Abstract

A method for measuring the focal length of lenses using the Talbot effect and the moiré technique is described. The test lens is placed in front of a set of two gratings. The first grating illuminated by the light passing through the test lens produces the magnified Talbot image. The moiré fringe is generated by superimposing this Talbot image on the second grating. The tilt angle of the moiré fringe is a measure of the focal length of the lens. In the experiments, the focal lengths of positive, negative, and power-distributed lenses are measured.

© 1985 Optical Society of America

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References

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  1. J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford U.P., London, 1956).
  2. S. Yokozeki, “Moiré Fringes,” Opt. Laser Eng. 2, 13 (1981).
    [CrossRef]
  3. J. Zimmerman, “A Method for Measuring the Distortion of Photographic Objectives,” Appl. Opt. 2, 759 (1963).
    [CrossRef]
  4. D. W. Swift, “A Simple Moiré Fringe Technique for Magnification Checking,” J. Phys. E 7, 164 (1974).
    [CrossRef]
  5. G. Oster, M. Wasserman, C. Zwerling, “Theoretical Interpretation of Moiré Patterns,” J. Opt. Soc. Am. 54, 169 (1964).
    [CrossRef]
  6. O. Kafri, “Noncoherent Method for Mapping Phase Objects,” Opt. Lett. 5, 555 (1980).
    [CrossRef] [PubMed]
  7. Z. Karny, O. Kafri, “Refractive-Index Measurements by Moiré Deflectometry,” Appl. Opt. 21, 3326 (1982).
    [CrossRef] [PubMed]
  8. H. Uozato, S. Kamiya, M. Saishin, “A Method for Measuring the Distribution of Lens Power Using the Moiré Technique,” Folia Ophthalmol. Jpn. 35, 951 (1984), in Japanese.
  9. Y. Nakano, K. Murata, “Measurements of Phase Objects Using the Talbot Effect and Moiré Techniques,” Appl. Opt. 23, 2296 (1984).
    [CrossRef] [PubMed]
  10. F. Talbot, “Facts Relating to Optical Science. No. IV,” Philos. Mag. 9, 401 (1936).
  11. S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opt. 10, 1575 (1971).
    [CrossRef] [PubMed]
  12. S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter. Part 2,” Appl. Opt. 10, 1690 (1971).
    [CrossRef] [PubMed]
  13. A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971).
    [CrossRef]
  14. A. W. Lohmann, D. E. Silva, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
    [CrossRef]
  15. D. E. Silva, “Talbot Interferometer for Radial and Lateral Derivatives,” Appl. Opt. 11, 2613 (1972).
    [CrossRef] [PubMed]
  16. J. M. Cowley, A. F. Moodie, “Fourier Images: Part I—The Point Source,” Proc. Phys. Soc. London Sect. B 70, 486 (1957).
    [CrossRef]
  17. G. L. Rogers, “A Simple Method of Calculating Moiré Patterns,” Proc. Phys. Soc. London 73, 142 (1959).
    [CrossRef]

1984 (2)

H. Uozato, S. Kamiya, M. Saishin, “A Method for Measuring the Distribution of Lens Power Using the Moiré Technique,” Folia Ophthalmol. Jpn. 35, 951 (1984), in Japanese.

Y. Nakano, K. Murata, “Measurements of Phase Objects Using the Talbot Effect and Moiré Techniques,” Appl. Opt. 23, 2296 (1984).
[CrossRef] [PubMed]

1982 (1)

1981 (1)

S. Yokozeki, “Moiré Fringes,” Opt. Laser Eng. 2, 13 (1981).
[CrossRef]

1980 (1)

1974 (1)

D. W. Swift, “A Simple Moiré Fringe Technique for Magnification Checking,” J. Phys. E 7, 164 (1974).
[CrossRef]

1972 (2)

A. W. Lohmann, D. E. Silva, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
[CrossRef]

D. E. Silva, “Talbot Interferometer for Radial and Lateral Derivatives,” Appl. Opt. 11, 2613 (1972).
[CrossRef] [PubMed]

1971 (3)

1964 (1)

1963 (1)

1959 (1)

G. L. Rogers, “A Simple Method of Calculating Moiré Patterns,” Proc. Phys. Soc. London 73, 142 (1959).
[CrossRef]

1957 (1)

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

1936 (1)

F. Talbot, “Facts Relating to Optical Science. No. IV,” Philos. Mag. 9, 401 (1936).

Cowley, J. M.

