Coupled-soliton photonic logic gates: practical design procedures

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Coupled-soliton photonic logic gates: practical design procedures G. Cancellieri, F. Chiaraluce, E. Gambi, and P. Pierleoni Dipartimento di Elettronica ed Automatica, Universita` di Ancona, Via Brecce Bianche, 60131 Ancona, Italy Received October 10, 1994; revised manuscript received March 6, 1995 The feasibility of AND, OR, and EX-OR functions based on the interaction of optical solitons is proved by use of a five-layer dielectric structure with a nonlinear core. With the exception of the OR logic gate, the design of these devices is rather flexible, offering a wide variety of choices with respect to both the geometrical parameters and the input power levels.

1.

INTRODUCTION

The possibility of obtaining photonic switching, i.e., optically controlled directional couplers, was demonstrated with various Kerr-like nonlinear media.1 It is somewhat more difficult to realize true photonic logic gates2–4 —alloptical devices with two input ports and one output port— able to operate according to elementary logic functions, e.g., AND, OR, EX-OR. These components can find useful applications in the field of optical information processing,5,6 permitting the realization of logic operations at speeds unattainable by conventional electronics. The final goal should be realization of all-optical digital computers interfaced, without the need for optoelectronic converters, with soliton communication systems in which the data rates are so high that conventional electronic logic is too slow. Image addition and subtraction schemes also have obvious applications to optical logic. For example, the process of image subtraction is equivalent to an EX-OR, and various solutions have been used for demonstrating this. The theory of the devices that we propose can be developed by study of the interaction between spatial solitons.7 Starting from this approach, we consider a five-layer planar waveguide, uniform along the z axis, in which the only nonlinear layer is the central one. The structure is symmetric with respect to the yz plane. The two input ports are placed at z ­ 0 and correspond to the second and fourth layers. The output port is placed at z ­ L and corresponds to the third, central, layer. The purpose of this paper is to give an outline of the theoretical model of spatial soliton interaction along z that is capable of implementing such operations, taking into account different situations of amplitude and relative phase difference in the two launched waves. The fundamentals of the theory will be presented in Section 2, and as will be shown in Section 3, the design of logic gates is possible by the proper choice of the device parameters. In particular, maximum tolerances in the design parameter L will be evaluated, depending on the optical power density and taking into account the width of the layers. Such a study, whose main results are presented in Section 4, demonstrates how some logic functions ap0740-3224/95/071300-07$06.00

pear more critical than others with respect to practical realization.

2.

THEORETICAL MODEL

The structure examined is shown in Fig. 1, which also shows the refractive indices for the various layers. We have assumed that n2 . n3 $ n1 and have confined the nonlinearity to the core region, taking there a nonlinear index variation Dn ­ nnl jEj2 , where E is the electric field. Actually it is much more common to specify the structure in terms of squared refractive indices, so that, again looking at Fig. 1, we have 8 > > > n1 2 > > > > > > > > n2 2 > > > > < n2 sxd ­ > n3 2 1 ajEj2 > > > > > > n2 2 > > > > > > > 2 > : n1

w 1d 2 w w ,x# 1d 2 2 w w #x# 2 , 2 2 w w 2d#x,2 2 2 2 w 2d x,2 2 x.

(1)

where a ­ 2n3 nnl , d is the width of the second and fourth layers, and w is the width of the third, central, layer. The term nnl 2 jEj4 has been neglected because of the very small value of nnl , which is of the order of 2 3 10212 m2yV2 for the material considered in the present analysis, i.e., liquid-crystal g-methoxybenzylidene-g 0 -nbutylaniline (MBBA).8 At the same time, it should be noted that this material is characterized by a very high maximum change in the refractive index (Dnsat ø 1), which inhibits the appearance of any saturation effect for the electric-field intensities of practical interest.7 The proposed structure can be seen as consisting of two slabs with nonlinear cladding placed side by side, whose guiding and soliton emission characteristics are well known from previous studies.9,10 Denoting by Dn0 the refractive-index difference in the linear structure, in the presence of weak nonlinearities sDnyDn0 ,, 1d we can  1995 Optical Society of America

