9. Third-order nonlinear polarization

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Total Points: –/100 pts

### UNCOMMENT AND RUN THIS CELL IF USING GOOGLE COLAB
# !pip install ipympl -q
# from google.colab import output
# output.enable_custom_widget_manager()

Now that we know how to handle the the nonlinear suceptibility for different polarizations of light and different crystal symmetries, we can explore third-order nonlinear processes. For this notebook, we will concentrate on third-order processes in isotropic media.

Using the same notation we introduced for second-order processes, we can write the polarization \(P_i\) resulting from a third-order susceptibility as

\[ \begin{align} P_i(\omega_4) = \epsilon_0\sum_p\sum_{jkl}\chi^{(3)}_{ijkl}(\omega_4; \omega_1, \omega_2, \omega_3)E_j(\omega_1)E_k(\omega_2)E_l(\omega_3) \end{align} \]

Some examples of 3rd-order processes are

Third Harmonic Generation

\[ \begin{align} P_i(3\omega) = \epsilon_0\sum_{jkl}\chi^{(3)}_{ijkl}(3\omega; \omega, \omega, \omega)E_j(\omega)E_k(\omega)E_l(\omega) \end{align} \]

The DC Kerr Effect

\[ \begin{align} P_i(\omega) = 3\epsilon_0\sum_{jkl}\chi^{(3)}_{ijkl}(\omega; 0, 0, \omega)E_j(0)E_k(0)E_l(\omega) \end{align} \]

The Optical Kerr Effect

\[ \begin{align} P_i(\omega_1) = 3\epsilon_0\sum_{jkl}\chi^{(3)}_{ijkl}(\omega_1; \omega_2, -\omega_2, \omega_1)E_j(\omega_2)E^*_k(\omega_2)E_l(\omega_1) \end{align} \]

Intensity-Dependent Refractive Index

\[ \begin{align} P_i(\omega) = 3\epsilon_0\sum_{jkl}\chi^{(3)}_{ijkl}(\omega; \omega, -\omega, \omega)E_j(\omega)E^*_k(\omega)E_l(\omega) \end{align} \]

All these effects are the same as the ones we studied in notebook 3, and have the same corresponding energy level diagrams. The only difference is that we allow electric fields with different polarizations to interact and produce material polarization in different cartesion directions. Another important consequence is that we may now also consider waves traveling in different directions. For example, in a process known as degenerage four-wave mixing, four waves with the same frequency, but traveling in different directions can interact.

9.1. Exercises

1. (20 pts) Use python’s symbolic algebra to find the degeneracy factors for all the third-order processes listed above.

9.2. Symmetry in third-order processes

Just like in second-order processes, we can simply the number of terms we must use in the susceptibility tensor by appealing to symmetry. It’s a little more involved since there are 81 elements in the general third-order susceptibility tensor. However, if we limit ourselves to isotropic media, all cartesian directions are the same, meaning for example that

\[\begin{split} \begin{align} \chi_1 = \chi_{iiii} \\ \chi_{xxxx} = \chi_{yyyy} = \chi_{zzzz} \end{align} \end{split}\]

We thus give a label of \(\chi_1\) to any term \(\chi_{iiii}\). There are three more types, with six members each:

Type 2

\[\begin{split} \begin{align} \chi_2 = \chi_{jjkk} \\ \chi_{xxyy} = \chi_{yyzz} = \chi_{zzxx} = \chi_{yyxx} = \chi_{zzyy} = \chi_{xxzz} \end{align} \end{split}\]

Type 3

\[\begin{split} \begin{align} \chi_2 = \chi_{jkjk} \\ \chi_{xyxy} = \chi_{yzyz} = \chi_{zxzx} = \chi_{yxyx} = \chi_{zyzy} = \chi_{xzxz} \end{align} \end{split}\]

Type 4

\[\begin{split} \begin{align} \chi_2 = \chi_{jkkj} \\ \chi_{xyyx} = \chi_{yzzy} = \chi_{zxxz} = \chi_{yxxy} = \chi_{zyyz} = \chi_{xzzx} \end{align} \end{split}\]

Adding all these up we have a total of \(3 + 3\times6 = 21\) terms, and it turns out that all other terms must be zero!. Furthermore, in structurally isotropic media (not to be confused with optically isotropic!), nonlinear polarization must be the same in any coordinate system. With a little bit of linear algebra, one can show that this means that

\[ \begin{align} \chi_1 = \chi_2 + \chi_3 + \chi_4 \end{align} \]

The upshot of this realization is that there are actually only 3 independent terms in the third-order nonlinear susceptibility! That will be nice for doing calculations. For other optically isotropic media (such as cubic crystals) we can also reduce the number of independent terms down from 21, but there will be more. For example, cubic crystals have 4 independent terms.

