Abstract

Interferometry in grazing incidence can be used to test cylindrical mantle surfaces. The absolute accuracy of the resulting surface profiles is limited by systematic wavefront aberrations caused in the interferometer, in particular due to an inversion of the test wavefront in an interferometer using diffractive beam splitters. For cylindrical specimens, a calibration method using four positions has therefore been investigated. This test is combined with another method of optical metrology: the rotational averaging procedure. The implementation for grazing incidence is described and measurement results for hollow cylinders are presented. The gain in accuracy is demonstrated.

© 2006 Optical Society of America

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References

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  1. N. Abramson, "The Interferoscope: a new type of interferometer with variable fringe separation," Optik. (Stuttgart) 30, 56-71 (1969).
  2. K. G. Birch and F. J. Green, "Oblique incidence interferometry applied to nonoptical surfaces," J. Phys. E 6, 1045-1048 (1973).
    [CrossRef]
  3. S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, "Testing of rod objects by grazing incidence interferometry: experiment," Appl. Opt. 38, 121-125 (1999).
    [CrossRef]
  4. H. Nürge and J. Schwider, "Testing of cylindrical lenses by grazing incidence interferometry," Optik 111, 545-555 (2000).
  5. T. Dresel, S. Brinkmann, R. Schreiner, and J. Schwider, "Testing of rod objects by grazing incidence interferometry: theory," J. Opt. Soc. Am A 15, 2921-2928 (1998).
    [CrossRef]
  6. K. Mantel, N. Lindlein, and J. Schwider, "Simultaneous characterization of the quality and orientation of cylindrical lens surfaces," Appl. Opt. 44, 2970-2977 (2005).
    [CrossRef] [PubMed]
  7. S. Brinkmann, "Verfahren zur Prüfung stabförmiger technischer Objekte mittels computer-generierter Hologramme und Interferometrie in streifender Inzidenz," Ph.D. dissertation,Erlangen, (1999).
  8. D. Malacara, Optical Shop Testing (Wiley, 1991).
  9. K.-E. Elssner, R. Burow, J. Grzanna, and R. Spolaczyk, "Absolute sphericity measurement," Appl. Opt. 28, 4649-4661 (1989).
    [CrossRef] [PubMed]
  10. C. J. Evans, and R. N. Kestner, "Test optics error removal," Appl. Opt. 35, 1015-1021 (1996).
    [CrossRef] [PubMed]
  11. M. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001).
    [CrossRef]
  12. R. Schreiner, "Interferometric shape measurement of rough surfaces at grazing incidence," Opt. Eng. 41, 1570-1576 (2002).
    [CrossRef]
  13. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, "Digital wavefront measuring interferometer for testing optical surfaces and lenses," Appl. Opt. 13, 2693-2703 (1974).
    [CrossRef] [PubMed]
  14. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, "Digital wave-front measuring interferometry: some systematic error sources," Appl. Opt. 22, 3421-3432 (1983).
    [CrossRef] [PubMed]

2005

2002

R. Schreiner, "Interferometric shape measurement of rough surfaces at grazing incidence," Opt. Eng. 41, 1570-1576 (2002).
[CrossRef]

2001

M. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001).
[CrossRef]

2000

H. Nürge and J. Schwider, "Testing of cylindrical lenses by grazing incidence interferometry," Optik 111, 545-555 (2000).

1999

1998

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

1996

1989

1983

1974

1973

K. G. Birch and F. J. Green, "Oblique incidence interferometry applied to nonoptical surfaces," J. Phys. E 6, 1045-1048 (1973).
[CrossRef]

1969

N. Abramson, "The Interferoscope: a new type of interferometer with variable fringe separation," Optik. (Stuttgart) 30, 56-71 (1969).

Abramson, N.

N. Abramson, "The Interferoscope: a new type of interferometer with variable fringe separation," Optik. (Stuttgart) 30, 56-71 (1969).

Birch, K. G.

K. G. Birch and F. J. Green, "Oblique incidence interferometry applied to nonoptical surfaces," J. Phys. E 6, 1045-1048 (1973).
[CrossRef]

Brangaccio, D. J.

Brinkmann, S.

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

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

S. Brinkmann, "Verfahren zur Prüfung stabförmiger technischer Objekte mittels computer-generierter Hologramme und Interferometrie in streifender Inzidenz," Ph.D. dissertation,Erlangen, (1999).

Bruning, J. H.

Burow, R.

Dresel, T.

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

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

Elssner, K.-E.

Evans, C. J.

Gallagher, J. E.

Green, F. J.

K. G. Birch and F. J. Green, "Oblique incidence interferometry applied to nonoptical surfaces," J. Phys. E 6, 1045-1048 (1973).
[CrossRef]

Grzanna, J.

Herriott, D. R.

Kestner, R. N.

Küchel, M.

M. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001).
[CrossRef]

Lindlein, N.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 1991).

Mantel, K.

Merkel, K.

Nürge, H.

H. Nürge and J. Schwider, "Testing of cylindrical lenses by grazing incidence interferometry," Optik 111, 545-555 (2000).

Rosenfeld, D. P.

Schreiner, R.

R. Schreiner, "Interferometric shape measurement of rough surfaces at grazing incidence," Opt. Eng. 41, 1570-1576 (2002).
[CrossRef]

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

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

Schwider, J.

Spolaczyk, R.

White, A. D.

Appl. Opt.

