Nonlinear Beam Propagation

We developed an iterative nonlinear beam propagation method, providing ray diagrams which are physically more intuitive. At the same time, wave effects and coherent properties are preserved, meaning that this method not only takes the wave effects, such as diffraction and interferences, into account, but also works in both coherent and partially coherent regime. Iterative method could be applied to the investigation of nonlinear metamaterials and devices.

The proposed iterative nonlinear beam propagation method is based on Hamiltonian ray tracing and the Wigner distribution function (WDF).

The method consists of three steps in one iteration. In the first step, the initial rays for the input illumination are defined based on the WDF. The WDF defines the generalized radiance of every position and momentum. Thus, every discretized point on the WDF plane corresponds to one ray, with the position and momentum defined. The generalized radiance carried by the ray is defined by the value of WDF of the corresponding point.

In the second step, Hamiltonian ray tracing is applied to each of the ray defined in the previous step. Hamiltonian ray tracing solves the two differential equations governing the ray trajectories.

In the third step, the optical intensity of each point of the nonlinear media is calculated through the projection in the momentum direction on the WDF of the corresponding plane, which is gener­ated based on the ray tracing results. Also, an updated refractive index is calculated based on the intensity through the Kerr effect relationship, where refractive index changes linearly to the optical intensity.

After one iteration, another ray tracing is applied to the new refractive index profile and another updated version of refractive index is generated. Finally the iterations will converge and the final intensity profile is the beam propagation results we seek.

This method provides a ray picture to the nonlinear beam propagation which is physically more intuitive, offering an insight into the radiance evolution in the nonlinear media. Also, it has the potential application in optical system design softwares such as ZEMAX, where ray tracing is gener­ally used. With this iterative method, while dealing with nonlinear element, it could be consistent with the ray tracing, avoiding the transition between rays and waves. Furthermore, it has been proved that Hamiltonian ray tracing, under the locally periodic assumption, is valid in photonic nanostructures. Thus, this iterative method could also be applied to nonlinear metamate­rials and nanophotonics devices, providing a systematic approach to nonlinear beam propagation simulation in these devices.

The proposed iterative method is validated with self-focusing and spatial soliton phe­nomena in nonlinear Kerr effect media with totally coherent illumination. The results match well with either those provided by the BPM method, or analytical results.