Advanced Discretization Techniques for Paraxial Laser Propagation

Jeffrey Banks (14-ERD-032)


We examined long-term research and development needs for high-fidelity laser beam analysis and simulation. Improved understanding of the operation of high-intensity lasers, such as Livermore's National Ignition Facility, are critical to optimizing their performance and expanding their utility as experimental facilities. High-resolution simulations are a major contributor to such understanding, and the scale of problems to be addressed continues to grow. Relevant questions span the range from highly resolved simulations of small sub-regions through whole-beam simulations to in-line shot modeling with multiple beams. Computational aspects such as high orders of accuracy, parallel scalability, adaptive meshing, and reduced-order modeling will need to be considered to address problems across this range and at increasing fidelity.

Background and Research Objectives

The efficiency of high-intensity lasers can be improved, and their use as experimental facilities expanded, through an improved understanding and optimization of laser components. For instance, understanding and controlling the deleterious edge-diffraction effects caused by beam-line obstructions requires the resolution of small features in transmission masks. Furthermore, shot planning would benefit from the ability to perform whole-beam modeling of fratricide from optics defects. High-resolution simulations are a major contributor to such studies, but many problems remain to be addressed. In this project, considering the long-term research needs for high-fidelity beam analysis and simulation, we proposed to investigate the paraxial model of light propagation (i.e., at a small angle to the optical axis of the system) and adopting new discretization techniques (transforming continuous attributes into discrete ones) for the paraxial wave equation with high-order finite differences.

We studied light propagation in the paraxial approximation, which assumes that there is a primary propagation direction, z, and that the structure transverse to that direction is band-limited to small angular content. The paraxial wave equations describe the evolution of the complex electric field in the x and y plane along the propagation direction z (a time-like direction). Most current tools for laser beam analysis use uniform spectral (Fourier) discretization in the x–y plane and a second-order-accurate splitting in z. While this approach works well for many applications, it is difficult to apply on non-periodic domains and can be challenging to adapt efficiently to massively parallel computing architectures. These methods can also prove challenging when small-scale features, such as optical flaw blockers and apodizers, are present in the system being simulated. The technical goals of this project were significantly modified from the original plan because of a shortened timescale, with the principal investigators departure for an academic position, but we included development of explicitly integrated discretizations for the paraxial model with a perfectly matched layer (artificial absorbing layer for wave equations) with orders of accuracy from two to eight. The perfectly matched layer strongly absorbs outgoing waves from the interior of a computational region without reflecting them back into the interior. In addition, our revised scope included an investigation of the possibility of using alternating-direction implicit formulations either based on a standard defect-correction scheme or through the use of the so-called spectral deferred correction. The alternating-direction implicit formulation is a finite-difference method for solving parabolic, hyperbolic, and elliptic partial differential equations. It is most notably used to solve the problem of heat conduction or solving the diffusion equation in two or more dimensions.

Scientific Approach and Accomplishments

Our technical approach with this project was to first derive high-order conservative finite-difference algorithms for the variable-coefficient Schrödinger equation, which has applications in a variety of fundamental science scenarios, including atomic structure, the Bose–Einstein condensate state of matter, and ocean wave propagation. In this case, variable coefficients are used to implement an absorbing-boundary condition with a perfectly matched layer, which was used to relax the standard assumption of periodicity implied by spectral discretizations. Thus, one research task was to calibrate the perfectly matched layer and catalogue its performance in terms of reflectivity. For example, the figure below shows the effect of the perfectly matched layer on an outgoing Gaussian beam profile. A Gaussian beam is a beam of monochromatic electromagnetic radiation with specific transverse magnetic and electrical field amplitude profiles, which describes the intended output of most lasers. In order to validate the numerical approach, we also derived exact solutions to the paraxial model in restricted situations, plane waves, and cylindrical symmetry.

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Figure 1. Effect of a absorbing-boundary condition using a perfectly matched layer on an outgoing Gaussian photon-beam profile.

With the initial explicit finite-difference discretizations in hand, we turned to the task of investigating alternating-direction implicit approaches to time-integration computations. Our idea was to achieve the stability of an implicit method, but relaxing the need to solve a single large implicit matrix equation for all degrees of freedom. In the alternating-direction implicit approach, we need only solve implicit systems along grid lines in each coordinate direction, a process that can be made extremely efficient.

The technique was straight forward, but as expected, the method introduced second-order discretization errors. Thus, the final research goal we considered for the project was the elimination of these second-order errors using a deferred-correction strategy. In this task we considered both a simple defect correction as well as the so-called spectral deferred correction, which was undertaken by our collaborators at Lawrence Berkeley National Laboratory. In both cases, some success was seen in that over the domain interior, the error from splitting (evolution in a time interval by separating into two steps) can be eliminated from the alternating-direction implicit time stepper. However, for both approaches, the presence of non-smooth errors near the domain boundaries from the splitting resulted in either a reduced order of accuracy or significant computational cost on the same order as solving the original fully coupled system. There are promising ideas to address these difficulties, similar to those used to address time-dependent boundary conditions in Runge–Kutta integrators (used to construct a numerical solution as linear combinations of approximations) that should be pursued.

Impact on Mission

Research for this project helped advance the Laboratory’s core competency in lasers and optical science and technology as well as high-performance computing, simulation, and data science by enhancing our ability to perform large-scale, high-fidelity simulations that will improve our understanding of laser components in the near term, enable whole-beam modeling of the effects of optic defects in the medium term, and advance short-pulse applications in the long term by predicting the effects on beam characteristics of phenomena such as spectral dispersion.


We have begun an investigation into the possibility of using novel splitting techniques for discretizations used in computational simulations, such as the alternating-direction implicit approach, as tools to increase scalability of complex codes while using deferred-correction strategies to retain high orders of accuracy applicable to laser beam analysis. The project showed that there are significant gains that can be made with this approach—continued careful computational effort is needed if the full promise is to be achieved. In addition, the work performed with this project can serve as the basis for future beam-propagation codes that are important for a variety of research missions relevant to the National Ignition Facility. The basic discretization strategies we developed are shown to be highly effective and competitive in certain regimes with more traditional Fourier-based representations, while being vastly more computationally flexible.