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Fig. 1. Five-layer dielectric waveguide with nonlinearity in the central layer.

study propagation by resorting to coupled-mode theory and separately considering the nonlinear field configurations of the two slabs.11 An analytical result can even be reached in these cases. Coupled-mode theory, however, becomes unreliable in the presence of strong nonlinearities sDnyDn0 . 0.1d, for which the propagating fields can be greatly different from those of the linear structure and the adoption of an almost purely numerical approach becomes practically obliged. Since our interest is focused on just the latter situation we have developed a suitable computer program able to solve the following modified nonlinear Schr¨odinger equation, which describes the beam-shape evolution along the five-layer dielectric waveguide, here seen as a whole: ≠2 A ≠A 2 k0 2 fne 2 2 n2 sxdgA ­ 0 , 2 2jne k0 ≠x2 ≠z

(2)

where ne is the normalized modal index that is determined through the solution of the dispersion equation for the TE modes of the linear dielectric structure, k0 is the vacuum wave number, and j represents the imaginary unit. A is a slowly varying complex phasor related to the electromagnetic-field amplitude E by the following expression: Asx, zd ­ Esx, zdexpsj v0 t 2 jne k0 zd ,

(3)

where v0 is the circular frequency and t is the time. Obviously we have jEj ­ jAj. Equation (2) is a bidimensional equation describing propagation in two spatial dimensions, assumed here to be z and x. In a real device, however, it is not often desirable to excite a broad slab mode since the optical power is then spread out and the amplitude is reduced. Hence some kind of confinement structure in the y dimension is usually introduced, with the aim of limiting diffraction in the direction normal to the xz plane. To preserve soliton features, however, one must design the structure so that the beam maintains a nearly constant shape in the y direction.12 This goal can be reached, for example, by use of an etched ridge whose index distribution is sufficiently robust with respect to the index changes induced by nonlinearity. In this way the soliton reshaping effect in the x direction can be observed before substantial changes occur in the beam shape in the y direction. In this sense the propagation of the field is basically determined by Eq. (2), and there is no need for more involved approaches,13–15 at least for the present application. We solve Eq. (2) by considering an input distribution Asx, 0d ­ A1 sx, 0d 1 A2 sx, 0d of the type depicted in Fig. 2.

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Each elementary slab is supposed to be excited by the TE0 mode of the corresponding linear structure [denoted by usx 2 wy2 2 dy2d and usx 1 wy2 1 dy2d in the figure]. ai si ­ 1, 2d represents a Boolean variable specifying the presence sai ­ 1d or the absence sai ­ 0d of the ith elementary source, while u is a relative phase difference. In our simulations we have set u ­ 0 (in phase excitation) or u ­ p (opposite phase excitation). The input power per unit length P(0) can be computed analytically as ne Z 1` P s0d ­ jAsx, 0dj2 dx , (4) 2Z0 2` where Z0 represents the characteristic impedance of free space. For the sake of convenience, however, the results will be expressed as a function of the power Ps s0d supplied by the single excitation. When simultaneously present, both the excitations are characterized by the same amplitudes, so that in this case we can set Ps s0d ø P s0dy2. If a1 ­ 0 or a2 ­ 0, instead, we obviously have Ps s0d ­ P s0d.

3.

