Dual-input hybrid acousto-optic set reset flip-flop and its nonlinear dynamics

Transcription

1 Dual-input hybrid acousto-optic set reset flip-flop and its nonlinear dynamics Shih-Tun Chen and Monish R. Chatterjee The characteristics of a dual-input hybrid acousto-optic device are investigated numerically and experimentally. The device, which operates as a set reset flip-flop, uses the well-known bistable acousto-optic device with feedback to which two input beams are applied. The resulting flip-flop is analyzed numerically by use of nonlinear dynamical and nonlinear circuit-modeling techniques, and some of its properties are demonstrated experimentally Optical Society of America The authors are with the Department of Electrical Engineering, State University of New York at Binghamton, Binghamton, New York Received 18 October 1996; revised manuscript received 13 January $ Optical Society of America 1. Introduction The dynamic and hysteretic behavior of a hybrid acousto-optic bistable device has been studied for more than a decade. 1 6 Actual practical applications of the device, however, have been scarce or unavailable. In this paper, we investigate an optical set reset flip-flop that combines a hybrid acoustooptic modulator with feedback with a dual-input scheme for its operation. Unlike conventional methodologies, in this configuration one may add an additional state to the available states by manipulating the dc bias to the rf modulator. The availability of the additional state can be illustrated by means of a nonlinear dynamic analysis. The additional state may be used as one of the equilibrium states of the device. Since this state is shown to have no bifurcation characteristics, the device will not undergo chaotic oscillations around this state. It is admittedly difficult to assess the usefulness of this optical logic device in physical systems, especially in terms of comparisons with competing devices and techniques. Our purpose here is simply to introduce a feasible logical application of the hybrid acousto-optic bistability effect that has been studied esoterically for more than a decade. In Section 2 of this paper, we present the theoretical background to the dual-input hybrid device. In Section 3, we carry out a nonlinear dynamical analysis of the device. In Section 4, we present a nonlinear circuit model of this device along the lines of Ref. 4, and, in Section 5, we describe some experimental results. In Section 6 we offer conclusions with respect to this paper. 2. Theoretical Background A. Dual-Input Acousto-Optic Bragg Modulator Suppose we consider a Bragg cell on which two input light beams are incident symmetrically. Let the two beams of amplitudes E inc0 and E inc1 be incident at negative and positive Bragg angles, respectively, as shown in Fig. 1. Intuitively, the two output fields in this case will receive contributions from each of the incident beams. To determine the diffracted outputs, we must use the well-known coupled differential equations for two Bragg orders and apply the boundary conditions for the E fields at z 0 to accommodate the two incident beams. Thus, the coupled differential equations for E 0 and E 1 within the Bragg cell are given in terms of the normalized longitudinal distance z L where L is the so-called interaction width by 7 : with the boundary conditions de 0 d j 2 E 1, de 1 d j 2 E 0, (1) E 0 z 0 E inc0, E 1 z 0 E inc1. (2a) (2b) 10 May 1997 Vol. 36, No. 14 APPLIED OPTICS 3147

2 Fig. 1. Dual-input acousto-optic cell, with the two beams incident symmetrically relative to the horizontal axis at B, the Bragg angle. AOM, acousto-optic modulator. The solutions may be obtained as E 0 E inc0 cos 2 je inc1 sin 2, E 1 E inc1 cos 2 je inc0 sin 2. (3) The above E fields may be expressed in intensity form as I 0 E 2 inc0 cos 2 2 E 2 inc1 sin 2 2 I inc0 cos 2 2 I inc1 sin 2 2, I 1 E 2 inc1 cos 2 2 E 2 inc0 sin 2 2 I inc1 cos 2 2 I inc0 sin 2 2. (4) We note from the above solutions Eqs. 3 and 4 that complete switching of the two beams occurs when and 1. Figure 2 shows the switching effect in a dual-input Bragg cell observed at z L. At z 0, I 0 and I 1 have been given the normalized intensities 1.0 and 0.25, respectively. As seen from Fig. 1, when is zero, i.e., no sound wave is applied, the output intensities remain unchanged. However, when changes from 0 to, the two beams exchange their energies. B. Dual-Input Acousto-Optic Device with Feedback The switching properties of the above dual-input scheme combined with feedback will result in a set reset optical flip-flop. Such a flip-flop may have two possible configurations, viz, a zeroth order or a first order. 1. Zeroth-Order Flip-Flop The diffracted output E 0 of the dual-input acoustooptic modulator may be fed back to the acoustic driver as typically connected for obtaining acousto-optic Fig. 2. Normalized intensity versus showing the optical energy exchange between light beams in a dual-input acousto-optic modulator. hysteresis. A schematic diagram of this setup is shown in Fig. 3. The zeroth order E 0 at the output is picked up by a photodetector and fed back through a time-delay element and an effective feedback path gain to the acoustic driver. The open-loop bias to the acoustic driver, represented by 0, is combined with the feedback signal to produce the equivalent bias for the acoustic driver. Thus, the input to the acoustic driver becomes 0 I 0 t TD t TD 0 E 2 inc0 cos 2 2 t TD E 2 inc1 sin 2 2, (5) where TD is the time delay in the feedback loop and I 0 is the zeroth-order intensity. We note that Eq. 5 may be simplified to that for a regular single-input acousto-optic bistable device if E inc0 is set to zero. If E inc0 0 and 0 is varied, the output E 0 will exhibit Fig. 3. Zeroth-order acousto-optic flip-flop. AOM is the acoustooptic modulator; and 0 are the acoustic bias drives; TD is the time delay in the feedback loop; PD is the photdetector; is the amplifier gain; and Q and Q are the output states of the flip-flop. TRIG and RESET are the set and reset input optical signals, respectively APPLIED OPTICS Vol. 36, No May 1997

