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SPATIAL CHAOS AND OPTICAL MEMORY
W. Firth
To cite this version:
W. Firth. SPATIAL CHAOS AND OPTICAL MEMORY. Journal de Physique Colloques, 1988, 49
(C2), pp.C2-451-C2-454. �10.1051/jphyscol:19882107�. �jpa-00227617�
SPATIAL CHAOS AND OPTICAL MEMORY
W.J. FIRTH
Department o f P h g s i c s and A p p l i e d P h y s i c s , U n i v e r s i t y o f S t r a t h c l y d e , GB-Glasgow G 4 ONG, S c o t l a n d , G r e a t B r i t a i n
R&sum&
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Un re'seau cl'e'l6ments bistable peut &tre utilisd c o m e mdmoire optique. Les klbents doivent ope'rer de facon indGpendante, ce qui limite leur densitd en raison de l'interaction possible causge par le recourrement attributable d la diffusion. Considdrante un re'seau unidimensionnel, l'auteur exploite une analogie avec un systhe dynamique non-lin8aire et d6montre que le chaos spatial est ndcessaire si l'on veut presdrver l'indgpendance des BlGments. Le pas de r6seau minimum tolerable est alors estim6.Abstract - Optical memories consisting of arrays of bistable pixels are considered.
Attainable pixel density is limited by cross-talk, so that there is a minimum pixel separation for which all pixels may be independently held "on" or "off". This problem is analysed for diffusive cross-talk between members of a linear array of pixels. A nonlinear dynamics analogy links pixel independence to spatial chaos, and enables estimates of minimum pixel density. The stability of pixel patterns is governed by a form of Schrodinger equation. More realistic models should show similar features.
1 - OPTICAL MEMORY
One of the major motivations for the investigation of optical bistability [l-31 has been its possible application as a two-state optical memory device. This basic idea is developed here with a view to isolating and analysing the fundamental requirements for a functional optical memory, capable of recording and retaining many bits of information whether as an image or as digital data or program memory. Cost and thermal load considerations favour devices with low power per pixel but a high pixel density, and the main objective of this work is to establish criteria by which the maximum pixel density for a fully functional optical memory can be established. While the case analysed is highly simplified, namely a linear pixel array with diffusive coupling [4], it is hoped that the approach and ideas used will find wider application. In particular, adaptation of techniques developed for the study of nonlinear dynamics and chaos provides powerful methods for demonstrating the existence and stability of arbitrary complex pixel patterns.
Perhaps the simplest bistable optical memory is a uniformly illuminated nonlinear etalon held in its bistable region. One might hope to record an image by augmenting the holding beam with an image beam, which would switch the etalon in bright regions, leaving it in the low response state in darker regions. This simple scheme does not work in general, because any cross-talk in the transverse dimension, e.g. by diffraction or by diffusion of the medium excitation responsible for the optical nonlinearity, makes either the "on" or the "off" state unstable against propagation of a switching wave [4] which pulls down the "high" regions or vice versa, depending on the bias conditions. Only exceptionally, at a unique bias condition, is the switching wave static, and therefore only in this exceptional case can information be encoded on the device - one has, in effect, one single, large, pixel.
The switching wave can be pinned and made harmless by spatially modulating either the holding beam or the material parameters in such a way as to produce effectively isolated pixels.
This brings a significant advantage in terms of holding power per unit area [4] but at the expense of image fidelity or data density. One would therefore like to pack the pixels as densely as is consistent with independent pixel operation. This, then, is the main question addressed in this paper: having attained the power advantage of an array of narrow beamlets, calculate the minimum pixel separation consistent with independent operation of these pixels.
