What Does the Third-Rank Tensor Represent in Nonlinear Optics?

[DanSandberg]

In nonlinear optics the polarization of a molecule can be represented as a power series:

P(t)=X₁E(t)+X₂E²(t)+X₃E³(t)+...

Where X₁E(t) is the linear response and everything after is nonlinear. The polarization, P(t) and field strength E(t) are both vectors and X₂ is a third-rank tensor, X₃ is a fourth-rank, etc.

My question is... why is X₂ a third-rank tensor? I'm having difficulty getting a straight answer as to what the indices represent. So in a paper where they say $$\beta$$_xxx what does that mean? (Where $$\beta$$ is commonly used in place of X₂)

 

[xGAME-OVERx]

My supervisor told me this:

Polarisability $$\alpha_{ij}$$, which is X₁ in the power series, is the amplitude of the electric field induced in the molecule in the i direction given a unit amplitude field in the j direction, hence

$$E_i^{induced} = \alpha_{ij} E_j^{incident}$$
(observing summation convention)

Extending this physical interpretation, $$\beta_{ijk}$$ (X₂) is the amplitude of the electric field induced in the i direction given an unit incident field in the j direction applied after unit incident field in the k direction has already been applied, so

$$E_i^{induced} = \beta_{ijk} E_j^{incident2} E_k^{indicent1}$$

Did I explain that in a way that makes sense?

Thanks
Scott

 

[DanSandberg]

My supervisor told me this:

Polarisability $$\alpha_{ij}$$, which is X₁ in the power series, is the amplitude of the electric field induced in the molecule in the i direction given a unit amplitude field in the j direction, hence

$$E_i^{induced} = \alpha_{ij} E_j^{incident}$$
(observing summation convention)

Extending this physical interpretation, $$\beta_{ijk}$$ (X₂) is the amplitude of the electric field induced in the i direction given an unit incident field in the j direction applied after unit incident field in the k direction has already been applied, so

$$E_i^{induced} = \beta_{ijk} E_j^{incident2} E_k^{indicent1}$$

Did I explain that in a way that makes sense?

Thanks
Scott

Yes that makes a lot of sense and is a great start in the right direction. For the linear response, I understand it completely. For for the second order response, there is only one applied (incident) field in the experiment. So when we talk about the direction of the applied field we are talking about the polarization of the incident light?