Abstract

Grazing-incidence interferometry that makes use of diffractive axicons for the measurement of cylindrical mantle surfaces has already been reported. However, measurement of concave rod structures poses a severe problem because these structures are subject to spurious fringes caused by parasitic diffraction orders of the diffractive axicons. By breaking the symmetry of the interferometric setup it is possible to obtain unique interferograms of the inner mantle surfaces of hollow cylinders as cages for roller bearings or other workpieces produced on lathe machines that have a suitable surface finish. Special design issues for the computer-generated holograms and the interferometric setup are discussed, and test examples are given.

© 2002 Optical Society of America

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References

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  1. G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 13, pp. 95–167.
    [CrossRef]
  2. K.-E. Elssner, R. Burow, J. Grzanna, R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. 28, 4649–4661 (1989).
    [CrossRef] [PubMed]
  3. See also J. Schwider, K.-E. Elssner, J. Grzanna, R. Spolaczyk, “Results and error sources in absolute sphericity measurement,” in Proceedings of the 1st Symposium Budapest, T. Kemény, K. Havrilla, eds., Vol. 14 of International Measurement Confederation Technical Committee Series (Nova Science, Commack, New York, 1987), pp. 93–103.
  4. D. Malacara, Optical Shop Testing (Wiley, New York, 1992).
  5. J. Schwider, “Verfahren und Anordnung zur Prüfung beliebiger Mantelflächen rotationssymmetrischer Festkörper mittels synthetischer Hologramme,” German patentDDR WP 106 769 (filed 4Jan.1972).
  6. S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).
  7. N. Lindlein, R. Schreiner, S. Brinkmann, T. Dresel, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces: systematic error assessment,” Appl. Opt. 36, 2791–2795 (1997).
    [CrossRef] [PubMed]
  8. S. Brinkmann, R. Schreiner, T. Dresel, J. Schwider, “Interferometric testing of plane and cylindrical workpieces with computer generated holograms,” Opt. Eng. 37, 2508–2511 (1998).
    [CrossRef]
  9. T. Dresel, S. Brinkmann, R. Schreiner, J. Schwider, “Testing of rod objects by grazing incidence interferometry: theory,” J. Opt. Soc. Am. A 15, 2921–2928 (1998).
    [CrossRef]
  10. S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Testing of rod objects by grazing incidence interferometry: experiment,” Appl. Opt. 38, 121–125 (1999).
    [CrossRef]

1999 (1)

1998 (2)

S. Brinkmann, R. Schreiner, T. Dresel, J. Schwider, “Interferometric testing of plane and cylindrical workpieces with computer generated holograms,” Opt. Eng. 37, 2508–2511 (1998).
[CrossRef]

T. Dresel, S. Brinkmann, R. Schreiner, J. Schwider, “Testing of rod objects by grazing incidence interferometry: theory,” J. Opt. Soc. Am. A 15, 2921–2928 (1998).
[CrossRef]

1997 (1)

1996 (1)

S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

1989 (1)

Brinkmann, S.

Burow, R.

Dresel, T.

Elssner, K.-E.

K.-E. Elssner, R. Burow, J. Grzanna, R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. 28, 4649–4661 (1989).
[CrossRef] [PubMed]

See also J. Schwider, K.-E. Elssner, J. Grzanna, R. Spolaczyk, “Results and error sources in absolute sphericity measurement,” in Proceedings of the 1st Symposium Budapest, T. Kemény, K. Havrilla, eds., Vol. 14 of International Measurement Confederation Technical Committee Series (Nova Science, Commack, New York, 1987), pp. 93–103.

Grzanna, J.

K.-E. Elssner, R. Burow, J. Grzanna, R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. 28, 4649–4661 (1989).
[CrossRef] [PubMed]

See also J. Schwider, K.-E. Elssner, J. Grzanna, R. Spolaczyk, “Results and error sources in absolute sphericity measurement,” in Proceedings of the 1st Symposium Budapest, T. Kemény, K. Havrilla, eds., Vol. 14 of International Measurement Confederation Technical Committee Series (Nova Science, Commack, New York, 1987), pp. 93–103.

Lindlein, N.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1992).

Schreiner, R.

Schulz, G.

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 13, pp. 95–167.
[CrossRef]

Schwider, J.

S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Testing of rod objects by grazing incidence interferometry: experiment,” Appl. Opt. 38, 121–125 (1999).
[CrossRef]

S. Brinkmann, R. Schreiner, T. Dresel, J. Schwider, “Interferometric testing of plane and cylindrical workpieces with computer generated holograms,” Opt. Eng. 37, 2508–2511 (1998).
[CrossRef]

T. Dresel, S. Brinkmann, R. Schreiner, J. Schwider, “Testing of rod objects by grazing incidence interferometry: theory,” J. Opt. Soc. Am. A 15, 2921–2928 (1998).
[CrossRef]

N. Lindlein, R. Schreiner, S. Brinkmann, T. Dresel, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces: systematic error assessment,” Appl. Opt. 36, 2791–2795 (1997).
[CrossRef] [PubMed]

S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

See also J. Schwider, K.-E. Elssner, J. Grzanna, R. Spolaczyk, “Results and error sources in absolute sphericity measurement,” in Proceedings of the 1st Symposium Budapest, T. Kemény, K. Havrilla, eds., Vol. 14 of International Measurement Confederation Technical Committee Series (Nova Science, Commack, New York, 1987), pp. 93–103.