More precisely, estimate the maximum pixel density at which one can guarantee the existence of stable pixel patterns representing all possible configurations of "off" and "on" states
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in fact, all binary digit sequences. For simplicity, the present analysis concentrates on a very simple response function [5], with some reference to more general responses [6].2
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BASIC MODELBistability frequently arises from nonlinear coupling between the input optical intensity
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19882107
C2-452 JOURNAL DE PHYSIQUE
Iin and some material parameter @ (photocarrier density, temperature, atomic inversion
etc.) [l--31. In general @ will evolve in space and time: we assume that it diffuses and decays according to the following equation:
Here PD is the (transverse) diffusion length, T the excitation decay time, and f(@) the response function governing the driving of by Iin. Here and throughout we consider only transverse spatial variation, and it is the transverse diffusion of
+
which is responsible for cross-talk in this model.If Iin is constant in space and time, the corresponding uniform material states obviously obey:
Roots of this equation correspond to the intersections of the straight line @/Iin with the graph vs. @ of the response function. If f(Q) is suitably nonlinear then multiple roots, and thus bi- or multi-stability, exist for Ii, large enough. If f(@) has a step-like character we get a very simple form of bistability, corresponding to "optical brstability due to increasing absorption", as, e.g., if Q is the temperature and f(O) is the absorptivity and increases with temperature.
To form an array, however, Iin and/or f(@) must be spatially modulated, and (1) must be solved in full, which generally requires numerical solution [4]. If we approximate f(@) by a piecewise linear function (1) is also piecewise linear and analytic progress can be made.
Consider the "edge filter function" (EFF):
which is just about the simplest response function showing bistability.
Solution of (2) is trivial, and in the bistable region we have solutions
with bistability for ~<I~,<T-~.
We simplify .further by assuming that the array structure is implemented by spatial modulation of Iin, e.g. as an array of gaussian beams, and by examining only one-dimensional arrays 141. The steady states of (I) are then given by
The stability of these steady states can be examined by linearisation:
with 9 small and thus bounded: then
v(r)
is a solution of a Schrodinger-like equation [ 7 ] :where V(r) = -Iin(r) .f'(@,(r)) : E = -(I+&).
Thus @, is stable only if the ground state of (6) has E
>
-1. Hlhile this is generally a difficult problem, we note that for the EFF the "potential" V(g) is identically zero, and (6) trivially soluble. l'he only bounded solutions are superpositions of complex exponentials for which 6)O. It follows that glJ steady states of the EFF are stable (due to the vertical step at@=1, which strictly should have finite width). For the EFF we thus need only consider the existence of solutions; of ( 5 ) - their stability is guaranteed. In general, there will be steady states with "middle branch" pixels, some of which are "born" by bifurcation in conjunction with
corresponding "upper" or "lower" branch states. It can be shown
[a]
that all such states are3
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STEADY-STATE ANALYSISThere is an evident analogy between (5) and a Newtonian mechanics problem, that of a driven
"hyperbolic oscillator". This problem is non-autonomous and generically non-integrable for non-uniform Tin, with the possibility of chaotic orbits, which here correspond to spatially-chaotic pixel response patterns.
It has been shown in an analogous problem in dispersive optical bistability [9] that optical power requirements are reduced if the pump beam width is much less than the diffusion length, while pixel independence may be expected to demand a spacing of order QD. It is thus a natural, as well as a useful, approximation to write Iin as a sum of delta functions
This corresponds to a "kicked" dynamics problem, for which (5) can be easily integrated.
Defining An. as the value of
eS
on the nth site, the solution is conveniently expressed as a"stroboscopic" mapping:
This map can be interpreted as a two-term recurrence relation expressing a dependence of the state of the nth pixel on its nearest neighbours: other systems and other forms of cross-talk can be expected to give rise to similar maps with qualitatively similar properties.
This map, for any f(@), is area-preserving and invertible (by B-A). At fixed points
which is a simple generalisation of (2) to finite spacing.
For the EFF, the map is linear and rather simple. While, as shown above, the fixed points (Ap, AU) are dynamically stable, they are mapping unstable, due to the area preservation. The characteristic equation is
with eigenvalues e*L and eigenvectors
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(efL, 1). Both fixed points are thus hyperbolic, and the stable and unstable manifolds are just straight lines.It is clear that most points eventually escape to infinity under the map, which is physically unacceptable. We must therefore try to identify and obtain the "bounded set" S of points in the (A,B) plane for which An remains finite for all positive and negative n. For the present case, it can be shown that the bounded set must lie within the rhombus formed by the
intersections of the stable and unstable manifold lines.
