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Unformatted text preview: Lecture 8 The Wave Equation for Nonlinear Optical Media Laser field induces polarization at new frequency components. Generation of new components are described by Maxwell’s equations. dipole radiation pattern 1 ϖ 2 ϖ 1 2 ϖ ϖ + 1 2 ϖ ϖ + 3 1 2 ϖ ϖ ϖ = + 1 N atoms: if the relative phasing of the dipoles is correct, the field radiation by each dipole will add constructively in the forward direction, leading to radiation in the form of a welldefined beam. Acts as a phased array of dipoles if the phasematching condition is satisfied. Forward intensity is N 2 larger than single atom. 1 2 ϖ ϖ + Consider a secondorder NL process such as sumfrequency generation (SFG). Maxwell’s Equations (omit the arrow sign, for simplicity) (1) (2) (3) (4) D B B E t D H J t ρ ∇⋅ = ∇⋅ = ∂ ∇× =  ∂ ∂ ∇× = + ∂ ɶ ɶ ɶ ɶ ɶ ɶ ɶ = 2 with no free charges and no free currents the material is nonmagnetic (5) Polarization vector ( ) nonlinearly depends on (6) substituting (4), then (5) (7) J ρ = = ɶ B H μ = ɶ ɶ r D E P E ε ε ε = + = ɶ ɶ ɶ ɶ P ɶ E ɶ ( 29 or B E E B t t ∂ ∂ ∇×∇× = ∇×  ∇×∇× =  ∇× ∂ ∂ ɶ ɶ ɶ ɶ 2 2 2 2 2 2 2 2 2 2 2 1 1 1 E D D E P t c t c t c t μ ε ε ∂ ∂ ∂ ∂ ∇×∇× =  =  =  ∂ ∂ ∂ ∂ ɶ ɶ ɶ ɶ ɶ remember that: but since so so (7) becomes (8) If we define where all nonlinear contributions are in 2 2 2 2 2 2 2 1 1 E E P c t c t ε ∂ ∂ ∇ =...
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 Fall '10
 Shalaev
 Frequency, wave equation, Laplace operator, Helmholtz equation

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