Abstract

A scanning probe consisting of a source and receive fiber pair is used to measure the phase difference between wave fronts scattered from the front and rear surfaces of an aspheric optic. This system can be thought of as a classical interferometer with an aperture synthesized from the data collected along the path of the probe. If the form of either surface is known, the other can be deduced. In contrast with classical interferometers, the method does not need test or null plates and has the potential to be integrated into the manufacturing process.

© 2003 Optical Society of America

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  1. E. Heynacher, “Aspheric optics. How they are made and why they are needed,” Phys. Technol. 10 (3), 124–131 (1979).
    [CrossRef]
  2. D. M. G. Stevens, “The application of optical techniques in aspheric surface assessment,” Int. J. Mach. Tools Manufact. 32 (1), 19–25 (1992).
    [CrossRef]
  3. G. Doughty, J. Smith, “Microcomputer-controlled polishing machine for very smooth and deep aspherical surfaces,” Appl. Opt. 26, 2421–2426 (1987).
    [CrossRef] [PubMed]
  4. R. O. Maschmeyer, C. A. Andrrysick, T. W. Geyer, H. E. Meissner, C. J. Parker, L. M. Sanford, “Precision molded-glass optics,” Appl. Opt. 22, 2410–2412 (1983).
    [CrossRef] [PubMed]
  5. D. G. Burns, “Null test for hyperbolic convex mirrors,” Appl. Opt. 22, 12–13 (1983).
    [CrossRef]
  6. B. Dorband, H. J. Tiziani, “Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors,” Appl. Opt. 24, 2604–2611 (1985).
    [CrossRef] [PubMed]
  7. M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1078 (1993).
    [CrossRef]
  8. P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
    [CrossRef]
  9. W. Quandou, Z. Zhongyu, Z. Xuejun, “Novel profilometer with dual digital length gauge for large aspherics measurement,” in Advanced Optical Manufacturing and Testing Technology 2000, L. Yang, H. M. Pollicove, Q. Xi, J. C. Wyant, eds., Proc. SPIE4231, 39–46 (2000).
    [CrossRef]
  10. T.-H. Tsai, K.-C. Fan, J.-I. Mou, “A variable-resolution optical profile measurement system,” Meas. Sci. Technol. 13, 190–197 (2002).
    [CrossRef]
  11. K. Ehrmann, A. Ho, K. Schindhelm, “A 3D optical profilometer using a compact disc reading head,” Meas. Sci. Technol. 9, 1259–1265 (1998).
    [CrossRef]
  12. T. Dawei, “In-process sensor for surface profile measurement applying a common-mode rejection technique,” Opt. Laser Technol. 27, 351–353 (1995).
    [CrossRef]
  13. M. Born, E. Wolf, “Elements of the theory of interference and interferometers,” in Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997), Chap. 7, pp. 256–367.
  14. J. P. Fitch, Synthetic Aperture Radar (Springer-Verlag, Berlin, 1987).
  15. C. Bradley, J. Jeswiet, “An optical surface texture sensor suitable for integration into a coordinate measuring machine,” Ann. CIRP 48, 459–462 (1999).
    [CrossRef]
  16. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” Appl. Opt. 11, 107–117 (1972).
  17. J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 9 (1), 112–147 (1998).
  18. C. C. Paige, M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM (Assoc. Comput. Mach.) Trans. Math. Software 8, 43–71 (1982).
    [CrossRef]

2002 (1)

T.-H. Tsai, K.-C. Fan, J.-I. Mou, “A variable-resolution optical profile measurement system,” Meas. Sci. Technol. 13, 190–197 (2002).
[CrossRef]

1999 (1)

C. Bradley, J. Jeswiet, “An optical surface texture sensor suitable for integration into a coordinate measuring machine,” Ann. CIRP 48, 459–462 (1999).
[CrossRef]

1998 (2)

K. Ehrmann, A. Ho, K. Schindhelm, “A 3D optical profilometer using a compact disc reading head,” Meas. Sci. Technol. 9, 1259–1265 (1998).
[CrossRef]

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 9 (1), 112–147 (1998).

1995 (1)

T. Dawei, “In-process sensor for surface profile measurement applying a common-mode rejection technique,” Opt. Laser Technol. 27, 351–353 (1995).
[CrossRef]

1993 (1)

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1078 (1993).
[CrossRef]

1992 (1)

D. M. G. Stevens, “The application of optical techniques in aspheric surface assessment,” Int. J. Mach. Tools Manufact. 32 (1), 19–25 (1992).
[CrossRef]

1987 (1)

1985 (1)

1984 (1)

P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
[CrossRef]

1983 (2)

1982 (1)

C. C. Paige, M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM (Assoc. Comput. Mach.) Trans. Math. Software 8, 43–71 (1982).
[CrossRef]

1979 (1)

E. Heynacher, “Aspheric optics. How they are made and why they are needed,” Phys. Technol. 10 (3), 124–131 (1979).
[CrossRef]

1972 (1)

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” Appl. Opt. 11, 107–117 (1972).

