“Finding” a Pulse Shape

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Presentation transcript:

“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 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

Kerr-Lens Mode Locked Oscillator

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

Mode Locking

Artificial Fast Saturable Absorbers [Hau00]

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

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

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

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

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

Separate into Linear and Non-Linear Operators where

Simulation

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

Gain, SAM, GVD, and SPM pulse width frequency

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

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

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

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

Complex Ginzburg-Landau Equation general CGLE

Lyapunov Function A good approximate Lyapunov function is known for a 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

Simplify Equation ignore gain depletion, BW filtering and SAM

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

Numerical Confirmation Try Lyapunov function in simulator.

NLSE Lyapunov function on CGLE with gain saturation

Gain, SAM, GVD, and SPM pulse width frequency

Zoom in on Hump

Zoom in on 2nd Hump

Zoom in on 3rd Hump

Zoom in on 4th Hump

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?