Abstract

Tilted-wave interferometry (TWI) is a novel optical measurement principle for the measurement of aspherical surfaces. For the reconstruction of the wavefront and the surface under test, respectively, perturbation methods are applied, which require the calculation of the Jacobian matrix. For the practical use of the instrument, a fast and exact calculation of the Jacobian matrices is crucial, since this strongly influences the calculation times of the TWI. By applying appropriate approaches in optical perturbation methods we are able to calculate the required Jacobian matrices analytically when the nominal optical path through the system is given. As a result, calculation times for the TWI can be considerably reduced. We finally illustrate the improved TWI procedure and apply methods of optimal design to determine optimal positions of the surface under test. For such applications the fast calculation of the Jacobian matrices is essential.

© 2014 Optical Society of America

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References

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  6. M. F. Kuechel, “Absolute measurement of rotationally symmetrical aspheric surfaces,” OSA Optical Fabrication and Testing, Rochester, OFTuB5 (2006).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2013 (4)

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” J. Europ. Opt. Soc. Rap. Public. 8, 13074 (2013).
[Crossref]

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Comparison of alignment errors in asphere metrology between an interferometric null-test measurement and a non-null measurement with the tilted-wave-interferometer,” Proc. SPIE 8884, Optifab 2013, 88840T (2013).
[Crossref]

P. D. Lin, “Design of optical systems using derivatives of a rays: derivatives of variable vector of spherical boundary surfaces with respect to system variable vector,” Appl. Opt. 52, 7271–7287 (2013).
[Crossref] [PubMed]

P. D. Lin, “Analysis and design of prisms using the derivatives of a ray. Part II: the derivatives of boundary variable vector with respect to system variable vector,” Appl. Opt. 52, 4151–4162 (2013).
[Crossref] [PubMed]

2011 (2)

C. Tian, Y. Yang, T. Wei, and Y. Zhuo, “Nonnull interferometer simulation for aspheric testing based on ray tracing,” Appl. Opt. 50, 3559–3569 (2011).
[Crossref] [PubMed]

A. Wiegmann, M. Stavridis, M. Walzel, F. Sievert, T. Zeschke, M. Schulz, and C. Elster, “Accuracy evaluation for sub-aperture interferometry measurements of a synchrotron mirror using virtual experiments,” Precision Engineering 35, 183–190 (2011).
[Crossref]

2009 (2)

D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17, 7025–7035 (2009).
[Crossref] [PubMed]

E. Garbusi and W. Osten, “Perturbation methods in optics: application to the interferometric measurement of surfaces,” J. Opt. Soc. Am. A 26, 122538–2549 (2009).
[Crossref]

2008 (1)

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 242973– 2975 (2008).
[Crossref]

2004 (3)

2003 (1)

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces, ” Proc. SPIE 5188, 296–307 (2003).
[Crossref]

2001 (1)

1997 (1)

1988 (1)

1985 (1)

1982 (1)

1970 (1)

1968 (1)

1957 (1)

Andersen, T. B.

Baer, G.

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” J. Europ. Opt. Soc. Rap. Public. 8, 13074 (2013).
[Crossref]

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Comparison of alignment errors in asphere metrology between an interferometric null-test measurement and a non-null measurement with the tilted-wave-interferometer,” Proc. SPIE 8884, Optifab 2013, 88840T (2013).
[Crossref]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Sensitivity analysis of tilted-wave interferometer asphere measurements using virtual experiments,” in Modeling Aspects in Optical Metrology IV, B. Bodermann, K. Frenner, and R. M. Silver, eds., Proc. SPIE8789, 878907 (2013).
[Crossref]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Results of a Sensitivity Analysis for the Tilted-Wave Interferometer,” in Fringe 2013, 7th International Workshop on Advanced Optical Imaging and Metrology, W. Osten, ed., (Springer, 2014), pp. 701–706.

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics, (Dover Publications, Inc., 1993), pp. 8–11.

Burden, R. L.

R. L. Burden and J. D. Faires, Numerical Analysis, 9th edition, (Brooks/Cole, 2011), Chap. 4.1.

Dörband, B.

Dumas, P.

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces, ” Proc. SPIE 5188, 296–307 (2003).
[Crossref]

Elster, C.

