What Do the Symbols in Nonlinear Optics Equations Represent?

[n0_3sc]

Homework Statement

a) What does the tensor nature of susceptibility mean? Ie. the [tex] xxxx [/tex] in:
[tex] \overline{P}=\epsilon_0\chi_{xxxx}\overline{E} [/tex]

b) What does the 3 vertical dots mean in: [tex] \chi^{(2)}\vdots\overline{E} [/tex]?

The Attempt at a Solution

a) I understand that [tex] \overline{P}=\epsilon_0\chi_{xx}\overline{E} [/tex] means the first x corresponds to the component of P and the second x relates to the polarization of the Electric Field. But what do the other x's mean?

b) No clue...

 

[Jhero]

a) The tensor nature of susceptibility refers to the fact that the susceptibility, represented by the symbol \chi, is a tensor quantity. This means that it has multiple components, each of which represents the susceptibility in a different direction. In the equation \overline{P}=\epsilon_0\chi_{xxxx}\overline{E}, the four x's represent the four different components of the susceptibility tensor. So, for example, \chi_{xyxy} would represent the susceptibility in the x-y plane.

b) The three vertical dots, represented by the symbol \vdots, are used to indicate the dot product between the tensor \chi^{(2)} and the vector \overline{E}. The dot product is a mathematical operation that results in a scalar value, and in this context, it represents the interaction between the second-order susceptibility tensor and the electric field vector.