Abstract

Cylindrical specimens may be tested advantageously by using grazing-incidence interferometry. A multiple positions test in combination with rotational averaging has recently been used to separate the surface deviations of the specimen from the interferometric aberrations. To reduce the measuring time and to check whether the results are reliable, a second procedure is now investigated, which uses the principle of the multiple positions test to determine quantities proportional to the difference quotients of the surface deviations. After numerical integration, the results can be compared with those obtained previously by rotational averaging. The measurement principle is described, and calibration results are presented.

© 2006 Optical Society of America

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References

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  1. K. G. Birch and F. J. Green, "Oblique incidence interferometry applied to nonoptical surfaces," J. Phys. E 6, 1045-1048 (1973).
    [CrossRef]
  2. 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]
  3. R. Schreiner, "Interferometric shape measurement of rough surfaces at grazing incidence," Opt. Eng. 41, 1570-1576 (2002).
    [CrossRef]
  4. 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]
  5. Fabricated by HELLMA Optik, Jena.
  6. K.-E. Elssner, R. Burow, J. Grzanna, and R. Spolaczyk, "Absolute sphericity measurement," Appl. Opt. 28, 4649-4661 (1989).
    [CrossRef] [PubMed]
  7. T. Blümel, K.-E. Elssner, and G. Schulz, "Absolute interferometric calibration of toric and conical surfaces," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 370-378 (1997).
  8. K. Mantel, J. Lamprecht, N. Lindlein, and J. Schwider, "Absolute calibration in grazing-incidence interferometry via rotational averaging," Appl. Opt. 45, 3740-3745 (2006).
    [CrossRef] [PubMed]
  9. C. J. Evans and R. N. Kestner, "Test optics error removal," Appl. Opt. 35, 1015-1021 (1996).
    [CrossRef] [PubMed]
  10. D. L. Fried, "Least-square fitting a wavefront distortion estimate to an array of phase difference measurements," J. Opt. Soc. Am. 67, 370-375 (1977).
    [CrossRef]
  11. B. R. Hunt, "Matrix formulation of the reconstruction of phase values from phase differences," J. Opt. Soc. Am. 69, 393-399 (1979).
    [CrossRef]
  12. H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
    [CrossRef]
  13. G. Schulz, "Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte," J. Mod. Opt. 14, 375-388 (1967).
  14. F. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001).
    [CrossRef]

2006 (1)

2005 (1)

2002 (1)

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

2001 (1)

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

1998 (1)

1997 (1)

T. Blümel, K.-E. Elssner, and G. Schulz, "Absolute interferometric calibration of toric and conical surfaces," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 370-378 (1997).

1996 (1)

1994 (1)

H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
[CrossRef]

1989 (1)

1979 (1)

1977 (1)

1973 (1)

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

1967 (1)

G. Schulz, "Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte," J. Mod. Opt. 14, 375-388 (1967).

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]

Blümel, T.

T. Blümel, K.-E. Elssner, and G. Schulz, "Absolute interferometric calibration of toric and conical surfaces," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 370-378 (1997).

Brinkmann, S.

Burow, R.

Dresel, T.

Elssner, K.-E.

T. Blümel, K.-E. Elssner, and G. Schulz, "Absolute interferometric calibration of toric and conical surfaces," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 370-378 (1997).

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

Evans, C. J.

Falkenstörfer, O.

H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
[CrossRef]

Fried, D. L.

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.

Hunt, B. R.

Kestner, R. N.

Küchel, F.

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

Lamprecht, J.

Lindlein, N.

Mantel, K.

Schreiner, R.

Schulz, G.

T. Blümel, K.-E. Elssner, and G. Schulz, "Absolute interferometric calibration of toric and conical surfaces," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 370-378 (1997).

G. Schulz, "Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte," J. Mod. Opt. 14, 375-388 (1967).

Schwider, J.

Sickinger, H.

H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
[CrossRef]

Spolaczyk, R.

Appl. Opt. (4)

J. Mod. Opt. (1)

G. Schulz, "Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte," J. Mod. Opt. 14, 375-388 (1967).

J. Opt. Soc. Am. (2)

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

J. Phys. E (1)

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

Opt. Eng. (2)

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

H. Sickinger, O. Falkenstörfer, N. Lindlein, and J. Schwider, "Characterization of microlenses using a phase-shifting shearing interferometer," Opt. Eng. 33, 2680-2686 (1994).
[CrossRef]

Optik (1)

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

Other (2)

Fabricated by HELLMA Optik, Jena.

T. Blümel, K.-E. Elssner, and G. Schulz, "Absolute interferometric calibration of toric and conical surfaces," in Optical Manufacturing and Testing II, H. P. Stahl, ed., Proc. SPIE 3134, 370-378 (1997).