LOGIC FUNCTIONS

To derive simple but efficient rules for the design of photonic logic gates, it is important to collect information about the interaction along the structure of the fields excited at its input ports. Some examples are given in this section, whose objective is to demonstrate the feasibility of the logic functions mentioned in the Introduction by means of a suitable choice of the parameters involved. In all the cases considered, power levels are high enough sDnyDn0 ø 2d to produce the emission of solitons, whose interaction we are interested in studying as a function of z. Material losses are neglected, mainly as a consequence of the very short lengths (of the order of 0.1 –0.2 mm at most) necessary for the devices. With reference to the notation previously introduced, let us consider a structure operating at a wavelength l ­ 1.064 mm, with n1 ­ n3 ­ 1.55, n2 ­ 1.57, a ­ 6.37 3 10212 m2yV2 , d ­ 2 mm, and w ­ 3.4 mm. The value of a, in particular, is that resulting from the adoption of liquidcrystal MBBA, already mentioned in Section 2. Both the input ports are excited sa1 ­ a2 ­ 1d by two equal fields su ­ 0d, each supplying a specific power Ps s0d ­ 65 Wym. The evolution of the total field amplitude in such a structure is shown in Fig. 3, in which for the sake of clarity the boundaries of the layers have been marked by the dashed lines. The same representation will be adopted in the subsequent plots. As Fig. 3 shows, when the two emitted solitons, which are attracted by the nonlinear medium, collide, a strong peak appears (here reported in normalized units) on the symmetry plane. If instead of consid-

Fig. 2. ports.

Amplitude distribution of the excitation at the input

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Fig. 3. Amplitude evolution when d ­ 2 mm, w ­ 3.4 mm, a1 ­ a2 ­ 1, u ­ 0, and Ps s0d ­ 65 Wym.

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tons, with exchange of the relative position, an example of which is shown in Fig. 6. After each crossing, part of the propagating energy comes back into regions II and IV, thus restoring the initial conditions favorable to soliton emission. When losses are negligible, this phenomenon is repeated along the device with the same characteristics. The graphs that we have just presented, together with similar ones obtained with respect to single excitation, give an idea of the possibility of employing the proposed five-layer dielectric structure for realizing logic functions, i.e., AND, OR, or EX-OR, by a suitable dimensioning of the geometrical and electrical parameters. These devices will now be introduced with reference to a number of explanatory cases, whereas a more quantitative discussion of the design problems (especially the sensitivity to the choice of the parameters) will be object of Section 4. A. AND Gate Let us consider the structure and the excitation conditions used for Fig. 3. We have already seen that, by equally feeding the two input ports, we can obtain at a certain distance a strong peak amplitude in the waveguide central region. In other words, setting a1 ­ a2 ­ 1, we obtain a3 ­ 1, having denoted by a3 the Boolean variable identifying the presence or absence of power at the device output port. The decision on the value of a3 can be

Fig. 4. Amplitude evolution when d ­ 2 mm, w ­ 3.4 mm, a1 ­ a2 ­ 1, u ­ p, and Ps s0d ­ 65 Wym.

ering u ­ 0 we assume u ­ p (antipodal input waves), we obtain the behavior shown in Fig. 4. The two propagating envelopes tend to cancel each other in the central region that no longer contains power sufficient for the emission of solitons. The result appears to be the propagation of two undistorted pulses, but in reality this is the effect of an involved interaction that cancels solitons where they would otherwise be present if the input ports were individually excited (a1 ­ 1 and a2 ­ 0 or a1 ­ 0 and a2 ­ 1). On the other hand, the behavior shown in Fig. 4 is somewhat dependent on the specific value assumed for Ps s0d. For increasing input powers, in fact, soliton emission comes back to be clearly evident, but, maintaining the hypothesis of u ­ p, the solitons do not combine in such a way to form a central peak, as in the case when u ­ 0. As Fig. 5 shows, they tend to cancel on the plane yz, thus leaving a minimum in the amplitude distribution at x ­ 0. An interesting feature, which emerges only in part from the previous figures but is well known in soliton interaction theory, is that of the periodic nature of the pulse propagation along z. In practice, after a short transient length the field amplitude reaches a definite distribution that repeats itself at regular distances. The qualitative interpretation of this behavior is particularly simple when we have an effective crossing of the two interacting soli-

Fig. 5. Amplitude evolution when d ­ 2 mm, w ­ 3.4 mm, a1 ­ a2 ­ 1, u ­ p, and Ps s0d ­ 180 Wym.

Fig. 6. Amplitude evolution when d ­ 2 mm, w ­ 2 mm, a1 ­ a2 ­ 1, u ­ 0, and Ps s0d ­ 40 Wym.

