Abstract

When testing aspheric surfaces with a computer-generated hologram, some problems should be considered. In this paper, first, we compare two types of hologram: Lohmann and interference. The phase error in the Lohmann type hologram is estimated, and a method of compensating the error is described. Second, we discuss the relation between the shape of the required wavefront and the number of resolution cells of the hologram. Since testing smaller f number optical elements increases the required number of resolution cells of the hologram, we propose the aberration balancing method to reduce the number of resolution cells. The optimum values of the defocus aberration are calculated. Especially, it is shown that the number of resolution cells in the hologram is capable of being reduced to 25%. Third, we discuss the error due to incorrect hologram size and due to misalignment of the optical system when the aberration balancing method is applied. Finally, an experimental example for testing an aspheric mirror 150 mm in diameter and 300 mm in focal length is given.

© 1978 Optical Society of America

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References

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  1. A. J. MacGovern, J. C. Wyant, Appl. Opt. 10, 619 (1971).
    [CrossRef] [PubMed]
  2. K. G. Birch, F. J. Green, J. Phys. D. 5, 1982 (1972).
    [CrossRef]
  3. A. F. Fercher, M. Kriese, Optik 35, 168 (1972).
  4. T. Yatagai, K. Yasuda, H. Saito, Jpn. J. Opt. 3, 132 (1974).
  5. W. Witz, Optik 42, 287 (1975).
  6. T. Takahasi, K. Konno, M. Kawai, M. Isshiki, Appl. Opt. 15, 546 (1976).
    [CrossRef]
  7. R. S. Sirohi, H. Blume, K. J. Rosenbruch, Opt. Acta 23, 229 (1976).
    [CrossRef]
  8. A. F. Fercher, Opt. Acta 23, 347 (1976).
    [CrossRef]
  9. J. C. Wyant, V. P. Bennett, Appl. Opt. 11, 2833 (1972).
    [CrossRef] [PubMed]
  10. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  11. M. Faulde, A. F. Fercher, R. Torge, R. N. Wilson, Opt. Commun. 7, 363 (1973).
    [CrossRef]
  12. E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, London, 1963), p. 58.
  13. B. R. Brown, A. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
    [CrossRef]
  14. M. Schwartz, Information Transmission, Modulation, and Noise (McGraw-Hill, New York, 1970), p. 228.
  15. W. H. Lee, Appl. Opt. 13, 1677 (1974).
    [CrossRef] [PubMed]
  16. S. D. Conte, Elementary Numerical Analysis (McGraw-Hill, New York, 1965), p. 71.
  17. Since the error of measuring interference fringes is considered to be 1/10 of a fringe, it is expected that the reconstructed wavefront error is far less than 1/10 of a fringe in fact.
  18. E. Isaaccson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966), p. 85.

1976 (3)

T. Takahasi, K. Konno, M. Kawai, M. Isshiki, Appl. Opt. 15, 546 (1976).
[CrossRef]

R. S. Sirohi, H. Blume, K. J. Rosenbruch, Opt. Acta 23, 229 (1976).
[CrossRef]

A. F. Fercher, Opt. Acta 23, 347 (1976).
[CrossRef]

1975 (1)

W. Witz, Optik 42, 287 (1975).

1974 (2)

W. H. Lee, Appl. Opt. 13, 1677 (1974).
[CrossRef] [PubMed]

T. Yatagai, K. Yasuda, H. Saito, Jpn. J. Opt. 3, 132 (1974).

1973 (1)

M. Faulde, A. F. Fercher, R. Torge, R. N. Wilson, Opt. Commun. 7, 363 (1973).
[CrossRef]

1972 (3)

K. G. Birch, F. J. Green, J. Phys. D. 5, 1982 (1972).
[CrossRef]

A. F. Fercher, M. Kriese, Optik 35, 168 (1972).

J. C. Wyant, V. P. Bennett, Appl. Opt. 11, 2833 (1972).
[CrossRef] [PubMed]

1971 (1)

1969 (1)

B. R. Brown, A. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
[CrossRef]

1967 (1)

Bennett, V. P.

Birch, K. G.

K. G. Birch, F. J. Green, J. Phys. D. 5, 1982 (1972).
[CrossRef]

Blume, H.

R. S. Sirohi, H. Blume, K. J. Rosenbruch, Opt. Acta 23, 229 (1976).
[CrossRef]

Brown, B. R.

B. R. Brown, A. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
[CrossRef]

Conte, S. D.