A. Wiegmann, M. Stavridis, M. Walzel, F. Sievert, T. Zeschke, M. Schulz, and C. Elster, “Accuracy evaluation for sub-aperture interferometry measurements of a synchrotron mirror using virtual experiments,” Precision Engineering 35, 183–190 (2011).
[Crossref]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Results of a Sensitivity Analysis for the Tilted-Wave Interferometer,” in Fringe 2013, 7th International Workshop on Advanced Optical Imaging and Metrology, W. Osten, ed., (Springer, 2014), pp. 701–706.

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Sensitivity analysis of tilted-wave interferometer asphere measurements using virtual experiments,” in Modeling Aspects in Optical Metrology IV, B. Bodermann, K. Frenner, and R. M. Silver, eds., Proc. SPIE8789, 878907 (2013).
[Crossref]

Faires, J. D.

R. L. Burden and J. D. Faires, Numerical Analysis, 9th edition, (Brooks/Cole, 2011), Chap. 4.1.

Feder, D. P.

Fleig, J.

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces, ” Proc. SPIE 5188, 296–307 (2003).
[Crossref]

Forbes, G. W.

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces, ” Proc. SPIE 5188, 296–307 (2003).
[Crossref]

Fortmeier, I.

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Results of a Sensitivity Analysis for the Tilted-Wave Interferometer,” in Fringe 2013, 7th International Workshop on Advanced Optical Imaging and Metrology, W. Osten, ed., (Springer, 2014), pp. 701–706.

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Sensitivity analysis of tilted-wave interferometer asphere measurements using virtual experiments,” in Modeling Aspects in Optical Metrology IV, B. Bodermann, K. Frenner, and R. M. Silver, eds., Proc. SPIE8789, 878907 (2013).
[Crossref]

Garbusi, E.

E. Garbusi and W. Osten, “Perturbation methods in optics: application to the interferometric measurement of surfaces,” J. Opt. Soc. Am. A 26, 122538–2549 (2009).
[Crossref]

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 242973– 2975 (2008).
[Crossref]

Grappinger, R. O.

Greivenkamp, J. E.

Gross, H.

H. Gross, Handbook of Optical Systems, Volume 3: Aberration Theory and Correction of Optical Systems, (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2007), Chap. 32, pp. 307–309.

Kidger, M. J.

M. J. Kidger, Fundamental Optical Design, (SPIE Press, 2002), pp. 4–5.

Koliopoulos, C.

Kuechel, M. F.

M. F. Kuechel, “Absolute measurement of rotationally symmetrical aspheric surfaces,” OSA Optical Fabrication and Testing, Rochester, OFTuB5 (2006).
[Crossref]

Lawrence, G.

Lin, P. D.

Lindlein, N.

Liu, D.

Liu, Y.

Luo, Y.

Murphy, P. E.

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces, ” Proc. SPIE 5188, 296–307 (2003).
[Crossref]

Osten, W.

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Comparison of alignment errors in asphere metrology between an interferometric null-test measurement and a non-null measurement with the tilted-wave-interferometer,” Proc. SPIE 8884, Optifab 2013, 88840T (2013).
[Crossref]

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” J. Europ. Opt. Soc. Rap. Public. 8, 13074 (2013).
[Crossref]

E. Garbusi and W. Osten, “Perturbation methods in optics: application to the interferometric measurement of surfaces,” J. Opt. Soc. Am. A 26, 122538–2549 (2009).
[Crossref]

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 242973– 2975 (2008).
[Crossref]

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, “Computer-generated holograms in interferometric testing,” Optical Engineering,  43, 2534–2540 (2004).
[Crossref]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Results of a Sensitivity Analysis for the Tilted-Wave Interferometer,” in Fringe 2013, 7th International Workshop on Advanced Optical Imaging and Metrology, W. Osten, ed., (Springer, 2014), pp. 701–706.

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Sensitivity analysis of tilted-wave interferometer asphere measurements using virtual experiments,” in Modeling Aspects in Optical Metrology IV, B. Bodermann, K. Frenner, and R. M. Silver, eds., Proc. SPIE8789, 878907 (2013).
[Crossref]

Pfund, J.

