Abstract

Several conventional methods for absolute sphericity testing of optical surfaces are known. One method has been used previously for real time interferometry, but a detailed investigation has not been carried out so far. In this work the known conventional methods are reviewed and assessed for suitability in real time interferometry. One method has been applied to phase stepping interferometry and examined. Adjustment peculiarities, experimental results of deviation measurements on normal surfaces, and application of normals are reported. Measuring errors, especially coherent noise, are analyzed and a special subsequent digital spatial filtering technique for diminution of noise is described.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Schulz, J. Schwider, “Interferometric Testing of Smooth Surfaces,” Prog. Opt. 13, 93–167 (1976).
    [CrossRef]
  2. Ref. 1, p. 136.
  3. A. E. Jensen, “Absolute Calibration Method for Laser Twyman-Green Wave-Front Testing Interferometers,” J. Opt. Soc. Am. 63, 1313A (1973);J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), p. 409.
  4. J. S. Harris, “The Universal Fizeau Interferometer,” Ph.D. Thesis, U. Reading, U.K. (1971);H. H. Hopkins, “Applied Optics at Reading,” Opt. Laser Technol. 2, 158–163 (1971).
  5. G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheits-pruefung laengs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
    [CrossRef]
  6. G. Schulz, “Interferentielle Absolutpruefung zweier Flaechen,” Opt. Acta 20, 699–706 (1973).
    [CrossRef]
  7. K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutpruefung von Sphaerizitaetsnormalen,” Opt. Acta 27, 563–580 (1980).
    [CrossRef]
  8. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital Wave-Front Measuring Interferometry: Some Systemtic Error Sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  9. J. Schwider et al., “Echtzeitinterferometer fuer die Optikprue-fung,” Opt. Appl. Wroclaw 15, 395–412 (1985).
  10. J. Schwider, K. E. Elssner, J. Grzanna, R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in Proceedings, IMEKO TC-14 Symposium, T. Kemeny, K. Havrilla, Eds. (Nova Science Publishing, Commack, NY, 1987), pp. 93–103.
  11. M. O. Freeman, B. E. A. Saleh, “Optical Location of Cen-troids of Nonoverlapping Objects,” Appl. Opt. 26, 2752–2759 (1987).
    [CrossRef] [PubMed]
  12. M. Vygodsky, Mathematical Handbook (Mir Publishers, Moscow, 1971), p. 692.
  13. K. H. Carnell, M. J. Kidger, “Some Experiments on Precision Lens Centering and Mounting,” Opt. Acta 21, 615–629 (1974).
    [CrossRef]
  14. J. C. Wyant, Ed., Precision Surface Metrology, Proc. Soc. Photo-Opt. Instrum. Eng.429 (1983), pp. 114, 187, 199;L. M. Cohen, Ed., Structural Mechanics of Optical Systems, Proc. Soc. Photo-Opt. Instrum. Eng.450 (1983), pp. 59, 81, 88, 131.
  15. E. C. Kintner, R. M. Sillitto, “A New Analytic Method for Computing the Optical Transfer Function,” Opt. Acta 23, 607–619 (1976).
    [CrossRef]
  16. K. H. Womack, “A Frequency Domain Description of Interferogram Analysis,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 166–173 (1983).
  17. H. C. Burger, P. H. van Cittert, “Wahre und scheinbare Intensitaetsverteilung in Spektrallinien,” Z. Phys. 79, 722–730 (1932).
    [CrossRef]
  18. F. M. Kuechel, T. Schmieder, H. J. Tiziani, “Beitrag zur Verwendung von Zernike-Polynomen bei der automatischen Interferenzstreifenauswertung,” Optik 65, 123–142 (1983).

1987

1985

J. Schwider et al., “Echtzeitinterferometer fuer die Optikprue-fung,” Opt. Appl. Wroclaw 15, 395–412 (1985).

1983

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital Wave-Front Measuring Interferometry: Some Systemtic Error Sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

K. H. Womack, “A Frequency Domain Description of Interferogram Analysis,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 166–173 (1983).

