Theory for higher order nonlinear dielectric microscopy

1 Theory for higher order nonlinear dielectric microscopy 

Equation (16.2) is a polynomial expansion of the electric displacement 3 as a function of electric field Es. 

and 33333 correspond to linear and nonlinear dielectric constants and are tensors of rank 2nd, 3rd, 4th and 5th, respectively. Even-ranked tensors including linear dielectric constant 33 do not change with polarization inversion, whereas the sign of the odd-ranked tensors reverses. Therefore, information regarding polarization can be elucidated by measuring odd-ranked nonlinear dielectric constants such as 333 and 33333. 

Considering the effect up to EA, the ratio of the alternating variation of capacitance ACs underneath the tip to the static capacitance Cso is given by 

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