Pruss, C.

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” J. Europ. Opt. Soc. Rap. Public. 8, 13074 (2013).
[Crossref]

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Comparison of alignment errors in asphere metrology between an interferometric null-test measurement and a non-null measurement with the tilted-wave-interferometer,” Proc. SPIE 8884, Optifab 2013, 88840T (2013).
[Crossref]

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 242973– 2975 (2008).
[Crossref]

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, “Computer-generated holograms in interferometric testing,” Optical Engineering,  43, 2534–2540 (2004).
[Crossref]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Sensitivity analysis of tilted-wave interferometer asphere measurements using virtual experiments,” in Modeling Aspects in Optical Metrology IV, B. Bodermann, K. Frenner, and R. M. Silver, eds., Proc. SPIE8789, 878907 (2013).
[Crossref]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Results of a Sensitivity Analysis for the Tilted-Wave Interferometer,” in Fringe 2013, 7th International Workshop on Advanced Optical Imaging and Metrology, W. Osten, ed., (Springer, 2014), pp. 701–706.

Pukelsheim, F.

F. Pukelsheim, Optimal Design of Experiments (Classics in Applied Mathematics, Vol. 50), (Society for Industrial and Applied Mathematics, 2006).

Reichelt, S.

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, “Computer-generated holograms in interferometric testing,” Optical Engineering,  43, 2534–2540 (2004).
[Crossref]

Rimmer, M.

Schindler, J.

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Comparison of alignment errors in asphere metrology between an interferometric null-test measurement and a non-null measurement with the tilted-wave-interferometer,” Proc. SPIE 8884, Optifab 2013, 88840T (2013).
[Crossref]

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” J. Europ. Opt. Soc. Rap. Public. 8, 13074 (2013).
[Crossref]

Schulz, G.

G. Schulz, Aspherical Surfaces, (North-Holland Physics Publishing, Amsterdam, 1988), Chap. IV, pp. 349–415.

Schulz, M.

A. Wiegmann, M. Stavridis, M. Walzel, F. Sievert, T. Zeschke, M. Schulz, and C. Elster, “Accuracy evaluation for sub-aperture interferometry measurements of a synchrotron mirror using virtual experiments,” Precision Engineering 35, 183–190 (2011).
[Crossref]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Results of a Sensitivity Analysis for the Tilted-Wave Interferometer,” in Fringe 2013, 7th International Workshop on Advanced Optical Imaging and Metrology, W. Osten, ed., (Springer, 2014), pp. 701–706.

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Sensitivity analysis of tilted-wave interferometer asphere measurements using virtual experiments,” in Modeling Aspects in Optical Metrology IV, B. Bodermann, K. Frenner, and R. M. Silver, eds., Proc. SPIE8789, 878907 (2013).
[Crossref]

Schwider, J.

Sievert, F.

A. Wiegmann, M. Stavridis, M. Walzel, F. Sievert, T. Zeschke, M. Schulz, and C. Elster, “Accuracy evaluation for sub-aperture interferometry measurements of a synchrotron mirror using virtual experiments,” Precision Engineering 35, 183–190 (2011).
[Crossref]

Stavridis, M.

A. Wiegmann, M. Stavridis, M. Walzel, F. Sievert, T. Zeschke, M. Schulz, and C. Elster, “Accuracy evaluation for sub-aperture interferometry measurements of a synchrotron mirror using virtual experiments,” Precision Engineering 35, 183–190 (2011).
[Crossref]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Sensitivity analysis of tilted-wave interferometer asphere measurements using virtual experiments,” in Modeling Aspects in Optical Metrology IV, B. Bodermann, K. Frenner, and R. M. Silver, eds., Proc. SPIE8789, 878907 (2013).
[Crossref]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Results of a Sensitivity Analysis for the Tilted-Wave Interferometer,” in Fringe 2013, 7th International Workshop on Advanced Optical Imaging and Metrology, W. Osten, ed., (Springer, 2014), pp. 701–706.

Stone, B. D.

Tian, C.

Tiziani, H. J.