F. M. Kuechel, T. Schmieder, H. J. Tiziani, “Beitrag zur Verwendung von Zernike-Polynomen bei der automatischen Interferenzstreifenauswertung,” Optik 65, 123–142 (1983).

1980

K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutpruefung von Sphaerizitaetsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

1976

G. Schulz, J. Schwider, “Interferometric Testing of Smooth Surfaces,” Prog. Opt. 13, 93–167 (1976).
[CrossRef]

E. C. Kintner, R. M. Sillitto, “A New Analytic Method for Computing the Optical Transfer Function,” Opt. Acta 23, 607–619 (1976).
[CrossRef]

1974

K. H. Carnell, M. J. Kidger, “Some Experiments on Precision Lens Centering and Mounting,” Opt. Acta 21, 615–629 (1974).
[CrossRef]

1973

A. E. Jensen, “Absolute Calibration Method for Laser Twyman-Green Wave-Front Testing Interferometers,” J. Opt. Soc. Am. 63, 1313A (1973);J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), p. 409.

G. Schulz, “Interferentielle Absolutpruefung zweier Flaechen,” Opt. Acta 20, 699–706 (1973).
[CrossRef]

1967

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheits-pruefung laengs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

1932

H. C. Burger, P. H. van Cittert, “Wahre und scheinbare Intensitaetsverteilung in Spektrallinien,” Z. Phys. 79, 722–730 (1932).
[CrossRef]

Burger, H. C.

H. C. Burger, P. H. van Cittert, “Wahre und scheinbare Intensitaetsverteilung in Spektrallinien,” Z. Phys. 79, 722–730 (1932).
[CrossRef]

Burow, R.

Carnell, K. H.

K. H. Carnell, M. J. Kidger, “Some Experiments on Precision Lens Centering and Mounting,” Opt. Acta 21, 615–629 (1974).
[CrossRef]

Elssner, K. E.

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital Wave-Front Measuring Interferometry: Some Systemtic Error Sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutpruefung von Sphaerizitaetsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

J. Schwider, K. E. Elssner, J. Grzanna, R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in Proceedings, IMEKO TC-14 Symposium, T. Kemeny, K. Havrilla, Eds. (Nova Science Publishing, Commack, NY, 1987), pp. 93–103.

Freeman, M. O.

Grzanna, J.

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital Wave-Front Measuring Interferometry: Some Systemtic Error Sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutpruefung von Sphaerizitaetsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

J. Schwider, K. E. Elssner, J. Grzanna, R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in Proceedings, IMEKO TC-14 Symposium, T. Kemeny, K. Havrilla, Eds. (Nova Science Publishing, Commack, NY, 1987), pp. 93–103.

Harris, J. S.

J. S. Harris, “The Universal Fizeau Interferometer,” Ph.D. Thesis, U. Reading, U.K. (1971);H. H. Hopkins, “Applied Optics at Reading,” Opt. Laser Technol. 2, 158–163 (1971).

Jensen, A. E.

A. E. Jensen, “Absolute Calibration Method for Laser Twyman-Green Wave-Front Testing Interferometers,” J. Opt. Soc. Am. 63, 1313A (1973);J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), p. 409.

Kidger, M. J.

K. H. Carnell, M. J. Kidger, “Some Experiments on Precision Lens Centering and Mounting,” Opt. Acta 21, 615–629 (1974).
[CrossRef]

Kintner, E. C.

E. C. Kintner, R. M. Sillitto, “A New Analytic Method for Computing the Optical Transfer Function,” Opt. Acta 23, 607–619 (1976).
[CrossRef]

Kuechel, F. M.

F. M. Kuechel, T. Schmieder, H. J. Tiziani, “Beitrag zur Verwendung von Zernike-Polynomen bei der automatischen Interferenzstreifenauswertung,” Optik 65, 123–142 (1983).

Merkel, K.

Saleh, B. E. A.

Schmieder, T.

