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SPONTANEOUS SPATIAL SYMMETRY BREAKING IN PASSIVE OPTICAL FEEDBACK SYSTEMS
A. Ouarzeddini, H. Adachihara, J. Moloney, D. Mclaughlin, A. Newell
To cite this version:
A. Ouarzeddini, H. Adachihara, J. Moloney, D. Mclaughlin, A. Newell. SPONTANEOUS SPATIAL
SYMMETRY BREAKING IN PASSIVE OPTICAL FEEDBACK SYSTEMS. Journal de Physique
Colloques, 1988, 49 (C2), pp.C2-455-C2-458. �10.1051/jphyscol:19882108�. �jpa-00227618�
SPONTANEOUS SPATIAL SYMMETRY BREAKING IN PASSIVE OPTICAL FEEDBACK SYSTEMS
A. OUARZEDDINI, H. ADACHIHARA, J.V. MOLONEY', D.W. McLAUGHLIN and A.C. NEWELL
Department of Mathematics, University of Arizona, Tucson, AZ 85721, U.S.A.
" ~ e p a r t m e n t of Physics, Heriot-Watt University, Riccarton, GB-Edinburgh EH14 4AS, Scotland, Great-Britain
R6sum6
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On montre qu' une formule d'instabilit6 modulationelle derive6 auparavant est universellement ap- plicable aux systbmes optiques non lin6aires feedback. Un nouveau type de structure d'6volution complexe en temps et en espace est identifik & deux dimensions d7espace transverse et est associe avec un recurrent 6crasement des larges amplitudes de filaments satur6s dans des structures d'anneaux et de cretes.Abstract - We show that a previously derived modulational instability formula is universally applicable to nonlinear optical feedback systems. A new type of complex spatio-temporal pattern evolution is identified in two-transverse space dimensions and is associated with the recurrent collapse of large amplitude saturated filaments into ring-like and ridge-like structures.
1
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INTRODUCTION-
In an earlier work [I] a formula describing a modulational instability on the plane or quasi-plane wave background of the field in a passive nonlinear optical ring cavity was derived. This formula was applied to the specific case of the eigenvalue of the linearization of the fixed point going through -1 signifying a period doubling bifurcation. The Ikeda plane wave instability analysis was shown to be invalid, indicating that such fixed points are unstable to transverse fluctuations. It is shown that this formula is universally applicable to nonlinear optical feedback systems exhibiting strong nonlinear dispersion. We recover, as a special case, the recent mean field result of Lugiato et al. 121 in the case where the above eigenvalue approaches +I, signifying a saddle-node bifurcation. The modulational instability is of widespread occurrence even in situations where the plane wave solution (K = 0) is strongly damped. It explains the initiation of upper bistable branch solitary wavetrains [3] and shows that the dynamical switching from a low to high transmission state with transverse spatial rings occurs via nonlinear generation of higher harmonics in K space. We demonstrate that the initial instability growth is the same, irrespective of whether it grows on an apertured plane wavefront or on a broad Gaussian envelope. The final asymptotic states differ significantly, however, between the plane wave and Gaussian beam cases. This can be understood simply by realizing that only a finite number of nonlinear modes can grow from a plane wavefront when seeded with a sinusoidal modulation whereas a continuum of modes exist in K - space for a Gaussian beam. We will illustrate this instability for both Kerr and saturable nonlinearities for the three separate cases of (i) a single-valued response curve where the plane wave solution is strongly damped, (ii) a critical region showing infinite gain where one of the linearized eigenvalues of the plane wave solution is about to cross the unit circle along the positive real axis, and (iii) a hysteretic response region where this latter eigenvalue has passed through the unit circle, signifying a saddle-node bifurcation...-. r
Fig. 1 - Plane wave (g vs a) response curve and accompanying stability curve b ( p , T ) for transverse perturbations. The b(p, T) curve is shown for the three values 191 located a t the positions labened 1, 2 and 3 on the response curve. The horizontal dashed line represents the critical value 6,. Dashed portions of the stability curves define the oscillatory band.
Parameters are p = 2, = 1.2 and R = 0.9.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19882108
C2-456 JOURNAL
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PHYSIQUEThese examples serve t o elnphasize that the basic modulational instability discussed in [I] is in no way dependent on hysteresis effects but is a universal phenomenon associated with strongly dispersive nonlinear feedback systems. The instability for optical beams is intrinsically two-dimensional. Our numerical study of the two-transverse dimensional problem points to an extremely rich and complex spatio-temporal pattern evolution which we will briefly discuss below.
