Results of a Sensitivity Analysis for the Tilted-Wave Interferometer

Abstract

IntroductionTilted-wave interferometry (TWI) is a novel measurement technique for the highly accurate optical measurement of aspheres and freeform surfaces [1][2][3][4][5][6][7]. It combines interferometric measurements with ray tracing and mathematical reconstruction procedures. The first results of a sensitivity analysis were given in [8]. The results support the feasibility of the basic TWI principle, and good reconstruction results were obtained for surfaces whose deviations from the design topography are described by Zernike polynomials. The TWI reconstruction procedure is divided into two parts. The first part reconstructs the long-wave deviation of the specimen from its design form. The second part aims at the reconstruction of high spatial frequencies [7]. This paper investigates the robustness of the long-wave TWI reconstruction method. Therefore, we extend the sensitivity analysis to surfaces whose deviation from their design topography no longer corresponds to a Zernike polynomial. We utilize a simulation environment that was developed at PTB for assessing optical measuring systems [8,9]. Reconstruction accuracies are then investigated in dependence on the variation of the specimen's topography and the measurement noise. Different variations of the specimen's topography with different spatial frequencies are studied. Furthermore, the influence of the deviation of the vertex radius of the design topography is investigated. Throughout all simulations no errors in the characterization of the TWI were assumed to be present which corresponds to a perfect calibration. Mathematical Reconstruction ProcedureThe TWI reconstruction principle for the basic form of the specimen utilizes measurements of the optical path length (OPL) in order to calculate the deviation of the surface under test from its design topography. The idea is that small changes of the surface topography lead to small changes in the OPLs. The long-wave

Development of a metrological reference system for the form measurement of aspheres and freeform surfaces based on a tilted-wave interferometer

“…Therefore, in the approach presented here, the model parameters α sought in equation ( 4) contain the coefficients of the two wavefront manipulators E Q and E P (currently 3696 parameters), the deviations of the sphere positions in x-, yand z-direction (currently 139 × 3 = 417 parameters), and the unknown patch offsets O (typically in the range of several hundred parameters). This high-dimensional nonlinear inverse problem of the model correction is solved by means of a trustregion-reflective algorithm using the Jacobian matrix that contains the partial derivatives of the OPDs with respect to the parameters to be optimized and is calculated analytically for each of the parameters mentioned from the ray path data as shown in [8].…”

“…In this way, the distance measured by the DMI between the CP and the MP at the center point of the SUT can be applied as a constraint (known surface position) and is not shifted during surface reconstruction. To solve the inverse reconstruction problem, the partial derivatives of the OPDs of the source-pixel pairs chosen with respect to the Zernike polynomial coefficients are analytically calculated from the ray path data as shown in [8,10]. Additionally, the position of the SUT is optimized by using the partial derivatives of the OPDs with regard to the position and orientation of the SUT.…”

“…In 2011, the NMI of Germany, the Physikalisch-Technische Bundesanstalt (PTB), began to develop a metrological reference system for the optical form measurement of aspheres and freeform surfaces that would allow traceable, contact-free form measurements with low uncertainty in the range of some ten nanometers [7][8][9][10][11]. The system at PTB is based on the optical tilted-wave interferometer (TWI) proposed by scientists at the University of Stuttgart and further developed by them within the scope of different projects, as well as by PTB and Mahr GmbH.…”

“…Topics addressed in previous work with respect to tiltedwave interferometry include the demonstration and investigation of the basic idea of the TWI [7,8,[13][14][15][16], the analytical calculation of the Jacobian matrices of the numerical model for the optical setup of the TWI [17], improvements to the model correction (calibration) of the interferometer [18,19], development of a new method of generating the point light sources [20], reconstructing the high-frequency components [21], investigations on the improvement of absolute form measurements [22], a method for simultaneous topography reconstruction and elimination of calibration errors [23], and the investigation of different error sources on the reconstruction result [19,[24][25][26]. Furthermore, an initial concept for the development of traceable optical form measurements based on tilted-wave interferometry was presented in [11].…”