Abstract

This paper describes the digital retrieval method of the wave aberration of an optical system from the point spread function or the optical transfer function data of the system. The Gerchberg-Saxton algorithm is used as the iterative retrieval technique of the wave aberration with additional information about the type of aberration. The usefulness of the proposed method is confirmed by computer simulations. Several problems associated with the retrieval procedure are indicated by elementary experiments with a computer generated hologram as well as by computer simulations.

© 1981 Optical Society of America

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  1. R. A. Gonsalves, J. Opt. Soc. Am. 66, 961 (1976).
    [CrossRef]
  2. R. W. Gerchberg, W. O. Saxton, Optik 34, 275 (1971).
  3. R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).
  4. D. C. Ehn, J. Opt. Soc. Am. 67, 1364 (1977).
  5. A. J. Devaney, R. A. Gonsalves, R. Chidlaw, J. Opt. Soc. Am. 67, 1422 (1977).
  6. A. J. Devaney, R. Chidlaw, R. A. Gonsalves, J. Opt. Soc. Am. 68, 1386 (1978).
    [CrossRef]
  7. W. H. Southwell, J. Opt. Soc. Am. 67, 396 (1977).
    [CrossRef]
  8. W. H. Southwell, N. A. Massie, J. S. Hartlove, J. Opt. Soc. Am. 69, 1468 (1979).
    [CrossRef]
  9. R. Barakat, J. Opt. Soc. Am. 69, 1467 (1979).
    [CrossRef]
  10. S. R. Robinson, R. Barakat, J. Opt. Soc. Am. 69, 1468 (1979).
  11. R. H. Boucher, J. Opt. Soc. Am. 69, 1442 (1979).
  12. R. H. Boucher, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130 (1980).
  13. R. W. Gerchberg, W. O. Saxton, in Image Processing and Computer-Aided Design in Electron Optics, P. W. Hawkes, Ed. (Academic, New York, 1973), p. 66.
  14. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  15. J. R. Fienup, Opt. Lett. 3, 27 (1978).
    [CrossRef] [PubMed]
  16. M. C. Gallagher, B. Liu, Appl. Opt. 12, 2328 (1973).
    [CrossRef] [PubMed]
  17. J. Gassmann, Optik 48, 347 (1977).
  18. J. R. Fienup, Opt. Eng. 19, 297 (1980).
    [CrossRef]
  19. K. Murata, Prog. Opt. 5, 199 (1966).
    [CrossRef]
  20. Y. Ishii, J. Maeda, K. Murata, Opt. Acta 26, 969 (1979).
    [CrossRef]

1980

R. H. Boucher, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130 (1980).

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

1979

Y. Ishii, J. Maeda, K. Murata, Opt. Acta 26, 969 (1979).
[CrossRef]

W. H. Southwell, N. A. Massie, J. S. Hartlove, J. Opt. Soc. Am. 69, 1468 (1979).
[CrossRef]

R. Barakat, J. Opt. Soc. Am. 69, 1467 (1979).
[CrossRef]

S. R. Robinson, R. Barakat, J. Opt. Soc. Am. 69, 1468 (1979).

R. H. Boucher, J. Opt. Soc. Am. 69, 1442 (1979).

1978

J. R. Fienup, Opt. Lett. 3, 27 (1978).
[CrossRef] [PubMed]

A. J. Devaney, R. Chidlaw, R. A. Gonsalves, J. Opt. Soc. Am. 68, 1386 (1978).
[CrossRef]

1977

J. Gassmann, Optik 48, 347 (1977).

D. C. Ehn, J. Opt. Soc. Am. 67, 1364 (1977).

A. J. Devaney, R. A. Gonsalves, R. Chidlaw, J. Opt. Soc. Am. 67, 1422 (1977).

W. H. Southwell, J. Opt. Soc. Am. 67, 396 (1977).
[CrossRef]

1976

1973

1972

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

1971

R. W. Gerchberg, W. O. Saxton, Optik 34, 275 (1971).

1966

K. Murata, Prog. Opt. 5, 199 (1966).
[CrossRef]

Barakat, R.

S. R. Robinson, R. Barakat, J. Opt. Soc. Am. 69, 1468 (1979).

R. Barakat, J. Opt. Soc. Am. 69, 1467 (1979).
[CrossRef]

Boucher, R. H.

R. H. Boucher, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130 (1980).

R. H. Boucher, J. Opt. Soc. Am. 69, 1442 (1979).

Chidlaw, R.