2 - THEORY - Our model is an optical bistable ring resonator for which the electromagnetic field circulation assuming good cavity conditions is described by an infinite dimensional map [I],
G,+I(x, 0) = a(x)
+
Rexp(id0)G,(x, L), GO = 0. (2) Here Gn refers to the intra-cavity transverse Geld amplitude on the n-th circuit of the resonator; a(x) represents the (real) input pump laser beam amplitude, R is the input and output mirror intensity reflection coefficient (R=90% is assumed) and d o is the empty cavity detuning. The nonlinear optical response is described by the function N (I), I = GG*, where I is the intracavity field intensity. In the specific cases addressed here, N ( I ) = -1+
21 for a Kerr nonlinearity and N(I) = -1/(1+ 21) for a saturable nonlinearity. The parameter 7 = In2/4?rF measures the strength of linear diffraction, being inversely proportional t o the Fresnel number F while the parameter p measures the effective nonlinear medium propagation length. We are treating purely dispersive optical bistability under self-focussing conditions (p < 0). The modulational instability also occurs for our problem for self-defocussing nonlinearities. This map is iterated by specifying the input pump laser beam profile a(x) which, for our present purposes will be either a plane wave or a Gaussian spatial profile. This determines GI from (2) which now becomes the initial data for the propagation equation (1) over the filled resonator length L. Eqn(2) now determines the new initial data and the process is repeated until a final asymptotic state is reached. This final state may be stationary, periodic or chaotic. In particular, reference [I] addressed the case originally studied in a plane wave context by Ikeda [3] and showed that a new type of modulational chaos developes invalidating the plane wave analysis. In the present work we concentrate on the "stable" regime in a parameter window remote from the period doubling bifurcation. Stability of a plane wave t o perturbations with a transverse spatial dependence is investigated. We will demonstrate that these instability predictions carry over directly t o the case of a broad Gaussian beam.Fig. 2 - Response curve and accompanying stability diagram for a saturable nonlinearity near the infinite gain region.
The parameter
+,,
has been changed t o 0.8.A linear stability analysis of the map ((1) and (2)) about the p l a e vrave fixed point is carried out by assuming G,(x, z) = (g+yn(z, z))exp(ipN(lg12)z/2), where g is the plane azve fixed point and y,(r, z ) is a small perturbation with the explicit dependence,
eor[aneiK.~ + bne-iK.x] + e-oz[CneiK.~ + dne-iK.x]
Yn (x, 2 ) =
As discussed in reference [I] there are two distinct regimes of instability behavior in transverse K-space. The instability growth curve for finite wavelength transverse spatial modes and for a general optical nonlinearity N(I), is given by,
for T
>
2pNf [ the oscillatory band] andderivative of N with respect to I. Notice that r is proportional to the actual squared magnitude of the transverse wavenumber I( with the scaling parameter y inversely proportional to the Fresnel number F . This scaling shows explicitly that the wavelength of the most unstable mode decreases with increasing Fresnel number as expected.
3 - RESULTS - In Figure 1, we graph the response curve as a function of the input laser field amplitude a ( x ) for a saturable nonlinearity. The curve b(p, r ) is plotted as a function of the transverse wavenumber T for the three representative points marked on the response curve in Figure 1. Solid regions of the b(p, r ) curves correspond the Benjamin-Feir band and dashed regions to the oscillatory band. The horizontal line representing the critical line above which transverse modes are unstable is given by b, = (R
+
R-l)/2. This corresponds to an eigenvalue of the linearization of the map (eqns(1) and (2)) crossing the unit circle along the positive real axis. A similar critical line exists a t b, = -(R+
R-*)I2 where the eigenvalue crosses the unit circle along the negative real axis signifying a period doubling bifurcation. This latter case was treated in reference [I]. The plots in Figure 1 correspond to the case where we are remote from any bifurcation (saddle- node or period-doubling) of the plane wave map (K = 0). The finite band in r lying above b, represents a continuum of wavenumbers which are unstable to growth from an infinite plane wave background at the value of g indicated on the response curve. Here plane wave perturbations ( r = 0) are strongly damped. The instability growth rate which is proportional t o Ib(,u, 7)- b,1
in the unstable band, is significantly larger for the Kerr case. In practice we will be considering an apertured plane wave in one transverse dimension so the the mode spectrum will be discrete. Our aperture width is chosen to match approximately the characteristic width of an input Gaussian pump beam allowing us to establish that the plane wave instability analysis is equally applicable to wide Gaussian beams. Figure 2 shows the response curve in the infinite gain region with the accompanying b(p, T ) curves on the right for the saturable nonlinearity. Here, as expected, a plane wave perturbation is marginally stable with strong instability growth evident at finite wavenumbers. Dashed lines denote regions where r = 0 is unstable irrespective of the behavior of other modes and a dotted curve regions in which r = 0 is stable but finite wavenumber modes are unstable. The pictures shown in Figure 3 represents the hysteresis region with r = 0 strongly unstable in the negative sloped region. The corresponding curves for the Xerr case are essentially the same except that instability growth rates are typically larger. Our main objective is to show that all of the phenomena leading to the emergence of transverse spatial patterns whether on plane wavefronts or broad Gaussian beams can be understood by reference to tlie above b(p, r ) curves.Fig. 3 - Response curve and accompanying stability diagram for a saturable nonlinearity in the hysteresis region. The parameter +o = 0.6.
