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Rahul Alex Panicker Department of Electrical Engineering Stanford Univeristy

Compensation of Modal Dispersion in Multimode Fiber Systems using Adaptive Optics via Convex Optimization. Rahul Alex Panicker Department of Electrical Engineering Stanford Univeristy. Playing an Electromagnetic F Major on an Optical Fiber. Multimode Fiber Systems – what, why?.

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Rahul Alex Panicker Department of Electrical Engineering Stanford Univeristy

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  1. Compensation of Modal Dispersion in Multimode Fiber Systems using Adaptive Opticsvia Convex Optimization Rahul Alex Panicker Department of Electrical Engineering Stanford Univeristy

  2. Playing an Electromagnetic F Major on an Optical Fiber

  3. Multimode Fiber Systems – what, why? • Ubiquitous short range communication medium

  4. Multimode Fiber Systems – what, why? • Ubiquitous short range communication medium • Local area networks, campus area networks

  5. Multimode Fiber Systems – what, why? • Ubiquitous short range communication medium • Local area networks, campus area networks • Lots of installed fiber – can we make the best use of this? (remember telephone lines and DSL?)

  6. Multimode Fiber Systems – what, why? Ethernet Roadmap

  7. Multimode Fiber Systems – what, why? Performance – bit rate (Mbps/Gbps) and transmission distance (km) – currently limited by modal dispersion.

  8. Adaptive transmission scheme involving optical dispersion compensation. Contributions of this Thesis • Created novel adaptive algorithms for real-time implementation. • New comprehensive mathematical formulation. • Performance maximization posed as an optimization problem. • Globally optimal solution computed. • Experimental demonstration of 10 Gbps and 100 Gbps transmissions.

  9. Outline • Multimode fiber essentials • Adaptive transmission scheme • Optimization problem • Adaptive algorithms • Experimental results

  10. Outline • Multimode fiber essentials • Adaptive transmission scheme • Optimization problem • Adaptive algorithms • Experimental results

  11. Modes of a Multimode Fiber • Many natural phenomena involve modes: • Solar oscillations • Swaying buildings • Vibrating strings • Ripples in a pond • Molecular vibrations • Light in an optical fiber

  12. Modes of a Multimode Fiber And God said… …and then there was light.

  13. Modes of a Multimode Fiber Ideal Modes • Mutually orthogonal solutions of wave equation having well-defined propagation constants. • Propagate without cross-coupling in ideal fiber. • Typical multimode fiber supports of order 100 modes.

  14. Modes of a Multimode Fiber Mode Coupling • Bends and imperfections couple modes over distances of the order of meters. • Coupling varies on time scale of seconds.

  15. t Transmitted Received t Modes of a Multimode Fiber Modal Dispersion • Different modes have different delays. • Single pulse in – many pulses out.

  16. Modes of a Multimode Fiber Principal modes • PMs are linear combinations of ideal modes. • Form a basis over all propagating modes. • Vary from fiber to fiber. • Single pulse in – single pulse out (well defined group delay). S. Fan and J. M. Kahn, Optics Letters, vol. 30, no. 2, pp. 135-137, January 15, 2005.

  17. Good eye Bad eye Poor eye Eye Diagram Indicates how discernable 1-bits and 0-bits are.

  18. Outline • Multimode fiber essentials • Adaptive transmission scheme • Optimization problem • Adaptive algorithms • Experimental results

  19. Fourier Lens Iout(t) Photo-Detector AdaptiveAlgorithm Clock & DataRecovery Rec.Data Multimode Fiber Spatial LightModulator Iin(t) ISIEstimation Trans.Data OOKModulator ISI ObjectiveFunction Transmitter Receiver Low-Rate Feedback Channel +0.1 Adaptive Transmission Scheme E. Alon, V. Stojanovic, J. M. Kahn, S. P. Boyd and M. A. Horowitz, Proc. of IEEE Global Telecommun. Conf., Dallas, TX, Nov. 29-Dec. 3, 2004.

  20. ky y kx x MMF SLM Spatial Light Modulator • 2-D array of mirrors. • Reflectance each mirror (vi) can be controlled.

  21. Fourier Lens Iout(t) Photo-Detector AdaptiveAlgorithm Clock & DataRecovery Rec.Data Multimode Fiber Spatial LightModulator Iin(t) ISIEstimation Trans.Data OOKModulator ISI ObjectiveFunction Transmitter Receiver Low-Rate Feedback Channel +0.8 +0.4 Adaptive Transmission Scheme +0.1 -0.3

  22. Outline • Multimode fiber essentials • Adaptive transmission scheme • Optimization problem • Adaptive algorithms • Experimental results

  23. Optimization Problem maximize eye opening subject to physical constraints

  24. Optimization Problem The impulse response is given by The pulse response is, therefore, given by and the eye opening is given by

  25. Optimization Problem • Not in any standard form. • For example, not convex. R. A. Panicker, S. P. Boyd, and J. M. Kahn, subm. Journal of Lightwave Technology

  26. Optimization Problem • Convex! (Second order cone program)

  27. Optimization Problem • Can compute globally optimal solution. • Efficient algorithms exist. • Roughly same complexity as solving a linear program of same size.

