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ECE615_Lecture06

ECE615_Lecture06 - Lecture 6 Symmetry Properties of the...

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Lecture 6 Symmetry Properties of the Nonlinear Susceptibility 1 Consider mutual interaction of three waves: (1) Then need to determine six tensors. and 6 additional terms where each frequency is negative. Indices i,j,k can independently take on values x,y,z. Each of 12 tensors consist of 27 Cartesian components. Thus, 324 complex numbers to fully describe interaction. Restrictions and simplifications relate components of => Need far fewer in practice. (2) 0 ( ) ( ) ( , , ) ( ) ( ) i n m ijk m n n m j n k m jk nm P E E ϖ ϖ ε χ ϖ ϖ ϖ ϖ ϖ ϖ + = + ∑∑ 1 2 3 1 2 , , ϖ ϖ ϖ ϖ ϖ = + 1 3 2 2 3 1 ( ; ) ϖ ϖ ϖ ϖ ϖ ϖ = - = - (2) (2) (2) 1 3 2 1 2 3 2 3 1 (2) (2) (2) 2 1 3 3 1 2 3 2 1 ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ijk ijk ijk ijk ijk ijk χ ϖ ϖ ϖ χ ϖ ϖ ϖ χ ϖ ϖ ϖ χ ϖ ϖ ϖ χ ϖ ϖ ϖ χ ϖ ϖ ϖ - - - - (2) χ

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1. Reality of the Fields Nonlinear polarization of sum-frequency response for input is purely real (2) also (3) From (2) and (3) it follows that: (4) , m n ϖ ϖ ( , ) ( ) exp[ ( ) ] ( ) exp[ ( ) ] i i n m n m i n m n m P r t P i t P i t ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ = + - + + - - + ɶ ɶ ɶ ɶ ( , ) i P r t ɶ ɶ * ( ) ( ) i n m i n m P P ϖ ϖ ϖ
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