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

Guild, J.

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford U.P., London, 1956).

Kafri, O.

Kamiya, S.

H. Uozato, S. Kamiya, M. Saishin, “A Method for Measuring the Distribution of Lens Power Using the Moiré Technique,” Folia Ophthalmol. Jpn. 35, 951 (1984), in Japanese.

Karny, Z.

Lohmann, A. W.

A. W. Lohmann, D. E. Silva, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
[CrossRef]

A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971).
[CrossRef]

Moodie, A. F.

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

Murata, K.

Nakano, Y.

Oster, G.

Rogers, G. L.

G. L. Rogers, “A Simple Method of Calculating Moiré Patterns,” Proc. Phys. Soc. London 73, 142 (1959).
[CrossRef]

Saishin, M.

H. Uozato, S. Kamiya, M. Saishin, “A Method for Measuring the Distribution of Lens Power Using the Moiré Technique,” Folia Ophthalmol. Jpn. 35, 951 (1984), in Japanese.

Silva, D. E.

A. W. Lohmann, D. E. Silva, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
[CrossRef]

D. E. Silva, “Talbot Interferometer for Radial and Lateral Derivatives,” Appl. Opt. 11, 2613 (1972).
[CrossRef] [PubMed]

A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971).
[CrossRef]

Suzuki, T.

Swift, D. W.

D. W. Swift, “A Simple Moiré Fringe Technique for Magnification Checking,” J. Phys. E 7, 164 (1974).
[CrossRef]

Talbot, F.

F. Talbot, “Facts Relating to Optical Science. No. IV,” Philos. Mag. 9, 401 (1936).

Uozato, H.

H. Uozato, S. Kamiya, M. Saishin, “A Method for Measuring the Distribution of Lens Power Using the Moiré Technique,” Folia Ophthalmol. Jpn. 35, 951 (1984), in Japanese.

Wasserman, M.

Yokozeki, S.

Zimmerman, J.

Zwerling, C.

Appl. Opt. (6)

Folia Ophthalmol. Jpn. (1)

H. Uozato, S. Kamiya, M. Saishin, “A Method for Measuring the Distribution of Lens Power Using the Moiré Technique,” Folia Ophthalmol. Jpn. 35, 951 (1984), in Japanese.

J. Opt. Soc. Am. (1)

J. Phys. E (1)

D. W. Swift, “A Simple Moiré Fringe Technique for Magnification Checking,” J. Phys. E 7, 164 (1974).
[CrossRef]

Opt. Commun. (2)

A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971).
[CrossRef]

A. W. Lohmann, D. E. Silva, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
[CrossRef]

Opt. Laser Eng. (1)

S. Yokozeki, “Moiré Fringes,” Opt. Laser Eng. 2, 13 (1981).
[CrossRef]

Opt. Lett. (1)

Philos. Mag. (1)

F. Talbot, “Facts Relating to Optical Science. No. IV,” Philos. Mag. 9, 401 (1936).

Proc. Phys. Soc. London (1)

G. L. Rogers, “A Simple Method of Calculating Moiré Patterns,” Proc. Phys. Soc. London 73, 142 (1959).
[CrossRef]

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

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

Other (1)

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford U.P., London, 1956).

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Figures (6)

Fig. 1
Fig. 1

Optical arrangement for measuring the focal length of a lens.

Fig. 2
Fig. 2

Experimental setup. An inverse telescope L1L2 produces a collimated light beam which passes through a test lens (T.L). The first negative Talbot image length is zk=1. The moiré fringe patterns can be viewed on an observing screen (O.S) placed behind g2.

Fig. 3
Fig. 3

Relationship between focal length f of the test lens and tilt angle of the moiré fringe.

