Nonlinear Material

This section describes nonlinear materials.

In theory, all materials have the possibility to produce nonlinear phenomena and nonlinearity is a property of materials.

In the software, nonlinear materials consist of base materials and nonlinear coefficients. The software supports two nonlinear algorithms: Chi2 nonlinear and Chi3 Raman/Kerr nonlinear.

For nonlinear materials, the dielectric polarization density in the frequency domain is expressed as:

$P(\omega) = \varepsilon_0\chi_e^{(1)}(\omega)E(\omega) + \varepsilon_0\chi_e^{(2)}(\omega)E^2(\omega) + \varepsilon_0\chi_e^{(3)}(\omega)E^3(\omega) +...$

Wherein, the first term refers to the linear response of dielectric polarization density; and the following terms refer to the nonlinear response of the material. Specifically, $\chi_e^{(2)}$ represents the coefficient of the second-order nonlinear response, which is applied to second harmonic generation, parameter mixing, etc.; and $\chi_e^{(3)}$ represents the coefficient of the third-order nonlinear response, which is applied to third-harmonic generation, Kerr effect, Raman scattering, two-photon absorption, etc.

For nonlinear materials, the material parameter window will display data of a base material if one has been added. If no base material has been added, default refractive index is set to 1 (i.e., vacuum).

Only the refractive index data of the base material is displayed on the refractive index monitor.

Nonlinear materials should be defined by two components: Base material and Nonlinear data.

Second-Order Nonlinear Material

The polarization of a second-order nonlinear material is expressed as:

$P(\omega) = \varepsilon_0\chi^{(1)}(\omega)E(\omega) + \varepsilon_0\chi^{(2)}(\omega)E^2(\omega)$

For second-order nonlinear materials, the input parameters include:

Name	Range	Default	Description
$\chi^{(1)}$	Real number, $\chi^{(1)} \geq1$	0	$\chi^{(1)}$ is the first-order linear polarization coefficient.
$\chi^{(2)}$	Real number, $\chi^{(2)} \geq0$	0	$\chi^{(2)}$ is the second-order nonlinear polarization coefficient.

Material Setting

In the Material library window, you can add a second-order nonlinear material model by selecting Add Material>Add new material>Add chi2 nonlinear, and modify material parameters of the nonlinear model in the pop-up editing interface to create the desired nonlinear material model.

To add a second-order nonlinear material with diagonal anisotropy, you need to enable the Anisotropy (Diagonal) option and define the polarization coefficient for each direction.

Third-Order Nonlinear Material

For Kerr effect or Raman scattering, polarization of a third-order nonlinear material is expressed as:

$P(t)= \varepsilon_0\chi^{(1)}E(t) +\varepsilon_0\chi^{(2)}E^2(t)+ \varepsilon_0 \alpha \chi^{(3)}E^3(t) + \varepsilon_0(1-\alpha)\chi^{(3)}E(t)\frac{\omega_R^2}{\omega_R^2-\omega^2+2j\omega\delta_R}*E^2(t)$

where, $\omega_R= \sqrt{\frac{\tau_1^2 + \tau_2^2}{\tau_1^2 \tau_2^2 }}$，$\delta_R= \frac{1}{\tau_2 }$.

For Kerr effect or Raman scattering, the input parameters of a third-order nonlinear material include:

Name	Range	Default	Description
$\chi^{(1)}$	Real number, $\chi^{(1)} \geq1$	0	$\chi^{(1)}$ is the first-order linear polarization coefficient.
$\chi^{(2)}$	Real number, $\chi^{(2)} \geq0$	0	$\chi^{(2)}$ is the second-order nonlinear polarization coefficient.
$\chi^{(3)}$	Real number, $\chi^{(3)} \geq0$	0	$\chi^{(3)}$ represents the third-order nonlinear polarization coefficient.
$\alpha$	Real number, $0 \leq\alpha \leq 1$	0	$\alpha$ represents the ratio of Kerr intensity to the total intensity from Kerr effect and Raman scattering.
$\tau_1$	Real number, $\tau_1 \geq0$	0	$1/\tau_1$ represents the characteristic frequency.
$\tau_2$	Real number, $\tau_2 \geq0$	0	$\tau_2$ represents the damping time constant.

Material Setting

You can add a third-order nonlinear material model by selecting Add material>Add new material>Add chi3 Raman/Kerr nonlinear, and modify material parameters of the nonlinear model in the pop-up editing interface to create the desired nonlinear material model.

To add a third-order nonlinear material with diagonal anisotropy, you need to enable the Anisotropy(Diagonal) option and define material parameters of the nonlinear model for each direction.

More Information

When simulating nonlinear materials, the following should be considered:

• As known above, the magnitude of incident light intensity greatly affects the observation of nonlinear effects. It is recommended to set a larger light source amplitude to produce significant nonlinear effects, provided the numerical stability is not affected;
• Nonlinear effect refers to the energy exchange between light and medium at a certain frequency, which changes the optical properties such as the frequency of transmitted light. Therefore, an accurate pulse shape is crucial to getting the system excited;
• The frequency range of the monitor is different from the source because there are more frequencies to be excited due to nonlinear effects;
• The FDTD pluse normalization of the monitor should be set No normalization, because the nonlinear effect excites light outside the source frequency range, and normalized calculations would yield incorrect results.

Case: Nonlinear Effects

Materials with second-order nonlinear coefficient can produce nonlinear effects when illuminated by strong lasers. See Nonlinear Effects.