The Dynamics of Delay Coupled Optoelectronic Devices. A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College

Transcription

1 The Dynamics of Delay Coupled Optoelectronic Devices A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College In Partial Fulfillment of the Requirements for the Degree Bachelor of Arts Gregory Hoth May 2010

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3 Approved for the Division (Physics) Lucas Illing

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5 Acknowledgements My thesis project could not have succeeded without the help of many people. In particular, I would like to thank Greg Eibel for his brilliant solution to my measurement problem. I am also indebted to Chris May for his work on the systems that are at the heart of my thesis. I would like to thank Lucas for the opportunity to work in the Nonlinear Optics Lab and for all of the good advice. Finally, I would like to thank my friends and family for their love and support.

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7 Table of Contents Introduction Chapter 1: Devices and Components The Mach Zehnder Modulator The Single Loop System The Cross Coupled System Methods for Observing the Dynamics Components Used to Construct the Delay Coupled Systems Connections Laser Diode Polarization Controller The Mach Zehnder Modulator Revisited Optical Attenuator Photodetector, Splitter and Amplifier Concluding Remarks Chapter 2: Methods and Results: Single Loop System Setting the Free Parameters The Bias Point of the MZ The Laser Current Measuring the Delay Observed Dynamics Chapter 3: Methods and Results: Cross Coupled System Setting the Free Parameters Observed Dynamics Chapter 4: Dynamical Systems and the Hopf Bifurcation Dynamical Systems The Geometric Approach Phase Space Attractors Bifurcations A Simple Model for the Onset of Oscillations The Hopf Bifurcation

8 4.5 Fixed Points and Linear Stability Analysis Concluding Remarks Chapter 5: Theory and Modeling: Single Loop System Deriving the Model Reproducing the Observed Dynamics A Method for Solving Delay Differential Equations Numerically Numerical Solutions Linear Stability Analysis The Frequency at the Onset of Oscillations Chapter 6: Theory and Modeling: Cross Coupled System Deriving the Model Numerical Solution Linear Stability Analysis Positive Round Trip Gain Negative Round Trip Gain Conclusion Appendix A: Midpoint Runge-Kutta for Delay Differential Equations 65 Appendix B: The Model for the Cross Coupled System References

9 List of Tables 1.1 Specified Cutoff Frequencies for Electronic Components Parameters Used to Numerically Model the Single Loop System Parameters Used to Numerically Model the Cross Coupled System.. 58

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11 List of Figures 1 Abstract Structure of Delay Coupled Systems Comparison of the abstract model for an MZ with data and Schematic Diagram of an MZ Schematic Diagram of the Single Loop System Schematic Diagram of the Cross Coupled System Intensity as a function of Current for the Laser Diode Drift of the MZ s bias point Calibration of the Optical Attenuators Gain of the Photodetector and Amplifier Measuring the Delay of the Single Loop System General Delay Measurements Amplitude of Oscillation in the Single Loop System as a Function of α Diagram for the Cross Coupled System used to Derive the Symmetry Conditions Measuring the Linear Feedback Gain used to Set the Laser Current Amplitude of Oscillations in the Cross Coupled System as Function of α Amplitude of Oscillations in the Cross Coupled System as a Function of α 1 α Observed Scaling Law in the Cross Coupled System Phase Portraits for the Damped Harmonic Oscillator Phase Portrait for the Hopf Normal Form Amplitude of Oscillations as a Function of µ for the Hopf Normal Form Phase Portraits of Another System with a Hopf Bifurcation Amplitude of Oscillation as a Function of the Bifurcation Parameter for the Hopf Bifurcation Comparison of Model and Measured Transfer Function Schematic Diagram of the Single Loop System Used to Derive IDDE Model Numerical Results on the Single System Bifurcation Diagram for the Single Loop System

12 6.1 Numerical Results for the Cross Coupled System Bifurcation Diagrams for the Cross Coupled System Comparison of the Predicted Frequency and the Observed Frequency for the Cross Coupled System with Positive Round Trip Gain Comparison of Predicted Frequency and Observed Frequency for the Cross Coupled System with Negative Round Trip Gain Three Device System B.1 Diagram Used to Derive the IDDE model for the Cross Coupled system 68

13 Abstract This thesis considers the dynamics of a non-linear oscillator with delayed feedback and the dynamics of a pair of delay-coupled oscillators. Experimentally, I show that these systems undergo a sharp onset of oscillations at a critical value of the coupling strength and that there is a scaling law that governs the relationship between the coupling strength and the amplitude of oscillations. Theoretically, these dynamics correspond to a Hopf bifurcation. I construct integro-differential delay equations that model these systems, and show numerically that these models capture the observed dynamics close to the onset of oscillations. I also show analytically that the onset of oscillations in the model is due to a Hopf bifurcation.

