The scalar problem of reconstruction of an unknown refractive index of an inhomogeneous solid is considered. The original boundary value problem for the Helmholtz equation with an unknown refractive index is reduced to the source-type integral equation. The solution to the inverse problem is obtained in two steps. First, the integral equation of the first kind is solved in the inhomogeneity domain using measurements of the total field outside the domain. The uniqueness of a solution to the integral equation of the first kind is proved in a special class of functions. Second, the sought-for refractive index is explicitly expressed via the found solution and the total field. The two-step method was verified by solving a test problem with a given refractive index. Procedures for refining approximate solutions were proposed and implemented. Efficiency of the proposed method was approved by comparison between the exact solution and the approximate ones.

中文翻译：

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求解标量逆 3D 衍射问题的非迭代两步法

考虑重建非均匀固体的未知折射率的标量问题。未知折射率的亥姆霍兹方程的原始边值问题简化为源型积分方程。求逆问题的解分两步。首先，使用域外总场的测量在非均匀域中求解第一类积分方程。在一类特殊的函数中证明了第一类积分方程解的唯一性。其次，寻求的折射率通过找到的解决方案和总场明确表达。通过解决具有给定折射率的测试问题来验证两步法。提出并实施了改进近似解的程序。