A rigorous theory for the synthesis of offset dual reflector antennas is presented. The synthesis problem is posed as a three dimensional vector field which becomes the surface normals to one of the reflectors. The ray trace which defines the vector field is calculated from the reflection law at one reflector and a constant path length requirement. Implicit in the definition of the vector field is the mapping from the input wavefront to the output wavefront. The Pfaffian integrability condition is applied to the vector field which leads to a partial differential equation (PDE) for the input to output mapping. Integrating this PDE coupled to the PDE for the shape of the first reflector yields the desired solution. The method allows for a wide choice of the input to output mapping and combined with a judicious choice of geometry produces practical designs which outperform traditional conic section designs. The use of an explicit existence theorem in the problem formulation is innovative and leads to simpler software which give excellent results. Unlike previous work, much of the algebra is done without reference to coordinate frames, simplifying the presentation and relegating algebraic details to symbolic software. A novel method for integrating the coupled PDE’s is introduced.

中文翻译：

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基于Pfaffian可积性条件的双反射器成形

提出了用于合成偏置双反射器天线的严格理论。合成问题被提出为三维矢量场，该三维矢量场成为反射器之一的表面法线。根据一个反射镜的反射定律和恒定的路径长度要求，可以计算出定义矢量场的光线轨迹。向量场的定义中隐含了从输入波前到输出波前的映射。Pfaffian可积性条件应用于矢量场，该矢量场导致输入到输出映射的偏微分方程（PDE）。对于第一反射器的形状，将耦合到PDE的PDE集成在一起可以产生所需的解决方案。该方法允许输入到输出映射的广泛选择，并与明智的几何形状选择相结合，可产生优于传统圆锥截面设计的实用设计。在问题表述中使用显式存在定理是创新的，并导致产生更佳结果的简单软件。与以前的工作不同，大部分代数是在不参考坐标系的情况下完成的，从而简化了表示过程，并将代数细节移交给了符号软件。介绍了一种集成耦合PDE的新颖方法。大部分代数是在不参考坐标系的情况下完成的，从而简化了表示过程，并将代数细节移交给了符号软件。介绍了一种集成耦合PDE的新颖方法。大部分代数是在不参考坐标系的情况下完成的，从而简化了表示过程，并将代数细节移交给了符号软件。介绍了一种集成耦合PDE的新颖方法。