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Unformatted text preview: Lecture 4 Nonlinear Susceptibility of an Anharmonic Oscillator Approach: Lorentz (harmonic) oscillator: A good approximation of the linear response of an atom to an incident electric field. Add additional nonlinear restoring force term => Differential equation for electron displacement (not directly solvable) 1 Employ perturbation analysis to obtain approximate solution. Lorentz Harmonic Oscillator (1) where is the applied field add nonlinear term (2) damping force is directed against restoring force is ( 29 2 2 ( ) x x x E t e m γ ϖ + + =  ɺɺ ɺ ɶ ɶ ɶ ɶ ( ) E t ɶ ( 29 2 2 2 ( ) x x x ax E t e m γ ϖ + + + =  ɺɺ ɺ ɶ ɶ ɶ ɶ ɶ 2 m x γ ɺ ɶ x ɺ ɶ 2 2 restoring F m x max ϖ =  + ɶ ɶ Potential energy is integral of restoring force over displacement (3) Note potential is not symmetric in => implies noncentrosymmetric medium (for centrosymmetric media, must hold, so only even powers of appear.) Symmetric potential of Lorentz is a parabola. Nonlinear potential is asymmetric. ssume incident field 2 2 3 1 1 2 3 r U F dx m x max ϖ =  = + ∫ ɶ ɶ ɶ ɶ x ɶ x ɶ x ɶ ˆ ˆ ( ) ( ) U x U x = 2 Assume incident field and solve for displacement of (2) for For a small applied field, then and (2) can be solved using a perturbation expansion....
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This note was uploaded on 01/10/2011 for the course ECE 615 taught by Professor Shalaev during the Fall '10 term at Purdue.
 Fall '10
 Shalaev

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