MODELLING OF CROSS TALK IN 2D BISTABLE ARRAYS AND NOISE ON AN InSb ETALON

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MODELLING OF CROSS TALK IN 2D BISTABLE ARRAYS AND NOISE ON AN InSb ETALON

Erika Abraham, H. Richardson

To cite this version:

Erika Abraham, H. Richardson. MODELLING OF CROSS TALK IN 2D BISTABLE ARRAYS AND NOISE ON AN InSb ETALON. Journal de Physique Colloques, 1988, 49 (C2), pp.C2-447-C2-450.

�10.1051/jphyscol:19882106�. �jpa-00227616�

JOURNAL DE PHYSIQUE

Colloque C2, Suppl6ment au n06, Tome 49, juin 1988

MODELLING OF CROSS TALK IN 2D BISTABLE ARRAYS AND NOISE ON AN InSb ETALON

E. ABRAHAM and H.J. RICHARDSON

Department of Physics, Heriot-Watt University, Riccarton, GB-Edinburgh EH14 4AS, Scotland. Great-Britain

Abstract

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We present a theoretical model based on the diffusion equation for the single-pass nonlinear phase shift and apply it to an InSb etalon irradiated by a two-dimensional array of beams. Independent operation of the individual elements depends on a critical spacing which we determine as a function of the holding power. When the plane-wave limit of our model is taken, the resulting equation can be likened to that of an overdamped particle subject to a conservative force. This enables us to describe a mechanism for noise-induced transitions.

1

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OPTICAL PIXELLATION

Consider an InSb etalon irradiated by a 2D array of Gaussian beams of radii w. Since InSb possesses an electronic nonlinearity, there will be a change in carrier concentration which will subsequently diffuse away from the excited region. If the array of elements, which we will assume bistable throughout, is not sufficiently spaced out, their coupling through carrier diffusion will be the mechanism for cross talk. In this section we work out quantitatively the conditions for cross talk and how to suppress it.

Our starting point is the equation for the nonlinear phase shift +(z-,t) /1/ in the two transverse dimensions x and y,

where LD is the diffusion length, TR is the recombination time, F is proportional to the finesse, $, is the initial cavity detuning and Iin is a normalsied input

intensity of the array of beams assumed non-interfering,

where W w/lD,

E

E g/i!~ and

3

^E $/XD is the position of the jth beam centre.

In what follows, all coordinates are normalised to 1~ and the time to TR.

It is convenient to transform (1) into an integral equation by means of the Green function

appropriate for (1) and corresponding to a 2D homogeneous infinite medium. Then the formal solution of (1) reads,

Under the conditions W

<

1 or W -.

-

(plane-wave approximation) the phase shift $(R8,t*) is relatively constant over the spot size. As a result, the variation'of $ with

E'

can be

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19882106

JOURNAL DE PHYSIQUE

ignored /2/ in the spatial integration of (4) and we can put $(g,t) = $(s,t) E $j(t).

Then substituting (2) and (3) in ( 4 ) integrating over space variables and assuming steady-state conditions, we get

where dlj = (gi-cj) a .

To study cross talk we consider two elements with identical bistable characteristics and holding intensities Ihold initially in the lower branch. If element El is switched on, carriers will diffuse into the second element E2 effectively detuning it and thus its switching intensity Is is reduced to a new value I,'. Whether or not E2 switches

depends on a critical separation dc as follows

a) if dl,

>

dc then Is' > Ihold and E2 remains in the OFF state;

b) if dl,

<

dc then Isv

>

Ihold and E2 switches on.

However, if dl, = dc, then Is' = Ihold and the switching is not defined but this serves as a condition to define dc from the relation

a~,/a+, =

o

(I, = I ~ ~ ~ ~ ) (7)

which we obtain from (5). As El is ON, any changes in E2 will virtually have no effect on El; hence it suffices to consider one equation from (5) namely for i = 2. Therefore when condition (7) is applied we obtain an equation for the critical $,, termed $,, which is independent of dc. The resulting +c is substituted back in (5) and then the dc

found is such that it renders the equation consistent. The resulting dc will be a function of Ihold.