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15 Introduction This thesis is a study of the dynamics of two delay-coupled nonlinear optoelectronic devices with bandpass filtering. The structure of the devices considered is illustrated schematically in Fig. 1. In this representation, the circles correspond to nonlinear devices with one input port and one output port, while the arrows represent links between these devices. Each link is characterized by a coupling strength α and a delay τ. Physically, the coupling strengths represent how much a signal is amplified (or diminished) as it travels from one nonlinearity to another, and the delays represent the amount of time it takes a signal to travel between two of the nonlinearities. I say that the devices are delay-coupled to emphasize that we will be studying systems where these travel times are comparable to the timescale of the dynamics. The links between the nonlinearities are also characterized by certain speed limits. Signals which change too slowly or too quickly do not travel over the link. To say the same thing more formally, signals propagating between the devices are bandpass filtered. Physically, this filtering arises from the intrinsic limits of the components used to construct the devices we will be studying. Broadly speaking, the goal of my thesis is to characterize how the output of the nonlinearities in these devices is influenced by the coupling strengths. Of course, this problem is much too hard to solve in general, but it can be approached experimentally by observing how the behavior of these devices change as their parameters are varied. My thesis is focused on the relationship between the amplitude of oscillations in these devices and the coupling strengths, but that is getting ahead of the story. It is natural to wonder why anyone would be interested in the dynamics of these rather esoteric systems. Perhaps the most compelling reason is that these systems provide a nice, clean environment in which to explore how significant delay times affect dynamics. Long time delays occur naturally in many physical and biological systems, but there are not many mathematical tools for working with models that incorporate them [1]. Systems like the ones considered here provide test cases that can illustrate how to model time delays in messier contexts. These systems can also serve as physical implementations of abstract models with long time delays. For example, Kim et al. [2] show that a system of cross coupled laser diodes can be used to study a model of how disease spreads between two populations coupled by migration. Finally, these systems also have some technological applications (see the introduction to [1] for some discussion). These observations provide motivation for studying the dynamics of delay coupled systems generally, but this thesis is rather far removed from any of these applications.

16 2 Introduction Single Loop System: Α, Τ Cross Coupled System: Α 1, Τ 1 Α 2, Τ 2 Figure 1: Schematic representation of the systems studied in this thesis. The first system is a nonlinearity with time delayed feedback, and the second system is a pair of cross coupled nonlinearities. Personally, I chose to work on this project because the systems considered here provide an interesting environment to explore how ideas from nonlinear dynamics can be used to understand and model experimentally observed behavior. In my work on this project, I have consulted only a small section of the literature on delay-coupled systems. My experimental work follows that of Peil et al. [1], who give a detailed study of an optoelectronic circuit that is essentially identical to the one I use to implement the single loop structure, and Kim et al. [2] who study a system with the cross coupled structure. My theoretical work has been guided by a general framework for analyzing the dynamics of the delay-coupled systems described by Illing and Gauthier [3]. This thesis can be roughly divided into two parts. In the first part, which consists of chapters 1-3, I will introduce the optoelectronic devices that are represented by the abstract structures shown in Fig. 1 and discuss the dynamics that can be observed experimentally. In the second part, chapters 4-6, I will introduce some ideas from nonlinear dynamics and then show how they can be used to model the observed dynamics.

17 Chapter 1 Devices and Components In this chapter I will introduce the optoelectronic devices that this thesis is centered around. Most of the components used in these devices are familiar and intuitive, but the Mach-Zehnder modulators (MZs) that are the nonlinear hearts of these devices require some explanation. After discussing how these modulators work, I ll show you how they can be used to implement the delay coupled structures shown in Fig The Mach Zehnder Modulator In abstract, an MZ is a three terminal device. It has two inputs light of intensity I in and a voltage V and one output light with intensity I out. Quantitatively, the output is related to the inputs by I out = ζi in sin 2 (bv + φ) (1.1) where ζ, b, and φ are intrinsic parameters of the device. In practice, we will hold I in constant, so it is convenient to introduce the parameter a = ζi in to simplify Eq. 1.1 to I = a sin 2 (bv + φ). (1.2) One can experimentally verify that Eq. 1.2 provides an excellent model for these devices, as the data shown in Fig. 1.1a illustrate. Physically, an MZ creates this transformation by exploiting the electro-optic effect. In a material that exhibits this effect, the index of refraction changes when a voltage is applied. An MZ is essentially an electro-optic crystal that has waveguides cut into it with the structure shown schematically in Fig. 1.1b. When light is sent into an MZ, it travels through these waveguides smoothly until it reaches the first y-junction where it splits in two. The splitting ratio depends on the polarization of the incoming light. After the junction, part of the original beam travels in a path where a voltage has been applied to the crystal. Because of the electro-optic effect, this beam experiences a higher index of refraction than its companion in the other path and so it travels at a slower speed. As a result, a relative phase shift develops between the two beams, which modifies the intensity of the recombined beam at the output of the modulator.

18 4 Chapter 1. Devices and Components " /01,-. 20 &'+! &'* &') &'( &'! &'" " #$! "#$! " "!! #$ % Figure 1.1: (a) Comparison of the abstract model (black curve) for an MZ with data (grey points) obtained by measuring the intensity output by an MZ as a function of the voltage applied to it. The parameters for the black curve were determined by using a LabView program to fit Eq. 1.2 to the measured data. The best fit parameters are a=0.63, b=0.39, and c=0.73. (b) The structure of a Mach Zehnder modulator is illustrated schematically. This interference effect can be modeled quantitatively by considering two plane waves that are identical except for a relative phase shift of θ. For simplicity, we will take the planes waves to be propagating in the z-direction and polarized in the x-direction. Using complex notation, these plane waves can be described by Ẽ 1 = E 1 e i(kz ωt)ˆx and Ẽ 2 = E 2 e i(kz ωt+θ)ˆx. (1.3) If we combine these two fields, we obtain another plane wave, given by Ẽ 3 = Ẽ1 + Ẽ2 = (E 1 + E 2 e iθ )e i(kz ωt)ˆx. (1.4) Using an identity from Griffiths [4] (pp. 382), the intensity of the combined fields is I = 1 2 ɛ 0cRe (Ẽ3 ) Ẽ 3 = 1 2 ɛ 0c ( E E E 1 E 2 cos θ) ) = 1 2 ɛ 0c ( (E 1 E 2 ) 2 + 4E 1 E 2 sin 2 (θ/2 + π/2) ). In the special case where E 1 = E 2, we have I = 2ɛ 0 ce 2 1 sin 2 (θ/2 + π/2), (1.5) which has the same form as Eq. 1.1 if we assume that the phase shift θ induced by the modulator is linearly related to the applied voltage. This derivation explains the origins of the sin 2 function in the abstract model of an MZ. It also shows that Eq. 1.1 only provides the correct model for an MZ if the