F. M. Kuechel, T. Schmieder, H. J. Tiziani, “Beitrag zur Verwendung von Zernike-Polynomen bei der automatischen Interferenzstreifenauswertung,” Optik 65, 123–142 (1983).

Schulz, G.

K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutpruefung von Sphaerizitaetsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

G. Schulz, J. Schwider, “Interferometric Testing of Smooth Surfaces,” Prog. Opt. 13, 93–167 (1976).
[CrossRef]

G. Schulz, “Interferentielle Absolutpruefung zweier Flaechen,” Opt. Acta 20, 699–706 (1973).
[CrossRef]

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheits-pruefung laengs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

Schwider, J.

J. Schwider et al., “Echtzeitinterferometer fuer die Optikprue-fung,” Opt. Appl. Wroclaw 15, 395–412 (1985).

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital Wave-Front Measuring Interferometry: Some Systemtic Error Sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

G. Schulz, J. Schwider, “Interferometric Testing of Smooth Surfaces,” Prog. Opt. 13, 93–167 (1976).
[CrossRef]

J. Schwider, K. E. Elssner, J. Grzanna, R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in Proceedings, IMEKO TC-14 Symposium, T. Kemeny, K. Havrilla, Eds. (Nova Science Publishing, Commack, NY, 1987), pp. 93–103.

Sillitto, R. M.

E. C. Kintner, R. M. Sillitto, “A New Analytic Method for Computing the Optical Transfer Function,” Opt. Acta 23, 607–619 (1976).
[CrossRef]

Spolaczyk, R.

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital Wave-Front Measuring Interferometry: Some Systemtic Error Sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

J. Schwider, K. E. Elssner, J. Grzanna, R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in Proceedings, IMEKO TC-14 Symposium, T. Kemeny, K. Havrilla, Eds. (Nova Science Publishing, Commack, NY, 1987), pp. 93–103.

Tiziani, H. J.

F. M. Kuechel, T. Schmieder, H. J. Tiziani, “Beitrag zur Verwendung von Zernike-Polynomen bei der automatischen Interferenzstreifenauswertung,” Optik 65, 123–142 (1983).

van Cittert, P. H.

H. C. Burger, P. H. van Cittert, “Wahre und scheinbare Intensitaetsverteilung in Spektrallinien,” Z. Phys. 79, 722–730 (1932).
[CrossRef]

Vygodsky, M.

M. Vygodsky, Mathematical Handbook (Mir Publishers, Moscow, 1971), p. 692.

Womack, K. H.

K. H. Womack, “A Frequency Domain Description of Interferogram Analysis,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 166–173 (1983).

Appl. Opt.

J. Opt. Soc. Am.

A. E. Jensen, “Absolute Calibration Method for Laser Twyman-Green Wave-Front Testing Interferometers,” J. Opt. Soc. Am. 63, 1313A (1973);J. H. Bruning, “Fringe Scanning Interferometers,” in Optical Shop Testing, D. Malacara, Ed. (Wiley, New York, 1978), p. 409.

Opt. Acta

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheits-pruefung laengs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

G. Schulz, “Interferentielle Absolutpruefung zweier Flaechen,” Opt. Acta 20, 699–706 (1973).
[CrossRef]

K. E. Elssner, J. Grzanna, G. Schulz, “Interferentielle Absolutpruefung von Sphaerizitaetsnormalen,” Opt. Acta 27, 563–580 (1980).
[CrossRef]

K. H. Carnell, M. J. Kidger, “Some Experiments on Precision Lens Centering and Mounting,” Opt. Acta 21, 615–629 (1974).
[CrossRef]

E. C. Kintner, R. M. Sillitto, “A New Analytic Method for Computing the Optical Transfer Function,” Opt. Acta 23, 607–619 (1976).
[CrossRef]

Opt. Appl. Wroclaw

J. Schwider et al., “Echtzeitinterferometer fuer die Optikprue-fung,” Opt. Appl. Wroclaw 15, 395–412 (1985).

Optik

F. M. Kuechel, T. Schmieder, H. J. Tiziani, “Beitrag zur Verwendung von Zernike-Polynomen bei der automatischen Interferenzstreifenauswertung,” Optik 65, 123–142 (1983).