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case a1 ­ a2 ­ 0, which, however, produces the trivial result a3 ­ 0; no power exists in the central region if the waveguide is not excited. B. EX-OR Gate Let us consider the structure used for Fig. 4 but assume an input power Ps s0d ­ 90 Wym. We have already seen —although for a lower power level —that, by the antipodal feeding of the two input ports, the amplitude distribution exhibits a null, independent of z, at x ­ 0. In other words, setting a1 ­ a2 ­ 1, we have a3 ­ 0 for any value of L. This result can be interpreted as a row of the truth table of an EX-OR gate, which must give a3 ­ 1 only in case of single excitation of one input port. Then, the device length L has to be designed in such a way as to find an intense peak amplitude at the output port when a1 ­ 1 and a2 ­ 0 or a1 ­ 0 and a2 ­ 1. As confirmed by Fig. 8, a possible choice is L ­ 70 mm. C. OR Gate Let us consider a five-layer waveguide with n1 ­ n3 ­ 1.55, n2 ­ 1.57, a ­ 6.37 3 10212 m2yV2 (liquid-crystal MBBA), d ­ 2 mm, w ­ 2 mm, and operating at l ­ 1.064 mm. It is interesting to note that, with respect to the previously examined structures, the width of the central layer has been reduced, thus causing a stronger coupling between fields propagating in regions II and IV. When the goal is to realize an OR gate, it is necessary

Fig. 7. Example of an a1 ­ 1, a2 ­ 0.

AND

gate when (a) a1 ­ a2 ­ 1 and (b)

made by comparison of the detected amplitude, at x ­ 0, with a suitable threshold. This seems to be a point criterion, hard to implement in practice. Nevertheless, it is evident that we can easily transform it into a power (or energy) criterion by taking into account the broadening properties of the detected pulse. The amplitude criterion is preferred here for the sake of simplicity only. We are interested in choosing a device length so as to ensure the maximum amplitude at the output port. For the specific example considered here, this occurs at L ­ 85 mm. A so-designed structure can be used as an AND gate if, in accordance with the excitation at one input port only (a1 ­ 1 and a2 ­ 0 or a1 ­ 0 and a2 ­ 1), at the same distance L we have an amplitude low enough (under threshold) to permit the conclusion that a3 ­ 0. This clearly happens for the present case, as shown in Fig. 7, which summarizes the device behavior; the amplitude at x ­ 0 and z ­ L is practically null in Fig. 7(b) and is double, with respect to each input pulse, in Fig. 7(a). This large dynamic range suggests, in particular, that the choice of the threshold value should be not critical. At the same time, it is possible to verify that, in the case of a1 ­ a2 ­ 1, the energy in the central layer remains sufficiently high along z several micrometers about the maximum. For these reasons the design of an AND gate seems to exhibit wide margins of tolerance, as will be quantitatively shown below. For completeness we must note that the AND’s truth table must be completed for the

Fig. 8. Example of an a1 ­ 1, a2 ­ 0.

EX-OR

gate when (a) a1 ­ a2 ­ 1 and (b)

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to ensure that, at z ­ L, we have a3 ­ 1, in accordance with the excitation of one input port only. This requirement recalls the behavior already shown in Fig. 8(b), with respect to the EX-OR device. At the same time, however, in contrast to Fig. 8(a), it is necessary to obtain a3 ­ 1 when both the input ports are excited. The latter constraint forces the assumption that u ­ 0, whereas the main difficulty in the design lies in the assumption of a power level that permits us to verify both the conditions at the same distance L. A possible solution is that shown in Fig. 9, in which Ps s0d ­ 75 Wym and L ­ 114 mm have been assumed. Here the periodicity feature of the propagation has been exploited. The device length is fixed by the single-port excitation condition [see Fig. 9(b)], but L must be a multiple of a finite number of periods for the propagating wave in the case of double excitation. In the example of Fig. 9(a), two intermediate peaks appear at x ­ 0 when a1 ­ a2 ­ 1, but they do not affect the device operation. It is qualitatively comprehensible that the OR gate is rather critically dependent on the tolerances in the values of the geometrical and electrical parameters. Let it be enough, in this sense, to observe that the variability of the amplitude distribution along z is much faster here than for the previous devices; at very short distances (,10 mm) from the solitons’ crossing points the amplitude can become so small that it inhibits a correct detection.