Fig. 4
Fig. 4

Moiré fringe pattern observed without a test lens.

Fig. 5
Fig. 5

Moiré fringe pattern observed through test lenses: (a) positive lens: α = 17°, f = 4050 mm; (b) negative lens: α = −13°, f = −4126 mm; (c) positive lens: α = 51°, f = 913 mm.

Fig. 6
Fig. 6

Moiré fringe pattern observed through a power distributed lens. Part of the parallel moiré fringe shows no power. Part of the curved moiré fringe shows a power distribution. In the small circle the tilt angle α = 68° is observed, and the f = 450-mm focal length is obtained.

Equations (16)

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g 1 ( ξ, η ) = 1 2 [ 1 + cos ( 2 π p ξ ) ] .
E 0 ( ξ , η ) = exp [ π i ( ξ 2 + η 2 ) λ f ] .
E 1 ( ξ , η ; z = 0 ) = 1 2 exp [ π i λ f ( ξ 2 + η 2 ) ] + 1 4 [ exp ( 2 π i p ξ ) + exp ( 2 π i p ξ ) ] × exp [ π i ( ξ 2 + η 2 ) λ f ] .
E 2 ( x , y , z ) = ( i 2 λ z ) exp ( + 2 π i z λ ) exp [ π i λ z ( x 2 + y 2 ) ( 1 m ) ] × + exp [ B ( η m y ) 2 ] d η × { + exp [ B ( ξ m x ) 2 ] d ξ + 1 2 exp ( π i λ z p 2 · m ) exp ( 2 π i x p · m ) × + exp [ B ( ξ + λ z p x p · m ) 2 ] d ξ + 1 2 exp ( π i λ z p 2 · m ) exp ( 2 π i x p · m ) × + exp [ B ( ξ λ z + p x p · m ) 2 ] d ξ } ,
B = π i λ ( 1 z 1 f ) , m = f f z .
E 2 ( x , y , z ) = A 1 2 [ 1 + exp ( π i m λ z p 2 ) cos ( 2 π m x p ) ] .
z = z k = k p 2 f λ f + k p 2 ,
exp ( π i λ z k p 2 · f f z k ) = 1 ,
g 1 ( x , y ) = E 2 ( x , y ; z k ) = A 1 2 [ 1 + cos ( 2 π x p · f f z k ) ] ,
g 2 ( x , y ) = 1 2 { 1 + cos ( 2 π p ( x cos θ + y sin θ ) ) } .
I ( x , y ; z k ) = 9 64 A 1 2 + 3 16 A 1 2 cos ( 2 π x m p ) + 3 64 A 1 2 cos ( 4 π x m p ) + 3 16 A 1 2 cos { 2 π p [ ( x cos θ + y sin θ ) ] } + 3 64 A 1 2 cos { 4 π p [ ( x cos θ + y sin θ ) ] } + 1 8 A 1 2 cos { 2 π p [ ( m + cos θ ) x + y sin θ ] } + 1 8 A 1 2 cos { 2 π p [ ( m cos θ ) x y sin θ ] } + 1 32 A 1 2 cos { 2 π p [ ( m + 2 cos θ ) x + 2 y sin θ ] } + 1 32 A 1 2 cos { 2 π p [ ( m 2 cos θ ) x 2 y sin θ ] } + 1 32 A 1 2 cos { 2 π p [ ( 2 m + cos θ ) x + y sin θ ] } + 1 32 A 1 2 cos { 2 π p [ ( 2 m cos θ ) x y sin θ ] } + 1 128 A 1 2 cos { 4 π p [ ( m + cos θ ) x + y sin θ ] } + 1 128 A 1 2 cos { 4 π p [ ( m cos θ ) x y sin θ ] } .
I M ( x , y ; z k ) = 1 8 A 1 2 cos { 2 π p [ ( f f z k cos θ ) x y sin θ ] } .
tan α k = f f z k cos θ sin θ .
f = 1 sin θ tan α k + cos θ 1 · k p 2 λ .
W = p ( f z k ) z k .
f = k p W λ .

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