19 1.2. The Single Loop System 5 Laser Diode Polarization Controller Mach Zehnder Modulator Amplifier Delay Τ Α Variable Attenuator To Oscilloscope Splitter Photodetector Figure 1.2: A schematic diagram of the system used to implement the single loop structure. Each element of the system is represented by a labeled icon. These icons will be used to represent these components throughout this thesis. incoming light is split into two identical beams at the first y-junction. Consequently, we will have to ensure that the light source we use to seed the MZs in our experiments has just the right polarization so that Eq. 1.1 holds. Finally, it is interesting to note that the other parameters in Eq. 1.1 can also be accounted for physically. The prefactor ζ represents the fact that light loses some of its energy as it travels through the electro-optic crystal. The parameters b and φ express the fact that the phase shift induced by the electro-optic effect is linearly related to the applied voltage. 1.2 The Single Loop System In abstract, the single loop system is just a nonlinear device with delayed self-feedback. This structure can be created with a Mach Zehnder modulator by using a photodetector to convert the intensity output by the MZ into a voltage, which can then be fed into the MZ s electrical input. Of course, we also have to provide the MZ with light of the correct polarization. In our system, we use a current controlled laser diode with an external polarization controller to seed the MZ. For experimental convenience, we add several more components to this basic feedback loop. In particular, we add a variable attenuator to gain control over the coupling strength, a splitter so that we can observe the system s behavior, and an amplifier to boost the strength of the feedback. The whole system is schematically illustrated in Fig. 1.2.

20 6 Chapter 1. Devices and Components MZ 1 MZ 2 To Oscilloscope Circulators To Oscilloscope Α 2 Α 1 Figure 1.3: A schematic diagram of the system used to implement the cross coupled structure. The components of the system are represented by the icons used in Fig The arrows indicate the path traveled by light leaving MZ 1 as it propagates to the input of MZ The Cross Coupled System The cross coupled structure consists of two copies of the single loop system with the output of the nonlinearity in one loop connected to the input of the nonlinearity in the other. In our system, these two copies of the single loop system are coupled using optical circulators. These devices have three ports which channel light in the following way: light that enters port 1 exits port 2, and light that enters port 2 exits port 3. The structure of the cross coupled system is illustrated schematically in Fig Methods for Observing the Dynamics The Nonlinear Optics lab is equipped with two instruments that can be used to observe the output of these systems: a fast real time oscilloscope (12 GHz analog bandwidth, 40 GBit/s sampling rate) and a spectrum analyzer (1.5 GHz bandwidth). Using the oscilloscope, one views the signal output by the system as a function of time, which is ideal for developing an intuitive picture of the dynamics. The oscilloscope can also be used to quantify various properties of the waveform output by the system, but these measurements are hampered by the large amount of noise in the system. In contrast, the spectrum analyzer allows one to view the output of the system as a power spectrum. In this representation the high frequency noise is separated from the signal, enabling more precise measurements. In the next chapters, I will discuss the dynamics that can be observed with these instruments. The rest of this chapter is devoted to the components used to construct these systems in the laboratory.

21 1.5. Components Used to Construct the Delay Coupled Systems Components Used to Construct the Delay Coupled Systems Connections All the optical devices in these systems are fiber coupled with single mode fiber and FC/APC connectors. In practice, this means that each device has fiber leads that have connectors attached to their ends which can be screwed together to connect various devices. One can also insert extra lengths of optical fiber between two devices, which makes it possible to adjust the delays in these systems experimentally. These connectors are delightfully easy to use, but there are a few difficulties to be aware of. First, the connections between devices are not perfect so there will be some losses at every junction. By itself, this is not so bad, it just reduces the gain of the feedback loop by a small amount. However, it turns out that these losses change when the connections are broken and remade. This effect is usually small, but it can cause the losses in the system to change by 1-5%. For this reason, the connections should not be broken in the middle of an experiment. Second, it is possible for light to reflect back at the junctions and start propagating around the loop backwards. The connectors are cut at an angle to minimize these back reflections, and I have never had a problem due to back reflection but it is prudent to be aware that it is possible. The electronic devices in these systems are connected with SMA type connectors, which are specially designed to have a high cut-off frequency Laser Diode The light source for our system is a fiber coupled laser diode (Sumitomo Electric SLT5411-CC distributed feedback laser with optical isolator, λ = 1550 nm, P max 50 mw). The diode is held at constant temperature and constant current by electronic controllers. The temperature control stabilizes the diode s frequency, while the current control determines the diode s output intensity. The relationship between the driving current and the output intensity is illustrated in Fig We can account for the physical origins of this current-intensity relationship by considering an abstract, qualitative model of a laser consisting of a cavity with a perfect mirror on one end, a partially transparent mirror on the other end and a gain medium in the middle. The atoms in this gain medium have only two states a ground state and an excited state. An atom in the excited state can drop down to the ground state through either spontaneous emission or stimulated emission. In the first process, an excited atom emits a photon in a random direction. In the second process, a photon passing by an excited atom stimulates the atom to emit a second photon, which is identical to the first. Ordinarily, almost all of the atoms in the gain medium will be in the ground state, and there will be no light in the cavity. In order to generate light, we have to put atoms into the excited state by pumping energy into the gain medium. At first, these excited atoms will emit photons through spontaneous emissions. Most of these photons will simply be absorbed by the cavity walls, but some of them will bounce

