# Hyperpolarizability tensor

1. Aug 20, 2010

### 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)

2. Aug 20, 2010

### 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

3. Aug 21, 2010

### DanSandberg

Yes that makes ALOT 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?