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, “Computer-generated holograms in interferometric testing,” Optical Engineering,  43, 2534–2540 (2004).
[Crossref]

B. Dörband and H. J. Tiziani, “Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors,” Appl. Opt. 24, 2604–2611 (1985).
[Crossref] [PubMed]

Walzel, M.

A. Wiegmann, M. Stavridis, M. Walzel, F. Sievert, T. Zeschke, M. Schulz, and C. Elster, “Accuracy evaluation for sub-aperture interferometry measurements of a synchrotron mirror using virtual experiments,” Precision Engineering 35, 183–190 (2011).
[Crossref]

Wang, L.

Wei, T.

Wiegmann, A.

A. Wiegmann, M. Stavridis, M. Walzel, F. Sievert, T. Zeschke, M. Schulz, and C. Elster, “Accuracy evaluation for sub-aperture interferometry measurements of a synchrotron mirror using virtual experiments,” Precision Engineering 35, 183–190 (2011).
[Crossref]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Results of a Sensitivity Analysis for the Tilted-Wave Interferometer,” in Fringe 2013, 7th International Workshop on Advanced Optical Imaging and Metrology, W. Osten, ed., (Springer, 2014), pp. 701–706.

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Sensitivity analysis of tilted-wave interferometer asphere measurements using virtual experiments,” in Modeling Aspects in Optical Metrology IV, B. Bodermann, K. Frenner, and R. M. Silver, eds., Proc. SPIE8789, 878907 (2013).
[Crossref]

Yang, Y.

Zeschke, T.

A. Wiegmann, M. Stavridis, M. Walzel, F. Sievert, T. Zeschke, M. Schulz, and C. Elster, “Accuracy evaluation for sub-aperture interferometry measurements of a synchrotron mirror using virtual experiments,” Precision Engineering 35, 183–190 (2011).
[Crossref]

Zhuo, Y.

Appl. Opt. (10)

B. Dörband and H. J. Tiziani, “Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors,” Appl. Opt. 24, 2604–2611 (1985).
[Crossref] [PubMed]

Y. Liu, G. Lawrence, and C. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. 27, 4504–4513 (1988).
[Crossref] [PubMed]

R. O. Grappinger and J. E. Greivenkamp, “Iterative reverse optimization procedure for calibration of aspheric wavefront measurements on a nonnull interferometer,” Appl. Opt. 43, 5152–5161 (2004).
[Crossref]

J. E. Greivenkamp and R. O. Grappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt. 43, 5143–5151 (2004).
[Crossref] [PubMed]

C. Tian, Y. Yang, T. Wei, and Y. Zhuo, “Nonnull interferometer simulation for aspheric testing based on ray tracing,” Appl. Opt. 50, 3559–3569 (2011).
[Crossref] [PubMed]

J. Pfund, N. Lindlein, and J. Schwider, “Nonnull testing of rotationally symmetric aspheres: a systematic error assessment,” Appl. Opt. 40, 439–446 (2001).
[Crossref]

T. B. Andersen, “Optical aberration functions: derivatives with respect to axial distances for symmetrical systems,” Appl. Opt. 21, 1817–1823 (1982).
[Crossref] [PubMed]

P. D. Lin, “Design of optical systems using derivatives of a rays: derivatives of variable vector of spherical boundary surfaces with respect to system variable vector,” Appl. Opt. 52, 7271–7287 (2013).
[Crossref] [PubMed]

P. D. Lin, “Analysis and design of prisms using the derivatives of a ray. Part II: the derivatives of boundary variable vector with respect to system variable vector,” Appl. Opt. 52, 4151–4162 (2013).
[Crossref] [PubMed]

M. Rimmer, “Analysis of Perturbed Lens Systems,” Appl. Opt. 9, 533–537 (1970).
[Crossref] [PubMed]

J. Europ. Opt. Soc. Rap. Public. (1)

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” J. Europ. Opt. Soc. Rap. Public. 8, 13074 (2013).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

B. D. Stone, “Perturbations of optical systems,” J. Opt. Soc. Am. A 14, 2837–2849 (1997).
[Crossref]

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[Crossref]

Opt. Express (1)

Opt. Lett. (1)

E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 242973– 2975 (2008).
[Crossref]

Optical Engineering (1)