For more general f(A), e.g. a Lorentzian, there are three fixed points, the lower and upper being hyperbolic, as for the EFF, while closed orbits may exist around the middle fixed point [6]. Again the bounded set must lie within the "diamond" formed by the stable and unstable manifolds of the upper and lower fixed points 181.
It is convenient, and in the spirit of "symbolic dynamics [lo], to represent each member of S by a string of binary digits, the nth digit being "0" if An is "low" and "1" if An is "high".
Pixel independence implies that each digit can be independently set "low" or "high" and thus possible strings of binary digits must be represented in S.
On physical grounds, it has been guessed [4] that the hardest such pattern to maintain has just one pixel in the opposite state to all others, symbolically ---- 00001000---- or its converse. In mapping terms, this corresponds to a homoclinic orbit, where A maps out on the
C2-454 JOURNAL DE PHYSIQUE
unstable manifold of Ap to intersect and then follow the stable manifold of Ap. The simplicity of the EFF allows an exact solution for this homoclinic orbit (or "point defect"). The minimum possible spacing, independent of P and T, is
i.e. pixel independence is incomplete unless the array spacing exceeds 1.1 diffusion lengths.
Smooth response functions f($) typically yield somewhat larger critical spacings, in the range 2-3 diffusion lengths.
In nonlinear dynamics, homoclinic orbits are closely associated with chaos [lo], which in symbolic terms means that the dynamics possesses orbits describable by arbitrary strings of binary digits. This association supports the intuitive notion that point defects imply pixel independence, as discussed above: it also suggests that an array can be an effective memory or processor only if its state equation allows spatially chaotic solutions. In this sense, we have a case in which chaos is actually essential for applications purposes.
Another diagnostic for chaos in nonlinear dynamics is the presence of "horseshoes" in the phase portrait [lo], which for a mapping means that key areas of the phase plane have folded images, giving rise to a Cantor set structure for what we have called the bounded set S. Since the members of a Cantor set can be placed in 1:l correspondence with strings of binary digits, a Cantor set struc:ture for S guarantees pixel independence as before.
Similar considerations apply to smooth response functions except that the middle branch generates a third stripe and thus a larger Cantor set [ll]: as noted above, the associated members of S correspond to dynamically unstable patterns [ 8 ] .
In this paper we have argued, and demonstratcd explicitly in a simple case, that pixel independence in an optically bistable array is closely associated with chaos in dynamical systems - chaos being in fact necessary for ideal array performance. Fully isolated pixels are trivially independent, but in the case analysed pixel independence does not require elimination of cross-talk
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in fact cross-talk is about 30% for the maximum allowable pixel density, which suggests that allowable pixel densities in real systems may be higher than might be expected.We have considered only the case of a homogeneous medium with pixels defined by the input beam pattern. Material pixellisation has been demonstrated [12] and analysed [13], with a view to inhibiting transverse carrier diffusion. Thermal and optical cross--talk will st.ill exist, however, as will remanent diffusion. The beauty of the mapping scheme (7) is that it can probably be adapted even to such complex forms of cross-talk, and our qualitative conclusions should retain some validity. Further, more serious, complications will ensue when two-dimensional arrays are analysed. In that case (7) will look like an interacting spin system, and analogies with domain growth and stability calculations may prove fruitful.
This research was supported in part by a Twinning Contract under the European Community Stimulation Action Program.
REFERENCES
1. "Optical Bistability - Controlling Light with Light", H.M. Gibbs (Academic, 1985).
2. "From Optical Bistability Towards Optical Computing, P. Mandel, S.D. Smith and B.S.
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3. Optical Bistability 111, H.M. Gibbs, P. Mandel, N. Peyghambarian and S.D. Smith, eds.
(Springer, 1986).
4. W.J. Firth and I. Galbraith, IEEE J. Quant. Elsc., QE-21, 1399-1403 (1985).
5. W.J. Firth, Phys. Lett.
u,
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7. N.N. Rosanov, Sov. Phys. JETP, 53, 47-53 (1981).
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10. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. (Springer, 1983).
11. W.J. Firth in "Instabilities and Chaos in Quantum Optics", N.B. Abraham, F.T. Arecchi and L.A. Lugiato (eds. ) : Plenum (in press).
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