Modeling and Results for a Mach-Zehnder Chaotic System Karl Schmitt Jim Yorke, Rajarshi Roy, and Tom Murphy,Adam Cohen, Bhargava Ravoori, University of Maryland April 3, 9 / 4
Motivation Why bother? Reduce Experimentation time Comfirm Theories Direct Research Isolate errors Find new results / 4
System Overview Motivation Laser Diode MZM Bias Photoreceiver Amplifier Delay (τ) V(t) Low-pass High-pass x(t) + τẋ(t) + t θ t o x(s)ds = βcos [x(t T ) + φ] x(t) is normalized RF voltage τ is the low pass filter time constant θ is the high pass filter time constant T is the time delay in the loop β is the feedback strength φ is the phase offset in nonlinearity [Kouomou, Thesis] 3 / 4
Motivation Modeling the system β =.5 β =. β =.45 Traditional numeric methods can be applied Generates a variety of dynamics β = 3.5 β = 4.3 What can improve? Speed Versatility of Model This Gets us: Ability to explore wider parameter choices New avenues of research New results 3 4 t (ns) [Cohen et. al., PRL] 4 / 4
Details of Improvements Model Changes Improving Speed Change Integration Discrete Map Incorporate matrix math vs. multi-steps Improving Versatility Allow more filter types/orders Explicit variable differation Simple expansion to multiple systems 5 / 4
State-Space Model Comparing Differentials vs. State-Space Differential Form x(t) + τẋ(t) + t θ t o x(s)ds = y(t) Laser Diode MZM Bias Photoreceiver Amplifier Delay (τ) V(t) Low-pass High-pass Discrete State-Space Form u [n + ] = Au [n] + B cos (Cu [n k] + φ) 6 / 4
State-Space Model Equations for Coupled Mach-Zehnder Loops (one Mach-Zehnder loop) u [n + ] = Au [n] + B cos (Cu [n k] + φ) (coupled Mach-Zehnder loops) u [n + ] = Au [n] + ( γ ) B cos (Cu [n k ] + φ ) +γ B cos (Cu [n k ] + φ ) u [n + ] = Au [n] + ( γ ) B cos (Cu [n k ] + φ ) +γ B cos (Cu [n k ] + φ ) 7 / 4
Milestones State-Space Model Implementation and Verification of individual simulations Implementation and Verification of final simulation Generation of new results Further Expansion and Use of Code November st (complete) December st (complete) Feburary st (complete) Ongoing 8 / 4
Milestones State-Space Model Implementation and Verification of individual simulations Implementation and Verification of final simulation Generation of new results Further Expansion and Use of Code November st (complete) December st (complete) Feburary st (complete) Ongoing 9 / 4
Milestones State-Space Model Implementation and Verification of individual simulations Implementation and Verification of final simulation Generation of new results Further Expansion and Use of Code November st (complete) December st (complete) Feburary st (complete) Ongoing / 4
Milestones State-Space Model Implementation and Verification of individual simulations Implementation and Verification of final simulation Generation of new results Further Expansion and Use of Code November st (complete) December st (complete) Feburary st (complete) Ongoing / 4
Validation Plan Validation Stage : Single Loop Characteristic Curves from Kouomuo[] Bifurcation Points from Kouomuo Stage : Coupled Lorenz Conditions by Anishchenko [4] Stage 3: Coupled Mach-Zehnders Open Loop: Argysis [5] Symmetric 5/5 coupling: Piel [6] Success Partial Success Success Success Success / 4
Validation Comparision of Analytic to Simulated Results x 4 Frequency Bifurcation for Positive Beta.5.5 Success Success Frequency.5 3 Failure Failure 3.5 4 3 4 4.5 5....3.4.5.6.7.8.9 Beta Kouomou predicts the following bifurcations: Hopf Bifurcation Points β k = ( ) k+ [ + (ɛr k π ) k π R ] ω k = k π R 3 / 4.
Validation Comparision of Experimental to Simulated Time Series Experiment Simulation y 4 4 y β=.4 β=. β=.8 4 4 5 Time(ms) 4 4 4 4 5 Time(ms) 4 4 4 4 5 Time(ms) 4 / 4
Validation Comparision of Experimental to Simulated Bifurcation β 3 4 5 3 4 5 Experimental 3 Normalized Voltage 3 4 3 Simulation 3 Color indicates frequency of value 5 / 4
Validation Synchronization of Coupled Systems.5 Curves = Simulation Normalized RMS Difference.5 β = 6 Dots = Experimental β = 8 5/5 Coupling [Piel]...3.4.5.6.7.8.9 Coupling Strength 6 / 4
New Results What We Want Looking for: Affects of measurement error Exploration of new variables Analysis of larger parameter space 7 / 4
New Results Equations for Coupled Mach-Zehnder Loops (one Mach-Zehnder loop) u [n + ] = Au [n] + B cos (Cu [n k] + φ) (coupled Mach-Zehnder loops) u [n + ] = Au [n] + ( γ ) B cos (Cu [n k ] + φ ) +γ B cos (Cu [n k ] + φ ) u [n + ] = Au [n] + ( γ ) B cos (Cu [n k ] + φ ) +γ B cos (Cu [n k ] + φ ) 8 / 4
Mismatch New Results.5 β Mismatch Synchronization.5 β = -5% β = -% β = - 5% β = %...3.4.5.6.7.8.9 γ β Mismatch, Pos. vs. Neg..5 Synchrnoization.5 β = -5% β = 5% β = %...3.4.5.6.7.8.9 γ 9 / 4
Matching the Experiment New Results.5 Simulation vs. Experiment β = 6 β = 6 β = 6% φ = % k =.5 Normalized RMS Difference.5...3.4.5.6.7.8.9 γ / 4
Voltage Bifurcations in Delay (k) New Results Delay 3 4 5 6 7 8 9 3 β= - - -3 Normalize Voltage 3 - - -3 β=3 3 - - -3 β=4 3 4 5 6 7 8 9 Delay / 4
New Results γ vs γ γ.9.8.7.6.5.4.3.. Synchronization Error.5.5..75.5.5...3.4.5.6.7.8.9 γ / 4
Acknowledgements New Results Dr. Zimin and Dr. Yorke for feedback and improvements Dr. Roy and Dr. Murphy for initial problem Adam Cohen and Bhargava Ravoori for experimental results DoD for the grant to do this research 3 / 4
References New Results Kouomou, Yanne. Nonlinear Dynamics of Semiconductor Laser systems with feedback Doctoral Thesis Cohen, Adam, et. al. Using synchronization for prediction of high-dimensional chaotic dynamics Phys. Rev. Let., Publication Pending Murphy, Thomas. Personal Communications Anischchenko, V. S. et. al, Mutual synchronization and desynchronization of Lorenz systems Tech. Phys. Lett., Vol. 4, Nmb 4, Apr 998, Amer. Inst. Phys. Argysis, Apostolos, et. al. Chaos Based Communications at high bit-rates using commercial fibre-optics Nature, 5 Nature Publishing Piel, Michael, et. al, Versatile and robust chaos synchronization phenomena imposed by delayed shared feedback coupling Phys. Rev. E Vol 76, 7 Amer. Phys. Soc. 4 / 4