Nonlinear Dynamical Behavior in a Semiconductor Laser System Subject to Delayed Optoelectronic Feedback

Transcription

1 Nonlinear Dynamical Behavior in a Semiconductor Laser System Subject to Delayed Optoelectronic Feedback Final Report: Robert E. Lee Summer Research 2000 Steven Klotz and Nick Silverman Faculty Adviser: Professor David Sukow Washington and Lee University Physics and Engineering Department September 11, 2000 Abstract We study the nonlinear dynamical behavior of a semiconductor laser subject to delayed optoelectronic feedback. We perform this study by designing a simple electronic circuit whose behavior is governed by the same equations that govern the laser system. The circuit offers important experimental advantages, since it operates with much slower speeds, and has easily tunable parameters. Furthermore, it can be investigated systematically using automated data acquisition and control of system parameters. The dynamics we observe are in good agreement with numerical and theoretical predictions. We also model certain unavoidable limitations of the real laser system by adding a filter to the idealized circuit. The filter induces some changes to the global dynamics, but does not affect the fundamental oscillation frequency of the system. Introduction Semiconductor lasers subjected to delayed feedback produce complex nonlinear dynamical behavior. By studying the dynamical behavior we are able to better understand

2 and improve the performance of these semiconductor lasers in applications where feedback occurs, such as telecommunications, compact disk technology, and bar code readers, to name a few. Figure 1 One particular and important configuration is the laser subject to delayed optoelectronic feedback. As illustrated in Figure 1, light from a semiconductor laser enters a detector which generates an electrical signal that then is summed back into the power source that drives the laser. This process causes a delayed feedback due to the time it takes the electrical and optical signals to travel around the feedback loop. Although the signals travel at the speed of light, the delay time is significant enough to create the nonlinear dynamics we have studied this summer. Characterizing these laser systems experimentally is difficult due to the high natural frequencies at which such systems operate (gigahertz). For this reason, we have developed an analog computer to mimic the laser system and the nonlinear dynamical behavior it produces. The advantages in using this form of an analog computer rather than the actual system are numerous. First, the circuit is comparatively inexpensive. Second, the circuit operates at a much slower time scale, which allows us to explore the dynamical behavior more thoroughly. Last, the circuit s parameters are easily varied so that we can find the most suitable realms in which to work. We have developed a circuit based on the differential equations that govern the laser system, as derived in previous theoretical 1 work. Our circuit represents these differential

3 equations, reproduces the same nonlinear dynamics seen in the laser system, and allows us to vary the parameters with relative ease. Using this experiment, we investigate the effects of filtering high frequency oscillations in the laser intensity by using a low pass filter. This filter is unavoidable in a real semiconductor laser but not considered in the theoretical predictions. This is important because such filtering has been hypothesized as the origin of certain unexplained effects in the laser system. Theoretical Model The differential equations that model the laser system are as follows 1 : y '= x + xy ξ x ' = y ξx xy + η+ ηy( s τ). 2 The parameter η represents the feedback strength and τ represents the delay time. We used η and τ as bifurcation parameters in our data acquisition to form the two separate bifurcation diagrams. The other parameter ξ is proportional to the pump parameter. Once we found a suitable pump parameter we held its value constant for all of the data acquisition. The variable x represents the semiconductor carrier number and the y variable is the laser intensity and the main interest in our data acquisition. Based on previous theoretical 1 and numerical 2 work we expected to observe multiple branches in our bifurcation diagrams, and simple periodic, quasiperiodic, and chaotic waveforms. This coexistence of branches is dependent on both the delay time and the pump parameter. As these values vary we expected to see the waveforms change from simple periodic to quasiperiodic and finally into chaos. An oscillating envelope and three distinct frequencies should be seen when the waveform is in its quasiperiodic state. We developed our circuit to experimentally test these theoretical and numerical predictions. Through our analog computer and the data acquisition program built in