3 3 Reset: The initial state may be restored by application of a reset signal to the input E inc1. Note that E inc1 remains high cw during operation. If the cw light beam E inc1 is momentarily interrupted, the output beam E 0 will go to zero. The feedback in turn will go to zero, and eventually the acoustic grating inside the acousto-optic cell will disappear. The device now returns to its original state E 0 0. Fig. 4. First-order acousto-optic flip-flop. The labels are as for Fig. 3. hysteresis with respect to 0. For the zeroth-order flip-flop, 0 is set to zero. Since E 0 is used to generate the feedback signal, we define this configuration as a zeroth-order flip-flop. We explain its operation in Subsection 2.C. 2. First-Order Flip-Flop An alternative use of the same setup is to adjust 0 to and use the diffracted output E 1 as the source of negative feedback, as shown in Fig. 4. The corresponding system will have a mathematical model similar to that of the zeroth-order flip-flop. In this case, since E 1 is used to generate the feedback signal, the device is called a first-order flip-flop. The operation of the device is explained in Subsection 2.C. C. Heuristic Analysis 1. Zeroth-Order Flip-Flop The initialization, triggering, and reset operations of the zeroth-order flip-flop shown in Fig. 3 may be described as follows: 1 Initialization: E inc0 is set to 0 and E inc1 is set to 1, initially. Because 0 is also initially set to zero, there will be no grating formed inside the acoustooptic modulator. Therefore, the incident light beam E inc1 goes through the modulator without any diffraction. Thus, the photodetector does not detect any signal at the output end. Consequently, remains clamped at zero. The state of the device remains in the equilibrium state E Triggering: An optical impulse E inc0 is applied to change the state of the device. The converted electrical signal from the undiffracted optical impulse generates a feedback signal. As the feedback signal is fed through the feedback amplifier to the acoustic driver, a sound wave is delivered into the cell, forming an acoustic grating inside the acoustooptic cell. E inc1 is then diffracted by the acoustic grating in the E 0 direction. The diffracted light is intercepted by the photodetector. The detected signal further increases and thereby enhances the grating strength and the diffracted field E 0. The device continues to increase the feedback and the diffracted field until diffraction reaches its maximum saturation. The device maintains the new state E 0 1 after saturation is reached. In summary, since the state of the device is controlled by the application of E inc0 as a trigger signal to initiate a change of state, E inc0 is designated as the trigger signal TRIG. E inc1 is designated as the reset signal RESET. In an analogy to conventional flip-flop outputs, E 0 is similar to Q and E 1 to Q. 2. First-Order Flip-Flop With reference to Fig. 4, the operations of a first-order flip-flop may be described as follows: 1 Initialization: E inc0 is set to zero and E inc1 is set to one, initially. Since 0 has been set to, a strong acoustic grating exists in the acousto-optic cell and causes strong Bragg diffraction at the output. Most of the E inc1 light beam is therefore initially diffracted in the E 0 direction to the beam stop, and only a small portion arrives at the photodetector. Therefore, an insignificant amount of feedback is subtracted from 0, so that remains close to, and the device remains in the equilibrium state E Triggering: An optical impulse applied to E inc0 is strongly diffracted by the existing acoustic grating in the E 1 direction and is intercepted by the photodetector. The feedback increases and causes a reduction in. Consequently, the acoustic-wave amplitude decreases, resulting in weaker diffraction, so that the detected field E 1 is stronger. The feedback is continuously reinforced, and the device eventually reaches a saturation point at which the feedback is a maximum and the diffraction into E 0 is a minimum E Reset: The E inc1 input has to be turned off for a period of time to restore the original state. This diminishes the feedback, so that a strong acoustic grating is formed again inside the acousto-optic cell as a result of the value 0. When E inc1 is turned back on, the device returns to the original state E 1 0. In the first-order acousto-optic flip-flop, we note that E inc0 is designated as the trigger signal TRIG, and E inc1 is designated as the reset signal RESET. At the flip-flop output, E 1 is designated as Q, and E 0 is Q. 3. Iteration-Map Analysis of the Acousto-Optic Bistable Device Acousto-optic bistable devices have been analyzed by use of nonlinear dynamical analysis, numerical analysis, and nonlinear circuit modeling. 1 5 The iteration-map method is used in nonlinear dynamic analyses to provide a visual interpretation of possible stable and unstable states of the integrated bistable system. Chrostowski 4 has analyzed the bifurcation 10 May 1997 Vol. 36, No. 14 APPLIED OPTICS 3149