C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, “Computer-generated holograms in interferometric testing,” Optical Engineering,  43, 2534–2540 (2004).
[Crossref]

Precision Engineering (1)

A. Wiegmann, M. Stavridis, M. Walzel, F. Sievert, T. Zeschke, M. Schulz, and C. Elster, “Accuracy evaluation for sub-aperture interferometry measurements of a synchrotron mirror using virtual experiments,” Precision Engineering 35, 183–190 (2011).
[Crossref]

Proc. SPIE (2)

J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces, ” Proc. SPIE 5188, 296–307 (2003).
[Crossref]

G. Baer, J. Schindler, C. Pruss, and W. Osten, “Comparison of alignment errors in asphere metrology between an interferometric null-test measurement and a non-null measurement with the tilted-wave-interferometer,” Proc. SPIE 8884, Optifab 2013, 88840T (2013).
[Crossref]

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[Crossref]

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[Crossref]

I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Results of a Sensitivity Analysis for the Tilted-Wave Interferometer,” in Fringe 2013, 7th International Workshop on Advanced Optical Imaging and Metrology, W. Osten, ed., (Springer, 2014), pp. 701–706.

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Figures (6)

Fig. 1
Fig. 1 a) Optical design of the TWI. b) Two bundles of rays, each starting at a different point source, are traced through the optical system of the TWI. Due to the different tilts of the wavefronts leaving the objective lens, different parts of the specimen under test can be measured. The rays of the reference arm are not shown here.
Fig. 2
Fig. 2 (a) Schematic sketch of a small shift of a surface: Small perturbations Δs induce a path length change of ΔL = nr(exΔx + eyΔy + ezΔz) − n′r(e′xΔx + e′yΔy + e′zΔz), see also [26], [27]. (b) Tilt of a single surface: Definition of the rotation axes. (c) Schematic sketch of a ray path through an optical system consisting of several surfaces. Since a translation of the element group leads to the same perturbation of all surfaces of the element, the resulting OPL perturbation of the paths inside the element group cancel each other out.
Fig. 3
Fig. 3 Profile of specimen under test (design surface 1).
Fig. 4
Fig. 4 Deviation of the specimen from its design and residual of reconstruction result.
Fig. 5
Fig. 5 Comparison between the Jacobian matrices established by numerical differential quotients and by calculation from nominal ray path data.
Fig. 6
Fig. 6 Optimal specimen positioning: The uncertainty values for each estimated parameter (column) of the specimen reconstruction are shown for different surface positions (row) for design surface 1 (a) and design surface 2 (b). The color of each entry displays the uncertainty of an estimated Zernike coefficient for a defined specimen position.

Tables (3)

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Table 1 Design parameters of the aspherical surfaces under test used for the simulations.

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Table 2 Computing time for a simple example performed on a computational server for a typical TWI reconstruction problem (13,350 rays, 153 Zernike coefficients, design surface with parameters described in Tab. 1, no parallel jobs).

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Table 3 Computing time for a simple example performed on a computational server for a typical TWI calibration problem (63,152 rays, 1,368 parameters, surface under test: sphere with radius R = 10 mm, 259 sphere positions, no parallel jobs).

Equations (16)

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z ( x , y ) = T 0 ( x , y ) + j = 1 m c j Z j ( x , y ) .
Δ L = Jc ,
J i j = L i c j .
J i j = L i ( c j + Δ c j ) L i ( c j ) Δ c j .
Δ L = n r ( e x Δ x + e y Δ y + e z Δ z ) n r ( e x Δ x + e y Δ y + e z Δ z ) ,
( L x L y L z ) = ( n r e x n r e x n r e y n r e y n r e z n r e z ) .
L p k = L z z p k ,
Δ L = Δ s T n ( n r e T n n r e T n ) ,
L α = ( ( R α 1 α p ) T n ) ( n r e T n n r e T n ) ,
L α = l N ( L α ) l ,
L p k = i = 1 N L i p k .
L i c j = L i z z c j .
z c j = Z j ( x , y ) ,
χ 2 = ( J c ^ Δ L ) T V Δ L 1 ( J c ^ Δ L )
V c ^ = ( J T V Δ L 1 J ) 1 ,
V c ^ = σ 2 ( J T J ) 1 .

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