Proc. Soc. Photo-Opt. Instrum. Eng.

K. H. Womack, “A Frequency Domain Description of Interferogram Analysis,” Proc. Soc. Photo-Opt. Instrum. Eng. 429, 166–173 (1983).

Prog. Opt.

G. Schulz, J. Schwider, “Interferometric Testing of Smooth Surfaces,” Prog. Opt. 13, 93–167 (1976).
[CrossRef]

Z. Phys.

H. C. Burger, P. H. van Cittert, “Wahre und scheinbare Intensitaetsverteilung in Spektrallinien,” Z. Phys. 79, 722–730 (1932).
[CrossRef]

Other

J. C. Wyant, Ed., Precision Surface Metrology, Proc. Soc. Photo-Opt. Instrum. Eng.429 (1983), pp. 114, 187, 199;L. M. Cohen, Ed., Structural Mechanics of Optical Systems, Proc. Soc. Photo-Opt. Instrum. Eng.450 (1983), pp. 59, 81, 88, 131.

J. Schwider, K. E. Elssner, J. Grzanna, R. Spolaczyk, “Results and Error Sources in Absolute Sphericity Measurement,” in Proceedings, IMEKO TC-14 Symposium, T. Kemeny, K. Havrilla, Eds. (Nova Science Publishing, Commack, NY, 1987), pp. 93–103.

Ref. 1, p. 136.

J. S. Harris, “The Universal Fizeau Interferometer,” Ph.D. Thesis, U. Reading, U.K. (1971);H. H. Hopkins, “Applied Optics at Reading,” Opt. Laser Technol. 2, 158–163 (1971).

M. Vygodsky, Mathematical Handbook (Mir Publishers, Moscow, 1971), p. 692.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (26)

Fig. 1
Fig. 1

Absolute testing of spheres according to Harris,4 employing a modification of the method for plane surfaces given by Schulz.5 The figure shows the side view of three two-by-two combinations of spherical surfaces A,B,C in a Fizeau interferometer.

Fig. 2
Fig. 2

Detailed illustration of the AB combination in Fig. 1. A and B are spherical surfaces with deviations x and y.

Fig. 3
Fig. 3

Absolute testing of spheres according to Schulz and Schwider2 and Jensen3: (i) basic position, (ii) rotated position, (iii) cat's-eye position.

Fig. 4
Fig. 4

Scheme of the interferometer9: CRS, PZT, driven reference surface; CDM, capacitive distance meter; PO, polarizer; λ/2, plate; λ/4, plate; He–Ne, laser.

Fig. 5
Fig. 5

Scheme of the adjustment device for basic and rotated positions containing eight degrees of mechanical freedom. The arrows indicate translational and rotational stages, respectively.

Fig. 6
Fig. 6

Adjustment device for the basic and rotated positions.

Fig. 7
Fig. 7

Adjustment device for the cat's-eye position.

Fig. 8
Fig. 8

Cat's-eye device in the test arm. From the left: beam splitting unit, beam shaping optics holder, cat's-eye device, basic/ rotated position device.

Fig. 9
Fig. 9

Degrees of mechanical freedom of the mirror in the cat's-eye position.

Fig. 10
Fig. 10

General state of misalignment of the surface to be tested: M, center of curvature; V, vertex of sphere; F, focus of beam shaping optics; F′, image of F.

Figure 11
Figure 11

Examples of absolutely determined deviations of normal surfaces made of (a) BK7, and (b) low expansion glass ceramic: rms, root mean square deviation; PV, peak-to-valley deviation; CD, contour line distance; D, R, diameter and radius of curvature of the surface. The whole surface (100%) is shown.

Fig. 12
Fig. 12

Absolute deviations of a surface measured with different methods: left, according to Refs. 2 and 3, right, according to Refs. 6 and 7.