4 Table 1. Notation Relations between the Symbols Herein and Those of Chrostowski a Chrostowski s Notation X n Our Notation n 2 E 2 inc 2, Gk a Chrostowski s notation is from Ref. 4. in acousto-optic bistability in a single-input device using such a method. A deterministic map not shown here of the bistable device may be drawn according to the governing equation X n 1 1 sin 2 X n 0.5, (6) where X n and X n 1 are the inputs to the acoustic driver rf modulator at different times, I L Gk is the bifurcation parameter, I L is the input laser intensity, G is a geometric factor, k is the light voltage conversion ratio of the photodetector, and is the feedback-amplifier gain. It may be shown that a first bifurcation occurs around 0.32, generating two states, 4 and, at around 0.63, a second bifurcation occurs that causes a multiple-state system. The bistable device uses the two-state region that is defined by the range For the sake of notational consistency, we make the connections, shown in Table 1, between the symbols in Chrostowski s study and those presented in this paper. An iteration-map analysis may be applied to the zeroth-order optical flip-flop of Fig. 3. We first define the system equation according to the system schematic. The triggering signal is ignored in the analysis since it appears as only a transient to promote a perturbation in the tracing path. The state after the introduction of the perturbation is more interesting. As depicted in Fig. 3, 0 is set to zero, is designated as the input to the acoustic driver rf modulator, and the diffracted field resulting from the signal RESET is intercepted by the photodetector. The intensity I 0, which is detected by the photodetector, may be deduced from Eqs. 4, and the system equation may be written as n 1 f n E 2 inc1 sin 2 n 2, (7) where n and n 1 are the inputs to the acoustic driver at different times. The deterministic map, as shown in Fig. 5 for, has three intersection points. Although the point in the center is unstable, the two outer ones are stable. Note that the amplitude of the sine-squared curve is proportional to. It may be shown that, when 0.9, the sine-squared curve of f n falls below the unit slope straight line; there is thus no intersection between f n and n 1, except for the origin. Therefore, in such cases all tracing paths converge at the only intersection point 0, 0, i.e., the origin, as depicted in Fig. 6 a, where is set to 0.6. Fig. 5. Deterministic map of the zeroth-order optical flip-flop. We may designate the state of the system at 0.6 as monostable in view of the single stable point at the origin. When is increased to, as shown in the iteration map of Fig. 6 b and the asymptotic attractor graph of Fig. 7, there are two points of convergence: for n 1.57 and 0 for n This value of corresponds to a point in the bistable region which extends from approximately 2.7 through 3.9. For 4.0, as shown in Figs. 6 c and 7, the system has three stable points: two self-looping or oscillating points at 3.4 and 4.0 for n 1.13 and the third at 0 n The oscillations, which represent a bifurcation in the attractor graph, can also be seen in the circuit simulation, which is presented in Section 4. Finally, when is increased to 1.7, as shown in Figs. 6 d and 7, the system is essentially in chaos, even though the zero or ground state persists. This effect is also verifiable by circuit simulation, as is also shown below. Note that, in the asymptotic attractor graph shown in Fig. 7, an equilibrium state near zero, or the ground state, exists throughout the range of. The ground state and the other equilibrium state beginning at approximately 2.7 and before any bifurcation appears are used as the two logic states for the device the bistable range. Since in the chosen bistable range is further from the chaotic region than are the two bistable states in the first bifurcation region, operation of the flip-flop in such a range is expected to be more stable. An additional advantage of operation in this range is that the lower state is at ground or zero intensity, such that the flip-flop switches between dark and bright instead of between two brightness levels which may be harder to differentiate. We also note that the same analysis may be applied to the firstorder flip-flop. Indeed, the mathematical models for both flip-flops are similar. It may be shown that both share the same deterministic map. 4. Nonlinear Circuit Modeling and Simulation As discussed in Section 3, the acousto-optic bistable device may also be modeled by use of equivalent non APPLIED OPTICS Vol. 36, No May 1997