Andrrysick, C. A.

Born, M.

M. Born, E. Wolf, “Elements of the theory of interference and interferometers,” in Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997), Chap. 7, pp. 256–367.

Bradley, C.

C. Bradley, J. Jeswiet, “An optical surface texture sensor suitable for integration into a coordinate measuring machine,” Ann. CIRP 48, 459–462 (1999).
[CrossRef]

Burns, D. G.

Dawei, T.

T. Dawei, “In-process sensor for surface profile measurement applying a common-mode rejection technique,” Opt. Laser Technol. 27, 351–353 (1995).
[CrossRef]

Dorband, B.

Doughty, G.

Ehrmann, K.

K. Ehrmann, A. Ho, K. Schindhelm, “A 3D optical profilometer using a compact disc reading head,” Meas. Sci. Technol. 9, 1259–1265 (1998).
[CrossRef]

Fan, K.-C.

T.-H. Tsai, K.-C. Fan, J.-I. Mou, “A variable-resolution optical profile measurement system,” Meas. Sci. Technol. 13, 190–197 (2002).
[CrossRef]

Fitch, J. P.

J. P. Fitch, Synthetic Aperture Radar (Springer-Verlag, Berlin, 1987).

Geyer, T. W.

Ghiglia, D. C.

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” Appl. Opt. 11, 107–117 (1972).

Hariharan, P.

P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
[CrossRef]

Heynacher, E.

E. Heynacher, “Aspheric optics. How they are made and why they are needed,” Phys. Technol. 10 (3), 124–131 (1979).
[CrossRef]

Ho, A.

K. Ehrmann, A. Ho, K. Schindhelm, “A 3D optical profilometer using a compact disc reading head,” Meas. Sci. Technol. 9, 1259–1265 (1998).
[CrossRef]

Jeswiet, J.

C. Bradley, J. Jeswiet, “An optical surface texture sensor suitable for integration into a coordinate measuring machine,” Ann. CIRP 48, 459–462 (1999).
[CrossRef]

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 9 (1), 112–147 (1998).

Maschmeyer, R. O.

Mazzoni, A.

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1078 (1993).
[CrossRef]

Meissner, H. E.

Melozzi, M.

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1078 (1993).
[CrossRef]

Mou, J.-I.

T.-H. Tsai, K.-C. Fan, J.-I. Mou, “A variable-resolution optical profile measurement system,” Meas. Sci. Technol. 13, 190–197 (2002).
[CrossRef]

Oreb, B. F.

P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
[CrossRef]

Paige, C. C.

C. C. Paige, M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM (Assoc. Comput. Mach.) Trans. Math. Software 8, 43–71 (1982).
[CrossRef]

Parker, C. J.

Pezzati, L.

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1078 (1993).
[CrossRef]

Quandou, W.

W. Quandou, Z. Zhongyu, Z. Xuejun, “Novel profilometer with dual digital length gauge for large aspherics measurement,” in Advanced Optical Manufacturing and Testing Technology 2000, L. Yang, H. M. Pollicove, Q. Xi, J. C. Wyant, eds., Proc. SPIE4231, 39–46 (2000).
[CrossRef]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 9 (1), 112–147 (1998).

Romero, L. A.

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” Appl. Opt. 11, 107–117 (1972).

Sanford, L. M.

Saunders, M. A.

C. C. Paige, M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM (Assoc. Comput. Mach.) Trans. Math. Software 8, 43–71 (1982).
[CrossRef]

Schindhelm, K.

K. Ehrmann, A. Ho, K. Schindhelm, “A 3D optical profilometer using a compact disc reading head,” Meas. Sci. Technol. 9, 1259–1265 (1998).
[CrossRef]

Smith, J.

Stevens, D. M. G.

D. M. G. Stevens, “The application of optical techniques in aspheric surface assessment,” Int. J. Mach. Tools Manufact. 32 (1), 19–25 (1992).
[CrossRef]

Tiziani, H. J.

Tsai, T.-H.

T.-H. Tsai, K.-C. Fan, J.-I. Mou, “A variable-resolution optical profile measurement system,” Meas. Sci. Technol. 13, 190–197 (2002).
[CrossRef]

Wanzhi, Z.

P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, “Elements of the theory of interference and interferometers,” in Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997), Chap. 7, pp. 256–367.

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 9 (1), 112–147 (1998).

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 9 (1), 112–147 (1998).

Xuejun, Z.

W. Quandou, Z. Zhongyu, Z. Xuejun, “Novel profilometer with dual digital length gauge for large aspherics measurement,” in Advanced Optical Manufacturing and Testing Technology 2000, L. Yang, H. M. Pollicove, Q. Xi, J. C. Wyant, eds., Proc. SPIE4231, 39–46 (2000).
[CrossRef]

Zhongyu, Z.

