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的特性。深度扫描三维显微镜，如干涉显微镜、共聚焦显微镜和变焦显微镜，在不同的聚焦位置记录一系列显微图像，以重建表面的深度信息。因此，该图像序列的三维频率表示不仅包括图像的横向空间频率分量，还包括轴向空间频率系数。光学三维显微镜在空间频域的完整表征不仅导致轴向和横向分辨率的能力，而且使强度输出信号的模拟和表面形貌的重建。因此，三维传递函数对理解物理机制有很大的要求。此外，确定实际测量仪器的三维传递函数可以在实践中减少系统测量偏差，例如由光学像差引起的偏差。这是通过在3D空间频域应用滤波器实现的，提供一系列与理想无像差系统的结果相对应的滤波图像。然而，在应用这一概念时，出现了需要详细研究的科学问题，这将在本建议中加以解决。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。这就要求表面微形貌的局部最小曲率半径要远远大于光的波长。此外，假设被测表面的反射率与入射角无关，忽略反射和散射过程的偏振依赖性。最近，通过实验确定3D传递函数的方法需要使用直径为40-110µm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，则这导致一方面的低强度值，并且与有效的传递函数相比，测量的传递函数存在系统偏差。初步研究表明，在不同的倾斜角度下，平面镜可以获得相对于光轴的三维传递函数。然而，这种方法是耗时的，因此需要调查可用的表面校准标准的适用性。散射光场的有限元计算除了考虑角度和偏振相关的反射系数外，还考虑了表面上的边缘和单点散射。基于三维传递函数的系统测量偏差可以被理解，信号处理算法和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.