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where P is the polarization, E is the applied electric field, a is the linear or first-order polarizability term or susceptibility of the material, d is the first nonlinear polarizability term or the second-order susceptibility (quadratic polarizability), c is the third-order susceptibility, and so on. The subscripts i, ,j, k, l indicate the direction (x, y, z) of the nth component of the polarization or electric field. The susceptibility terms sum over these directions, i.e., aij is a 3 x 3 matrix with 9 terms, dijk is a 3 x 3 x 3 matrix with 27 terms, and cijkl is a 3 x 3 x 3 x 3 matrix with 81 terms).
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