Abstract

We present a concept of interferometric testing, believed to be novel, that can be applied to measuring aberrations of optical components that have rotational symmetry. The optical configuration uses two coherent, collimated wave fronts that are tilted to impinge upon the optical component being tested such that one beam is on axis and the other is off axis. For small tilt angles the two aberrated wave fronts can be considered to be carrying the same aberrations. Furthermore, the off-axis beam is displaced along a direction orthogonal to the optical axis of the component. Interference between the two aberrated wave fronts produces a fringe pattern that is similar to a lateral shear interference pattern. Moiré fringes are obtained by spatial beating of the interference pattern with a CCD TV camera array. Under such conditions it is possible to subtract most of the linear carrier that is intrinsically present in the resultant fringe pattern owing to the large defocus aberration and tilt.

© 2001 Optical Society of America

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References

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  1. G. W. R. Leibbrandt, G. Harbers, P. J. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt. 35, 6151–6161 (1996).
    [CrossRef] [PubMed]
  2. H. Nomura, T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed pattern,” Appl. Opt. 38, 2800–2807 (1999).
    [CrossRef]
  3. R. S. Sirohi, T. Eiju, T. H. Barnes, “Multiple-beam lateral shear interferometry for optical testing,” Appl. Opt. 34, 2864–2870 (1995).
    [CrossRef] [PubMed]
  4. D. Malacara, “Twyman–Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 2.
  5. M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 4.
  6. D. Malacara, “Radial, rotational, and reversal shear interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 5.
  7. M. Servin, D. Malacara, J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt. 35, 4343–4348 (1996).
    [CrossRef] [PubMed]
  8. S. Loheide, “Innovative evaluation method for shearing interferograms,” Opt. Commun. 141, 254–258 (1997).
    [CrossRef]
  9. A. Cornejo-Rodríguez, “Ronchi test,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 9.
  10. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), Sec. 8.3.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. M. Takeda, S. Kobayashi, “Lateral aberration measurements with a digital Talbot interferometer,” Appl. Opt. 23, 1760–1764 (1984).
    [CrossRef] [PubMed]
  14. K. Patorski, “Talbot interferometry with increased shear: further considerations,” Appl. Opt. 25, 1111–1116 (1986).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  16. J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12, 2057–2061 (1973).
    [CrossRef] [PubMed]
  17. P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
    [CrossRef]
  18. M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
    [CrossRef]
  19. M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
    [CrossRef]
  20. M. V. Klain, Optics (Wiley, New York, 1970), pp. 142–145.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  23. S. De Nicola, P. Ferraro, “A two-dimensional fast Fourier transform method for measuring inclination angle of parallel fringe pattern,” Opt. Laser Technol. 30, 167–173 (1998).
    [CrossRef]
  24. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  25. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Chap. 10, pp. 247–270.

1999 (2)

H. Nomura, T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed pattern,” Appl. Opt. 38, 2800–2807 (1999).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

1998 (1)

S. De Nicola, P. Ferraro, “A two-dimensional fast Fourier transform method for measuring inclination angle of parallel fringe pattern,” Opt. Laser Technol. 30, 167–173 (1998).
[CrossRef]

1997 (2)

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

S. Loheide, “Innovative evaluation method for shearing interferograms,” Opt. Commun. 141, 254–258 (1997).
[CrossRef]

1996 (2)

1995 (1)

1991 (1)

1987 (1)

1986 (1)

1985 (1)

1984 (2)

1980 (1)

1975 (1)

1974 (1)

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

1973 (1)

Barnes, T. H.

Bell, B. W.

Cornejo-Rodríguez, A.

A. Cornejo-Rodríguez, “Ronchi test,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 9.

de Angelis, M.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

De Nicola, S.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

S. De Nicola, P. Ferraro, “A two-dimensional fast Fourier transform method for measuring inclination angle of parallel fringe pattern,” Opt. Laser Technol. 30, 167–173 (1998).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

Eiju, T.

Engelhardt, K.

Ferraro, P.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

S. De Nicola, P. Ferraro, “A two-dimensional fast Fourier transform method for measuring inclination angle of parallel fringe pattern,” Opt. Laser Technol. 30, 167–173 (1998).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

Finizio, A.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

Glatt, I.

Harbers, G.

Hariharan, P.

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Kafri, O.

Keren, E.

Klain, M. V.

M. V. Klain, Optics (Wiley, New York, 1970), pp. 142–145.

Kobayashi, S.

Koliopoulos, C. L.

Kunst, P. J.

Leibbrandt, G. W. R.

Livnat, A.

Loheide, S.

S. Loheide, “Innovative evaluation method for shearing interferograms,” Opt. Commun. 141, 254–258 (1997).
[CrossRef]

Malacara, D.

M. Servin, D. Malacara, J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt. 35, 4343–4348 (1996).
[CrossRef] [PubMed]

D. Malacara, “Radial, rotational, and reversal shear interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 5.

D. Malacara, “Twyman–Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 2.

Mantravadi, M. V.

M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 4.

Marroquin, J. L.

Nomura, H.

Ohnishi, K.

Oreb, B. F.

Patorski, K.

K. Patorski, “Talbot interferometry with increased shear: further considerations,” Appl. Opt. 25, 1111–1116 (1986).
[CrossRef] [PubMed]

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), Sec. 8.3.

