Abstract

Wavefront shearing interferometers have inherent advantages over more conventional interferometers because they do not need a separate reference wavefront. However, the fringe patterns are less directly related to the wavefront shape. In this paper, a method is described that uses data obtained from two lateral shear interferograms sheared in orthogonal directions to describe a wavefront of any arbitrary shape. Analysis of the data defines the wavefront on a regular grid, using a least squares criterion to match the measured data to the reconstructed shear data. Because each point on the final wavefront is involved in at most four measurements, the matrices tend to have many zero elements, making them easily solvable by simple numerical techniques, even for several hundred points. An error analysis indicates that the accuracy of the final results can be as good as the accuracy of the measured data. The procedure is described, and results of a typical analysis are shown.

© 1974 Optical Society of America

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References

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  1. O. Bryngdahl, Applications of Shearing Interferometry, in Progressin Optics (North-Holland, Amsterdam, 1965), p. 39.
  2. J. B. Saunders, J. Res. Nat. Bur. Stand. 65B, 239 (1961).
  3. D. Dutton et al., Appl. Opt. 7, 125 (1968).
    [CrossRef] [PubMed]
  4. J. B. Saunders, R. J. Bruening, Aston. J. 73, 415 (1968).
    [CrossRef]
  5. M. L. Salvidori, M. L. Baron, Numerical Methods in Engineering (Prentice-Hall, Englewood Cliffs, N. J., 1961), p. 31.
  6. J. C. Wyant, Appl. Opt. 12, 2057 (1973).
    [CrossRef] [PubMed]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 464ff.

1973 (1)

1968 (2)

J. B. Saunders, R. J. Bruening, Aston. J. 73, 415 (1968).
[CrossRef]

D. Dutton et al., Appl. Opt. 7, 125 (1968).
[CrossRef] [PubMed]

1961 (1)

J. B. Saunders, J. Res. Nat. Bur. Stand. 65B, 239 (1961).

Baron, M. L.

M. L. Salvidori, M. L. Baron, Numerical Methods in Engineering (Prentice-Hall, Englewood Cliffs, N. J., 1961), p. 31.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 464ff.

Bruening, R. J.

J. B. Saunders, R. J. Bruening, Aston. J. 73, 415 (1968).
[CrossRef]

Bryngdahl, O.

O. Bryngdahl, Applications of Shearing Interferometry, in Progressin Optics (North-Holland, Amsterdam, 1965), p. 39.

Dutton, D.

Salvidori, M. L.

M. L. Salvidori, M. L. Baron, Numerical Methods in Engineering (Prentice-Hall, Englewood Cliffs, N. J., 1961), p. 31.

Saunders, J. B.

J. B. Saunders, R. J. Bruening, Aston. J. 73, 415 (1968).
[CrossRef]

J. B. Saunders, J. Res. Nat. Bur. Stand. 65B, 239 (1961).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 464ff.

Wyant, J. C.

Appl. Opt. (2)

Aston. J. (1)

J. B. Saunders, R. J. Bruening, Aston. J. 73, 415 (1968).
[CrossRef]

J. Res. Nat. Bur. Stand. (1)

J. B. Saunders, J. Res. Nat. Bur. Stand. 65B, 239 (1961).

Other (3)

O. Bryngdahl, Applications of Shearing Interferometry, in Progressin Optics (North-Holland, Amsterdam, 1965), p. 39.

M. L. Salvidori, M. L. Baron, Numerical Methods in Engineering (Prentice-Hall, Englewood Cliffs, N. J., 1961), p. 31.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), p. 464ff.

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Figures (7)

Fig. 1
Fig. 1

Schematic representation of one-dimensional shearing interferometry: (a) wavefront; (b) wavefront shifted by s; (c) difference of (a) and (b).

Fig. 2
Fig. 2

Schematic representation of two-dimensional shearing interferometry (four measurements): (a) wavefront; (b) shearing interferogram in x direction; (c) shearing interferogram in y direction.

Fig. 3
Fig. 3

Schematic representation of two-dimensional shearing interferometry (ten measurements): (a) wavefront; (b) shearing interferogram in x direction; (c) shearing interferogram in y direction.

Fig. 4
Fig. 4

Shearing interferograms of 10-cm focal length, f/7 lens: (a) on axis, x shear; (b) on axis, y shear; (c) 5° off axis, x shear; (d) 5° off axis, y shear; (e) 100 off axis, x shear; (f) 10° off axis, y shear.

Fig. 5
Fig. 5

Wavefront from 10-cm focal length, f/7 lens on axis (contour interval =0.2 wavelength): (a) evaluation of shearing interferograms; (b) evaluation of lens design.

Fig. 6
Fig. 6

Wavefront from 10-cm focal length f/7 lens, 5° off axis (contour interval =0.5 wavelength): (a) evaluation of shearing interferograms; (b) evaluation of lens design.

Fig. 7
Fig. 7

Wavefront from 10-cm focal length, f/7 lens, 10° off axis (contour interval =0.5 wavelength); (a) evaluation of shearing interferograms; (b) evaluation of lens design

Equations (9)

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V ( x , y ; s ) = W ( x + s , y ) - W ( x , y ) ,
a 1 = W 2 - W 1 , a 2 = W 4 - W 3 , b 1 = W 3 - W 1 , b 2 = W 4 - W 2 .
W 1 = 0 , W 2 = a 1 , W 3 = b 1 , W 4 = a 1 + b 2 ,
W 4 = b 1 + a 2 .
ϕ = ( W 2 - a 1 ) 2 + ( W 4 - W 3 - a 2 ) 2 + ( W 3 - b 1 ) 2 + ( W 4 - W 2 - b 2 ) 2 ,
2 W 2 - W 4 = a 1 - b 2 , 2 W 3 - W 4 = b 1 - a 2 , - W 2 - W 3 + 2 W 4 = a 2 + b 2 .
M W = d ,
[ 2 - 1 0 0 0 0 0 0 - 1 2 0 0 - 1 0 0 0 0 0 3 - 1 0 - 1 0 0 0 0 - 1 2 - 1 0 0 0 0 - 1 0 - 1 3 0 0 - 1 0 0 - 1 0 0 2 - 1 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 - 1 0 - 1 2 ] [ W 2 W 3 W 4 W 5 W 6 W 7 W 8 W 9 ] = [ a 1 - a 2 a 2 - b 3 b 1 - a 3 - b 2 a 3 - a 4 b 3 + a 4 - b 4 b 2 - a 5 a 5 - a 6 b 4 + a 6 ]
W = A + B x + C y + D ( x 2 + y 2 )

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