Three-dimensional (3D) transfer functions build the basis for a comprehensive characterization of optical imaging systems in the spatial frequency domain. Utilizing the projection-slice theorem, the 2D modulation transfer function of an incoherent imaging system can be derived from a 3D transfer function by integration with respect to the axial spatial frequency. For a diffraction limited microscope with homogeneous incoherent pupil illumination, the modulation transfer function equals the 2D autocorrelation function of a circular disc. However, until now to the best of our knowledge no 3D transfer function has been published, which exactly leads to the 2D modulation transfer function of a diffraction limited microscope in reflection mode. In this article, we derive a formula, which after integration with respect to the axial spatial frequency coordinate perfectly fits to the diffraction limited 2D modulation transfer function. The inverse three-dimensional Fourier transform of the 3D transfer function results in a complex-valued 3D point spread function, from which the depth of field, the lateral resolution and, in addition, the corresponding 3D point spread function of both, a conventional and an interference microscope, can be obtained.

中文翻译：

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光学显微镜在反射模式下的三维传递函数

三维 (3D) 传递函数为空间频域中光学成像系统的综合表征奠定了基础。利用投影切片定理，非相干成像系统的 2D 调制传递函数可以通过相对于轴向空间频率的积分从 3D 传递函数导出。对于具有均匀非相干光瞳照明的衍射极限显微镜，调制传递函数等于圆盘的 2D 自相关函数。然而，到目前为止，据我们所知，还没有发布 3D 传递函数，这恰好导致了反射模式下衍射受限显微镜的 2D 调制传递函数。在这篇文章中，我们推导出一个公式，其在相对于轴向空间频率坐标积分后完全适合衍射受限的二维调制传递函数。3D 传递函数的 3D 逆傅立叶变换产生复值 3D 点扩散函数，其中景深、横向分辨率以及相应的 3D 点扩散函数，常规和可以得到干涉显微镜。