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，如干涉显微镜、共焦显微镜和焦点变化显微镜，记录不同焦点位置的一系列显微图像，以重建表面的深度信息。因此，这种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率域上的完整表征不仅具有轴向和横向分辨率，而且能够模拟强度输出信号和重建表面形貌。因此，3D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3D传递函数能够在实践中减少例如由光学像差引起的系统测量偏差。这是通过应用在3D空间频域中的过滤器来实现的，该过滤器提供与理想的无像差系统的结果相对应的一系列过滤图像。然而，在应用这一概念时，会出现科学问题，需要进行详细的研究，这将在本提案中得到解决。到目前为止，基尔霍夫近似或物理光学近似是定义3D传递函数的理论基础。这就要求表面微地形的局部最小曲率半径远远大于光的波长。此外，假设被测表面的反射率与入射角无关，忽略了反射和散射过程的偏振相关性。目前确定三维传递函数的实验方法需要使用直径为40-110微米的反射微球来测量散射光强度，这导致一个微球上的光强值较低，另一方面，如果测量平面衬底上的衍射结构，则测量的传递函数与生效的传递函数相比存在系统偏差。初步研究表明，平面镜可以在不同的光轴倾角下获得三维传递函数。然而，这种方法非常耗时，因此需要研究现有的表面校准标准的适用性。除了角度和偏振相关的反射系数外，还将通过适当严格的散射光场有限元计算来考虑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.