W. Quandou, Z. Zhongyu, Z. Xuejun, “Novel profilometer with dual digital length gauge for large aspherics measurement,” in Advanced Optical Manufacturing and Testing Technology 2000, L. Yang, H. M. Pollicove, Q. Xi, J. C. Wyant, eds., Proc. SPIE4231, 39–46 (2000).
[CrossRef]

ACM (Assoc. Comput. Mach.) Trans. Math. Software (1)

C. C. Paige, M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM (Assoc. Comput. Mach.) Trans. Math. Software 8, 43–71 (1982).
[CrossRef]

Ann. CIRP (1)

C. Bradley, J. Jeswiet, “An optical surface texture sensor suitable for integration into a coordinate measuring machine,” Ann. CIRP 48, 459–462 (1999).
[CrossRef]

Appl. Opt. (5)

Int. J. Mach. Tools Manufact. (1)

D. M. G. Stevens, “The application of optical techniques in aspheric surface assessment,” Int. J. Mach. Tools Manufact. 32 (1), 19–25 (1992).
[CrossRef]

Meas. Sci. Technol. (2)

T.-H. Tsai, K.-C. Fan, J.-I. Mou, “A variable-resolution optical profile measurement system,” Meas. Sci. Technol. 13, 190–197 (2002).
[CrossRef]

K. Ehrmann, A. Ho, K. Schindhelm, “A 3D optical profilometer using a compact disc reading head,” Meas. Sci. Technol. 9, 1259–1265 (1998).
[CrossRef]

Opt. Acta (1)

P. Hariharan, B. F. Oreb, Z. Wanzhi, “Measurement of aspheric surfaces using a microcomputer controlled digital radial-shear interferometer,” Opt. Acta 31, 989–999 (1984).
[CrossRef]

Opt. Eng. (1)

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1078 (1993).
[CrossRef]

Opt. Laser Technol. (1)

T. Dawei, “In-process sensor for surface profile measurement applying a common-mode rejection technique,” Opt. Laser Technol. 27, 351–353 (1995).
[CrossRef]

Phys. Technol. (1)

E. Heynacher, “Aspheric optics. How they are made and why they are needed,” Phys. Technol. 10 (3), 124–131 (1979).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Optim. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 9 (1), 112–147 (1998).

Other (3)

M. Born, E. Wolf, “Elements of the theory of interference and interferometers,” in Principles of Optics (Cambridge U. Press, Cambridge, UK, 1997), Chap. 7, pp. 256–367.

J. P. Fitch, Synthetic Aperture Radar (Springer-Verlag, Berlin, 1987).

W. Quandou, Z. Zhongyu, Z. Xuejun, “Novel profilometer with dual digital length gauge for large aspherics measurement,” in Advanced Optical Manufacturing and Testing Technology 2000, L. Yang, H. M. Pollicove, Q. Xi, J. C. Wyant, eds., Proc. SPIE4231, 39–46 (2000).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

(a) Typical layout of a fizeau interferometer. (b) Schematic of the interference pattern.

Fig. 2
Fig. 2

Simplest form of a synthetic aperture interferometer.

Fig. 3
Fig. 3

Practical synthetic aperture interferometer.

Fig. 4
Fig. 4

Shown are the x coordinates of the ray intercept for a plano-spherical optic.

Fig. 5
Fig. 5

(a) Optical path length for front and rear paths. (b) Change in OPD.

Fig. 6
Fig. 6

Calculated deviation from form after one iteration.

Fig. 7
Fig. 7

Interference data for an optical flat.

Fig. 8
Fig. 8

Interference data for a grooved lens.

Fig. 9
Fig. 9

Gray-scale image of a lens surface.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

Ixp=I01+cos2πOPDxpλ+ϕ0,
OPD=OPD0+λ2πcos-1I/I0-1-ϕ0,
OPD=OPLf-OPLr.
OPLf=xp-xf1-d2+yp-yf121/2+xf1-xp-d2+yf1-yp21/2,
OPLr=xp-xf2-d2+yp-yf221/2+nxf2-xr2+yf2-yr21/2+nxr-xf32+yr-yf321/2+xf3-xp-d2+yf3-yp21/2,
OPLfxf1=OPLrxf1=OPLrxf2=OPLrxf3=0.
ΔOPD=Δyf1yf1-ypxp-xf1-d2+yp-yf12-1/2+yf1-ypxf1-xp-d2+yf1-yp2-1/2+Δyf2-yp-yf2xp-xf2-d2+yp-yf22-1/2+nyf2-yrxf2-xr2+yf2-yr2-1/2+Δyf3-nyr-yf3xr-xf32+yr-yf32-1/2+yf3-ypxf3-xp-d2+yf3-yp2-1/2.

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