Abstract

A new technique for computer-based Talbot interferometry is presented. Fast and automatic aberration measurement has been achieved with improved accuracy and sensitivity. Experimental results are given that agree with the results of calculation by ray tracing.

© 1984 Optical Society of America

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References

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  1. S. Yokozeki, T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10, 1575 (1971); “Shearing interferometer using the grating as the beam splitter. Part 2,” Appl. Opt. 10, 690 (1971).
    [CrossRef] [PubMed]
  2. A. W. Lohman, D. E. Silva, “An Interferometer Based on Talbot effect,” Opt. Commun. 2, 413 (1971); “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
    [CrossRef]
  3. D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1980 (1971).
    [CrossRef]
  4. D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613 (1972).
    [CrossRef] [PubMed]
  5. J. C. Fouere, D. Malacara, “Focusing errors in a collimating lens or mirror use of a moire technique,” Appl. Opt. 13, 1322 (1974).
    [CrossRef] [PubMed]
  6. S. Yokozeki, K. Ohnishi, “Spherical aberration measurement with shearing interferometer using Fourier imaging and moire method,” Appl. Opt. 14, 623 (1975).
    [CrossRef] [PubMed]
  7. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156 (1982).
    [CrossRef]
  8. S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation Method Using Fourier Imaging and Moire Techniques,” Opt. Commun. 14, 401 (1975).
    [CrossRef]
  9. See, for example, A. Cornejo-Rodriguez, Optical Shop Testing, D. Malacara, Ed. (Wiley-Interscience, New York, 1978), Chap. 9.
  10. J. C. Wyant, “Double frequency grating lateral shear interferometry,” Appl. Opt. 12, 2057 (1973).
    [CrossRef] [PubMed]
  11. Y. Ichioka, T. Suzuki,“Comparison of Aberrations of the Direct Optical System with Those of the Reversed One. I,” Technol. Rep.Osaka Univ.15, 167 (1965);Y. Ichioka, T. Suzuki, “Comparison of Aberrations of the Direct Optical System with Those of the Reversed One. II,” Technol. Rep.Osaka Univ.17, 249 (1967).
  12. M. Takeda, K. Takesi, “Modulation transfer function measurement of TV zoom lenses: influence of distortion on the equivalence of normal and inverse conjugate techniques,” Appl. Opt. 17, 2171 (1978).
    [CrossRef] [PubMed]
  13. J. M. Cowley, A. F. Moodie, “Fourier Images: I-The Point Source,” Proc. Phys. Soc. London Sect. B 70, 486 (1957).
    [CrossRef]
  14. See, for example, J. J. Downing, Modulation Systems and Noise (Prentice-Hall, Englewood Cliffs, New Jersey, 1964), Chap. 5.

1982 (1)

1978 (1)

1975 (2)

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

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation Method Using Fourier Imaging and Moire Techniques,” Opt. Commun. 14, 401 (1975).
[CrossRef]

1974 (1)

J. C. Fouere, D. Malacara, “Focusing errors in a collimating lens or mirror use of a moire technique,” Appl. Opt. 13, 1322 (1974).
[CrossRef] [PubMed]

1973 (1)

1972 (1)

1971 (3)

D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1980 (1971).
[CrossRef]

S. Yokozeki, T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10, 1575 (1971); “Shearing interferometer using the grating as the beam splitter. Part 2,” Appl. Opt. 10, 690 (1971).
[CrossRef] [PubMed]

A. W. Lohman, D. E. Silva, “An Interferometer Based on Talbot effect,” Opt. Commun. 2, 413 (1971); “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
[CrossRef]

1957 (1)

J. M. Cowley, A. F. Moodie, “Fourier Images: I-The Point Source,” Proc. Phys. Soc. London Sect. B 70, 486 (1957).
[CrossRef]

Cornejo-Rodriguez, A.

See, for example, A. Cornejo-Rodriguez, Optical Shop Testing, D. Malacara, Ed. (Wiley-Interscience, New York, 1978), Chap. 9.

Cowley, J. M.

J. M. Cowley, A. F. Moodie, “Fourier Images: I-The Point Source,” Proc. Phys. Soc. London Sect. B 70, 486 (1957).
[CrossRef]

Downing, J. J.

See, for example, J. J. Downing, Modulation Systems and Noise (Prentice-Hall, Englewood Cliffs, New Jersey, 1964), Chap. 5.

Fouere, J. C.

J. C. Fouere, D. Malacara, “Focusing errors in a collimating lens or mirror use of a moire technique,” Appl. Opt. 13, 1322 (1974).
[CrossRef] [PubMed]

Ichioka, Y.

