J. Eur. Opt. Society-Rapid Publ. 21, 27( 2025) 291
Figure 21. Projection of residual deviations for the azimuthal angular measurement. The inset shows the absolute azimuthal angle compared to the reference light-blue tangens-spiral.
Figure 22. Projection of residual deviations for the polar angular measurement. Blue circled points are below 1 ° tilt. The inset shows the absolute polar angle compared to the reference light-blue sine-cone.
Figure 21 shows the with equation( 20) reconstructed azimuthal surface angle and its residues from the ideal the ideal arctan2()-reference plane for all 360 °. For large angles, the observed spot on the camera is cut off, because of the limited size of the sensor. Significantly, the discontinuity of the arctan2() leads to a high sensitivity for small deviations of the reconstructed spot center for small angles. Assuming a static relationship for systmatic deviations, a surface polynomial was fitted and subtracted from the measurements. This leads to a remaining standard deviation for the reconstructed azimuthal angles of r a, RMS = 0.1 ° as shown in Figure 21. For small angles the uncertainty remains higher due to the discontinuity.
For the polar angle, Figure 22 shows the resulting reconstructed cone characterizing the local steepness of the substrate. The resulting a M were calculated from equation( 23). Surface slopes up to # max 6 ° can be acquired. Higher surface slopes lead to higher deviations from the reference arcsin()-plane. A surface polynomial had been subtracted to account for static systematic deviations, too. A remaining standard deviation of r # = 0.46 ° can be determined for the whole range as shown by the projection in Figure 22. This deviation significantly improves to r # = 0.16 ° for small surface slopes of # 1 °.
The improvement for smaller surface slopes fits to the target function to adjust the tool axis perpendicular to the local slope. It is not necessary to have a very precise signal for larger surface slopes, as long as it can be used to lead back to the more precise center measurement. Currently, the major contributor to systematic and random deviations is the spot on the camera itself. By aiming for higher resolution, the beam of about D 2 4 mm is imaged onto the camera chip. Thereby, interference patterns from the coherent laserdiodes are visible. These patterns are very sensitive to vibration and small surface features such as roughness. For the applied thresholding and centroid calculation this leads to shifts of the reconstructed centers, which subsequently lead to deviations for the reconstructed angles.
5 Discussion
The performed tests and their results confirm the successful integration of surface focus- and angular-control into a small optical exposure and measurement tool head. Thereby, the angular measurement with the evaluation of a highly coherent spot shows good robustness against axial defocus from the surface. This also stems from the axially separated focii of the red measurement lasers that are used for the angular measurement. Although the overlay of these two beams leads to a higher temporal incoherence, the resulting image still shows significant coherent diffraction patterns. Finding a common robust threshold for these turned out to be not feasible. Therefore, it is anticipated, that the angular measurement would improve drastically by adding another tube lens into the camera beam-path( comp. Fig. 12). The camera does not necessarily need to be in this lens’ focal plane, but only requires a smaller spot on the camera. This enlarges the angular measurement range and improves the angular resolution. The latter profits from a higher spatial incoherence and under-sampling of the diffraction pattern by the finite camera pixels.
Regarding the chromatic differential confocal measurement setup, the new design methodology has been proven useful. The setup shows single-digit axial resolution from a compact beam-path. This axial resolution was anticipated to be higher, because of the integrated lock-in-filters and the realized beam-path with axial confocal defocii close to the assumed optimum. For the simple difference D aperiodic signal fluctuations prevent higher axial resolution. Their origin lies within the employed fiber-coupled laserdiodes, as they are prone to back-reflections from e. g. their own fiber facet.
In other differential confocal setups, such power fluctuation can also appear. However, these setups usually only employ one light source. Subsequently, all defocused confocal detectors acquire the same fluctuation. The FES of equation( 2) then allows for ideal equal signal-processing,