Fig. 13
Fig. 13

Noise growth in the data of a single phase measurement after tilting a plane test surface: (a) data in parallel adjustment with A,B, as test surface tilt angles; (b) subtraction of a second consecutive single measurement without adjustment change; (c) data after three fringes of tilt have been introduced; (d) difference of (c) and (a).

Fig. 14
Fig. 14

Calibration of a Twyman-Green interferometer for testing spherical surfaces.

Fig. 15
Fig. 15

Surface deviations of a BK7 sample absolutely tested using a calibrated Twyman-Green interferometer.

Fig. 16
Fig. 16

Two-step testing of an optical system.

Fig. 17
Fig. 17

Wavefront aberration of a microscope objective of N.A. = 0.32 and focal length 15.6 mm.

Fig. 18
Fig. 18

Point spread function of the objective of Fig. 17.

Fig. 19
Fig. 19

Modulus of the (a) optical transfer function and (b) cross section along the x-axis of (a) of the objective of Fig. 17: IMTF, ideal modulation transfer function.

Fig. 20
Fig. 20

Pairs of neighboring pixels along the columns of the detector matrix. The detector matrix is a quadratic array of pixels with small detecting areas spaced with extended dead areas; υj represents distances from centroid M.

Fig. 21
Fig. 21

Curve a, modulus M of the Fourier transform of the relevant data and coherent noise (schematically); curve b, the sampling theorem adequate reconstruction function; and curve c, the interpolation function obtained. The quantity v = spatial frequency; v = d is the limiting frequency of relevant data; v = e is the repetition frequency due to sampling; and D = diameter of the surface to be tested. Since in the course of data processing any constant in the deviation data is eliminated the spectrum of the data always shows zero modulation at zero frequency.

Fig. 22
Fig. 22

Erroneous contributions in the spectrum due to linear interpolation (hatched).

Fig. 23
Fig. 23

Schematic of the spatial frequency spectrum of measured data and several noise reduction filter functions; a, measured data; g, rectangular ideal filter; k, sinc function; l, sinc2 function; m, optimal filter.

Fig. 24
Fig. 24

Surface deviations after filtering according to the noise reduction procedure described: (a) without any filtering and (b) filtering according to Fig. 23(1). For optimal filtering of these data see Fig. 11(b).

Fig. 25
Fig. 25

Smoothed difference (a) between original data (curve a in Fig. 23 and convolution with tri(x) tri(y) (curve b in Fig. 23) and (b) between original data and convolution with the optimal kernel according to Eq. (B1) [Fig. 11(b)]. The checker indicates a contour line plateau of zero height.

Fig. 26
Fig. 26

Power spectra of the measured data (a) and of the difference (b) between unfiltered data and data after optimal filtering has been applied. The latter shows that part of the power spectrum suppressed by the filtering operation.

Tables (1)

Tables Icon

Table I Maximum Permissible Alignment Tolerances for a Measuring Error Caused by Misalignment of λ/200 for the Wavefront and λ/400 for the Surface

Equations (7)

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

W 1 = W I + 2 n x ,
W 2 = W I + 2 ( n 1 ) x 2 y .
W = W 1 W 2 = 2 ( x + y ) ,
( i ) W i ( ζ , η ) = R ( ζ , η ) + O ( ζ , η ) + N ( ζ , η ) . ( ii ) W i i ( ζ , η ) = R ( ζ , η ) + O ( ζ , η ) + N ( ζ , η ) . ( iii ) W i i i ( ζ , η ) = R ( ζ , η ) + ( 1 / 2 ) [ O ( ζ , η ) + O ζ , η ) ] .
N ( ζ , η ) = ( 1 / 2 ) [ W i ( ζ , η ) + W i i ( ζ , η ) ] ( 1 / 2 ) [ W i i i ( ζ , η ) + W i i i ( ζ , η ) ] .
r = ( i = 1 4 n u i + j = 1 4 m υ j ) / 4 ( n + m ) ,
h ( x ) = 2 tri ( x ) tri ( x ) * tri ( x ) ,

Metrics