The Validity of a Paraxial Approximation in the Simulation of Laser Plasma Interactions

The design of high-power lasers such as those used for inertial confinement fusion demands accurate modeling of the interaction between lasers and plasmas. In inertial confinement fusion, initial laser pulses ablate material from the hohlraum, which contains the target, creating a plasma. Plasma density variations due to plasma motion, ablating material and the ponderomotive force exerted by the laser on the plasma disrupt smooth laser propagation, undesirably focusing and scattering the light. Accurate and efficient computational simulations aid immensely in developing an understanding of these effects. In this paper, we compare the accuracy of two methods for calculating the propagation of laser light through plasmas. A full laser-plasma simulation typically consists of a fluid model for the plasma motion and a laser propagation model. These two pieces interact with each other as follows. First, given the plasma density, one propagates the laser with a refractive index determined by this density. Then, given the laser intensities, the calculation of one time step of the plasma motion provides a new density for the laser propagation. Because this procedure repeats over many time steps, each piece must be performed accurately and efficiently. In general, calculation of the light intensities necessitates the solution of the Helmholtz equation with a variable index of refraction. The Helmholtz equation becomes extremely difficult and time-consuming to solve as the problem size increases. The size of laser-plasma problems of present interest far exceeds current capabilities. To avoid solving the full Helmholtz equation one may use a partial approximation. Generally speaking the partial approximation applies when one expects negligible backscattering of the light and only mild scattering transverse to the direction of light propagation. This approximation results in a differential equation that is first-order in the propagation direction that can be integrated accurately and efficiently even for large computational domains. This paper explores the domain of validity of a paraxial approximation in laser-plasma simulations. High-intensity lasers may create high-density plasmas and induce extremely large and abrupt plasma density variations. Such variations in high-density plasmas can reflect or scatter a significant fraction of the incident light. However, as stated, the paraxial approximation assumes negligible backscatter. Furthermore, interference of incident and scattered waves may produce regions of high-intensity light that the partial approximation fails to predict accurately. Certainly, the paraxial approximation serves as an excellent approximation in many problems. We hope to provide insight into when it accurately models the problem and when it does not