Abstract

Interferometry at grazing incidence allows one to perform macroscopic shape tests of rough surfaces with submicrometer precision. The highest measurement accuracy is achieved with a null test: the object wave front is adapted to the ideal surface under test through the use of diffractive optical elements. Deviations between the ideal and the real surface shape result in characteristic phase distributions at the interferometer output. Rod objects with rather arbitrary cross-section profiles are especially suitable for this type of measurement. The design of appropriate test wave fronts can be carried out by means of geometrical considerations. Expressions describing aberrations that are due to misalignments of the work piece can easily be derived. Misalignment aberrations and genuine surface deviations are separated by least-squares fitting.

© 1998 Optical Society of America

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References

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  1. T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
    [CrossRef] [PubMed]
  2. N. Abramson, “The interferoscope: a new type of interferometer with variable fringe separation,” Optik 30, 56–71 (1969).
  3. K. G. Birch, F. J. Green, “Oblique incidence interferometry applied to non-optical surfaces,” J. Phys. E 6, 1045–1048 (1973).
    [CrossRef]
  4. J. Schwider, “Verfahren und Anordnung zur Prüfung beliebiger Mantelflächen rotationssymmetrischer Festkörper mittels synthetischer Hologramme,” German patent WP106 769 (January4, 1972).
  5. T. Dresel, J. Schwider, A. Wehrhahn, S. Babin, “Graz-ing incidence interferometry applied to the measurement of cylindrical surfaces,” Opt. Eng. 34, 3531–3535 (1995).
    [CrossRef]
  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 technical surfaces with computer generated holograms,” in Optical Inspection and Micromeasurements II, C. Gareki, ed. Proc. SPIE3098, 83–89 (1997).
    [CrossRef]
  9. L. Denes, J. Salsbury, “Flatness, parallelism and other novel uses of grazing-incidence interferometry in precision engineering,” in Topical Conference on Microscopic Inspection of Macroscopic Structures (American Society for Precision Engineering, Raleigh, N. Car., 1995), pp. 20–23.
  10. T. Dresel, M. Beyerlein, J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
    [CrossRef] [PubMed]

1997

1996

T. Dresel, M. Beyerlein, J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
[CrossRef] [PubMed]

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

1995

T. Dresel, J. Schwider, A. Wehrhahn, S. Babin, “Graz-ing incidence interferometry applied to the measurement of cylindrical surfaces,” Opt. Eng. 34, 3531–3535 (1995).
[CrossRef]

1992

1973

K. G. Birch, F. J. Green, “Oblique incidence interferometry applied to non-optical surfaces,” J. Phys. E 6, 1045–1048 (1973).
[CrossRef]

1969

N. Abramson, “The interferoscope: a new type of interferometer with variable fringe separation,” Optik 30, 56–71 (1969).

Abramson, N.

N. Abramson, “The interferoscope: a new type of interferometer with variable fringe separation,” Optik 30, 56–71 (1969).

Babin, S.

T. Dresel, J. Schwider, A. Wehrhahn, S. Babin, “Graz-ing incidence interferometry applied to the measurement of cylindrical surfaces,” Opt. Eng. 34, 3531–3535 (1995).
[CrossRef]

Beyerlein, M.

Birch, K. G.

K. G. Birch, F. J. Green, “Oblique incidence interferometry applied to non-optical surfaces,” J. Phys. E 6, 1045–1048 (1973).
[CrossRef]

Brinkmann, S.

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).

S. Brinkmann, R. Schreiner, T. Dresel, J. Schwider, “Interferometric testing of technical surfaces with computer generated holograms,” in Optical Inspection and Micromeasurements II, C. Gareki, ed. Proc. SPIE3098, 83–89 (1997).
[CrossRef]

Denes, L.

L. Denes, J. Salsbury, “Flatness, parallelism and other novel uses of grazing-incidence interferometry in precision engineering,” in Topical Conference on Microscopic Inspection of Macroscopic Structures (American Society for Precision Engineering, Raleigh, N. Car., 1995), pp. 20–23.