J. Schwider, “Verfahren und Anordnung zur Prüfung beliebiger Mantelflächen rotationssymmetrischer Festkörper mittels synthetischer Hologramme,” German patentDDR WP 106 769 (filed 4Jan.1972).

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 13, pp. 95–167.
[CrossRef]

Spolaczyk, R.

K.-E. Elssner, R. Burow, J. Grzanna, R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. 28, 4649–4661 (1989).
[CrossRef] [PubMed]

See also J. Schwider, K.-E. Elssner, J. Grzanna, R. Spolaczyk, “Results and error sources in absolute sphericity measurement,” in Proceedings of the 1st Symposium Budapest, T. Kemény, K. Havrilla, eds., Vol. 14 of International Measurement Confederation Technical Committee Series (Nova Science, Commack, New York, 1987), pp. 93–103.

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

S. Brinkmann, R. Schreiner, T. Dresel, J. Schwider, “Interferometric testing of plane and cylindrical workpieces with computer generated holograms,” Opt. Eng. 37, 2508–2511 (1998).
[CrossRef]

Optik (1)

S. Brinkmann, T. Dresel, R. Schreiner, J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

Other (4)

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Vol. 13, pp. 95–167.
[CrossRef]

See also J. Schwider, K.-E. Elssner, J. Grzanna, R. Spolaczyk, “Results and error sources in absolute sphericity measurement,” in Proceedings of the 1st Symposium Budapest, T. Kemény, K. Havrilla, eds., Vol. 14 of International Measurement Confederation Technical Committee Series (Nova Science, Commack, New York, 1987), pp. 93–103.

D. Malacara, Optical Shop Testing (Wiley, New York, 1992).

J. Schwider, “Verfahren und Anordnung zur Prüfung beliebiger Mantelflächen rotationssymmetrischer Festkörper mittels synthetischer Hologramme,” German patentDDR WP 106 769 (filed 4Jan.1972).

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Figures (16)

Fig. 1
Fig. 1

Schematic of the grazing incidence interferometer, which uses two CGHs: MO, microscope objective; AO, astronomical objectives; PZT, piezoelectric transducer; PC, computer.

Fig. 2
Fig. 2

Grazing-incidence test of the outer cylinder mantle surface by use of diffractive axicons (diffraxicons).

Fig. 3
Fig. 3

Arrangement of two fully structured axicons producing three-beam interference fringes.

Fig. 4
Fig. 4

Test setup for small-diameter samples. The convergent 1st diffraction order is used as the object wave. We established pure two-beam interference by breaking the symmetry of the setup.

Fig. 5
Fig. 5

Test setup for large-diameter samples. The divergent 1st diffraction order is used as the object wave.

Fig. 6
Fig. 6

Diffractive axicons shown scattered light illuminated by a white-light source: left, CGH 1; right, CGH 2.

Fig. 7
Fig. 7

(a) Measured phase distribution of the roller bearing. (b) Misalignment aberration of an ideal cylinder as a result of fitting the measured data in (a) with a functional representing the movement, that is due to four degrees of freedom. Here and in figures that follow, P-V means peak to valley and Max and Min delimit the maximum and minimum values, respectively, of the property measured on the y axis.

Fig. 8
Fig. 8

Surface deviation data freed from alignment contributions in the normal perspective of the grazing-incidence interferometer. The z axis points in the radial direction. The inner rim is the front end of the cylinder and the outer rim is the back end, i.e., the end nearest the second axicon seen in the direction of the light.

Fig. 9
Fig. 9

Pseudo-three-dimensional plot of the unrolled surface deviations.

Fig. 10
Fig. 10

Three-dimensional plot of the surface deviations of the inner mantle surface (with exaggerated scaling).

Fig. 11
Fig. 11

Histogram of a difference measurement of the roller-bearing cage. The Gaussian distribution implies noise with a rms value of 0.04 µm.

Fig. 12
Fig. 12

Interference pattern seen by the interferometric setup in Fig. 4. The z axis lies in the radial direction. The cylinder surface has a 100-µm elliptical deviation from a circular cylinder.

Fig. 13
Fig. 13

Elliptical deviation from roundness.

Fig. 14
Fig. 14

Pseudo-three-dimensional plot of the surface deviations from a circular cylinder.

Fig. 15
Fig. 15

Radial deviation profile of the cylinder whose interference pattern is shown in Fig. 12.

Fig. 16
Fig. 16

Deviations from an elliptical cylinder. A four-foil error is the principal error that remains.

Equations (12)

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

|C0|2=η1|C1|2,
|C0|2=121+cos φ,|C1|2=2π21-cos φ,
φ=cos-11-4π2 η11+4π2 η1.
Wτ, z=2nτ, zΔτ, z,
xτ=rcos τ, sin τ.
tan ϑ=z-z0r,
Δτ=a cos τ, b sin τ-acos τ, sin τ=0, b-asin τ.
Wτ=b-a1-cos2τ.
Δτ=A-BBAa cos τb sin τ- acos τsin τ=Aa-acos τ-Bb sin τBa cos τ+Ab-asin τ,
Wτ=Aa+Ab-2a+Aa-bcos 2τ+Ba-bsin 2τ=T+C cos 2τ+S sin 2τ.
tan α=BA=SC,
a-b=C2+S2.

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