9.3. Third harmonic generation

Let’s calculate the third-order nonlinear polarization in a structurally isotropic medium using our new tensor skills.

\[ \begin{align} P_i(3\omega) = \epsilon_0\sum_{jkl}\chi^{(3)}_{ijkl}(3\omega; \omega, \omega, \omega)E_j(\omega)E_k(\omega)E_l(\omega) \end{align} \]

Since all the input fields have the same frequency, it must be the case that we can swap the input fields and not change the polarization. This implies that

\[ \begin{align} \chi_2 = \chi_3 = \chi_4 \end{align} \]

and since in at least one direction \(\chi_1\) must be the only effective term, and this must be the same for all direction, it follows that we only need to consider \(\chi_{xxxx}\) and the tensor nature of the susceptibility does not matter at all.

9.4. Exercises

1. (10 pts) Third harmonic generation is often achieved in glasses, which are optically and structurally isotropic. Take a look at this paper: Optical third-harmonic generation from some high-index glasses and answer the following questions:

2. (20 pts) Table 2 lists measured third-order susceptibility of various glasses in esu units. Convert these to SI.

3. (20 pts) Equation (3) is the intensity for the 3rd harmonic generation. Choose a glass and plot the result as a function of input intensities and crystal and coherence length. Note: this should feel familiar! 2a. Bonus: solve the nonlinear ODE numerically to reproduce the same result/

4. (10 pts) Why does tilting angle of incidence of the laser beam to the glass sample change the THG signal as shown in Fig. 2?

9.5. DC Kerr Effect

Now consider the DC Kerr effect, for which

\[ \begin{align} P_i(\omega) = 3\epsilon_0\sum_{jkl}\chi^{(3)}_{ijkl}(\omega; 0, 0, \omega)E_j(0)E_k(0)E_l(\omega) \end{align} \]

Let’s consider plane wave travelling in the \(z\) direction. Suppose we apply a DC field which is polarized in the \(y\) direction, we can conclude that the resulting material polarization in the \(x\) and \(y\) directions are

\[\begin{split} \begin{align} P_x(\omega) = 3\epsilon_0\chi^{(3)}_{xyyx}(\omega; 0, 0, \omega)E^2_y(0)E_x(\omega) = 3\epsilon_0\chi_4E^2_y(0)E_x(\omega) \\ P_y(\omega) = 3\epsilon_0\chi^{(3)}_{yyyy}(\omega; 0, 0, \omega)E^2_y(0)E_y(\omega) = P_y(\omega) = 3\epsilon_0\chi_1E^2_y(0)E_y(\omega) \end{align} \end{split}\]

We see from this that the polarization at frequency \(\omega\), which determines the refractive index, is different in the \(x\) and \(y\) directions:

\[ \begin{align} n_y - n_x = \frac{3(\chi_1 - \chi_4)E_y^2(0)}{2n} = \frac{3\chi_2E_y^2(0)}{n} = \lambda_0KE_y^2(0) \end{align} \]

where \(K\) is called the Kerr constant.

9.6. Exercises

1. (20 pts) Design a Kerr cell based on notrobenzene, with \(K = 4.4\) pm/V^2. Assume you apply a DC field in the y-direction using two electrodes spaced by 1 cm. You would like to rotate the plane of polarization of light by 90 degrees in a 10 propogation length. What voltage on the plates will be required?

9.7. Optical Kerr Effect

The optical Kerr effect is analogous to the DC Kerr effect, except now we have an additional field at \(\omega_2\) the changes the refractive index of the field propagating at \(\omega_1\). Take another look at the resulting material polarization:

\[ \begin{align} P_i(\omega_1) = 3\epsilon_0\sum_{jkl}\chi^{(3)}_{ijkl}(\omega_1; \omega_2, -\omega_2, \omega_1)E_j(\omega_2)E^*_k(\omega_2)E_l(\omega_1) \end{align} \]

If fields at \(\omega_1\) and \(\omega_2\) have the same polarization (say in the \(x\) direction), we have

\[ \begin{align} P_x(\omega_1) = 3\epsilon_0 \chi^{(3)}_{xxxx}(\omega_1; \omega_2, -\omega_2, \omega_1)|E_x(\omega_2)|^2 E_x(\omega_1) \end{align} \]

If instead the fields at \(\omega_1\) and \(\omega_2\) are cross-polarized, then we have

\[ \begin{align} P_i(\omega_1) = 3\epsilon_0\chi^{(3)}_{xyyx}(\omega_1; \omega_2, -\omega_2, \omega_1)|E_y(\omega_2)|^E_x(\omega_1) \end{align} \]

This is sometimes known as the AC Kerr effect.