In Fig. 1 we plot dc against the holding power Phold normalised to the

single-element switch-up power P,; the bars are experimental results /3/. Parameters:

w = 30 p,

lid

= 60 pn, F = 3.5 and $, = 0.45. Negligible differences in dc are obtained when either the finesse or the detuning are changed /3/. We also studied numerically the

changes in dc when the element under study, say E2, is surrounded by other elements forming a square array. In this case dc refers to a critical 'lattice constant'. It

Fig. 1. Critical separation between two elements vs. holding power.

emerges that a significant increase in dc occurs only when nearest neighbours

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all in the ON state

-

are added, but negligible for any additional ones. For a separation d = 300 pm (dc = 180 p) when w = 30 pm, approximately 2.5 x lo3 elements can be placed in a cm2, which for a cycle time of 1 ys gives a switch rate of 2.5 x lo9 gate Hz c m - l .

2

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NOISE-INDUCED TRANSITIONS : PARTICLE ANALOGY

In this section we present a pictorial interpretation of the mechanism by which noise induces switching which deterministically would not occur. To do this we make the

plane-wave approximation in (1) by dropping the diffusion term. Assuming a single beam of intensity I, we get

We can re-express the r.h.s. of ( 8 ) as

where the potential U is given by

Fig. 2. Potential curves for various inputs Fig. 3. Pulsed excitation: noise-assisted

shown in the inset. switching

Fig. 4. As Fig. 3. with no switching. Fig. 5. Critical slowing down plus noise

C2-450 JOURNAL

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PHYSIQUE

where for definiteness we take C = 0. Eqn. (9) is formally equivalent to the equation of motion of an overdamped particle subject to a conservative force. By changing I we obtain the family of curves of Fig. 2; the inset shows the corresponding 1's in relation to the bistable curve (parameters: F = 10, 4, = 1.5), since aU/aQ, = 0: the left-hand ones correspond to the lower branch whereas the right-hand ones to the upper branch. When I is within the bistable region U ( 9 : I ) has two minima.

If the 'particle' is slightly displaced from its equilibrium position, it relaxes back to it with a characteristic time r, = (a2U/3Q,l)-1 as obtained from a linear stability

analysis. As r, goes as the inverse of the curvature of U, the shallower regions of the lower branch relax much more slowly than their upper branch counterparts. Clearly, zero curvature occurs at threshold points.

When a pulse is applied that causes the device to go past Is, the motion of the particle evolves along a potential curve where au/aQ, < ⁰ so that dQ,/dt > 0. If the duration of the pulse is not long enough, when the particle is brought back to the original curve, then aU/a$I > 0 and it relaxes back to the minimum. However, the presence of noise can change this by successively shifting the evolution to curves where aU/a$

<

0 and therefore the device switches. This is illustrated in Fig. 3 where for Ihold = 0.9 Is we apply a pulse of height 0.1 Is. In Fig. 4 we repeat this very simulation which shows statistically opposite effect of noise. In both cases the noise level is 1%.

1n.Fi.g'. 4 we repeat the numerical experiment with a pulse of the same height but infinite duration. In a deterministic situation the device would switch after a long delay due to critical slowing down. Suppose the switching time is ts. The presence of noise causes the sign of aU/aQ,

-

as well as its numerical value

-

to change in general and this will cause a spread in the switching times around t,. Hence if we fix the time of

observation and perform statistics on Q, we obtain transient optical bimodality /4/ as predicted by the mean field theory.

A proper treatment of noise-induced transitions in the above system is presently underway.

We found an exact steady-state solution of the relevant Fokker-Planck equation from which we can work out exit (switch) times. This work, however, will be deferred for a future publication.

We are grateful to Professor Lugiato for supplying the numerical code which we used in our noise simulations.

REFERENCES

/1/ Firth, W.J., Galbratih, I. and Wright, E.M., J. Opt. Soc. Am., B2, 1005 (1985).

/2/ Richardson, H.J., Abraham, E. and Firth, W.J., Opt. Commun., 63, 199 (1987).

/3/ Young, J., Richardson, H.J., MacKenzie, H.A., Abraham, E. and Hagan, D.J., J. Opt.

SOC. Am., B5, 1 (1988).

/4/ Broggi, G. and Lugiato, L.A., Phil. Trans. Roy. Soc. Lond.,

a,

425 (1984).