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Eikonal decomposition methods for fast computations of beam propagations

机译：用于光束传播快速计算的有效分解方法

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Purpose - The purpose of this paper is to develop highly efficient decomposition finite difference methods for computing solutions of highly oscillatory beam propagation partial differential equations. Design/methodology/approach - Highly oscillatory optical wave equations, such as the multidimensional paraxial Helmholtz equation, have been used extensively in modelling propagation of the light from lens to the focal region in various engineering applications. Numerical approximations of solutions of such equations contain crucial light information in focal regions even when the/-number is small. However, it has been difficult to acquire highly oscillatory numerical solutions efficiently. This paper proposes two correlated eikonal decomposition strategies for fast computations of the oscillatory solutions. Structures of the numerical methods are designed via an eikonal, or exponential, transformation. The approach converts successfully the oscillatory problems to non-oscillatory subproblems. Therefore, the underlying beam simulation equations can be solved readily with great accuracy and stability. Findings - It is found that the two correlated eikonal transformation based decomposition methods effectively remove the highly oscillatory features of the wave equations. The coupled non-oscillatory subproblems resulted are easier to solve. Discretization steps in computations can be chosen to be relatively large and this ensures the efficiency of computations. The decomposed finite difference schemes are simple to use in different optical applications. Practical implications - The computational approach provides a valuable tool to practical applications, such as those in the defence industry. Originality/value - Although the eikonal transformation has been used in the theory of nonlinear optics, this is the first time it has been utilized for effective engineering computations.
机译：目的-本文的目的是开发一种高效的分解有限差分方法，用于计算高振荡光束传播偏微分方程的解。设计/方法/方法-在各种工程应用中，高度振荡的光波方程（例如多维近轴Helmholtz方程）已广泛用于模拟从透镜到焦点区域的光传播。这样的方程式的解的数值逼近即使在/数很小的情况下，仍在焦点区域包含关键的光信息。但是，很难有效地获得高度振荡的数值解。本文提出了两种相关的电子分解策略，用于快速求解振动解。数值方法的结构是通过常规或指数转换设计的。该方法成功地将振荡问题转换为非振荡子问题。因此，可以很容易地以较高的精度和稳定性求解基础的梁仿真方程。发现-发现基于两种相关的基于电子变换的分解方法有效地消除了波动方程的高度振荡特征。由此产生的耦合的非振荡子问题更容易解决。可以将计算中的离散化步骤选择为较大，这样可以确保计算效率。分解的有限差分方案易于在不同的光学应用中使用。实际意义-计算方法为诸如国防工业等实际应用提供了有价值的工具。独创性/价值-尽管在非线性光学理论中使用了eikonal变换，但这是首次将其用于有效的工程计算。

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