S. D. Conte, Elementary Numerical Analysis (McGraw-Hill, New York, 1965), p. 71.

Faulde, M.

M. Faulde, A. F. Fercher, R. Torge, R. N. Wilson, Opt. Commun. 7, 363 (1973).
[CrossRef]

Fercher, A. F.

A. F. Fercher, Opt. Acta 23, 347 (1976).
[CrossRef]

M. Faulde, A. F. Fercher, R. Torge, R. N. Wilson, Opt. Commun. 7, 363 (1973).
[CrossRef]

A. F. Fercher, M. Kriese, Optik 35, 168 (1972).

Green, F. J.

K. G. Birch, F. J. Green, J. Phys. D. 5, 1982 (1972).
[CrossRef]

Isaaccson, E.

E. Isaaccson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966), p. 85.

Isshiki, M.

Kawai, M.

Keller, H. B.

E. Isaaccson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966), p. 85.

Konno, K.

Kriese, M.

A. F. Fercher, M. Kriese, Optik 35, 168 (1972).

Lee, W. H.

Lohmann, A. W.

B. R. Brown, A. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
[CrossRef]

A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
[CrossRef] [PubMed]

MacGovern, A. J.

O'Neill, E. L.

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, London, 1963), p. 58.

Paris, D. P.

Rosenbruch, K. J.

R. S. Sirohi, H. Blume, K. J. Rosenbruch, Opt. Acta 23, 229 (1976).
[CrossRef]

Saito, H.

T. Yatagai, K. Yasuda, H. Saito, Jpn. J. Opt. 3, 132 (1974).

Schwartz, M.

M. Schwartz, Information Transmission, Modulation, and Noise (McGraw-Hill, New York, 1970), p. 228.

Sirohi, R. S.

R. S. Sirohi, H. Blume, K. J. Rosenbruch, Opt. Acta 23, 229 (1976).
[CrossRef]

Takahasi, T.

Torge, R.

M. Faulde, A. F. Fercher, R. Torge, R. N. Wilson, Opt. Commun. 7, 363 (1973).
[CrossRef]

Wilson, R. N.

M. Faulde, A. F. Fercher, R. Torge, R. N. Wilson, Opt. Commun. 7, 363 (1973).
[CrossRef]

Witz, W.

W. Witz, Optik 42, 287 (1975).

Wyant, J. C.

Yasuda, K.

T. Yatagai, K. Yasuda, H. Saito, Jpn. J. Opt. 3, 132 (1974).

Yatagai, T.

T. Yatagai, K. Yasuda, H. Saito, Jpn. J. Opt. 3, 132 (1974).

Appl. Opt. (5)

IBM J. Res. Dev. (1)

B. R. Brown, A. W. Lohmann, IBM J. Res. Dev. 13, 160 (1969).
[CrossRef]

J. Phys. D. (1)

K. G. Birch, F. J. Green, J. Phys. D. 5, 1982 (1972).
[CrossRef]

Jpn. J. Opt. (1)

T. Yatagai, K. Yasuda, H. Saito, Jpn. J. Opt. 3, 132 (1974).

Opt. Acta (2)

R. S. Sirohi, H. Blume, K. J. Rosenbruch, Opt. Acta 23, 229 (1976).
[CrossRef]

A. F. Fercher, Opt. Acta 23, 347 (1976).
[CrossRef]

Opt. Commun. (1)

M. Faulde, A. F. Fercher, R. Torge, R. N. Wilson, Opt. Commun. 7, 363 (1973).
[CrossRef]

Optik (2)

W. Witz, Optik 42, 287 (1975).

A. F. Fercher, M. Kriese, Optik 35, 168 (1972).

Other (5)

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley, London, 1963), p. 58.

M. Schwartz, Information Transmission, Modulation, and Noise (McGraw-Hill, New York, 1970), p. 228.

S. D. Conte, Elementary Numerical Analysis (McGraw-Hill, New York, 1965), p. 71.

Since the error of measuring interference fringes is considered to be 1/10 of a fringe, it is expected that the reconstructed wavefront error is far less than 1/10 of a fringe in fact.

E. Isaaccson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966), p. 85.

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Figures (9)

Fig. 1
Fig. 1

A typical interferometer for testing an aspheric mirror by using a computer-generated hologram. L: laser; B1,B2: beam splitter; M1,M2,M3. mirrors; H: computer-generated hologram; D: diverger lens; T: mirror under test; I: imaging lens; F: spatial filter; P: interference plane.

