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“Finding” a Pulse Shape

“Finding” a Pulse Shape. Jason Taylor May 16, 2005. “The laser cavity finds the pulse that minimizes loss--it's like magic.”. My Research. I investigate the usefulness of laser cavities and laser dynamics for information processing. Kerr Lens Mode-Locked laser cavities. Project Goals.

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“Finding” a Pulse Shape

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  1. “Finding” a Pulse Shape Jason Taylor May 16, 2005 “The laser cavity finds the pulse that minimizes loss--it's like magic.”

  2. My Research • I investigate the usefulness of laser cavities and laser dynamics for information processing. • Kerr Lens Mode-Locked laser cavities

  3. Project Goals • Discover what a KLM laser minimizes • Cavity loss? • Population Inversion? • Use results to predict good bit representation and/or logical operators • Space • Time • Phase • Power

  4. Kerr-Lens Mode Locked Oscillator

  5. The Equation SPM GVD This equation describes the evolution of a short pulse over one round trip in a KLM cavity.

  6. Mode Locking

  7. Artificial Fast Saturable Absorbers fast saturable absorber [Hau00]

  8. fast saturable absorber master equation: solution unbounded Siegman, Haus [Hau00]

  9. Kerr-Lens Mode Locked Laser Ti:Sapphire KLM oscillators are available commercially—where else would this graphic come from?

  10. Soliton Effects in Ultrashort Pulses New master equation with GVD and SPM: SPM GVD Group Velocity Dispersion (GVD) Self Phase Modulation (SPM) pulse width d0 = no SPM Dn = GVD

  11. Haus’ Master Equation master equation with GVD and SPM: SPM GVD Gain depletion where

  12. Numerical Simulation Here we look at the equation properties via numerical simulation.

  13. Separate into Linear and Non-Linear Operators where

  14. Simulation

  15. Gain, SAM, no GVD, no SPM pulse width frequency

  16. Gain, SAM, GVD, and SPM pulse width frequency

  17. What is minimized? • Cavity loss? • Gain medium population inversion? • Pulse width?

  18. Haus’ Master Equation master equation with GVD and SPM: SPM GVD Gain depletion where

  19. Complex Ginzburg-Landau Equation master equation with GVD and SPM: SPM GVD Gain depletion where

  20. Complex Ginzburg-Landau Equation master equation: soliton like pulse CW solution general CGLE

  21. Complex Ginzburg-Landau Equation general CGLE

  22. Lyapunov Function soliton like pulse CW solution A good approximate Lyapunov function is known for a CW stationary solution. No Lyapunov function is known for soliton solutions—too close to chaos. IMHO

  23. Simplify Equation ignore gain depletion, BW filtering and SAM

  24. Non-linear Schrodinger Equation Lyapunov Function: Looks a lot like minimizing the action.

  25. Numerical Confirmation Try Lyapunov function in simulator.

  26. NLSE Lyapunov function on CGLE with gain saturation

  27. Gain, SAM, GVD, and SPM pulse width frequency

  28. Zoom in on Hump

  29. Zoom in on 2nd Hump

  30. Zoom in on 3rd Hump

  31. Zoom in on 4th Hump

  32. Conclusions • We found a fractal. • The NLSE approximate Lyapunov function isn’t valid far away from the soliton solution. • Is a numerical stability analysis of the CGLE sufficient?

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