22 8 Chapter 1. Devices and Components Intensity t Figure 1.4: A plot showing the relationship between current and intensity for one of the diodes in the nonlinear optics lab. The black points represent measured data and the black line represents the line of best fit. For a quantitative discussion of the device physics that lead to the current/intensity relationship in Fig. 1.4 see Chris May s thesis, Appendix A and B [5]. off of the mirrors at the ends of the cavity and pass back through the gain medium. These photons will stimulate the excited atoms to emit more photons that will bounce off one of the mirrors and pass through the gain medium again, producing even more stimulated emission. Eventually, the system reaches an equilibrium where the number of atoms emitting photons through stimulated emission is just balanced by the rate at which atoms are pumped into the excited state. In this equilibrium, the cavity outputs a coherent, nearly monochromatic beam with constant intensity. As long as the gain medium is not saturated with excited atoms, this output intensity is proportional to the rate at which energy is pumped into the system. This accounts for the linear relationship between current and intensity in Fig The reason that the current has to exceed a threshold before any light is generated is that a real laser cavity has some intrinsic losses which means that photons have to be generated at a sufficiently high rate before the equilibrium described above exists Polarization Controller The polarization controller is needed to ensure that the light from the laser diode has the right polarization so that the MZ is accurately modeled by Eq In order to use the controller, one needs to have a way identify when the polarization is just right experimentally. Fortunately, the MZ has been engineered so that the correct polarization can be identified by adjusting the polarization controller to maximize the output of the MZ while holding all the other parameters constant. Physically, the polarization controller (Thor Labs FPC 560) is made up of three paddles with fiber wound around a cylinder of some fixed radius inside of them. This puts stress on the fiber and thereby changes the index of refraction in the fiber in the direction in which the stress is applied. As a result, one component of the electric field inside the fiber travels faster then the other component and so the polarization

23 1.5. Components Used to Construct the Delay Coupled Systems 9 II max Figure 1.5: These plots illustrate the percent change in the intensity output by the MZ over the course of an hour and a half. Since all the other parameters in the set up are constant, this change must be due to drift in the bias point of the MZ. As the plot on the right shows, the drift in bias point is much slower for smaller bias voltages. of the light changes. The paddles are set up so that they constitute a quarter wave plate, a half plate, and a second quarter wave plate. With this configuration, it is possible to produce an arbitrary polarization [5] The Mach Zehnder Modulator Revisited Recall that our abstract model for an MZ has three parameters, ζ, b, and φ. Of these parameters, ζ and b are fixed, but φ can be adjusted by applying a DC bias to the MZ (JDSU Z5). The DC voltage applied to the MZ is called the bias voltage and choosing φ is called setting the bias point of the MZ. It is important to remember that the bias voltage is distinct from the voltage V that appears in Eq The latter is called the driving voltage. Unfortunately, once the bias point has been set, the MZ starts to slowly drift away from it, and the greater the magnitude of the bias voltage, the faster the bias point drifts. This trend is illustrated in Fig Fortunately, the largest bias voltages needed for my experiments corresponds to the plot on the right in Fig. 1.5 so it is reasonable to hope that the drift in the bias point is negligible for the cases considered here. Later on, I will discuss some experimental observations that support this idea Optical Attenuator The variable attenuators (Thor Labs VOA-50-APC) provide the key control parameter for our experiments so their behavior needs to be carefully quantified. This seems like it should be an easy task, but it is difficult because of the way the attenuators have been packaged. When one of these devices comes out of its shipping crate, it is a black box with fiber leads and a screw on the side which controls the amount of light transmitted by the attenuator. Unfortunately, there is no way to precisely measure how far the screw

24 10 Chapter 1. Devices and Components '. /. -)# Α #), #)+ #)$ #)*!"# $## $"# "## %&'( Figure 1.6: (a) Shown is a picture of one of the attenuators with Greg s modification. (b) Shown is the calibration for one of the optical attenuators in the nonlinear optics lab. The points represent the average of five measured data sets while the curve shows the interpolating function constructed by Mathematica. I have performed several consistency checks on the calibration illustrated in Fig In all cases, the measured attenuation agrees with the predicted attenuation within 5%. These calibration checks also showed that the behavior of the attenuator is independent of the input intensity. has been turned, which makes it impossible to calibrate the attenuator. In order to solve this problem, I turned to our machinist Greg Eibel for help, and he found a brilliant, simple solution. He attached a screwdriver with a dial and a numerical indicator to the side of the attenuator. When dial is adjusted, it turns the screw and advances the numerical indicator so that is possible to associate a number with the rotation of the screw. With Greg s attachment in place, it is possible to quantify the behavior of the attenuators in terms of the quantity α = I I max, where I is the intensity output by the attenuator at a given dial reading and I max is the maximum intensity output by the attenuator, for a fixed input intensity. Note that 0 α 1, and that small α corresponds to significant attenuation (weak coupling) and large α corresponds to little attenuation (strong coupling). The calibration was done using the following procedure. First, the maximum intensity output by the attenuator was measured. Then, the dial was turned in increments of 10 and the output was measured at each dial reading until the intensity output by the attenuator was close to zero. For each measured intensity, the quantity α was calculated. From this data, a function that predicts the value of α for any dial reading in the measured range was constructed using Mathematica s ListInterpolation command. The results of this procedure are illustrated in Fig Experimentally, we will use α as a quantitative measure of the feedback strength in our study of the single loop system and the cross coupled system.