4.

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propagating power. Thus in principle we can compensate for the effect of, for example, halving a by doubling Ps s0d. In this sense the validity of the plots presented here can also be extended, within suitable limits, to other nonlinearity values. As threshold condition at the output port we have assumed equality with the amplitude of the corresponding input signal (at each port); if the detected amplitude is larger than this threshold, we set a3 ­ 1, a3 ­ 0 otherwise. Note that the graphs presented here do not refer to limit cases; on the contrary, we report only those combinations that lead to detected amplitudes considerably distant from the threshold condition. In this way the configurations that we show are quite reliable and robust with respect to any possible numerical inaccuracies; this is necessary for making simulations directly usable by the designer. A first example is shown in Fig. 10 for an AND gate. Fixing d ­ 2 mm, we have plotted the L – Ps s0d regions that correspond to a set of values for w. Let us consider one of these regions as a generic example. Once having fixed the geometrical characteristics in the transverse plane, we see that each possible input power permits the device length L to be free to vary within a rather large interval, maintaining the device capability of operating as an AND gate. For example, in the case of w ­ 1.5 mm the device length can be between 40 and 105 mm when the input power Ps s0d is equal to 30 Wym. For each region,

DESIGN ASPECTS

The realization of logic gates based on a five-layer structure is characterized by several degrees of freedom that the designer must optimize, among which are the input power levels, the device length, and the width of the dielectric layers. Thus it is interesting to find the limits of variability for such parameters compatible with the achievement of a desired logic function. For example, one may be interested in evaluating maximum tolerances in the device length L depending on the applied optical power and the layer widths. To make this evaluation immediately applicable, it is advisable to plot the results in multidimensional representations (the dimension being fixed by the number of parameters simultaneously considered variable) from which the regions of acceptability can be directly seen. In our analysis, attention has been focused on the AND and EX-OR logic functions because of the large flexibility they offer in the design. As previously stressed, the realization of an OR logic function is much more rigid. For the sake of clarity we have chosen to plot the results in bidimensional representations assuming as variable the input power and the device length, whereas the values of the layer widths have been fixed or parametrized. In this way we have identified continuous surfaces of acceptable pairs, L – Ps s0d, that implement the desired logic function in conjugation with discrete prefixed values of w and d. In any case we have assumed that l ­ 1.064 mm, n1 ­ n3 ­ 1.55, n2 ­ 1.57, and a ­ 6.37 3 10212 m2yV2 (liquid-crystal MBBA), but it is evident that quite similar plots can be derived assuming different wavelengths or structures. With respect to the choice made for a, in particular, it may be observed that the nonlinear response of the device is basically dependent on its product by the

Fig. 9. Example of an a1 ­ 1, a2 ­ 0.

OR

gate when (a) a1 ­ a2 ­ 1 and (b)

Cancellieri et al.

Fig. 10. Regions of acceptable L – Ps s0d pairs for an AND gate with d ­ 2 mm: (1) w ­ 3.7 mm, (2) w ­ 3.4 mm, (3) w ­ 2.7 mm, (4) w ­ 2 mm, (5) w ­ 1.5 mm.

Fig. 11. Regions of acceptable L – Ps s0d pairs for an AND gate with w ­ 3.4 mm: (1) d ­ 2 mm, (2) d ­ 1.5 mm, (3) d ­ 1 mm.

