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J. Eur. Opt. Society-Rapid Publ. 21, 19( 2025)
resolvable interference fringes without direct information about the form deviation of the specimen. Instead, to reconstruct the specimen from the interferograms, an inverse problem has to be solved. For this purpose, a digital twin of the measurement setup, including the interferometer and a virtual specimen is needed to simulate the interferograms for the design of the specimen. With this, optical path length differences are simulated by ray tracing and ray aiming [ 14 ]. The simulated data are then compared to the data that are extracted from the measured interferograms [ 15, 16 ]. The form of the surface under test is then reconstructed by solving an inverse problem minimizing the difference between the simulated and measured data. Since a model of the setup is used, the model has to be adapted to the real system. This is done in a special calibration procedure that utilizes well-known reference specimens in a large number of measurement positions. For a detailed description of the digital twin modeling, the model correction, and the reconstruction process that is utilized at the Physikalisch-Technische Bundesanstalt( PTB), please refere to [ 17 ].
However, as with interferometric form measurement in general, TWI measurements also suffer from ambiguities between surface form errors and misalignment of the specimen [ 18 ]. For TWI measurements, the form measurement accuracy is especially affected by misalignment along the optical axis of the interferometer [ 19 ]. In order to resolve this ambiguity, additional external positioning information can be used to better align the surface positioning between the experiment and the modeling. Previous simulation studies have shown that a single absolute length information with an accuracy of 100 nm or better could reduce the surface form measurement error contribution of the alignment down to a few nanometers root-mean-square( RMS) [ 19 ].
To achieve this, accurate and repeatable specimen positioning methods, especially for positioning along the optical axis( z), are required. Therefore, at PTB, a multi step approach for the specimen positioning in a TWI is developed: First, the specimen is aligned into the so-called Cat’ s Eye position, which is a common reference position in interferometric form measurement [ 20 ]. The Cat’ s Eye position is defined as the position in which the focus of the interferometer’ s objective coincides with the specimen’ s surface. Because of that, all light gets reflected in a single spot, making the Cat’ s Eye theoretically independent of the measurement surface form. Therefore, the Cat’ s Eye only depends on the optical system of the interferometer, which leads to a well-defined criterion for surface positioning. From there, the specimen is moved into the measurement position while the positioning stage is tracked by a distance measuring interferometer( DMI). This ties the positioning accuracy of the axial positioning in measurement position effectively to the positioning accuracy of the Cat’ s Eye position. In a next step, the measurement position is used to align the surface laterally( x and y) as well as its tilt( a and b, depending on the degrees of freedom of the surface under test). In measurement position, the sensitivity coefficients between the optical path length difference changes with regard to positioning can be used. This step will be investigated in future work in more detail.
Figure 1. Specimen positioning within the TWI.( a) Specimen positioned in the Cat’ s Eye reference position.( b) Specimen moved to the measurement position, tracked by a distancemeasuring interferometer( DMI).
Since a lateral repositioning also has influence on the axial position, after the positioning optimization in measurement position, the specimen is moved back to the Cat’ s Eye reference position and the axial positioning procedure is repeated without changing the lateral position. Finally, the specimen is moved to the measurement position while the positioning stage is tracked by the DMI. The two important positions are the Cat’ s Eye position and the measurement position illustrated in Figure 1.
In this work, we present a method of adjusting the specimen into the Cat’ s Eye position in the TWI and show its repeatability. In Section 2 the measurement system and the utilized methods are described. In Section 3, the results of the repeated measurements are presented and discussed. Finally, Section 4 sets the results of our approach into the context of the TWI form reconstruction and discusses future challenges.
2 Method
2.1 Tilted-wave interferometer setup
Both for analyzing the interferograms of specimens in or near the Cat’ s Eye reference position and for generating test data for the positioning algorithm, virtual interference images are generated utilizing a digital twin of the TWI at PTB. The digital twin consists of a ray tracing simulation that simulates every optical surface in the instrument. It is built in MATLAB and utilizes the in-house developed ray tracing and ray aiming library SimOptDevive [ 14 ]. For further details on the implementation of the TWI simulation and the TWI measurement procedure, please refer to [ 17 ]. The general optical setup of the TWI, and therefore also of the digital twin, is depicted in Figure 2a.
The optical setup of the TWI utilizes a Nd: YAG laser with a wavelength of k = 532.3 nm as illumination light source. The light is divided by a beamsplitter into a primary object wavefront and a reference wavefront. The reference wavefront is guided into an optical fiber, while the primary object wavefront is widened, collimated, and cast onto a microlens array [ 21 ]. The microlens array then generates