# High Resolution Imaging Using Micro-Objectives Fabricated by 2-Photon-Polymerization
## 3 Simulation of Imaging Properties  
For a computational time optimized simulation of the imaging properties of the investigated optical biconvex micro structures, the Finite Refractive Index Selective Propagation (FRISP) method [19] was extended by the optical properties of the microscope used for imaging. FRISP is based on scalar diffraction theory to determine the influence of a refractive micro structure on a wave field and provides a high correlation with the results of Finite Different Time Domain (FDTD) simulations and experiments for the determination of the focal properties of refractive micro structures [19].  
However, the simulation of the imaging properties of micro structures requires an extension of the FRISP method to include an object surface, the optical properties of the illumination and those of the microscope objective used for the high resolution imaging. In our case, the object surface is characterized by a binary amplitude mask \(T(\mathbf{x},\mathbf{y},\mathbf{0})\). This simplified implementation allows us a rapid adaption of the object surface according to the test structure used in the experiments.  
For the simulation of the illumination conditions used in the experiment, which satisfies all the characteristics of Kohler illumination, the illumination wavefront is divided into a series of plane waves with different angles of incidence.  
\[U_{j}(\mathbf{r})=A_{j}\cdot e^{j\cdot\mathbf{k_{j}}\cdot\mathbf{r}}. \tag{3}\]  
where \(j\) is the description of the wave field with respect to the angle of incidence, \(k_{j}\) is the wave vector and \(U_{j}(\mathbf{r})\) the complex amplitude at the position \(\mathbf{r}\), with \(\mathbf{r}=\sqrt{\mathbf{x}^{2}+\mathbf{y}^{2}+\mathbf{z}^{2}}\). The influence of the refractive micro structure on the wave field is determined for each of these illumination directions. For this purpose, the structure is divided into thin layers as described in [19] and their influence on the wave field is determined. The propagation starts in the illumination plane at \(\mathbf{z}=\mathbf{z_{i}}\) and stops in the object plane at \(\mathbf{z}=\mathbf{0}\). In the object plane, which is typically located directly below the refractive micro structure, the wave field \(U_{j}(\mathbf{x},\mathbf{y},\mathbf{0})\) is multiplied by the previously defined binary amplitude mask \(T(\mathbf{x},\mathbf{y},\mathbf{0})\) and the influence of the object on the wave field is taken into account by  
\[U_{j}^{\prime}(\mathbf{x},\mathbf{y},\mathbf{0})=U_{j}(\mathbf{x},\mathbf{y}, \mathbf{0})\cdot T(\mathbf{x},\mathbf{y},\mathbf{0}). \tag{4}\]  
representing the illumination wave field \(U_{j}(\mathbf{x},\mathbf{y},\mathbf{0})\) modeled with the object information and is used as the input wave field for the back propagation through the refractive micro structure.  
In order to determine the imaging properties of the refractive micro structure, the wave field \(U_{j}^{\prime}(\mathbf{x},\mathbf{y},\mathbf{0})\) serves as the input wave field for the second propagation through the refractive micro structure. This wave field is again propagated through the micro structure using FRISP in the input plane of the illumination wave field. Finally, to obtain a focused image of the object surface, the wave fields \(U_{j}^{\prime}(\mathbf{x},\mathbf{y},\mathbf{z}^{\prime})\) are individually propagated into the image plane \(\mathbf{z}^{\prime}\) of the microscope objective for each illumination direction and the intensities are summarized incoherently by  
\[I(\mathbf{x},\mathbf{y},\mathbf{z}^{\prime})=\sum U_{j}^{\prime}(\mathbf{x}, \mathbf{y},\mathbf{z}^{\prime})\cdot U_{j}^{\prime}(\mathbf{x},\mathbf{y}, \mathbf{z}^{\prime})^{*}, \tag{5}\]  
which allows us to determine the intensity distribution \(I(\mathbf{r})\) of the high-resolution image recorded by the microscope in the experiment using a refractive micro structure on the surface of the object.