4 LabVIEW we were able to support the predictions made above and accurately reproduce the dynamics of the semiconductor laser system. Furthermore, since there has not been any theoretical work to predict the effects of adding the low pass filter that inherently exists in the laser system, we added the filter to the idealized system for which predictions have been made. We expected to see some significant differences in the global behavior of the system due to the loss of dynamic range in the feedback. Experimental Apparatus There were two parts to our experimental setup. The analog computer was used to generate the feedback dynamics and consists of a circuit built to compute the laser equations. Acquisition of data and automation of the analog computer parameters was controlled by LabVIEW. The Circuit An analog computer is built from components known as operational amplifiers or op amps and other integrated circuits. With an appropriate combination of resistors and capacitors op amps can be used to add or integrate. We refer to these combinations by the function they perform, respectively, summer and integrator. We can use integrators to solve differential equations with no closed form solution. The laser system we modeled is governed by such a set of differential equations, as presented earlier. The pump parameter, η, is controlled by a variable voltage source. The delay parameter, τ, is modeled by a programmable delay unit or PDU. The PDU simply digitizes the signal from the circuit, stores the signal for a set amount of time, and then sends an analog signal back to the circuit. The length of the delay can be manipulated from our workstation.

5 LabVIEW The digital oscilloscope we used, the LeCroy 9384M, is able to communicate with and be controlled by our workstation. We use a program called LabVIEW to interact with both the LeCroy 9384M and the PDU. We programmed LabVIEW to set τ, retrieve waveforms from the oscilloscope, and write waveforms and their peaks to disk. LabVIEW s control of τ enabled full automation of half of our data acquisition and thus we took considerably more waveforms when varying τ than when varying η, which was done by hand. Data We analyzed two different versions of our circuit. The first version corresponds to Figure 2, and the second version contains a lowpass filter. There were two parameters that we varied, τ and η. In a given data set one parameter was held constant while the other parameter was varied. We displayed our data in a bifurcation diagram, which plots the peaks associated with the intensity variable waveform against the value of a parameter at the time the waveform was measured. This removes a considerable amount of redundant information and allows us to observe global trends in the data. Note that in all of our measurements the parameter ξ=1.0. The two branches seen in Figure 3 are generated by setting τ=7.0x10-4 s and varying η. The first bifurcation point is found on the lower branch and indicates the start of simple periodic oscillation at η= We also see a bifurcation from simple periodic oscillation to quasiperiodicity at η= It is important to note that the upper branch coexists with the lower branch. We were able to move between branches by randomizing initial conditions. As mentioned before considerably more data was gathered while varying τ (Figure 4) and setting η=0.08. As τ is increased we see multiple branches which indicate simple periodic waveforms with increasing amplitude. We then see hints of quasiperiodicity leading back to simple periodic oscillations. The switch into chaos is distinct. These results agree with numerical predictions 2.

6 Having obtained reference data with no filter present, we now add a low-pass filter to the circuit. We set τ=7.0x10-4 s and vary η. The basic structure, as seen in Figure 5, changes little with the introduction of the low pass filter. Again, notice the coexistence, especially where the quasiperiodic regions overlap. The upper branch starts oscillating at a feedback strength of η=0.055 and bifurcates to quasiperiodicity at η= The lower branch begins oscillating at η=0.136 and bifurcates to quasiperiodicity at η= Adding the filter shifted the first bifurcation point from η=0.013 to η= The first bifurcation to quasiperiodicity was also shifted from η=0.071 to η= In Figure 6 we vary τ and set η=0.08. The dynamics proved interesting beyond the point where the system without the filter fell into chaos, so we recorded data at longer delays. The filter seems to have stabilized our circuit such that it is easier to see the dynamics. It also seems to have imposed a larger structure upon the dynamics that is difficult to characterize. Noticeable in the first half is the alternation between three quick changes in amplitude, seen as steep arcs, and a slow change in amplitude, seen as a band. A similar phenomenon occurs in the second half where the general trend is quasiperiodic. Conclusion Through our nine weeks of summer research we have taken a set of differential equations that govern a semiconductor laser system and built an analog computer to mimic the behavior of the laser system. We tested the dynamical behavior and acquired data to compare with theoretical and numerical predictions. Our data support the predictions for an ideal system with no frequency filtering. We added a low pass filter to the circuit to mimic the real laser system more accurately. Our results did not illustrate a significant difference in the global bifurcation structure, however, the values of η for corresponding bifurcation points increased and the characteristic oscillation frequency was unchanged. These results suggest that the low pass filter is not responsible for causing the change in oscillation frequency that was observed in the experimental data of the real laser system 2.

7 References 1 D. Pieroux, T. Erneux, A. Gavrielides, and V. Kovanis, Hopf bifurcation subject to a large delay, preprint (2000). 2 D. S. Tang and J.M. Liu, Chaotic pulsing and route to chaos in a semiconductor laser with delayed optoelectronic feedback, preprint (2000).

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