1. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Schematic layout of the interferometer setup. (a) T: test beam; R: reference beam; PSA: point source array; BT and BR: computer- controlled shutters; BS: beamsplitter; L1: collimation lens; L2: reference-beam focusing lens; TS: transmission sphere; TO: test object; B: interferometer aperture; L3: imaging lens; C: camera. (b) Tilted test beams, showing the extended test angular spectrum of the interferometer (experimental). Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

2. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Problem definition. The test surface must be brought to a predefined measurement position with respect to the interferometer. The coordinate system of the positioning stage is given by {xS,yS,zS}, and that of the interferometer by {xI,yI,zI}. An air bearing positioning stage corrects the xyz position of the test object, while the tilt is controlled by a tip-tilt stage. Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

3. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Lateral displacement of the surface. (a) Camera image for the test object on the interferometer axis. Pattern obtained for (b) 8-mm and (c) 15-mm lateral x displacement. Although a large deviation from the axial position is present, light from the source array reaches the camera. (d) Nearly centered asphere once the central source fills the camera. (e) and (f), same as (d) but with a lateral displacement of the surface. The lack of symmetry shows the misalignment of the surface. Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

4. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Flow chart of the positioning procedure. The first stage consists of a coarse positioning of the surface, based mainly on image processing. Once the coarse alignment is finished, the interferometric correction of the surface starts, based on Eqs.. A security protocol is necessary during the coarse alignment to avoid collisions with the transmission sphere. Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

5. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Local null-test condition on the aspherical surface A. (a) Normal to the surface with the test beam focused at the point O, which is the cat’s-eye position of the test beam; V: apex of the aspheric surface; P(xP,zP) intersection of the normal to the asphere through O and the aspherical surface; tP: tangent to the asphere at P; nP: normal to the asphere at P; RP: radius of the tangent circle to the asphere; CQ: interferometer test beam. (b) The aspheric deviation of the surface from the test beam CQ is given by the distance QP¯. Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

6. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Aspheric departure of the surface. (a) Aspheric departure from the test beam [distance QP¯ in Fig. ]. (b) Slope of the departure shown in (a). The shaded areas are vignetted at the interferometer aperture. (c) Curvature of the departure, indicating the sensitivity of the slope variation and thus the increase of fringe density as we move on the surface. (d) Annular interferogram with lateral misalignment of the test surface. Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

7. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Slope deviation between incident test beam and normal to the aspheric surface: n and t are the normal and tangent to the aspherical surface at P respectively; r is a ray incident at P but not necessarily normal to the surface. Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

8. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Slope deviation after the transmission sphere. The ray with deviation δ on the aspheric surface is imaged on the interferometer detector after the transmission sphere with a slope δk. Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

9. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Phase deviation (modulo 2π) seen on the camera as the test surface is (a) decentered or (b) displaced along the axis of the interferometer. Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

10. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Geometry definition for the two-dimensional alignment problem. Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

11. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Results of the computer-controlled positioning over 10 measurements. The plots depict the final position for 10 random different start positions of the test object in a volume of 15×20×5mm3. The reproducibility for the axes x, y, and z is shown in (a), (b), and (c), respectively. The maximum deviation (PV) amounts to 1.0μm. Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

12. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Results of the manual positioning over 10 measurements. The same start positions as in Fig. are employed, but now the final position shows a maximal dispersion of 25μm (x axis). Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051

13. Date of download: 6/3/2016 Copyright © 2016 SPIE. All rights reserved. Evolution of the position after each iteration. Approximately 10 to 15 iterations are required to reach the final position of the test surface. Figure Legend: From: Automated surface positioning for a non-null test interferometer Opt. Eng. 2010;49(9):095602-095602-12. doi:10.1117/1.3488051