Pierattini, G.

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

Sato, T.

Servin, M.

Sirohi, R. S.

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Chap. 10, pp. 247–270.

Steel, W. H.

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

Takeda, M.

Wyant, J. C.

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12, 2057–2061 (1973).
[CrossRef] [PubMed]

Yokozeki, S.

Appl. Opt. (10)

G. W. R. Leibbrandt, G. Harbers, P. J. Kunst, “Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer,” Appl. Opt. 35, 6151–6161 (1996).
[CrossRef] [PubMed]

H. Nomura, T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed pattern,” Appl. Opt. 38, 2800–2807 (1999).
[CrossRef]

R. S. Sirohi, T. Eiju, T. H. Barnes, “Multiple-beam lateral shear interferometry for optical testing,” Appl. Opt. 34, 2864–2870 (1995).
[CrossRef] [PubMed]

M. Servin, D. Malacara, J. L. Marroquin, “Wave-front recovery from two orthogonal sheared interferograms,” Appl. Opt. 35, 4343–4348 (1996).
[CrossRef] [PubMed]

S. Yokozeki, K. Ohnishi, “Spherical aberration measurement with shearing interferometer using Fourier imaging and moiré method,” Appl. Opt. 14, 623–627 (1975).
[CrossRef] [PubMed]

M. Takeda, S. Kobayashi, “Lateral aberration measurements with a digital Talbot interferometer,” Appl. Opt. 23, 1760–1764 (1984).
[CrossRef] [PubMed]

K. Patorski, “Talbot interferometry with increased shear: further considerations,” Appl. Opt. 25, 1111–1116 (1986).
[CrossRef] [PubMed]

J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12, 2057–2061 (1973).
[CrossRef] [PubMed]

K. Engelhardt, “Acquisition of 3-D data by focus sensing utilizing the moiré effect of CCD cameras,” Appl. Opt. 30, 1401–1407 (1991).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

Opt. Commun. (4)

P. Hariharan, W. H. Steel, J. C. Wyant, “Double grating interferometer with variable lateral shear,” Opt. Commun. 11, 317–320 (1974).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “A new approach to high accuracy measurement of the focal lengths of lenses using digital Fourier transform,” Opt. Commun. 136, 370–374 (1997).
[CrossRef]

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, G. Pierattini, “An interferometric method for measuring short focal length refractive lenses and diffractive lenses,” Opt. Commun. 160, 5–9 (1999).
[CrossRef]

S. Loheide, “Innovative evaluation method for shearing interferograms,” Opt. Commun. 141, 254–258 (1997).
[CrossRef]

Opt. Laser Technol. (1)

S. De Nicola, P. Ferraro, “A two-dimensional fast Fourier transform method for measuring inclination angle of parallel fringe pattern,” Opt. Laser Technol. 30, 167–173 (1998).
[CrossRef]

Opt. Lett. (3)

Other (7)

M. V. Klain, Optics (Wiley, New York, 1970), pp. 142–145.

A. Cornejo-Rodríguez, “Ronchi test,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 9.

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), Sec. 8.3.

D. Malacara, “Twyman–Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 2.

M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 4.

D. Malacara, “Radial, rotational, and reversal shear interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 5.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), Chap. 10, pp. 247–270.

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Figures (6)

Fig. 1
Fig. 1

Two plane wave fronts are incident upon the tested symmetrical lens (L) of focal length f. One is on axis and the other is off axis with angle α, and s is the lateral displacement suffered from the off-axis wave front at the observation plane (CCD array) at distance Δ.

Fig. 2
Fig. 2

Experimental setup: M, mirror; G, reflective grating; GP, glass plate; L, tested lens; f, focal length; Δ, distance of CCD array from the lens.

Fig. 3
Fig. 3

Finite moiré fringes between the interferogram of the tested lens (f = 38.1 mm) at distance Δ = 30.1 mm and the CCD array.

Fig. 4
Fig. 4

(a) Wrapped phase map mod. 2π and (b) unwrapped phase map of a reduced area, 190 × 190 pixels, of the interferogram of Fig. 3.

Fig. 5
Fig. 5

Moiré fringes of interferograms of the f = 38.1-mm lens recorded at (a) three distances from the lens; (b) a large linear term is visible in the paraxial region; (c) a residual linear term is observable; (d) the linear term has been completely subtracted.

Fig. 6
Fig. 6

Interferogram (without moiré fringes) of the f = 38.1-mm lens for a small off-axis angle α.

Equations (11)

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

δWΔx, y=WΔx, y-WΔx-s, y,
Wx, y=Ax2+y22+Dx2+y2,
Wx, y=Ax2+y22+Bxx2+y2+Dx2+y2+Ex,
WΔx, y=AΔx2+y22+DΔx2+y2,
WΔx, y=AΔx2+y22+BΔxx2+y2+DΔx2+y2+Ex,
δWΔx, y=x-E+2DΔs-3BΔs2+4AΔs3+3x2+y2BΔs-2AΔs2+xx2+y24AΔs-BΔ.
DΔ=12f-Δ,
E=-sin α-α.
ΔΦx, y2πx1pD+2πλ3x2+y2BΔs+xx2+y24AΔs-BΔ,
pD=pf-Δ/f
ΔΦeffx, y=2πx1pD-1pc+2πλ3x2+y2Bs+xx2+y24As-B.

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