  28. Simulation Results

  29. Outline • Multimode fiber essentials • Adaptive transmission scheme • Optimization problem • Adaptive algorithms • Experimental results

  30. Adaptive Algorithms • Optimal solution – fine when everything is known about the system. • In practice, we don’t know system parameters. • System can be time varying. • Need adaptive algorithms. • Can compute optimum without explicitly estimating system parameters.

  31. Adaptive Algorithms: Noiseless Amplitude-and-Phase SCA (APSCA): • Pick the ith SLM block • Optimize amplitude and phase of vi • i ← i+1 • Repeat

  32. Adaptive Algorithms: Noiseless • Quadratic in each block reflectance. • 4 real parameters to be estimated in a, b, and c. • Can be done with 4 measurements. • Objective function converges to global maximum. (on convergence, satisfies KKT conditions of convex problem)

  33. Adaptive Algorithms: Noiseless Continuous Phase SCA (CPSCA): • Pick the ith SLM block • Optimize phase of vi • i ← i+1 • Repeat

  34. Adaptive Algorithms: Noiseless • Linear in each block reflectance. • 3 real parameters to be estimated in b and d. • Can be done with 3 measurements. • Guaranteed to converge, but not to global optimum.

  35. Simulations Amplitude-and-Phase SCA: Opens a previously closed eye.

  36. Simulations

  37. Simulations Amplitude-and-Phase SCA, Continuous-Phase SCA, and 4-Phase SCA 0.6 Global maximum 0.4 0.2 0 1 pass over SLM Normalized objective function CPSCA -0.2 1 pass over SLM APSCA, 4PSCA APSCA CPSCA -0.4 4PSCA -0.6 -0.8 0 100 200 300 400 500 600 700 Number of SLM block flips

  38. Adaptive Algorithms: Noisy Amplitude-and-Phase SCA (APSCA): • Pick the ith SLM block • Estimate a, b, c. • Optimize amplitude and phase of vi • i ← i+1 • Repeat

  39. Adaptive Algorithms: Noisy Estimation done with p+q measurements, p ≥ 3, q ≥ 1. If noise has variance σ2, var(a) = σ2(1/p+1/q), var(Re(b)) = var(Im(b)) = σ2/p. In presence of Gaussian noise, these are ML estimates.

  40. Adaptive Algorithms: Noisy Continuous Phase SCA (CPSCA): • Pick the ith SLM block • Estimate b and d. • Optimize phase of vi • i ← i+1 • Repeat

  41. Adaptive Algorithms: Noisy Estimation done with p measurements, p ≥ 3. If noise has variance σ2, var(Re(b)) = var(Im(b)) = σ2/p. In presence of Gaussian noise, these are ML estimates.

  42. Simulations Convergence Plots: Amplitude-and-Phase SCA, Continuous-Phase SCA, and 4-Phase SCA Global maximum 0.4 0.2 0 Objective function -0.2 1 pass over SLM APSCA without noise -0.4 APSCA with noise CPSCA with noise 4PSCA with noise -0.6 -0.8 0 100 200 300 400 500 Number of SLM block flips

  43. Adaptation Time • Presently, 3–4 minutes in lab setup. • Objective function estimation time can be reduced to 25 ms • SLM switching time can be reduced to 100 ms • Overall adaptation (60 blocks, 4 phases) would require 30 ms

  44. Comparison with Electrical Equalization Electrical Equalization • Optimal equalizer is MLSD – complexity exponential in bit-rate and length. • Linear equalizers have noise enhancement. • DFE has error propagation at low SNR. • EE needs to be done per channel in WDM systems. • Steady power consumption

  45. Comparison with Electrical Equalization Optical Equalization • Complexity independent of bit-rate and length – only depends on mode structure. • No noise enhancement. • Can compensate over multiple channels in WDM systems. • After adaptation, no steady power consumption.

  46. Outline • Multimode fiber essentials • Adaptive transmission scheme • Optimization problem • Adaptive algorithms • Experimental results

  47. Transmission Scheme X. Shen, J. M. Kahn and M. A. Horowitz, Optics Letters, vol. 30, no. 22, pp. 2985-2987, Nov. 15, 2005.

  48. Transmission Scheme

  49. g(t) g(nT;t0) t - t0 0 T 2T 3T 4T 5T 6T Receivey(t) = Iout(t) * r(t) TransmitIin(t) yL-1 ymax yL y0 y-1 ymin t t t0-LT t0-T t0-(L+1)T t0 0 LT 2LT Estimation of the Objective Function Eye closed: transmit periodic square wave

  50. g(t) g(nT;t0) t - t0 0 T 2T 3T 4T 5T 6T Receivey(t) = Iout(t) * r(t) TransmitIin(t) y1 y0 t (mod T) t ( ( ) ) ˆ = - F g nT ; t y y 0 1 0 Estimation of the Objective Function Eye open: transmit data sequence

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