A. J. Devaney, R. Chidlaw, R. A. Gonsalves, J. Opt. Soc. Am. 68, 1386 (1978).
[CrossRef]

A. J. Devaney, R. A. Gonsalves, R. Chidlaw, J. Opt. Soc. Am. 67, 1422 (1977).

Devaney, A. J.

A. J. Devaney, R. Chidlaw, R. A. Gonsalves, J. Opt. Soc. Am. 68, 1386 (1978).
[CrossRef]

A. J. Devaney, R. A. Gonsalves, R. Chidlaw, J. Opt. Soc. Am. 67, 1422 (1977).

Ehn, D. C.

D. C. Ehn, J. Opt. Soc. Am. 67, 1364 (1977).

Fienup, J. R.

Gallagher, M. C.

Gassmann, J.

J. Gassmann, Optik 48, 347 (1977).

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

R. W. Gerchberg, W. O. Saxton, Optik 34, 275 (1971).

R. W. Gerchberg, W. O. Saxton, in Image Processing and Computer-Aided Design in Electron Optics, P. W. Hawkes, Ed. (Academic, New York, 1973), p. 66.

Gonsalves, R. A.

A. J. Devaney, R. Chidlaw, R. A. Gonsalves, J. Opt. Soc. Am. 68, 1386 (1978).
[CrossRef]

A. J. Devaney, R. A. Gonsalves, R. Chidlaw, J. Opt. Soc. Am. 67, 1422 (1977).

R. A. Gonsalves, J. Opt. Soc. Am. 66, 961 (1976).
[CrossRef]

Hartlove, J. S.

W. H. Southwell, N. A. Massie, J. S. Hartlove, J. Opt. Soc. Am. 69, 1468 (1979).
[CrossRef]

Ishii, Y.

Y. Ishii, J. Maeda, K. Murata, Opt. Acta 26, 969 (1979).
[CrossRef]

Liu, B.

Maeda, J.

Y. Ishii, J. Maeda, K. Murata, Opt. Acta 26, 969 (1979).
[CrossRef]

Massie, N. A.

W. H. Southwell, N. A. Massie, J. S. Hartlove, J. Opt. Soc. Am. 69, 1468 (1979).
[CrossRef]

Murata, K.

Y. Ishii, J. Maeda, K. Murata, Opt. Acta 26, 969 (1979).
[CrossRef]

K. Murata, Prog. Opt. 5, 199 (1966).
[CrossRef]

Robinson, S. R.

S. R. Robinson, R. Barakat, J. Opt. Soc. Am. 69, 1468 (1979).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

R. W. Gerchberg, W. O. Saxton, Optik 34, 275 (1971).

R. W. Gerchberg, W. O. Saxton, in Image Processing and Computer-Aided Design in Electron Optics, P. W. Hawkes, Ed. (Academic, New York, 1973), p. 66.

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Southwell, W. H.

W. H. Southwell, N. A. Massie, J. S. Hartlove, J. Opt. Soc. Am. 69, 1468 (1979).
[CrossRef]

W. H. Southwell, J. Opt. Soc. Am. 67, 396 (1977).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

R. A. Gonsalves, J. Opt. Soc. Am. 66, 961 (1976).
[CrossRef]

W. H. Southwell, J. Opt. Soc. Am. 67, 396 (1977).
[CrossRef]

W. H. Southwell, N. A. Massie, J. S. Hartlove, J. Opt. Soc. Am. 69, 1468 (1979).
[CrossRef]

R. Barakat, J. Opt. Soc. Am. 69, 1467 (1979).
[CrossRef]

S. R. Robinson, R. Barakat, J. Opt. Soc. Am. 69, 1468 (1979).

R. H. Boucher, J. Opt. Soc. Am. 69, 1442 (1979).

D. C. Ehn, J. Opt. Soc. Am. 67, 1364 (1977).

A. J. Devaney, R. A. Gonsalves, R. Chidlaw, J. Opt. Soc. Am. 67, 1422 (1977).

A. J. Devaney, R. Chidlaw, R. A. Gonsalves, J. Opt. Soc. Am. 68, 1386 (1978).
[CrossRef]

Opt. Acta

Y. Ishii, J. Maeda, K. Murata, Opt. Acta 26, 969 (1979).
[CrossRef]

Opt. Eng.

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

Opt. Lett.

Optik

R. W. Gerchberg, W. O. Saxton, Optik 34, 275 (1971).

R. W. Gerchberg, W. O. Saxton, Optik 35, 237 (1972).

J. Gassmann, Optik 48, 347 (1977).

Proc. Soc. Photo-Opt. Instrum. Eng.

R. H. Boucher, Proc. Soc. Photo-Opt. Instrum. Eng. 231, 130 (1980).

Prog. Opt.