Dresel, T.

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]

T. Dresel, M. Beyerlein, J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
[CrossRef] [PubMed]

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

T. Dresel, J. Schwider, A. Wehrhahn, S. Babin, “Graz-ing incidence interferometry applied to the measurement of cylindrical surfaces,” Opt. Eng. 34, 3531–3535 (1995).
[CrossRef]

T. Dresel, G. Häusler, H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
[CrossRef] [PubMed]

S. Brinkmann, R. Schreiner, T. Dresel, J. Schwider, “Interferometric testing of technical surfaces with computer generated holograms,” in Optical Inspection and Micromeasurements II, C. Gareki, ed. Proc. SPIE3098, 83–89 (1997).
[CrossRef]

Green, F. J.

K. G. Birch, F. J. Green, “Oblique incidence interferometry applied to non-optical surfaces,” J. Phys. E 6, 1045–1048 (1973).
[CrossRef]

Häusler, G.

Lindlein, N.

Salsbury, J.

L. Denes, J. Salsbury, “Flatness, parallelism and other novel uses of grazing-incidence interferometry in precision engineering,” in Topical Conference on Microscopic Inspection of Macroscopic Structures (American Society for Precision Engineering, Raleigh, N. Car., 1995), pp. 20–23.

Schreiner, R.

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).

S. Brinkmann, R. Schreiner, T. Dresel, J. Schwider, “Interferometric testing of technical surfaces with computer generated holograms,” in Optical Inspection and Micromeasurements II, C. Gareki, ed. Proc. SPIE3098, 83–89 (1997).
[CrossRef]

Schwider, J.

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]

T. Dresel, M. Beyerlein, J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
[CrossRef] [PubMed]

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

T. Dresel, J. Schwider, A. Wehrhahn, S. Babin, “Graz-ing incidence interferometry applied to the measurement of cylindrical surfaces,” Opt. Eng. 34, 3531–3535 (1995).
[CrossRef]

J. Schwider, “Verfahren und Anordnung zur Prüfung beliebiger Mantelflächen rotationssymmetrischer Festkörper mittels synthetischer Hologramme,” German patent WP106 769 (January4, 1972).

S. Brinkmann, R. Schreiner, T. Dresel, J. Schwider, “Interferometric testing of technical surfaces with computer generated holograms,” in Optical Inspection and Micromeasurements II, C. Gareki, ed. Proc. SPIE3098, 83–89 (1997).
[CrossRef]

Venzke, H.

Wehrhahn, A.

T. Dresel, J. Schwider, A. Wehrhahn, S. Babin, “Graz-ing incidence interferometry applied to the measurement of cylindrical surfaces,” Opt. Eng. 34, 3531–3535 (1995).
[CrossRef]

Appl. Opt.

J. Phys. E

K. G. Birch, F. J. Green, “Oblique incidence interferometry applied to non-optical surfaces,” J. Phys. E 6, 1045–1048 (1973).
[CrossRef]

Opt. Eng.

T. Dresel, J. Schwider, A. Wehrhahn, S. Babin, “Graz-ing incidence interferometry applied to the measurement of cylindrical surfaces,” Opt. Eng. 34, 3531–3535 (1995).
[CrossRef]

Optik

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

N. Abramson, “The interferoscope: a new type of interferometer with variable fringe separation,” Optik 30, 56–71 (1969).

Other

S. Brinkmann, R. Schreiner, T. Dresel, J. Schwider, “Interferometric testing of technical surfaces with computer generated holograms,” in Optical Inspection and Micromeasurements II, C. Gareki, ed. Proc. SPIE3098, 83–89 (1997).
[CrossRef]

L. Denes, J. Salsbury, “Flatness, parallelism and other novel uses of grazing-incidence interferometry in precision engineering,” in Topical Conference on Microscopic Inspection of Macroscopic Structures (American Society for Precision Engineering, Raleigh, N. Car., 1995), pp. 20–23.