25 1.5. Components Used to Construct the Delay Coupled Systems 11 Table 1.1: This table lists the cutoff frequencies for the three electronic devices in the system. The values are taken from the datasheets for the devices. Low Cut Off High Cut Off Photodetector (Miteq DR125-G) 30 khz 13 GHz Splitter (Picosecond Pulse Labs Model 5331) 0 18 GHz Amplifier (JDSU H301 RF Modulator) 75 khz 10 GHz V out V V out V Figure 1.7: These plots show a measured input-output relationship for the photodetector and amplifier. For both devices, there is a linear regime where the gain is constant, but the gain starts to decrease at high input amplitudes Photodetector, Splitter and Amplifier These electronic devices are the source of the bandpass filtering in the system. The low pass portion of the bandpass filter arises because these devices all have some intrinsic resistance and capacitance and therefore a finite rise time. The high pass portion of the filter occurs because the photodetector and the amplifier are AC coupled. Quantitatively, the cutoff frequencies for these devices are listed in Table 1.1. There are a few additional issues to be aware of with these electronic devices. First, both the photodetectors and the amplifiers are inverting. This fact does not affect how the devices are used in these systems, but it is sometimes important in measurements used to quantify various parameters in the system. Second, the gain of both the amplifier and the photodetector starts to decrease at higher input powers. This saturation effect is illustrated in Fig In practice, saturation effects will not concern us experimentally because we will focus on a regime where the amplitude of the signal output by the photodetector is small. Nevertheless, it is good to be aware of the experimental limitations of the devices in the system.

26 12 Chapter 1. Devices and Components 1.6 Concluding Remarks For the purposes of the rest of this thesis, the most important thing to take away from this discussion is that we have four quantities that we can adjust experimentally: the delay, the laser current, the bias point of the MZ, and the attenuation. Of these quantities, we will hold the delay, the laser current, and the bias points fixed at values that are experimentally convenient. This leaves the variable attenuation as measured by α to play the central role in our quantitative description of the dynamics of these systems.

27 Chapter 2 Methods and Results: Single Loop System At this point, all the tools that we will use to study the dynamics of the single loop system experimentally are in place. All that remains is to set the free parameters in the system, namely the bias point of the MZ, the laser current, and the delay in the feedback loop. Then, we can examine the system s dynamics. 2.1 Setting the Free Parameters The Bias Point of the MZ The experiments reported here only consider the dynamics of the single loop system with φ = π or φ = π. The main reason to choose these bias points is that they 4 4 correspond to the simplest cases to analyze theoretically [6]. It is also interesting to consider both of these cases because it enables us to explore how the sign of the round trip gain influences the dynamics of the system. When the bias point is π, the MZ s nonlinearity is locally upward sloping and so 4 the round trip gain is positive. In contrast, when the bias point is π, the nonlinearity 4 is locally downward sloping which makes the round trip gain negative. For signals that are large enough to explore more than the local slope of the MZ s nonlinearity the distinction between positive round trip gain and negative round trip gain breaks down, but for small signals the regimes are distinct as we shall see. The MZ can be tuned to the ± π bias points using the following procedure. First, 4 adjust the DC bias until the output intensity of the MZ reaches its maximum value. Then, adjust the DC bias until the output intensity is half the maximum intensity. If the output intensity is a decreasing function of the bias voltage, then φ = π. If 4 the output intensity is an increasing function of the bias voltage, then φ = π The Laser Current The value chosen for the laser current is somewhat arbitrary, but there is an important tradeoff to consider. Since the laser current determines how much power enters the

28 14 Chapter 2. Methods and Results: Single Loop System Φ Π4 Function Generator Τ 1 Τ 2 Α Τ 3 Oscilloscope Figure 2.1: A schematic illustration of the set up used to measure the delay in the single loop system. The elements of the single loop system are represented by the symbols introduced in the previous chapter. In the laboratory, the oscilloscope and the function generator are connected to the single loop using BNC cables. It is important to note that when this experiment is performed in the lab, the unused splitter output must be terminated at 50 Ω. It is also best to set the bias point of the MZ to π to maximize the linear response. 4 system, it sets the maximum feedback strength. At higher laser currents, it is possible to explore the dynamics of the system at greater feedback strengths, but the system s dynamics also change more rapidly as a function of α. This is problematic because our ability to tune and measure α is limited. For this reason, it is best to choose the smallest laser current that allows one to access the dynamical regime of interest Measuring the Delay In my experiments, I have simply worked with whatever delay was most convenient experimentally. Despite this, it is still important to quantify the delay because it is a crucial parameter for modeling the dynamics of the system. It can be measured using the set up shown in Fig In this set up, the function generator produces a square wave which then propagates through the two paths shown in the figure. As long as the square wave output by the function generator is sufficiently small, the output of each path will also be a square wave. Note that the output of the upper path reaches the oscilloscope after a travel time τ 1 + τ s + τ 2, where τ s is the time it takes to propagate through the components of the single loop system, while the output from the lower path reaches the oscilloscope after a time τ 3. By adjusting the position of one of the traces until the rising edges of both outputs cross on the time axis of the oscilloscope, one can measure the time interval τ 1 + τ s + τ 2 τ 3 with a precision of about 100 picoseconds. In order to extract the delay due to the single loop system from this measurement, we have to somehow ensure that τ 1 +τ 2 = τ 3. This can be done using four BNC cables of equal length. One cable is used to connect the function generator to the system,