5 Fig. 6. Trace diagrams for several different cases: a 0.6 monostable, b bistable, c 4.0 bifurcation, and d 1.7 multistable, chaotic. linear circuit models and simulated with commonly available circuit-simulation software. 5 In Fig. 8 we show the SPICE circuit model of a dual-input zerothorder bistable device. The acousto-optic modulator is modeled with sine E 7 and cosine E 6 functions by means of nonlinear dependent sources with Taylor series expansions. The circuit time-delay element is represented by a transmission line T 1 with proper termination, and the photodetector is a squaringoperator element that uses another nonlinear dependent source, E 1. The simulation is carried out by use of the PSPICE circuit-simulation program. Figure 9 shows the simulation result for a zeroth-order acousto-optic flipflop. The RESET is set to 1 and the TRIG is 0 in the beginning. We apply an impulse as the TRIG signal at 2 ms. The output Q rises with a slope and maintains a level at approximately 1. At 6 ms, the RE- SET is applied with a negative impulse and is then forced to 0 during the pulse. The output Q returns to zero because of the consequent removal of the feedback caused by the negative impulse. The output Q rises to a peak value of approximately 1.0 V for close to 3.0. However, the simulation shows that, as increases toward 3.0, the peak amplitude moves closer to 1.0, as is expected from the iteration-map analyses. In addition, the rise time of the transition decreases as increases. It is also found that the device undergoes bifurcation and exhibits chaotic behavior when exceeds 4.0. Bifurcation and chaotic oscillations are shown in Figs. 10 a and 10 b. It is found that the period of oscillation increases with the circuit time delay. Simulation for a first-order acousto-optic flip-flop may similarly be carried out. 5. Experimental Verification An experimental setup for a zeroth-order acoustooptic flip-flop is shown in Fig. 11. Two lasers are used in the setup as the sources for the TRIG and 10 May 1997 Vol. 36, No. 14 APPLIED OPTICS 3151

6 Fig. 9. SPICE simulation of the zeroth-order acousto-optic flip-flop, where 3.0 and the time delay is TD 0.01 ms. Fig. 7. Asymptotic attractors of the zeroth-order bistable device. is varied from 1.5 to 5.5 to observe the range of behavior from monostability through chaos. Note that a ground state exists throughout the range of. RESET signals. The incident light beams from the lasers are controlled by two mechanical shutters that are operated manually. The beams are aimed at the acousto-optic cell through a beam splitter that is used to redirect the trigger beam. Each beam suffers attenuation while passing through the beam splitter. The beam stop blocks the undiffracted RESET beam and lets the TRIG beam go through. The sequence for control of the flip-flop requires that shutter S1 RESET be turned off and then on to return the system to the initial state. The device is then reset. The flip-flop is now ready for a change of state. As soon as the TRIG signal is applied i.e., as shutter S2 is activated to turn the shutter on momentarily, the state of the flip-flop immediately changes from 0 to 1. Shutter S1 has to be turned off and on again to reset the flip-flop. After incorporating an input adder, a feedback-loop amplifier circuit, and a rf generator to provide an amplitude-modulated 40-MHz signal to drive the acousto-optic modulator AOM, the optical flip-flop is ready for testing. The feedback-amplifier gain range is expected to be higher than that of a standard 0 versus I 1 hysteretic response test since the bias 0 is set to zero. In Fig. 12 a, the triggering signal and the detected output Q are recorded for the triggering operation where the feedback gain is set to 625. Note that is only the gain of the operational amplifier and not the effective gain of the overall feedback loop. The rise time for the flip-flop shown in Fig. 12 b is limited mainly by the bandwidth of the rf generator s modulation input and the bandwidth of the operational amplifier. The overall experimental rise time of the zeroth-order flip-flop measured with respect to the two time samples shown in Fig. 12 a is approximately 640 s, which indicates a fairly low effective bandwidth of the experimental system used. The feedback-amplifier gain for operation of the zeroth-order optical flip-flop is higher than that necessary for generating the bistable 0 I 1 characteristic of a standard bistable device. This difference may be explained by the fact that, in the standard Fig. 8. Equivalent circuit diagram of a zeroth-order acousto-optic flip-flop APPLIED OPTICS Vol. 36, No May 1997

