Abstract

As an extension of the knife-edge test, a noninterferometric method for inspecting circularly symmetric aspheres is proposed in which the test surface is illuminated by a spherical wave. When a small circular stop is placed around the curvature center of the best-fitting sphere, only rays characteristic for the deviation from a perfect sphere remain unobstructed. An image of the tested surface is thereby obtained, showing light and dark regions whose boundaries are correlated to the surface profile, the stop size, and the stop position along the symmetry axis. The experiment has been carried out with a paraboloid.

© 1998 Optical Society of America

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References

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  1. A. Offner, “A null corrector for paraboloidal mirrors,” Appl. Opt. 2, 153–155 (1963).
    [CrossRef]
  2. A. J. MacGovern, J. C. Wyant, “Using computer generated holograms to test aspheric wavefronts,” Appl. Opt. 11, 2833–2839 (1972).
    [CrossRef]
  3. F. Ruskanda, A. Handojo, “Holographic interferometer to test planoid aspherics,” Appl. Opt. 27, 3773–3775 (1988).
    [CrossRef] [PubMed]
  4. T. Yatagai, “Aspheric surfaces testing with shearing interferometer using fringe scanning detection,” Opt. Eng. 23, 357–360 (1984).
    [CrossRef]
  5. G. Z. Tschi, “Aspheric surface profiling using a lateral-shear interferometer,” M.S. thesis (Faculty of Applied Physics, Delft University of Technology, 1995).
  6. Y. M. Liu, G. N. Lawrence, C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. 27, 4504–4513 (1988).
    [CrossRef] [PubMed]
  7. M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
    [CrossRef]
  8. J. K. Sinha, H. V. Tippur, “Infrared interferometry for rough surface measurements: application to failure characterization and flaw detection,” Opt. Eng. 36, 2233–2239 (1997).
    [CrossRef]
  9. D. Malacara, Optical Shop Testing (Wiley, New York, 1978).
  10. R. E. Parks, L. Z. Shao, “Testing large hyperbolic secondary mirrors,” Opt. Eng. 27, 1057–1062 (1988).
    [CrossRef]
  11. E. C. Bruner, “Sensitive visual test for concave diffraction gratings,” Appl. Opt. 11, 1357–1359 (1972).
    [CrossRef] [PubMed]

1997 (1)

J. K. Sinha, H. V. Tippur, “Infrared interferometry for rough surface measurements: application to failure characterization and flaw detection,” Opt. Eng. 36, 2233–2239 (1997).
[CrossRef]

1993 (1)

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
[CrossRef]

1988 (3)

1984 (1)

T. Yatagai, “Aspheric surfaces testing with shearing interferometer using fringe scanning detection,” Opt. Eng. 23, 357–360 (1984).
[CrossRef]

1972 (2)

1963 (1)

Bruner, E. C.

Handojo, A.

Koliopoulos, C. L.

Lawrence, G. N.

Liu, Y. M.

MacGovern, A. J.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1978).

Mazzoni, A.

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
[CrossRef]

Melozzi, M.

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
[CrossRef]

Offner, A.

Parks, R. E.

R. E. Parks, L. Z. Shao, “Testing large hyperbolic secondary mirrors,” Opt. Eng. 27, 1057–1062 (1988).
[CrossRef]

Pezzati, L.

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
[CrossRef]

Ruskanda, F.

Shao, L. Z.

R. E. Parks, L. Z. Shao, “Testing large hyperbolic secondary mirrors,” Opt. Eng. 27, 1057–1062 (1988).
[CrossRef]

Sinha, J. K.

J. K. Sinha, H. V. Tippur, “Infrared interferometry for rough surface measurements: application to failure characterization and flaw detection,” Opt. Eng. 36, 2233–2239 (1997).
[CrossRef]

Tippur, H. V.

J. K. Sinha, H. V. Tippur, “Infrared interferometry for rough surface measurements: application to failure characterization and flaw detection,” Opt. Eng. 36, 2233–2239 (1997).
[CrossRef]

Tschi, G. Z.

G. Z. Tschi, “Aspheric surface profiling using a lateral-shear interferometer,” M.S. thesis (Faculty of Applied Physics, Delft University of Technology, 1995).

Wyant, J. C.

Yatagai, T.

T. Yatagai, “Aspheric surfaces testing with shearing interferometer using fringe scanning detection,” Opt. Eng. 23, 357–360 (1984).
[CrossRef]

Appl. Opt. (5)

Opt. Eng. (4)

T. Yatagai, “Aspheric surfaces testing with shearing interferometer using fringe scanning detection,” Opt. Eng. 23, 357–360 (1984).
[CrossRef]

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
[CrossRef]

J. K. Sinha, H. V. Tippur, “Infrared interferometry for rough surface measurements: application to failure characterization and flaw detection,” Opt. Eng. 36, 2233–2239 (1997).
[CrossRef]

R. E. Parks, L. Z. Shao, “Testing large hyperbolic secondary mirrors,” Opt. Eng. 27, 1057–1062 (1988).
[CrossRef]

Other (2)

D. Malacara, Optical Shop Testing (Wiley, New York, 1978).

G. Z. Tschi, “Aspheric surface profiling using a lateral-shear interferometer,” M.S. thesis (Faculty of Applied Physics, Delft University of Technology, 1995).

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Figures (6)

Fig. 1
Fig. 1

Principle of the method, with the test surface illuminated from its center of curvature C. A ray reflected from a perfect sphere (solid line) will propagate back to C, and the ray returning from an asphere Σ (dashed line) will not.

Fig. 2
Fig. 2

Optical setup. M is an adjustable mirror facilitating the alignment; L’s, lenses.

Fig. 3
Fig. 3

Configuration for the derivation of the height of a ray reflected by the test surface. Ray CQ is reflected by the surface as ray QP. Point C′ is the curvature center of the surface at Q, and point Q′ is the projection of Q upon the z axis.

Fig. 4
Fig. 4

Height h of a reflected ray as a function of the height r at the paraboloid. The parameter is the displacement z s of the stop with respect to the spherical curvature center C. The dashed lines represent the diametrical rims of a 20-μm stop extending between h = -10 μm and h = +10 μm. Similarly, the dotted-dashed lines give the rims of a 100-μm-diameter stop.

Fig. 5
Fig. 5

Dark–light boundaries expressed in the radial coordinate r of the paraboloid as a function of z s for a 100-μm stop diameter. The central area bordered by the curves represents the dark region.

Fig. 6
Fig. 6

Two photographs showing the surface image with the 100-μm stop (top) at z s = -1 mm and (bottom) at z s = 0.5 mm. Parts of the mirror rim are obscured by its holder.

Equations (12)

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

R 2 Δ ϕ = d / 2 cos   α ,
β = arctan d z / d r .
γ = α - β .
R r = 1 + d z d r 2 3 / 2 d 2 z d r 2 ,
α = arctan r R 0 - z r .
OPQ = β - γ ,
h = PQ ¯ - CQ ¯ - z s tan ( OPQ ) = r tan β - γ - r tan β - z s tan β - γ .
z r = r 2 / 4 f ,
tan β = r / 2 f ,
tan γ = r 3 / 16 f 3 + 2 fr 2 .
tan β - γ = 16 f 3 r / 32 f 4 + 4 f 2 r 2 + r 4 ,
h = 16 f 3 r r 2 2 f + r 4 16 f 3 - z s   / 32 f 4 + 4 f 2 r 2 + r 4 .

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