29 2.2. Observed Dynamics 15 Function Generator Τ Unknown Delay Τ Oscilloscope 2Τ Figure 2.2: A schematic illustration of the method used to measure delays in general. one is used to connect the output of the system to the scope, and the other two are used to connect the function generator directly to the scope. With this set up, the lag measured on the scope corresponds to the delay in the feedback loop of the system. In fact, almost any delay of interest can be measured using essentially the same set up. For example, one could measure the delay due to the attenuator alone by measuring the delay due to all of the components in the single loop system and then measuring the delay with the attenuator removed. The schematic structure of a delay measurement is shown in Fig Observed Dynamics With the free parameters set, the dynamics of the system can be observed qualitatively by connecting the system s output to the oscilloscope and varying the dial on the attenuator. For very weak feedback strengths, the system is essentially quiescent; the output consists of high frequency noise. As the feedback strength increases, sinusoidal oscillations suddenly appear, rising up out of the noise. For a while, these oscillations grow smoothly as the feedback strength increases, but before long the oscillations lose their sinusoidal shape and become square waves. If the feedback strength is increased still further, this square wave structure breaks down and a variety of complicated waveforms can be observed. These complex waveforms are fascinating, but they are also difficult to analyze theoretically. For this reason, my thesis concentrates on the dynamics of the system close to the onset of oscillations. The onset of oscillations in the system can be described quantitatively by measuring the amplitude of oscillations in the system as a function of the feedback strength. This can be accomplished by varying the dial on the attenuator in steps and using the spectrum analyzer to measure the amplitude of the tallest peak in the power spectrum at each step. Data obtained using this procedure are shown in Fig. 2.3 along with time series illustrating the different dynamical regimes. The black curve in Fig. 2.3 is a plot of V = A α α c, (2.1) where V stands for the amplitude of oscillations, α c represents the critical value of α,

30 16 Chapter 2. Methods and Results: Single Loop System *+,-./ )! (! '!!"#!"$!"%!"& Α Figure 2.3: This plot illustrates the relationship between the amplitude of oscillations and the feedback strength. The curve shown in the figure is a plot of Eq. 2.1 with A = 69.7 and α c = These parameters were estimated by fitting a line to a plot of V 2 vs. α. The data were gathered with the bias point of the system set to π 4, which required a bias voltage of v = 0.14 V. Since this bias voltage is small and the time required to perform the experiment is short, it is unlikely that drift in the bias point significantly affected these results. The delay was 89.2 ns. and A is an overall prefactor. As the figure illustrates, this functional form provides a good model for the relationship between the feedback strength and the amplitude of oscillations close to onset, but it fails for α much larger than α c. For now, this is merely an unmotivated observation, but later on we will develop a theoretical framework where this functional form arises naturally. To be precise, I should specify that the results plotted in Fig. 2.3 were obtained with the MZ biased at π. Surprisingly, it turns out that the relationship between 4 feedback strength and the amplitude of oscillations is essentially the same when the MZ is biased to π. In spite of this remarkable similarity, the two bias points are 4 distinct dynamically because the frequency of the oscillations is radically different in each case. When the MZ is biased to π, the frequency close to the onset of 4 oscillations is 275 khz. In contrast, when the bias point is π, the frequency is 16 4 MHz. Later on, we will have more to say about the fact that the relationship between feedback strength and the amplitude of oscillations is essentially the same in these two cases even though the frequency is wildly different. For now, I want to conclude our discussion of the single loop system s dynamics with a qualitative argument to account for the sharp onset of oscillation in the system.

31 2.2. Observed Dynamics 17 Consider the single loop system with very weak feedback. In this regime, we have seen that the system does not exhibit large amplitude oscillations, but there are still tiny, noisy oscillations in the system. Since these oscillations do not grow as they propagate around the loop, it must be the case that the round trip gain is less than 1 in this regime 1. Now, suppose that we start to increase the feedback strength. Eventually, it will happen that the round trip gain exceeds 1. When this occurs, the small oscillations in the system will begin to grow rapidly and large oscillations will suddenly appear. This implies that there will be a critical value of the feedback strength at which oscillations emerge, as there is in Fig Unfortunately, this simple argument cannot account for the shape of the waveform or the frequency of the oscillations. In order to gain some insight into those features of the dynamics, we will have to develop a more sophisticated theory. First though, I want to discuss the dynamics of the cross coupled system. 1 I should really say the round trip gain must have absolute value less than 1, but since we re not concerned with the sign of the gain at the moment I don t think it will cause any confusion to let that technical point slide.

32

33 Chapter 3 Methods and Results: Cross Coupled System When we consider the cross coupled system, several new complications arise. There are now two outputs to study, twice the number of free parameters, and, worst of all, we have to worry about how intrinsic differences between the devices used to construct the system will affect its dynamics. Fortunately, we can solve two of these problems at once by using our control parameters to counter the differences between the various devices in the system. In the next section, I will discuss how this can be accomplished, and then we will look at the dynamics of the system. 3.1 Setting the Free Parameters Our primary goal in setting the free parameters of the cross coupled system will be to make the nonlinearities and their linking elements as symmetric as possible. In order to clarify what is meant by symmetry, it is helpful to mentally divide the cross coupled system into two subunits. Let each subunit consist of an MZ and all the components between the output of that MZ and the input of the other MZ. The subunits will be symmetric if the following two conditions hold. First, the delay from one nonlinearity to the other should be the same in both subunits. This can be achieved experimentally by measuring the delays from one MZ to the other using the technique discussed in section 2.1.3, and inserting an optical fiber of the appropriate length into the subunit with the shorter delay. In practice, this requires splicing a fiber to just the right length, which can be a difficult task, but fortunately, the devices used to construct the system are similar enough that the delays are inherently matched within 0.1 ns. Second, a signal output by the MZ in subunit one should affect the MZ in subunit two the same way that an identical signal output by the second MZ would affect the first, unless we have deliberately made the MZs asymmetric with the attenuators. Unfortunately, this condition cannot be enforced exactly because we have no way to counteract differences in the frequency response characteristics of the devices used to construct the system. Nevertheless, it is possible to achieve a close approximation to

