This page provides some background information on the Kerr effect. We then consider how the nonlinear Kerr effect changes the focal length and beam shape in a high NA system. For more information on the Kerr material model see Advanced Material models section.

## Background information on the Kerr effect

The polarization field response **P** of a nonlinear material can be expressed as a power series of **E** in the following manner.

$$ \mathbf{P}(t)=\varepsilon_{0}\left(\chi^{(1)}\mathbf{E}(t)+\chi^{(2)}\mathbf{E}(t)\mathbf{E}(t) + \chi^{(3)}\mathbf{E}(t)\mathbf{E}(t)\mathbf{E}(t) + \cdots \right) $$

For a Kerr nonlinear material even order terms of the susceptibility drop out due to inversion symmetry of the media, yet \( \chi^{(3)} \) remains significant. In general the materials anisotropic polarizabillity must be represented by \( \chi^{(n)} \) a \( n+1 \) tensor. Crystal symmetry usually allows one to reduce the number of unique components that need to be solved for, and the situation can be further simplified by enforcing linear polarization of the **E**-field along one of the principle axes.

As a final simplification the electric susceptibility is usually assumed to be dispersionless in the bandwidth of interest. Thus we obtain the following expression for the polarization field.

$$ \mathbf{P}(t)\approxeq \varepsilon_{0}\left(\chi^{(1)}+\chi^{(3)}|\mathbf{E}(t)|^{2} \right) \mathbf{E}(t) $$

In the limit where \( \chi^{(3)}|\mathbf{E}(t)|^{2} << \chi^{(1)} \)

$$ \begin{aligned} n &=\left(\varepsilon_{r}+\chi^{(3)}|\mathbf{E}(t)|^{2}\right)^{1 / 2} \\ & \cong n_{0}\left(1+\frac{1}{2} \frac{\chi^{(3)}|\mathbf{E}(t)|^{2}}{\varepsilon_{r}}\right) \\ &=n_{0}+n_{2} I \end{aligned} $$

where n_{0} is the linear refractive index, n_{2} is the nonlinear refractive index and \( I \) is the beam intensity (power per unit area).

$$ I=n_{0} \sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}|\mathbf{E}(t)|^{2} $$

The nonlinear refractive index is related to the nonlinear susceptibility by

$$ n_{2}=\frac{1}{2} \frac{\chi^{(3)}}{n_{0}^{2}} \sqrt{\frac{\mu_{0}}{\varepsilon_{0}}} $$

## Simulation setup

We will consider a system of glass at 632 nm. We assume that the linear refractive index is 1.455 and the nonlinear index (n2) is approximately 6 times that of fused silica, or 6*3e-16 cm^{2}/W. Applying the conversion from n2 to χ^{(3)}, we obtain a value of χ^{(3)} = 6*3.37e-22 (m^{2}/V^{2}) = 2.0e-21 (m^{2}/V^{2}).

The NA of the system is 1.4, so the beam is rapidly focusing near the waist. We will model a three dimensional region of 6 microns around the focal plane with a cross-sectional area of 8x8 μm^{2}. The pulse length in FDTD is set to 100 fs, with a gaussian envelope.

## Results

The file *kerr1_linear.fsp* and *kerr1_nonlinear.fsp* contain the simulation setup. The simulations create mpg movies in the x-y plane at the beam waist, and in the y-z plane. Symmetry is used to reduce the computational burden which can be seen in the mpg movies of the simulation.

After running the simulations, the file *kerr1.lsf* performs an analysis comparing the linear to the nonlinear case.

The first figure compares the power(t) vs time measured at the beam waist.

Kerr Effect on Gaussian Beam

The script then calculates the source power in various ways, and displays the following text to the script window:

nonlinear: maximum measured from P(t) at waist = 60893.5 Watts maximum measured from P(w) at waist = 68430 Watts maximum calculated at from source = 79532.7 Watts Average power over simulation time = 15423.1 Watts Average power at 80MHz pulse rate = 0.606477 Watts linear: maximum measured from P(t) at waist = 75853.2 Watts maximum measured from P(w) at waist = 77394.4 Watts maximum calculated at from source = 79532.7 Watts Average power over simulation time = 16883.8 Watts Average power at 80MHz pulse rate = 0.663918 Watts

In FDTD, the source power is controlled by setting the maximum electric field amplitude. In this case, the source amplitude was set to 1e9 V/m. The power can be adjusted to any value (such as p0) by running the same simulation and adjusting the source amplitude to 1e9*sqrt(p0/0.166038) V/m.

The following figures show various cross sections of the beam comparing the linear and nonlinear cases. The nonlinear beam intensity is higher, and the beam is compressed along the y-axis (the beam is polarized along the x-axis).

The next figure compares the maximum value of

Finally, the script tries to estimate the radius of the beam along the x and y axes and plot it as a function of z. The calculation is performed by integrating the length where |E|^2 0.5*max(|E|^2). Thus, it is not valid far from the beam waist. The following figure shows the beam radius as a function of z.

We can see that the linear and nonlinear results are almost identical before the focal plane (because the field intensities are small) but are substantially different after the focal plane due to the nonlinear effects near the focal plane and near the beam center.

If we zoom on the graph, we can see

The above curve confirms what could be seen qualitatively in the cross sectional views: the beam is slightly compressed along the y axis but the radius is not substantially modified along the x axis. The depth of focus is slightly shorter for the nonlinear beams.