Aspheres and freeforms are surfaces that deviate significantly from their spherical base forms. Since their forms can be very complex, manufacturing and measuring them is a very challenging task. Nevertheless, the high number of possible form variations can be used to improve the imaging properties or to reduce the number of elements in an optical system. For this reason, aspheres have become key components of modern imaging systems. Due to the sophisticated measuring equipment required for production control, a strong demand exists in research and industry for traceability to national reference standards and reference systems that guarantee the correctness and comparability of measurement results.

To this end, a collaboration was initiated in 2011 between the Institute of Technical Optics (ITO) of the University of Stuttgart, the German company Mahr GmbH, and PTB; the goal of the collaboration is the joint further development of the tilted-wave interferometer. At PTB, working groups 4.21 (Form and Wavefront Metrology) and 8.42 (Data Analysis and Measurement Uncertainty) are involved in this project.

The principle of the tilted-wave interferometer was invented and patented at ITO in 2006 [1,2]. The tilted-wave interferometer is a special interferometer for the form measurement of aspheres and freeform surfaces that combines interferometric measurements with optical perturbation theory and mathematical evaluation procedures. A scheme of its basic setup is shown in Fig. 1. The setup is based on a Twyman-Green interferometer but differs from it in two main points: A microlens array (PLQA) is used in the object arm, thus allowing a 2D point light source array to be used for the illumination instead of a single-point light source. Due to this illumination scheme, differently tilted wavefronts are generated behind the collimator and then used for the illumination of the surface under test. Furthermore, an additional beam stop is used in the Fourier plane of the imaging optics. This beam stop acts as a filter and blocks all rays that would produce too high a fringe density at the camera, thereby preventing sub-sampling effects. By means of this concept, depending on the local slope of the surface under test, different sources produce evaluable measurement data on the camera.

In order to reconstruct the form of the surface under test (SUT) from this data, a special evaluation procedure is needed. The basic idea behind this procedure is based on the assumption that every deviation of the SUT leads to a characteristic change of the optical path length differences (OPDs) that are measured on the camera. To reconstruct the form of the SUT, the OPDs are measured by means of the interferometer setup described above and then compared to OPDs that were simulated for the design of the SUT using a calibrated model of the interferometer. The deviation of the SUT from its design is then determined from the difference between the measured and simulated OPDs. To do so, it is necessary to solve a nonlinear inverse problem.

Within the scope of the joint further development of the interferometer, Mahr GmbH is focusing on developing a commercial measurement system. Fig. 2 shows the first prototype of the measurement system, which is located at department 4.2 of PTB.

The main focus of PTB in this collaboration is to develop a metrological reference system for measuring aspheres and freeform surfaces, and to perform a complete uncertainty evaluation based on the knowledge of the hardware and software of the system.

Virtual experiments are particularly well suited for analyzing the uncertainty of such complex systems. They can be utilized to determine the main influence quantities separately from other parameters and disturbances in such a way that a stepwise analysis of the procedure is possible. To this end, a virtual TWI based on a physical modeling has been developed at PTB (see Fig. 3). This virtual TWI was modeled using the simulation tool SimOptDevice, which had been developed a few years previously in PTB working group 8.42 (Data Analysis and Measurement Uncertainty) and is based on object-oriented programming in Matlab. The simulation tool combines the geometric modeling of a measurement system with optical raytracing methods.

Initial sensitivity analyses of the basic TWI principle have shown promising results [3,4]. Furthermore, within the framework of PTB’s work, the TWI evaluation procedures were optimized: The method used to date for the numerical determination of the Jacobian matrices, which are necessary to solve the inverse problem, was replaced with a numerical calculation method [5,6]. Depending on the number of parameters, this method allows the inverse problem to be solved in considerably less time. This improvement not only means that the operator can save time, but also creates better possibilities for systematic investigations, which are very important for an evaluation of the measurement uncertainty.

[1]    E. Garbusi, C. Pruss, W. Osten, “Interferometer for precise and flexible asphere testing”, Opt. Lett. 33, (2008), pp. 2973-2975.
[2]    J. Liesener, E. Garbusi, C. Pruss, W. Osten, 2006: Verfahren und Messvorrichtung zur Vermessung einer optisch glatten Oberfläche, Deutsches Patentamt, DE102006057606.3, 2016.
[3]    I. Fortmeier,  M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, C. Elster,  Sensitivity analysis of tilted-wave interferometer asphere measurements using virtual experiments, Proceedings of SPIE 8789, (2013), pp. 878907.
[4]    I. Fortmeier,  M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, C. Elster, Results of a Sensitivity Analysis for the Tilted-Wave Interferometer, Osten, W. (ed.): Fringe 2013, Springer, (2014), pp. 701--706.
[5]    I. Fortmeier,  M. Stavridis, A. Wiegmann, Verfahren zum Optimieren eines simulierten optischen Systems,  Deutsches Patentamt, DE102014001323 B4, 2014.
[6]    I. Fortmeier,  M. Stavridis, A. Wiegmann, M. Schulz, W. Osten, C. Elster, Analytical Jacobian and its application to tilted-wave interferometry, Optics Express 22(18), (2014), pp. 21313--21325.