Modeling and determination of 3D transfer functions of high-resolution 3D optical microscopes for surface topography measurement

传递函数表征了光学成像系统在空间频域中的特性。深度扫描3D显微镜，例如干涉、共焦和焦点变化显微镜，记录在不同焦点位置处的一系列显微图像，以便重建表面的深度信息。因此，这种图像系列的三维频率表示不仅包括图像的横向空间频率分量，而且包括轴向空间频率系数。光学3D显微镜在空间频域中的完整表征不仅导致轴向和横向分辨率能力，而且还能够模拟强度输出信号和重建表面形貌。因此，3D传递函数建立了理解物理机制的实质性要求。此外，确定实际测量仪器的3D传递函数能够减少系统测量偏差，例如，G.实际上是由光学像差引起的。这是通过在3D空间频率域中应用的滤波器来实现的，提供了与理想的无像差系统的结果相对应的一系列滤波图像。然而，在应用这一概念的过程中，会出现需要进行详细研究的科学问题，这将在本提案中得到解决。 到目前为止，基尔霍夫或物理光学近似是定义三维传递函数的理论基础。这需要表面微观形貌的局部最小曲率半径远大于光的波长。此外，被测表面的反射率被假定为是独立的入射角和反射和散射过程的偏振依赖性被忽略。最近通过实验确定3D传递函数的方法需要使用直径为40-110 µm的反射微球测量散射光强度。另一方面，如果测量平面衬底上的衍射结构，则这一方面导致低强度值，另一方面导致所测量的传递函数与起作用的传递函数相比的系统偏差。 初步的研究表明，三维传递函数可以从一个平面镜在不同的倾斜角度相对于光轴。然而，这种方法是耗时的，使得可用的表面校准标准的适用性需要进行调查。除了角度和偏振相关的反射系数，还将通过散射光场的适当严格的有限元计算来考虑表面上的边缘和单点散射。基于3D传递函数，可以理解系统测量偏差，并且信号处理算法以及硬件仪器都将得到改进。

Transfer functions characterize the properties of optical imaging systems in the spatial frequency domain. Depth-scanning 3D microscopes such as interference, confocal and focus variation microscopes record a series of microscopic images at different focus positions in order to reconstruct the depth information of a surface. Thus, the three-dimensional frequency representation of such an image series includes not only the transversal spatial frequency components of the images but also axial spatial frequency coefficients. The complete characterization of optical 3D microscopes in the spatial frequency domain not only leads to the axial and lateral resolution capabilities but also enables the simulation of intensity output signals and the reconstruction of surface topography. Therefore, 3D transfer functions build a substantial requirement for understanding physical mechanisms. In addition, determining the 3D transfer function of an actual measuring instrument enables the reduction of systematic measurement deviations, e. g. caused by optical aberrations, in practice. This is achieved by a filter applied in the 3D spatial frequency domain, providing a series of filtered images corresponding to the result of an ideal aberration-free system. Nevertheless, in the application of this concept scientific issues occur that demand for elaborate research, which will be addressed in this proposal. So far, the Kirchhoff or physical optics approximation is the theoretical basis on which 3D transfer functions are defined. This requires the local minimum radii of curvature of the surface’s micro-topography to be much greater than the wavelength of light. In addition, the reflectivity of the measured surface is assumed to be independent of the incidence angle and the polarization dependence of reflection and scattering processes are neglected. Recent methods to determine 3D transfer functions experimentally require the measurement of the scattered light intensity using reflective micro-spheres with diameters of 40-110 µm. This leads to low intensity values on the one and to systematic deviations of the measured transfer function compared to the transfer function that takes effect, if diffractive structures on a plane substrate are measured, on the other hand. Preparatory investigations show that 3D transfer functions can be obtained from a plane mirror under different tilt angles with respect to the optical axis. However, this method is time-consuming so that the suitability of available surface calibration standards needs to be investigated. Besides angle and polarization-dependent reflection coefficients also edge and single point scattering on a surface will be considered by appropriate rigorous FEM computations of scattered light fields. Based on 3D transfer function systematic measurement deviations can be understood and both, signal processing algorithms as well as hardware instrumentation will be improved.