and has to be finely adjusted. Below, for the sake of simplicity, we restrict ourselves to the special case of zero bias phase. The steady-states of the system ( 1) ,( 2)) homogeneous on a transverse plane {rl}, are described by the expression for the field intensities:
(4)
When studying the stability of the steady states (3) we suppose that the polarization perturbation 6R, which is proportional to exp( Ar - iklri), takes the factor exp(i0) = exp( ik:l/k) (1 is the distance between the film and the mirror, k is the z-component of the wave vector) after the feedback loop. We admit here that the delay time 7 is much shorter than the relaxation times of the resonant transition. Also, we suppose the reflectivity of the mirror to be equal to unit. Linear stability analysis results in the following equation, which determines the boundaries of the instability area (A = 0): [1+2CW,(l
R,= (-1
> 27S2.
+-
+cosf3)12+
[S+2CWSsin812
lEoI* 1 + a2 - 8(CW,)*( 4Y
1 + cos0)
= o,
(5)
( I + 4CWS)2 + s2
where W, is the steady-state solution for the population difference. An analysis of Eq. (5) shows that at S = 0 the homogeneous perturbation with kl = 0 has the lowest threshold, which corresponds to an instability of a middle branch of S-shaped bistable characteristics. At S # 0 (regardless of the sign of 6) inhomogeneous perturbations with nonzero vector kl dominate. The most unstable wavevector value depends not only on the internal system parameters like in Ref. [ 41, but also on the intensity and frequency of the external pump Eo. The regions of instability of nonzero spatial harmonics, as well as the bistability domain, on the plane of control parameters (2C, 10 = I&l*) are drawn in Fig. 2. It is seen from Fig. 2 that spatially inhomogeneous perturbations develop below the bistability threshold C < Cl. Increasing of parameter C > Cl leads to an appearence of the bistability region (bounded by the dashed line in Fig. 2), which splits the domain of spatial instability on two subzones. One of them lies on the lower branch of the bistable curve
A.A. Afanas ‘ev et al. /Optics Communications 115 (1995) 559-562
561
26.00
IO 26.00
24.00
16.00
14.00 5550 2ci
Fig. 2. Domains 6 = 2.y
=
of instability
1. The
instability
of the plane-wave
2c solutions for
1519.5419 56
19.6 1_.1
19.56
19.62
of unhomogeneous perturbations external mtensity
occurs inside the bands U, L and the closed loop bounded by the solid line. The region of the bistability is bounded by the dashed
Fig. 3. Bistahle characteristic jEI’( lE#)
line. The region of the switching induced by patterns formation
ity regions (crosses). The arrow illustrates the switching induced
near the switch-up edge of the band L is shaded.
by patterns formation.
(band L in Fig. 2)) the other one on the upper branch (band U) .It should be noted that the width of the region of the spatial instability on the upper branch is quite narrow, and therefore is undistinguished in Fig. 2. The relative disposition of the regions of instability of spatially inhomogeneous perturbations on the bistable curve is illustrated in Fig. 3. Both these regions adjoin the switch-up and switch-down points. The numerical simulation of the Eqs. ( 1) ,(2) was carried out on a Cartesian grid 128 x 128 using the second-order Runge-Kutt method for the Bloch equations and the fast Fourier transform for the light propagation to the mirror and back. We started from a ground state ( W( t = 0) = - 1) of two-level centers, perturbated by a small additive transverse noise and we monitored the transverse distribution of output light intensity and its spectrum of transverse spatial frequencies. In order to eliminate a development of high spatial harmonics we suppressed them with a filter. Physically it can be ensured by a diaphragm or just by a finite size of the feedback mirror. The typical transverse light distribution after a longtime (hundreds relaxation times) evolution is presented in Fig. 4. The first stage of development of this hexagonal pattern from a nonexcited state was the establishement of a quasy-homogeneous solution cor-
with transverse instabil-
Fig. 4. Contour plot of the output transverse intensity distribution at S = 2.7 = I, C = 3.7, I,, = 19.6. The location of the defect is marked by the circle.