Figure 4 confirms that the initial instability growth is the same whether it occurs on a plane wavefront or a broad Gaussian beam. By broad we mean a beam whose characteristic waist is much wider than the unstable wavelength predicted from b(p,r). This amounts to making a WKB approximation. In Figure 4a we show the early stage of the spatial pattern growth for a plane wave and a Gaussian beam at a point on the response curve in Figure la. The plane wave amplitude Igl = .3 and the Gaussian peak amplitude IG,,,I = .45. As expected from the instability curves, the wavelength increases slightly as the intensity tapers off on the wings of the Gaussian. Below a certain local lane wave amplitude the profile is again stable to modulational instabilities. The corresponding apertured plane wave computation is shown immediately above illustrating the perfect agreement between both cases. Figure 4b shows the initial modulational growth on the switched-on portion of an initial Gaussian beam corresponding to the parameter values in Figure 3a. The plane wave computation is shown above for comparison 191 = .7. It is easy to see that the flat-topped portion of the 'on' solution IGm,,l
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.7 resembles an apertured plane wave. The final asymptotic states in the modulational pattern growth for the Gaussian cases have been identified in references [3,4] as solitons/solitary waves.C2-458 JOURNAL DE PHYSIQUE
The instability formula b ( p , ~ ) is equally applicable to two-transverse space dimensions. For a Kerr nonlinearity one encounters the well-known blow-up singularity which leads to uncontrolled filamentation. The saturable case is very different [5] however leading to an extremely rich spectrum of complex spatio-temporal pattern evolutions. All of this behavior appears in a parameter window where one-transverse dimensional or plane wave analysis suggests simple stable behavior. Figure 5 shows two such spatial patterns representing the output at the 400-th arid 600-th pass in the resonator.
The initial evolution of the cylindrical switched-on beam follows the 1D behavior [5] and leads to concentric solitary wave rings. A slow modulational instability on the flat-topped rings leads to the gradual formation of large amplitude saturated filaments. These filaments tend to collapse, transiently forming ring-like or ridge-like structures which are again
Fig. 4 - Spatial pattern growth on an apertured plane wave and an initial input Gaussian beam profile. (a) Pattern growth at an unstable point on the response curve in Figure 1. (b) Pattern growth on the upper branch region in Figure 3.
Fig. 5
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Two-transverse din~ensional intensity contour plots showing metastable spatio-temporal patterns a t the 400-th and 600th resonator pass.susceptible to a modulational instability (see Figure 5). The patterns which form tend to be metastable with the external low amplitude pump beam acting as an energy reservoir for the formation of new filaments. There is no evidence for any repetition of patterns; rather the system appears to be in a weakly turbulent state.
Acknowledgement: Support for this work has been provided by a grant from the U.S. Air Force AFOSR F4962086C0130.
JVM acknowledges a NATO grant (RG.0005186) as travel support for collaborative work with the Arizona group.
5 - REFERENCES
/ I / McLaughlin D.W., Moloney J.V. and Newell A.C., Phys. Rev. Lett., 54 681 (1985).
/2/ Lugiato L.A. and Lefever R., Phys. Rev. Lett., 58 2209 (1987).
/3/ McLaughlin D.W., Moloney J.V. and Newell A.C., Phys. Rev. Lett. 51 75 (1983); Moloney J.V. and Gibbs H.M., Phys. Rev. Lett., 48 1607 (1982).
/4/ Adachihara H., McLaughlin D.W., Moloney J.V. and Newell A.C., J. Math. Phys., (1988).
/5/ Moloney J.V., Adachihara H., McLaughlin D.W. and Newell A.C., in "Chaos, Noise and Fractals", eds. E.D.R. Pike and L.A. Lugiato (Adam-Hilger, Bristol) (1987); Moloney J.V., IEEE J. Quant. Electron., 21 1393 (1985).
