Nonlinear Luneburg Lens

We developed a Luneburg lens design where Kerr nonlinearity is used to compensate for the focal point shift caused by diffraction of a Gaussian source. We studied the relationship between the focal point positions, source types and source intensities. A modified Luneburg lens has been introduced to minimize the spherical aberration caused by the nonlinearity.

The Luneburg lens has a refractive index profile which is able to focus an incoming plane wave at a perfect geometric point at the opposite side of the lens. It is difficult to implement with bulk medium so it was designed with aperiodic subwavelength nanostructures. The unit cell is square and contains silicon rod immersed in air.

This Luneburg lens has been verified with both FDTD and iterative nonlinear beam propagation method. With a Gaussian beam illumination, due to diffraction, the focus point is to the right of the lens edge so the good property of Luneburg lens is vanished.

However, for Luneburg lens with Kerr nonlinearity, the refractive index changes with respect to the optical intensity. The focusing power of Kerr effect can compensate the focal shift caused by the diffraction. Therefore, the nonlinear Luneburg lens could have a focal point right at the edge with Gaussian illumination.

An important relationship exists between the positions of focal shift, types of sources and source intensities. For Gaussian illumination with different waist sizes, we can compensate the diffraction by changing the intensity of the source accordingly.

One problem exists for nonlinear Luneburg lens: due to nonlinearity, spherical aberrations appear at the focal point. In order to tackle to problem, a modified Luneburg lens was designed to minimize the spherical aberration. The refractive index distribution is similar to the original design but with high index gradient at outer part.