34 20 Chapter 3. Methods and Results: Cross Coupled System a 1, b 1, Φ 1 a 2, b 2, Φ 2 MZ 1 MZ 2 g g 1 s 2 s 1 p 2 Α 2 p 1 Α 1 Figure 3.1: A schematic diagram for the cross coupled system with each component labeled by the parameters that characterize its behavior. The arrows on the diagram indicate the path followed by a signal output by MZ 1. this symmetry condition by setting the laser currents appropriately, but it takes some work to figure out how to do it. We will begin by deriving equations that describe how the output of one MZ is related to the output of the other. For simplicity, we will neglect all of the filtering effects except for the AC coupling in the photodetector. To make it easier to follow the derivation, Fig 3.1 shows the parameters used to characterize each device in the cross coupled system. Briefly, each MZ is characterized by three parameters a, b, and φ, each attenuator is characterized by an α, each splitter is characterized by a splitting ratio s, each photodetector is characterized by a gain p, and each amplifier is characterized by a gain g. Of these parameters, we have control over a 1 and a 2 through the laser currents, φ 1 and φ 2 through the bias voltages of the MZ s, and α 1 and α 2 through the dials on the attenuators. Keep in mind though, we want to derive a method to make the two subunits of the system symmetric when α 1 = α 2. Suppose that MZ 1 outputs intensity I 1. As this signal propagates through the circulators and the attenuator, its amplitude will be reduced by a factor α 1. Then, the photodetector will convert the signal to a voltage, amplify it by a factor p 1 and cut off its DC component. In order to account for the AC coupling, let us write I 1 = I 1,DC + I 1,AC so that the effects of the attenuator and the photodetector are described by the transformation I 1 p 1 α 1 I 1,AC. After this transformation, the signal is reduced by a factor of s 1 due the splitter and then amplified by a factor g 1 and fed into the input of MZ 2. As a result, the output MZ 2 will be I 2 = a 2 sin 2 (b 2 g 1 s 1 p 1 α 1 I 1,AC + φ 2 ) (3.1) Similarly, if MZ 2 outputs an intensity I 2 then the output of MZ 1 will be I 1 = a 1 sin 2 (b 1 g 2 s 2 p 2 α 2 I 2,AC + φ 1 ) (3.2) By comparing Eqs. 3.1 and 3.2 and enforcing the requirement that α 1 = α 2, we

35 3.1. Setting the Free Parameters 21 see that the following conditions must hold for the subunits to be symmetric: a 2 = a 1 φ 2 = φ 1 b 1 g 2 s 2 p 2 = b 2 g 1 s 1 p 1. Unfortunately, we cannot enforce the third condition experimentally because it does not contain any of our control parameters. It is possible to work around this difficulty by limiting ourselves to the small signal regime where the sin 2 nonlinearity can be replaced by its linear approximation. In this case, Eq. 3.1 becomes I 2 a 2 sin 2 (φ 2 ) + a 2 b 2 g 1 s 1 p 1 α 1 sin (2φ 2 ) I 1,AC. (3.3) The constant term here is just the DC component of I 2, which will be eliminated by the photodetector. The AC component of the output of MZ 2 is given by I 2,AC = a 2 b 2 g 1 s 1 p 1 α 1 sin (2φ 2 ) I 1,AC. (3.4) Applying the same reasoning to Eq. 3.2, we obtain I 1,AC = a 1 b 1 g 2 s 2 p 2 α 2 sin (2φ 1 ) I 2,AC. (3.5) In this approximation, the subunits of the system will be symmetric when α 1 = α 2 if a 2 b 2 g 1 s 1 p 1 sin (2φ 2 ) = a 1 b 1 g 2 s 2 p 2 sin (2φ 1 ). (3.6) For convenience, I will call the quantity on the right hand side of this equation G 1 and the quantity on the left hand side G 2. In principle, the condition represented by Eq. 3.6 can be enforced experimentally by using the laser currents to adjust a 1 and a 2 appropriately. In order to accomplish this in the lab, we need to quantify the relationship between the laser currents and G 1 and G 2. This can be done using the set up shown in Fig. 3.2a. With this set up, one can measure the gain of a small signal that propagates through the components of the cross coupled system in Fig. 3.2a. But what does this have to do with the relationship between G 1 and the laser current seeding MZ 1? Well, let s calculate the gain that this set up allows us to measure. Suppose that the signal entering the upper branch in Fig. 3.2a is V in. The corresponding output of the photodetector will be V out = p 2 a 1 sin 2 (b 1 g 2 s 2 V in + φ 1 ) p 2 a 1 sin 2 (φ 1 ), (3.7) where the second term is due to the AC coupling of the detector. If the input signal is small, we can replace the sin 2 by its linear approximation to obtain V out = p 2 a 1 b 1 g 2 s 2 sin (2φ 1 ) V in. (3.8) Evidently, the gain of a small signal is approximately G 1.

