# High Resolution Imaging Using Micro-Objectives Fabricated by 2-Photon-Polymerization
## 2 Fabrication
To account for the observed deformations, the desired geometry must be adjusted based on the process parameters and voxel dimensions used in the 2 PP-fabrication process, see Figure 2b. The geometry \(S^{\prime}(r)\) of the fabricated micro structure can be described by a convolution of the design geometry \(S(r)\) with the voxel geometry \(h(r)\),  
\[S^{\prime}(r)=S(r)\mathbin{\raisebox{-1.29 pt}{\scalebox{0.8}{$\circ$}}}h(r). \tag{1}\]  
Because of the large aspect ratio of the voxels, it is generally true that \(S^{\prime}(r)\
eq S(r)\) holds. This means the geometry applied during the fabrication process \(D(r)\) must be adapted, so that  
\[S(r)=D(r)\mathbin{\raisebox{-1.29 pt}{\scalebox{0.8}{$\circ$}}}h(r). \tag{2}\]The adapted geometry \(D(r)\) must be found by deconvolution. However, for the deconvolution to be a well posed problem in the sense of Hadamard, (i) \(S(r)\) must be a steady function and (ii) \(\mathcal{F}\{h(r)\}\) must have no roots. The implementation of the deconvolution process can be elaborate. However, a determination of the adapted geometry \(D(r)\) via a deconvolution is not necessary for the fabrication of fully symmetric structures such as micro spheres or biconvex lenses. Therefore, we can use a semi-empiric workflow as follows: (i) A design geometry \(S(r)\) is created and the voxel shape \(h(r)\) for the parameters used in the fabrication process is determined. (ii) A slice of the thickness of the voxel height is removed from the middle of the design geometry. (iii) The upper and lower sides of the design geometry are merged together, where the center of the adapted geometry \(D(r)\) match the center of the original design geometry. Figure 2 shows a schematic visualization of these adaptations of the design geometry to the adapted geometry for fabrication. Thereby (a) shows the cross-section of the design geometry \(S(r)\), the geometry of a voxel \(h(r)\) and the section to be removed in the center of the design geometry. (b) shows the adapted geometry \(D(r)\) for the manufacturing process. To create a micro sphere with a minimum contact area to the substrate, the center of the adapted geometry must remain at the position of the center point of the design geometry. Figure 1: (**a**) CAD model of a micro-optical system consisting of four interconnected micro-spheres, each with a diameter of 10 \(\upmu\)m. (**b**) SEM image of the microsystem fabricated with the dimensions of the CAD model using 2 PP-lithography. The structure shows a clear distortion in the vertical direction. The section at the top right shows a schematic cross-section of one of the micro-spheres and the outer contour of the design geometry (dashed line). (**c**) SEM image of the microsystem produced with an adapted design geometry. The micro-spheres have a diameter of 10 \(\upmu\)m and a contact area of \(<\)1 \(\upmu\)m\({}^{2}\) to the substrate surface that was determined by SEM measurements and FIB cross-sections. The top right section shows a schematic cross-section of one of the micro-spheres and the outer contour of the design geometry (dashed line). (**d**) In the FIB cross-section, the shape fidelity and symmetry of the micro-sphere are evident. Achieving such precision requires detailed knowledge of the voxel dimensions so that the design can compensate for the inherent deformations introduced during fabrication. By incorporating this information into the design process, it is possible to anticipate 2 PP-related deformations and generate a geometry tailored to the manufacturing parameters. This approach enables the fabrication of shape fidelity micro structures having any desired geometry. In addition, other process-related deviations, such as material shrinkage, can be addressed by adjusting the design geometry, further improving the fidelity of the fabricated structures. Figure 1c shows an SEM image of the microsystem shown in Figure 1a fabricated using an optimized design geometry. The micro-spheres retain the intended diameter of 10 \(\upmu\)m as shown in Figure 1b.