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Figures (10)

Fig. 1
Fig. 1

Setup of an interferometer with grazing-incidence illumination.

Fig. 2
Fig. 2

Demonstration of the systematic aberrations of a grazing-incidence setup. The convex surface of a lens of high surface quality has been measured in a 0° and a 180° (flipped) position. The p.v.'s of the surface deviations shown are 0.24 μ m (left) and 0.29 μ m (right). A polynomial fit of 12° has been applied. The contour lines have a spacing of 0.03 μ m . If there are only small systematic aberrations, the surface deviations should be seen to rotate for 180°, which is not the case.

Fig. 3
Fig. 3

Principle of the four-positions test for hollow cylinders using rotational averaging. The setup is asymmetric to avoid three-beam interference.

Fig. 4
Fig. 4

Principle of the four-positions test for hollow cylinders via integration of difference measurements. The difference of the rotated and the shifted positions with the basic position effectively gives the difference quotients of the surface deviations in the φ and z directions. The flipped position is needed to reconstruct a conical deviation of the surface.

Fig. 5
Fig. 5

Typical interferogram and the mask for the field of interest. The radial direction in the interferogram corresponds to the symmetry axis z of the cylinder, which is parallel to the optical axis of the interferometer. The mask for the field of interest has to be aligned to the interferogram.

Fig. 6
Fig. 6

Calibration result. Top, surface deviations. Bottom, systematic aberrations of the interferometer. The p.v.'s are given in micrometers.

Fig. 7
Fig. 7

Reproducibility of the calibration procedure via integration of difference measurements. The rms value is given in micrometers.

Fig. 8
Fig. 8

Consistency of the calibration procedure via integration of difference measurements. Top, surface deviations of the specimen before rotation. Bottom, surface deviations after rotation. The rotation angle was approximately 90°. The p.v.'s are given in micrometers.

Fig. 9
Fig. 9

Difference of the surface deviations obtained by integration of difference measurements and by rotational averaging. The rms value is given in micrometers.

Fig. 10
Fig. 10

In this case the excess fraction δ directly determines the minimum frequency of the nondetectable part of the surface deviations.

Equations (24)

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λ eff 4 π Φ rot λ eff 4 π Φ bas = h ( z , φ + δ φ ) h ( z , φ ) ,
λ eff 4 π Φ shift λ eff 4 π Φ bas = h ( z + δ z , φ ) h ( z , φ ) .
h ( x , y ) = i = 0 D j = 0 D i u i j U i j ( x , y ) .
T i ( x , y ) = cos ( i   arccos   y ) .
U i j ( x , y ) = T i ( y ) F j ( x ) .
h ( x , y ) = i = 0 D j = 0 D i u i j U i j ( x , y ) = m = 1 M u m U m ( x , y ) ,
u m := u i j , U m := U i j ,
m = i 2 D i + 3 2 + j + 1 , M = ( D + 2 ) ( D + 1 ) 2 ,
h x ( x , y ) := h ( x + l x , y ) h ( x , y ) ,
h y ( x , y ) := h ( x , y + l y ) h ( x , y ) ,
g ( u ) := l = 1 N { [ Δ x h ( x ( l ) , y ( l ) ) h x ( l ) ] 2 + [ Δ y h ( x ( l ) , y ( l ) ) h y ( l ) ] 2 }
Δ x h ( x ( l ) , y ( l ) ) = m = 1 M u m { U m [ x ( l ) + l x , y ( l ) ] U m [ x ( l ) , y ( l ) ] } ,
Δ y h ( x ( l ) , y ( l ) ) = m = 1 M u m { U m [ x ( l ) , y ( l ) + l y ] U m [ x ( l ) , y ( l ) ] } .
Φ meas = Φ rot Φ bas = Φ ( R δφ 1 x ) Φ ( x ) .
Φ meas mis = Φ ( R δφ 1 x a ) Φ ( x a ) .
Φ meas mis Φ ( R δφ 1 x ) a Φ ( R δφ 1 x ) ( R δφ 1 x ) [ Φ ( x ) a Φ ( x ) x ] ,
Δ Φ = a Φ ( x ) x a Φ ( R δφ 1 x ) ( R δφ 1 x ) .
Δ Φ = 2 π 50 .
Φ shift mis = Φ ( [ r + δ r ] + Δ r , φ ) Φ ( r , φ ) Φ ( [ r + δ r ] , φ ) Φ ( r , φ ) + Δ r Φ ( r + δ r , φ ) ( r + δ r ) .
Δ Φ = Δ r Φ ( r + δ r , φ ) ( r + δ r ) .
Δ r = Δ z D L = 1 300   pixels.
Δ Φ = 2 π 30,000 .
Δ Φ = Δ φ Φ ( r , φ + δ φ ) ( φ + δ φ ) .
Δ Φ = 2 π 100,000 .

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