J. Opt. Soc. Am A

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

J. Phys. E

K. G. Birch and F. J. Green, "Oblique incidence interferometry applied to nonoptical surfaces," J. Phys. E 6, 1045-1048 (1973).
[CrossRef]

Opt. Eng.

R. Schreiner, "Interferometric shape measurement of rough surfaces at grazing incidence," Opt. Eng. 41, 1570-1576 (2002).
[CrossRef]

Optik

H. Nürge and J. Schwider, "Testing of cylindrical lenses by grazing incidence interferometry," Optik 111, 545-555 (2000).

M. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001).
[CrossRef]

Optik.

N. Abramson, "The Interferoscope: a new type of interferometer with variable fringe separation," Optik. (Stuttgart) 30, 56-71 (1969).

Other

S. Brinkmann, "Verfahren zur Prüfung stabförmiger technischer Objekte mittels computer-generierter Hologramme und Interferometrie in streifender Inzidenz," Ph.D. dissertation,Erlangen, (1999).

D. Malacara, Optical Shop Testing (Wiley, 1991).

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Figures (13)

Fig. 1
Fig. 1

Setup of a grazing incidence interferometer. The DOEs are used as null elements.

Fig. 2
Fig. 2

Aberrations in the incoming test wavefront may lead to an optical path difference (OPD) between object and reference wave even if the specimen is perfect.

Fig. 3
Fig. 3

Rotational averaging eliminates the noncircular partsh noncirc(z, ϕ) of the surface figure deviation, leaving the circular parts h circ(z) that vary only in the z direction.

Fig. 4
Fig. 4

Principle of the four-position test with rotational averaging.

Fig. 5
Fig. 5

Rotational averaging does not eliminate all the angular frequencies of the surface figure deviations. In this case, the number of averaging positions is N = 8, and therefore frequencies 8∕2π, 16∕2π, 24∕2π, etc., remain unaffected.

Fig. 6
Fig. 6

Central part of the grazing incidence setup.

Fig. 7
Fig. 7

Influence of misalignments on the result of a single measurement; all values are in micrometers. Left: repeatability of a single measurement, rms 0.0165 μm = λeff∕600. Right: difference between measurements in aligned and misaligned states, rms 0.1378 μm = λeff∕70. The misalignment that had been adjusted amounted to approximately four fringes.

Fig. 8
Fig. 8

Influence of a misalignment on rotational averaging. The gray-level values correspond to micrometers; the z axis is scaled in mm. Top: difference of the averaged normal and flip positions 〈ϕflip〉 − 〈ϕnorm〉 without additional alignment of the specimen. The φ dependence, which should have been eliminated by the averaging procedure, can be clearly seen. Bottom: the same difference. In each of the averaging positions the specimen was aligned to fluffed out fringes. The phase values depend only on the z direction.

Fig. 9
Fig. 9

Alignment of the mask for the field of interest (left). If the mask has not been correctly aligned to the interferogram, systematic errors are introduced during the evaluation. In this case, a rotationally averaged measurement that has been unrolled to xy coordinates shows a periodic variation (right). This variation can be taken as an indication of a misalignment of the mask.

Fig. 10
Fig. 10

Alignment of the flip position. Circular portions (without conus) of a specimen that has been measured by the absolute calibration procedure, with the normal and the flipped position as starting positions, respectively. The result of the second measurement has been flipped back. No relative translation of the two parts can be seen.

Fig. 11
Fig. 11

Calibration result. Top: surface deviations. Bottom: interferometer aberrations (obtained by subtracting the surface deviations from a single measurement without calibration). All values are given in micrometers.

Fig. 12
Fig. 12

Reproducibility measurement. The difference between two successive measurements is shown. All values are given in micrometers.

Fig. 13
Fig. 13

Difference between results obtained with the normal and with the flip position as starting positions for the calibration procedure (the result of the flip position has been flipped back). All values are given in micrometers.

Equations (14)

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h ( z , φ ) = h circ ( z ) + h noncirc ( z , φ ) = [ h con ( z ) + h ̃ circ ( z ) ] + h noncirc ( z , φ ) .
h noncirc ( z , φ ) = 0.
ϕ norm ϕ norm = 4 π λ eff h noncirc ( z , φ ) ,
ϕ shift ϕ norm = 4 π λ eff [ h ˜ circ ( z + δ z ) h ˜ circ ( z ) ] + 4 π λ eff ( a + b ) δ z ,
ϕ flip ϕ norm = 4 π λ eff [ h ˜ circ ( z ) h ˜ circ ( z ) ] 8 π λ eff a z ,
h ˜ circ ( z + δ z ) = i = 1 M m i T i ( z + δ z ) ,
h ˜ circ ( z ) = i = 1 M m i T i ( z ) ,
T i ( z ) = cos ( i arccos z ) ,
ϕ = 1 N n = 0 N 1 ϕ ( R n Δ Θ                 1 x ) ,
ϕ a = 1 N n = 0 N 1 ϕ ( R n Δ Θ                 1 x a ) .
ϕ a 1 N n = 0 N 1 [ ϕ ( R n Δ Θ                 1 x ) a ϕ ( R n Δ Θ               1 x ) ( R n Δ Θ                 1 x ) ] .
ϕ a 1 N n = 0 N 1 ϕ ( R n ΔΘ                 1 x ) a N n = 0 N 1 ϕ ( R n Δ Θ                 1 x ) ( R n Δ Θ                 1 x ) = ϕ Δϕ .
Δϕ a ϕ ( x ) ( x ) .
Δϕ 2 π 100 .

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