responding to a choosen set of control parameters in a plane-wave approximation. Then this solution was broken up because of excitation of off-axis perturbations with the most unstable wavevector kl,, resulting in a ring-like spatial spectrum, or a disordered set of spots in a near-field intensity distribution. The last stage was the ordering of spots into a hexagonal lattice
562
A.A. Afanas’ev et al. /Optics
due to an interplay between the off-axis components. The process for the hexagonalization is quite common and can be clarified carrying out a nonlinear stability analysis which keeps higher orders of perturbations. In Fig. 4 can clearly be seen the presence of pentahepta defects of the regular hexagonal structure. The total number and positions of defects are completely determined by the initial noise. Defects of a such kind were earlier observed in Ref. [ 51 in a simulation of pattern formation in a bistable ring cavity. Also, we have observed a spontaneous switching from one bistable state to another initiated by a pattern formation. This happens near the upper switchup edge of the bistability domain (shaded in Fig. 2) and is caused by a competition between homogeneous and inhomogeneous components. Temporarily established homogeneous solution, corresponding to a lower branch of the bistability curve (see Fig. 3)) is destroyed due to a development of off-axis perturbations. If a transverse modulation depth in a resulting pattern is sufficient to “hook” the upper branch, the homogeneous solution corresponding to a second bistable state, appears and begins to predominate. As a result, the pattern dies, have performed its part as a catalyst for the spontaneous switching. We detected only unidirectional spontaneous switching from lower branch to upper, which means the upper branch is globally more stable. A similar effect at the transmission of a gaussian beam through a ring cavity with a Kerr slice was described in Ref. [ 121. As a special feature of pattern formation in a thin film of two-level centers, one can point out the following. In contrast with the resonators schemes [5,13 1, here we have more simple relations for the incident, reflected and transmitted fields, and therefore, our threshold condition for pattern formation is independent of the sign of the detuning between the field and the two-state transition. It is not the case for all previously considered pattern formation schemes [ 25,12,13], where distinctions occur for focusing and defocusing media. This particularity of the thin film model will determine the special requirements for corresponding experiment. Because of the importance of the phase relations, change of the distance between the film and mirror in a part of the light wavelength may
Communications
115 (1995) 559-562
lead to the dissappearance of bistability and/or pattern formation (in contrast with a diffusive Kerr slice). Also special attention has to be given to the coherence length of the light source. In conclusion, we have shown that a thin film of twolevel centers with a wide-aperture feedback is capable to produce a nonlinear spatio-temporal complexity resulting in a formation of regular hexagon patterns, which can be perturbed by lattice defects. An interplay between formation of inhomogeneous structures and bistability effect lies in the basis of a new kind of switching, namely, spontaneous switching initiated by patterns formation. This work was partially supported by the Soros Grants awarded through the American Physical Society and the International Science Foundation.
References I I] N.V. Abraham and W.J. Firth, J. Opt. Sot. Am. B 7 ( 1990) 95 1. 21 R. Chang, W.J. Firth, R. Indik, J.V. Moloney and E.M. Wright, Optics Comm. 88 ( 1992) 167. 31 G. Grynberg, E. LeBihan, P Verkerk. P Simoneau, J.J.R. Leite, D. Bloch. S. Le Boiteaux and M. Ducloy, Optics Comm. 66 (1988) 321. 4) G. D’Alessandro and W.3. Firth, Phys. Rev. Len. 66 ( 1991) 2597. 51 W.J. Firth, A.J. Scroggie, L.A. Lugiato and G.S. McDonald, Phys. Rev. A 46 ( 1992) R3609. 61 L.A. Lugiato, in: Progress in Optics, Vol. 21, ed. E. Wolf (North-Holland, Amsterdam, 1984) p. 71. 71 M. Benedict and E.D. Trifonov, Phys. Rev. A 38 (1988) 2854. 181 SM. Zakharov and E.A. Manykin. Poverkhnost’. No. 2 (1988) 137. [9] A.M. Basharov, Sov. Phys. JETP 67 (1988) 1741. 1lo] Yu. A. Logvin and A.M. Samson Sov. Phys. JETP 75 ( 1992) 250. [ I1 ] Yu. A. Logvin. A.M. Samson and V.M. Volkov, to appear in Solitons, Chaos and Fractals, 1994. [ 121 W.J. Firth, Complexity and defects in passive nonlinear optical systems, in: Nonlinear Dynamics and Spatial Complexity in Optical Systems, eds. R.G. Harrison and J.S. Uppal (SUSSP & Inst. Phys. Publ., Bristol and Philadelphia. 1993). 1131 P Mandel, M. Georgiou and T. Emeux, Phys. Rev. A 47 (1993) 4277.