# Training Videos

## Coupled-wave analysis of nonlinear optical interactions

This is a continuation from the previous tutorial - nonlinear optical interactions. The coupled-wave theory developed in the coupled-wave theory tutorial is used for analysis of nonlinear optical interactions in a homogeneous medium. In nonlinear optical waveguides, coupled-mode theory can be used if there is no mixing between different optical frequencies, but a combination of coupled-wave and coupled-mode formalisms has to be used if there is frequency mixing in the interaction. In applying coupled-wave or coupled-mode theory to the analysis of a nonlinear interaction, the perturbing polarization, generally expressed as $$\Delta\mathbf{P}$$ in the coupled-wave theory tutorial, is identified as the characteristic...

## Nonlinear Optical Interactions

This is a continuation from the previous tutorial - nonlinear optical susceptibilities. Optical susceptibilities in the frequency domain are, in general, complex quantities. Following the same convention, used in the linear optical susceptibility tutorial and the material dispersion tutorial for complex linear susceptibility in the frequency domain, complex nonlinear susceptibilities in the frequency domain can be expressed as $$\boldsymbol{\chi}^{(2)}=\boldsymbol{\chi}^{(2)'}+\text{i}\boldsymbol{\chi}^{(2)''}$$ and $$\boldsymbol{\chi}^{(3)}=\boldsymbol{\chi}^{(3)'}+\text{i}\boldsymbol{\chi}^{(3)''}$$ to define their real and imaginary parts clearly. Similarly to the case of linear susceptibility discussed in the material dispersion tutorial, the imaginary part of a nonlinear susceptibility is always associated with the intrinsic resonances of a material. Such resonances signify...

## Nonlinear Optical Susceptibilities

This is a continuation from the previous tutorial - optical nonlinearity. The nonlinear optical properties of a material are characterized by its nonlinear optical susceptibilities. In this tutorial, the general properties of nonlinear optical susceptibilities are discussed. It can be seen from (9-3) and (9-4) [refer to the optical nonlinearity tutorial] that the space- and time-dependent nonlinear susceptibilities $$\boldsymbol{\chi}^{(n)}(\mathbf{r}-\mathbf{r}_1,t-t_1;\mathbf{r}-\mathbf{r}_2,t-t_2;\ldots;\mathbf{r}-\mathbf{r}_n,t-t_n)$$ are real tensors because both $$\pmb{P}^{(n)}(\mathbf{r},t)$$ and $$\pmb{E}(\mathbf{r},t)$$ are real vectors. Though $$\boldsymbol{\chi}^{(n)}(\mathbf{r}_1,t_1;\mathbf{r}_2,t_2;\ldots;\mathbf{r}_n,t_n)$$ is always a real function of space and time, its Fourier transform is generally complex. Therefore, the frequency-dependent nonlinear susceptibilities $$\boldsymbol{\chi}^{(n)}(\omega_q=\omega_1+\omega_2+\cdots+\omega_n)$$ defined in the frequency domain are generally...