Fig. 2
Fig. 2

Wavefront and aperture positions. ○: phase at sampling point; □: aperture position determined by the Lohmann method; △: corrected aperture position.

Fig. 3
Fig. 3

Resultant wavefronts given by an optimization method of aberration balancing. The parameter κ of the optimum ratio and optimization methods are summarized in Table I. The thin line with κ = 0 shows the no-balancing case for reference.

Fig. 4
Fig. 4

Error due to the incorrect hologram size in the case of ε = 0.01.

Fig. 5
Fig. 5

Maximum error due to the incorrect hologram size vs the magnification factor ε for various values of the optimum parameter κ.

Fig. 6
Fig. 6

Error due to misalignment of a computer-generated hologram in the interferometer. (a) κ = 0; (b) κ = −0.828; (c) κ = −1.5.

Fig. 7
Fig. 7

Two types of holograms and their interferograms. (a) Loh- mann hologram; (b) interference-type hologram; (c) and (d) inter- ferograms resulting from interfering a plane wavefront with the wavefront produced by the holograms (a) and (b), respectively.

Fig. 8
Fig. 8

Interferogram of the wavefront affected by the undersampling effect of the computer-generated hologram.

Fig. 9
Fig. 9

(a) Computer-generated hologram for testing the Cassegrain aspheric mirror 150 mm in diameter and 300 mm in focal length. (b) Interferogram of the wavefront from the aspheric mirror under test with the reference wavefront produced by the hologram of (a).

Tables (1)

Tables Icon

Table I Optimization Methods and Estimated Optimum Ratios. The Fraction of Samples is Retained

Equations (35)

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

W ( x , y ) = 2 [ A 2 ( x 2 + y 2 ) + A 4 ( x 2 + y 2 ) 2 + A 6 ( x 2 + y 2 ) 3 + . . . ] ,
A 2 = d / 2 R 2 ,
P mn = Δ / λ · mod [ W ( m Δ , n Δ ) ] λ ,
h ( x , y ) = 0.5 + 0.5 · cos { 2 π / λ [ A 1 x + W ( x , y ) ] } ,
A 1 x + W ( x , y ) = m λ ,
x = m Δ + Δ / λ · mod [ W ( m Δ , n Δ ) ] λ .
Δ W = ( Δ λ ) 2 W ( x , y ) W ( x , y ) x .
x = m Δ + Δ / λ · mod [ W ( x , y ) ] λ ,
z W ( x , y ) = 0 .
α = [ ( W ) / ( x ) ] , β = [ ( W ) / ( y ) ] , γ = ( 1 α 2 β 2 ) 1 / 2 .
ν x = α / λ = 1 λ · W x , ν y = β / λ = 1 λ · W y .
B x = 2 λ · ( | W x | ) max ,
ν c = A 1 / λ ,
ν c = 1 / Δ x = N / 2 r o ,
ν c > B x ,
A 1 > 2 ( | W x | ) max .
ν c > 3 / 2 · B x ,
A 1 > 3 ( | W x | ) max .
A 2 = 3 A 4 r 0 2 / 4 .
| W [ ( A 2 / A 4 ) 1 / 2 r o ] | = | W ( r o ) | .
A 2 = 2 ( 2 1 ) A 4 r o 2 .
A 2 = 4 A 4 r o 2 / 3
A 2 = 3 A 4 r o 2 / 2 ,
κ = A 2 / A 4 r o 2 ;
η = N / N ,
Δ W = W ( ρ ) W [ ρ / ( 1 + ɛ ) ] ( 1 1 1 + ɛ ) · W ρ ρ ,
Δ W = W ( x + Δ x ) W ( x ) W x Δ x .
Δ W = d / 16 λ f 2 .
x = m Δ + Δ / λ · W ( x ) .
Φ ( x ) = x m Δ Δ / λ · W ( x ) .
x s m Δ Φ ( m Δ ) / Φ ( m Δ ) x = m Δ + Δ λ , W ( m Δ ) 1 Δ λ · W x .
1 Δ λ · W ( m Δ ) x .
x s m Δ + Δ / λ · W ( m Δ ) + ( Δ λ ) 2 W ( m Δ ) W ( m Δ ) x .
x L = m Δ + Δ / λ · W ( m Δ ) .
Δ W = x s x L ( Δ λ ) 2 W ( m Δ ) W ( m Δ ) x .

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