7 Fig. 10. a SPICE simulation of the zeroth-order acousto-optic flip-flop, where 4.0 and the time delay is TD 0.01 ms. Note that the Q output is beginning to exhibit the bifurcation phenomenon. b Simulation where 5.0 and the time delay is TD 0.01 ms. Note that the Q output has entered the multistate or chaotic regime. Fig. 11. Experimental setup for a zeroth-order acousto-optic flipflop. Laser1 and laser2 are He Ne lasers; S1 is a toggle-mode shutter ON OFF ; S2 is a trigger-mode pulse shutter fastest speed is s ; STP1 is an optical beam stop; P1 is a cubic beam splitter. A Hewlett-Packard Model HP-606A rf generator Gen with AM modulation input MOD-in serves as the acoustic driver. Sum, input (bias) adder; A, amplifier gain. Fig. 12. a Triggering operation of the optical flip-flop, showing Q upper trace and Trigger lower trace. The applied trigger signal is s long. The amplifier gain is 625. b Rising edge of I 1 when the flip-flop is triggered. It takes approximately 640 s to rise from 10% to 90% of its maximum amplitude. device, an external 0 is always present as part of the bias drive of the acoustic source. Hence, here the feedback acts as only a relatively small perturbation of the bias. In the zeroth-order flip-flop, however, 0 is nominally absent. The absence of 0 in the zeroth-order flip-flop is essential because this ensures that there is no grating present in the device initially. Hence, the feedback signal, which provides the entire acoustic-driver bias, is no longer a small perturbation, since it must be large enough to generate a feasible acoustic grating in the sound cell. Therefore, a large amplifier gain is expected. It must also be noted that the experimental device did not exhibit any multistable or chaotic characteristics. This may be due to the fact that, even for fairly high effective feedback gains, the actual driver power or the discrete n that drives the sound cell is presumably lower than that necessary to enter the multistable or chaotic regimes. The three approaches to analyzing the behavior of the flip-flops, viz, by means of nonlinear dynamical analysis, nonlinear circuit simulation, and some corroborating experiments, have been useful for gaining further 10 May 1997 Vol. 36, No. 14 APPLIED OPTICS 3153

8 insight into the properties of such a nonlinear system. 6. Conclusion In this paper, the use of an acousto-optic modulator with feedback and a dual-beam input as an optical flip-flop has been proposed and investigated. The function of such a two-state device has been verified with SPICE simulations and an experiment. The simulation results are in reasonable agreement with the experimental results; the experimental results were somewhat limited by the available acoustic-driver power. Bifurcation and chaotic behavior have been explored by means of nonlinear dynamical analysis and circuit simulation. The authors wish to acknowledge T.-C. Poon for initiating their interest in acousto-optic bistability. The authors also thank the reviewers for their insightful comments and suggestions. References 1. J. Chrostowski, C. Delisle, and T. Tremblay, Oscillations in an acoustooptic bistable device, Can. J. Phys. 61, G. J. Yue and Z. Z. Ren, The stability analysis and the modulation effect on a Bragg acoustooptic bistable system, IEEE J. Quantum Electron. 26, H. Jerominek, J. Y. D. Pomerleau, R. Tremblay, and C. Delisle, An integrated acousto-optic bistable device, Opt. Commun. 51, J. Chrostowski, Noisy bifurcations in acousto-optic bistability, Phys. Rev. A 26, M. R. Chatterjee and J.-J. Huang, Demonstration of acoustooptic bistability and chaos by direct nonlinear circuit modeling, Appl. Opt. 31, T.-C. Poon and S. K. Cheung, Performance of a hybrid bistable device using an acoustooptic modulator, Appl. Opt. 28, A. Korpel and T.-C. Poon, Explicit formalism for acousto-optic multiple plane-wave scattering, J. Opt. Soc. Am. 70, APPLIED OPTICS Vol. 36, No May 1997