K. Murata, Prog. Opt. 5, 199 (1966).
[CrossRef]

Other

R. W. Gerchberg, W. O. Saxton, in Image Processing and Computer-Aided Design in Electron Optics, P. W. Hawkes, Ed. (Academic, New York, 1973), p. 66.

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

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Figures (13)

Fig. 1
Fig. 1

Optical coordinate system.

Fig. 2
Fig. 2

Schematic block diagram of the retrieval method of wave aberration from PSF measurements or from OTF measurements.

Fig. 3
Fig. 3

Block diagram of the iterative wave aberration retrieval algorithm.

Fig. 4
Fig. 4

Assumed spherical aberration Ŵ(ψ,η): (a) contour plot (at λ/50 intervals); (b) cross-sectional plot [Ŵ(ψ,0)], and (c) perspective plot.

Fig. 5
Fig. 5

Section of PSF intensity distribution calculated from the assumed spherical aberration.

Fig. 6
Fig. 6

Retrieved result of spherical aberration from the PSF data without the effect of noise. The MSE is 0.003%.

Fig. 7
Fig. 7

Retrieved results of spherical aberration from the noisy PSF data. Ratios of noise to PSF mean value are (a) 3%, (b) 5%, (c) 10%, and (d) 20%; the MSEs are (a) 0.08%, (b) 0.16%, (c) 0.35%, and (d) 0.71%.

Fig. 8
Fig. 8

Influence of defocusing on the algorithm. (a), (b), (c), and (d) show the assumed spherical aberrations for D = −λ/8, λ/8, λ/4, and 3λ/8, respectively, and (a′), (b′), (c′), and (d′) are the corresponding retrieved results from the PSF data. The MSEs are (a′) 0.005%, (b′) 0.003%, (c′) 0.002%, and (d′) 0.131%.

Fig. 9
Fig. 9

Cross section and perspective view of the MTF calculated from the assumed spherical aberration.

Fig. 10
Fig. 10

Retrieved result of spherical aberration from the OTF data without the effect of noise; the MSE is 0.09%.

Fig. 11
Fig. 11

Retrieved results of spherical aberration from the noisy OTF data. Ratios of noise to OTF mean value are (a) 3%, (b) 5%, and (c) 10%; the MSEs are (a) 0.55%, (b) 0.89%, and (c) 1.73%.

Fig. 12
Fig. 12

The CGH with third-order spherical aberration.

Fig. 13
Fig. 13

Retrieved result of spherical aberration from the measured PSF data of the CGH; the MSE is 4.87%.

Tables (1)

Tables Icon

Table I MSE Decay for the Retrieval from PSF Measurements in the Case of Spherical Aberration

Equations (18)

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x 0 = ( X 0 / λ ) n 0 sin α 0 , y 0 = ( Y 0 / λ ) n 0 sin α 0 ,
x = ( X / λ ) n i sin α i , y = ( Y / λ ) n i sin α i ,
ξ = a / R ,             η = b / R .
f ( ξ , η ) = f ( ξ , η ) exp [ i k W ( ξ , η ) ] ,
h a ( x , y ) = - f ( ξ , η ) exp [ 2 π i ( x ξ + y η ) ] d ξ d η .
h ( x , y ) = h a ( x , y ) 2 .
H ( s , t ) = - h ( x , y ) exp [ - 2 π i ( s x + t y ) ] d x d y - h ( x , y ) d x d y ,
s = λ u n i sin α i ,             t = λ v n i sin α i .
f 1 ( ξ , η ) = f ( ξ , η ) exp [ i k W 0 ( ξ , η ) ] .
F 1 ( x , y ) = F 1 ( x , y ) exp [ i Φ ( x , y ) ] .
F 1 ( x , y ) = h a ( x , y ) exp [ i Φ ( x , y ) ] .
f 1 ( ξ , η ) = f 1 ( ξ , η ) exp [ i k W 1 ( ξ , η ) ] .
f 2 ( ξ , η ) = f ( ξ , η ) exp [ i k W 1 ( ξ , η ) ] .
= - f ( ξ , η ) - f n ( ξ , η ) 2 d ξ d η - f ( ξ , η ) 2 d ξ d η ,
f ( ξ , η ) = { 1 , ξ 2 + η 2 1 0 , ξ 2 + η 2 > 1
W ^ ( ξ , η ) = S ( ξ 2 + η 2 ) 2 ,
k W 0 ( ξ , η ) = - π ( ξ 2 + η 2 ) 2 .
f ( ξ , η ) = exp { i k [ S ( ξ 2 + η 2 ) 2 + D ( ξ 2 + η 2 ) ] } ,

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