Y. Ichioka, T. Suzuki,“Comparison of Aberrations of the Direct Optical System with Those of the Reversed One. I,” Technol. Rep.Osaka Univ.15, 167 (1965);Y. Ichioka, T. Suzuki, “Comparison of Aberrations of the Direct Optical System with Those of the Reversed One. II,” Technol. Rep.Osaka Univ.17, 249 (1967).

Ina, H.

Kobayashi, S.

Lohman, A. W.

A. W. Lohman, D. E. Silva, “An Interferometer Based on Talbot effect,” Opt. Commun. 2, 413 (1971); “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
[CrossRef]

Malacara, D.

J. C. Fouere, D. Malacara, “Focusing errors in a collimating lens or mirror use of a moire technique,” Appl. Opt. 13, 1322 (1974).
[CrossRef] [PubMed]

Moodie, A. F.

J. M. Cowley, A. F. Moodie, “Fourier Images: I-The Point Source,” Proc. Phys. Soc. London Sect. B 70, 486 (1957).
[CrossRef]

Ohnishi, K.

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

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation Method Using Fourier Imaging and Moire Techniques,” Opt. Commun. 14, 401 (1975).
[CrossRef]

Patorski, K.

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation Method Using Fourier Imaging and Moire Techniques,” Opt. Commun. 14, 401 (1975).
[CrossRef]

Silva, D. E.

D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11, 2613 (1972).
[CrossRef] [PubMed]

D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1980 (1971).
[CrossRef]

A. W. Lohman, D. E. Silva, “An Interferometer Based on Talbot effect,” Opt. Commun. 2, 413 (1971); “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
[CrossRef]

Suzuki, T.

S. Yokozeki, T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10, 1575 (1971); “Shearing interferometer using the grating as the beam splitter. Part 2,” Appl. Opt. 10, 690 (1971).
[CrossRef] [PubMed]

Y. Ichioka, T. Suzuki,“Comparison of Aberrations of the Direct Optical System with Those of the Reversed One. I,” Technol. Rep.Osaka Univ.15, 167 (1965);Y. Ichioka, T. Suzuki, “Comparison of Aberrations of the Direct Optical System with Those of the Reversed One. II,” Technol. Rep.Osaka Univ.17, 249 (1967).

Takeda, M.

Takesi, K.

Wyant, J. C.

Yokozeki, S.

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

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation Method Using Fourier Imaging and Moire Techniques,” Opt. Commun. 14, 401 (1975).
[CrossRef]

S. Yokozeki, T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10, 1575 (1971); “Shearing interferometer using the grating as the beam splitter. Part 2,” Appl. Opt. 10, 690 (1971).
[CrossRef] [PubMed]

Appl. Opt. (2)

J. C. Fouere, D. Malacara, “Focusing errors in a collimating lens or mirror use of a moire technique,” Appl. Opt. 13, 1322 (1974).
[CrossRef] [PubMed]

S. Yokozeki, T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10, 1575 (1971); “Shearing interferometer using the grating as the beam splitter. Part 2,” Appl. Opt. 10, 690 (1971).
[CrossRef] [PubMed]

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation Method Using Fourier Imaging and Moire Techniques,” Opt. Commun. 14, 401 (1975).
[CrossRef]

Opt. Commun. (1)

A. W. Lohman, D. E. Silva, “An Interferometer Based on Talbot effect,” Opt. Commun. 2, 413 (1971); “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
[CrossRef]

Proc. Phys. Soc. London Sect. B (1)

J. M. Cowley, A. F. Moodie, “Fourier Images: I-The Point Source,” Proc. Phys. Soc. London Sect. B 70, 486 (1957).
[CrossRef]

Other (3)

See, for example, J. J. Downing, Modulation Systems and Noise (Prentice-Hall, Englewood Cliffs, New Jersey, 1964), Chap. 5.

See, for example, A. Cornejo-Rodriguez, Optical Shop Testing, D. Malacara, Ed. (Wiley-Interscience, New York, 1978), Chap. 9.

Y. Ichioka, T. Suzuki,“Comparison of Aberrations of the Direct Optical System with Those of the Reversed One. I,” Technol. Rep.Osaka Univ.15, 167 (1965);Y. Ichioka, T. Suzuki, “Comparison of Aberrations of the Direct Optical System with Those of the Reversed One. II,” Technol. Rep.Osaka Univ.17, 249 (1967).

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Figures (6)

Fig. 1
Fig. 1

Generation of aberrated Talbot image.

Fig. 2
Fig. 2

Spatial frequency spectra of aberrated Talbot image. Only one spectrum (dotted) satisfying nm = 1 is selected by filtering operation.

Fig. 3
Fig. 3

Schematic diagram of spherical aberration measuring system.

Fig. 4
Fig. 4

Irradiance profile of aberrated Fourier image measured along the line y = 0.