Fig. 12. Regions of acceptable L – Ps s0d pairs for an EX-OR gate with d ­ 2 mm: (1) w ­ 3.4 mm, (2) w ­ 2.7 mm, (3) w ­ 2 mm.

however, the input power cannot be arbitrarily large or, conversely, arbitrarily small. The lower limit on Ps s0d is obviously due to the nonlinearity itself; if the input power is too small, the refractive-index variation in the central region is not large enough to deviate the propagating energy from its original direction. The upper limit, on the other hand, is related to the device logic function; if the input power is too high, soliton emission also takes place in the case of single excitation, which risks having a3 ­ 1 for inappropriate excitation conditions. Limits also exist on w that, in the example considered, must vary between 1.5 and 3.7 mm (precisely the extrema assumed in Fig. 10). For w . 3.7 mm the coupling between fields in regions II and IV is too low to permit the design of efficient and reliable devices, whereas for

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w , 1.5 mm it becomes so strong that it inhibits the detection of a3 ­ 0 when a single input port is excited. Some points must be made regarding the correct use of Fig. 10. To this end, let us suppose, for example, that the object is the synthesis of an AND gate L ­ 70 mm long. From the graph we can see that it is possible to choose w ­ 1.5, 2, or 2.7 mm, using input powers that vary approximately between 30 and 70 Wym. It is obviously reasonable to think that, though not considered in Fig. 10, all the values of w ranging at least between 1.5 and 2.7 mm are equally acceptable for this particular design problem. In some cases, wide overlapping zones among the regions exist, further enlarging the set of possible choices. For example, to obtain an AND logic function at L ­ 40 mm, using input powers between 50 and 55 Wym, we can arbitrarily assume that w ­ 1.5 or 2 mm. Another interesting feature is the possibility of extending the useful regions considered here, taking into account the periodicity of the interaction between solitons along the structure. In practice, once the period ZL has been identified, all the lengths L 1 hZL , where h is an integer, are equally acceptable provided that such length is not so great as to make unsatisfied the hypothesis of negligible losses. In other words, a family of regions could be plotted on those of Fig. 10; the latter in turn yield the shortest devices for a certain input power. Extending Fig. 10, we see that it is obviously possible to fix the width of the central layer and vary that of the second and fourth layers. In this way, further design opportunities come about. An example is shown in Fig. 11, in which we have set w ­ 3.4 mm. With a reduction of the value of d, soliton emission is possible for smaller input powers, thus permitting the realization of shorter devices with the other conditions remaining the same. Finally, in Fig. 12 we show an example of a plot useful for the design of an EX-OR gate. Here we have set d ­ 2 mm, while w plays the role of parameter. With respect to the previous figures the regions of acceptability for the L – Ps s0d pairs are narrower, closer together, and largely overlapped. This is a consequence of the increased complexity in the logic function, which requires a null of the field on the symmetry plane, corresponding to the simultaneous excitation of the input ports. The shaded regions are, however, globally extended, offering a wide choice of different configurations, especially with respect to the values of the input powers that, for any w, can vary within very large intervals (approximately 100 – 300 Wym). In fact, the upper bound on Ps s0d, though necessary for correct device operation, is much more relaxed here than in the AND case, so that the EX-OR gate also performs well for input powers that are particularly high. L – Ps s0d graphs could now be plotted for the OR logic device also. Owing to the increased criticality in the performance control, however, for any value of w and d the plots should be reduced to lines instead of surfaces. Since our main purpose is to discuss the tolerances in the design, we do not show here curves of this type.

5.

CONCLUSIONS

We have demonstrated the possibility of exploiting the spatial solitons’ interaction in a nonlinear dielectric struc-

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ture for realizing directly simple logic gates at optical level. The proposed solution represents a valid alternative to some previous proposals for the same objective, basically based on the frequency shift induced in a laser by one or more injected beams. With respect to the latter one, the use of active devices is avoided here, and the design exhibits wide tolerances in the assumption of the device length, the applied optical power, and the layer widths. We have focused our attention on AND, OR, and EX-OR; however, by the proper combination of these elementary gates other logical functions can easily be built up. For example, by use of the output of an OR (AND) gate for feeding, at one port, an EX-OR gate whose second input is maintained at high level, a NOR (NAND) operation can be achieved. In this particular configuration, in fact, EX-OR acts as a NOT gate.

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