J. Schwider, “Verfahren und Anordnung zur Prüfung beliebiger Mantelflächen rotationssymmetrischer Festkörper mittels synthetischer Hologramme,” German patent WP106 769 (January4, 1972).

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Figures (7)

Fig. 1
Fig. 1

Basic setup of a grazing incidence interferometer: diffractive optical elements, realized as computer-generated holograms, serve as beam splitters and as optical references for the object under test. Each class of objects requires its specific holograms that contain the shape information of a mathematically ideal object.

Fig. 2
Fig. 2

Rod objects can generally be described by specifying their xy cross section as a parameterized curve x(τ).

Fig. 3
Fig. 3

Classification of rod objects according to their xy cross-section profile.

Fig. 4
Fig. 4

Any object displacement can approximately be described as a τ- and z-dependent shift vector field Δ(τ,z) in the xy plane.

Fig. 5
Fig. 5

Simulated interferograms according to certain misalignment aberrations of the profile shown in Fig. 2.

Fig. 6
Fig. 6

A piecewise-defined profile consisting of three cylindrical segments.

Fig. 7
Fig. 7

Simulated interferogram containing certain misalignment aberrations of the profile shown in Fig. 6.

Equations (34)

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λeff=λcos θ.
v(τ)=vx(τ)vy(τ)  dxdτ=x˙y˙,
n(τ)  1(x˙2+y˙2)1/2y˙(τ)-x˙(τ).
d2xdl2=κ(l)n(l),
κ(τ)n(τ)=1x˙2+y˙2x¨y¨-x˙x¨+y˙y¨(x˙2+y˙2)2x˙y˙.
κ(τ)=1x˙2+y˙2d2xdτ2,n(τ),
x(τ, σ)  x(τ)+σn(τ).
z=σ[(p/λ)2-1]1/2.
Φ(τ, σ)=keffσ=2πpσ,
zmax=1|κmin|[(p/λ)2-1]1/2.
W(τ, z)=2n(τ), Δ(τ, z).
Φ(τ, σ)=keffW(τ, σ)=2πpW(τ, σ).
Δtx(τ)=tx10,
Δty(τ)=ty01,
Δωx(τ, z)=ωx0z,
Δωy(τ, z)=ωyz0,
Δωz(τ)=ωzy(τ)-x(τ).
Wtx(τ)=2txnx(τ),
Wty(τ)=2tyny(τ),
Wωx(τ, z)=2ωxzny(τ),
Wωy(τ, z)=2ωyznx(τ),
Wωz(τ)=2ωz[y(τ)nx(τ)-x(τ)ny(τ)].
V  x, yΦ(x, y)keff-txTx(x, y)-tyTy(x, y)-ωxΩx(x, y)-ωyΩy(x, y)-ωzΩz(x, y)2
Tx(τ)  2nx(τ),
Ty(τ)  2ny(τ),
Ωx(τ, z)  2zny(τ),
Ωy(τ, z)  2znx(τ),
Ωz(τ)  2[y(τ)nx(τ)-x(τ)ny(τ)].
TxTxTxTyTxΩxTxΩyTxΩzTyTxTyTyTyΩxTyΩyTyΩzΩxTxΩxTyΩxΩxΩxΩyΩxΩzΩyTxΩyTyΩyΩxΩyΩyΩyΩzΩzTxΩzTyΩzΩxΩxΩyΩzΩztxtyωxωyωz
=1keffΦmTxΦmTyΦmΩxΦmΩyΦmΩz,
AB  x, yA(x, y)B(x, y).
x(τ)=A(τ)cos τsin τ,
A(τ)=R[1+110cos(2τ)+210cos(3τ)].
x(τ)=03R+4Rcos τsin τ:τ  [-π+arcsin(3/4),-π/2]Rcos τsin τ:τ  [-π/2, π/2].0-3R+4Rcos τsin τ:τ  [π/2,π-arcsin(3/4)]

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