36 22 Chapter 3. Methods and Results: Cross Coupled System "4 54! 3 4 " 3 4 Φ 3 12(3 $ 5 % 5! !"#$%&'#()*#*+,%'+ # 5 -.$&//'.$'0* 0. -./.- 0/!"#$%&'(%%$)* +, Figure 3.2: (a) The set up used to quantify the relationship between G 1 and the laser current is shown. Each component from the cross coupled system is labeled with the parameter value that characterizes its behavior. These are the same parameter values used in Fig Note that the detector connected to the output of MZ 1 in this measurement is connected to the output of MZ 2 in the actual system. The same set up is used to measure G 2 with all the indices swapped. (b) The results obtained using this set up are shown. With this result in sight, it is possible to quantify the relationship between G 1 and the laser current by varying the current and measuring the gain. Results obtained using this procedure are shown in Fig. 3.2b along with a line of best fit. This linear relationship arises because a 1 is proportional to the intensity seeding MZ 1, which is linearly related to the laser current. Now, we can describe a procedure for setting the laser currents to enforce Eq. 3.6 experimentally. First, the relationship between G 1 and G 2 and the corresponding laser currents must be quantified using the technique described above. Then, the line of best fit for each relationship should be determined. Once this is done, a value can be chosen for G 1 and G 2, and the laser currents corresponding to the chosen value can be estimated using the equations for the lines of best fit. The value chosen for G 1 and G 2 is somewhat arbitrary, but the same tradeoff that arose in setting the laser current in the single loop system also applies here. As a rule of thumb, it is best to choose the smallest value for G 1 and G 2 that allows one to study the dynamical regime of interest. Of course, this procedure only makes the subunits of the system symmetric for oscillations that are small enough for the approximation we used to derive Eq. 3.6 to hold. It is possible to determine what small enough means experimentally by varying the amplitude of the signal sent into the system in the set up used to measure G 1 and G 2. If the input signal is too large, the nonlinear behavior of the MZ will come into play and the shape of the photodetector s output will be distorted. Using this criteria, one can obtain a quantitative bound on the validity of the linear approximation. I have found that with the MZs biased to π, the linear approximation is very 4 good for oscillations with an amplitude up to roughly 70 mv (rms), which is much greater than the amplitude observed experimentally close to the onset of oscillations

37 3.2. Observed Dynamics 23 (see Fig. 3.3). In summary, it is possible to make the subunits of the cross coupled system symmetric by making the delay in each subunit equal and by setting the laser currents to enforce Eq This leaves us free to set the delay in both subunits, the value of G 1 and G 2 and the bias points of both MZs. As in the single loop system, we will hold the delay fixed at whatever value is most convenient experimentally. The rule of thumb for choosing G 1 and G 2 has already been discussed. The bias points of the MZs will be set to either π or π. The round trip gain is positive when the 4 4 bias points for both MZs have the same sign and negative when the bias points have opposite signs. 3.2 Observed Dynamics Just as we did for the single loop system, we will quantify the dynamics of the cross coupled system by measuring the amplitude of oscillations as a function of the coupling strengths. In order to study the influence of both coupling strengths systematically, we will hold α 2 fixed and study the amplitude of oscillations as a function of α 1, and then repeat this process for several values of α 2. It turns out that at a fixed α 2, the dynamics of the cross coupled system look essentially the same as the dynamics of the single loop system. For small α 1, the system is quiescent, but as α 1 increases, sinusoidal oscillations appear at a critical value of α 1. When α 1 increases further, these sinusoids grow into square waves and then develop more complicated wave forms. This pictures describes the dynamics for both positive and negative round trip gain, but, as in the single loop system, the frequency of the oscillations is strongly affected by the sign of the round trip gain. When the round trip gain is positive, the system oscillates at 45 MHz, and, when the round trip gain is negative, it oscillates at 40 khz. Later on, we will try to account for these frequencies and those observed in the single loop system, but for now, we will continue our survey of the cross coupled system s dynamics. An interesting twist enters the story when we start to vary α 2. The dynamics remain qualitatively the same, but the critical value of α 1 changes. This is illustrated in Fig. 3.3, which shows the amplitude of oscillations as a function of α 1 for four different values of α 2. Looking at Fig. 3.3, it is natural to wonder whether it is possible to determine how the onset of oscillations depends on both α 1 and α 2. Following the same line of reasoning that we used to account for the sharp onset of oscillations in the single loop system, we expect the onset of oscillations to occur when the round trip gain is essentially 1. Since the round trip gain is proportional to α 1 α 2, this suggests that oscillations will appear in the system when α 1 α 2 exceeds a critical value.

38 24 Chapter 3. Methods and Results: Cross Coupled System! ' () '!! &! %! $! #!!!"!!"#!"$!"%!"& '"! Α ' Figure 3.3: On the left, the plot shows the amplitude of oscillations as a function of α 1 for four different value of α 2. For the circles, α 2 = 1, for the squares α 2 = 0.82, for the diamonds α 2 = 0.69, and for the triangles α 2 = The amplitude of oscillations was measured by connecting the spectrum analyzer to the output of the system in the same subunit as α 1. On the right, screenshots from the oscilloscope illustrate the qualitative behavior of both outputs of the cross coupled system. Note that independent of whether the oscillations are sinusoidal or square waves, the outputs are just scaled and phase shifted versions of each other. Since the two outputs are always related in this simple way in the regime that we are studying, it is sufficient to measure one of the two outputs. These data were taken with φ 1, φ 2 π with bias 4 voltages v 1 = 0.6 V and v 2 = 0.4 V. The delay from one nonlinearity to the other was 43.9 ns ± 0.1 ns. The laser currents were set so that G 1 G 2 2. This idea can be tested by plotting the amplitude of oscillation as a function of α 1 α 2. When this is done, the four curves in Fig. 3.3 all sprout from the same point 1 on the α 1 α 2 axis, providing convincing evidence that oscillations appear at a critical 1 Actually, if you look closely at Fig. 3.4a, you can see that there is some horizontal spread in the four curves. I attribute this spread to drift in the MZ bias point. Since it takes several hours to gather all of the data, it is certainly possible that there is significant drift in the bias point even though the bias voltages are relatively small. In order to investigate the effect of the drift, I measured the amplitude of oscillations as a function of α 1 and then left the system on for about 3 hours. After this time elapsed, I measured the amplitude as a function of α 1 again. I found that leaving the bias voltages on for several hours noticeably shifted the critical value of α 1. This effect of drift in the bias point shifting the critical value of α was also seen numerically by Lauren Shareshian.