Fig. 5
Fig. 5

Spatial frequency spectra of aberrated Talbot image. Only one spectrum (pointed by an arrow) satisfying nm = 1 is selected and shifted down to zero frequency.

Fig. 6
Fig. 6

Lateral spherical aberration (mm): measured —; calculated by ray tracing - - - - - -.

Equations (22)

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u 0 ( Q ) = A 0 exp { i k [ R 0 Q ¯ + W ( x , y ; z ) ] } ,
R 0 Q ¯ = z 0 - [ x 2 + y 2 + ( z 0 - z ) 2 ] 1 / 2 .
{ x n = x cos θ n - z sin θ n , y n = y , z n = x sin θ n + z cos θ n .
u n ( Q ) = A n exp { i k [ R n Q ¯ + W ( x n , y n ; z n ) ] } ,
R n Q ¯ = z 0 - [ x n 2 + y n 2 + ( z 0 - z n ) 2 ] 1 / 2 = z 0 - { ( z 0 - z ) 2 - 2 z 0 [ x sin θ n - z ( 1 - cos θ n ) ] + x 2 + y 2 } 1 / 2 .
R n Q ¯ z + z 0 [ x sin θ n - z ( 1 - cos θ n ) ] z 0 - z - x 2 + y 2 2 ( z 0 - z ) + z 0 2 [ x sin θ n - z ( 1 - cos θ n ) ] 2 2 ( z 0 - z ) 3 .
sin θ n = n λ d ,             1 - cos θ n ½ · ( n λ d ) 2
R n Q ¯ z + ( z 0 z 0 - z ) · ( n λ x d ) - ( z 0 z z 0 - z ) · ( n 2 λ 2 2 d 2 ) - x 2 + y 2 2 ( z 0 - z ) .
W ( x n , y n ; z n ) = W ( x cos θ n - z sin θ n , y ; z cos θ n + x sin θ n ) W ( x - n λ z / d , y ; z ) W ( x , y ; z ) - ( n λ z d ) · W ( x , y ; z ) x + ½ · ( n λ z d ) 2 · 2 W ( x , y ; z ) x 2 ,
u ( x , y ; z ) = exp { i k [ z - x 2 + y 2 2 ( z 0 - z ) + W ( x , y ; z ) ] } × n = - A n exp { 2 π i [ ( z 0 z 0 - z ) · n x d - ( z 0 z z 0 - z ) · n 2 λ 2 d 2 - ( n z d ) · W x + λ 2 · ( n z d ) 2 · 2 W x 2 ] } .
u ( x , y ; z ) 2 = n = - m = - A n A m * exp ( 2 π i { ( z 0 z 0 - z ) · ( n - m d ) · [ x - ( z z 0 ) · ( z 0 - z ) · W x ] - ( z 0 z z 0 - z ) · ( n 2 - m 2 ) λ 2 d 2 + λ 2 · ( z d ) 2 · ( n 2 - m 2 ) · 2 W x 2 } ) .
lim λ 0 u ( x , y ; z ) 2 = n = - m = - A n A m * exp { 2 π i [ ( z 0 z 0 - z ) · ( n - m d ) ] · [ x - ( z z 0 ) · ( z 0 - z ) · W x ] } ,
u ( x , y ; 0 ) 2 = n = - m = - A n A m * exp [ 2 π i · ( n - m d ) x ]
( z 0 z T z 0 - z T ) · ( λ 2 d 2 ) = N ( integer ) ,
u ( x , y ; z T ) 2 = n = - m = - A n A m * v n , m ( x , y ; z T ) · exp [ 2 π i ( n - m ) f 0 x ] ,
f 0 = z 0 ( z 0 - z T ) d ,
v n , m ( x , y ; z T ) = exp { - 2 π i ( n - m ) z T d · [ W x - λ z · ( n + m ) z T d · 2 W x 2 ] } .
U ( f , y ; z T ) = - u ( x , y ; z T ) 2 · exp ( - 2 π i f x ) d x = n = - m = - A n A m * · V n , m [ f - ( n - m ) f 0 , y ; z T ] ,
q ( x , y ; z T ) = n = - A n A n - 1 * · v n , n - 1 ( x , y ; z T ) .
q ( x , y ; z T ) = A 0 A - 1 * · v 0 , - 1 ( x , y ; z T ) + A 1 A 0 * · v 1 , 0 ( x , y ; z T ) = 1 π · cos ( π λ z T 2 d 2 · 2 W x 2 ) · exp ( - 2 π i z T d · W x ) .
log [ q ( x , y ; z T ) ] = log [ 1 π cos ( π λ z T 2 d 2 · 2 W x 2 ) ] - i · ( 2 π z T d ) · W x .
W ( x , y ; z T ) x = - d 2 π z T · Im { log [ q ( x , y ; z T ) ] }

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