Abstract

An unconventional autocorrelation method is described for measuring the transfer function of optical systems. The interference takes place between the scattered waves obtained from two laterally sheared correlated partial diffusers. The output of a detector responding only to an extremely narrowband of spatial frequencies is proportional to the autocorrelation of the system pupil function. An automatic display of the transfer function is obtained by continuously varying the shear between the diffusers. We present the theory and some experimental results of this simple and inexpensive device. A study of various parameters affecting the performance of the instrument is also given.

© 1980 Optical Society of America

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References

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  1. H. H. Hopkins, Opt. Acta 2, 23 (1955).
    [CrossRef]
  2. L. R. Baker, Proc. Phys. Soc. London Sect. B: 68, 871 (1955).
    [CrossRef]
  3. D. Kelsall, Proc. Phys. Soc. London 73, 465 (1959).
    [CrossRef]
  4. P. Hariharan, D. Sen, Proc. Phys. Soc. London 75, 434 (1960).
    [CrossRef]
  5. A. J. Montgomery, J. Opt. Soc. Am. 54, 191 (1964).
    [CrossRef]
  6. A. Lohmann, Optik 14, 510 (1957).
  7. T. Tsuruta, Appl. Opt. 2, 371 (1963).
    [CrossRef]
  8. D. Kelsall, Appl. Opt. 12, 1398 (1973).
    [CrossRef] [PubMed]
  9. J. C. Wyant, Opt. Commun. 19, 120 (1976).
    [CrossRef]
  10. J. C. Wyant, Appl. Opt. 14, 1613 (1975).
    [CrossRef] [PubMed]
  11. F. T. Arecchi, M. Bassan, S. F. Jacobs, G. Molesini, Appl. Opt. 18, 1247 (1979).
    [CrossRef] [PubMed]
  12. C. P. Grover, Opt. Commun. 24, 113 (1978).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 462.
  14. H. H. Hopkins, Proc. R. Soc. London Ser. A: 231, 98 (1955).
  15. S. S. Trivedi, C. P. Grover, IEEE/OSA Conference on Laser Engineering and Applications, Washington, D.C. (1979), paper 17.11.
  16. C. P. Grover, H. M. van Driel (in preparation).

1979 (1)

1978 (1)

C. P. Grover, Opt. Commun. 24, 113 (1978).
[CrossRef]

1976 (1)

J. C. Wyant, Opt. Commun. 19, 120 (1976).
[CrossRef]

1975 (1)

1973 (1)

1964 (1)

1963 (1)

1960 (1)

P. Hariharan, D. Sen, Proc. Phys. Soc. London 75, 434 (1960).
[CrossRef]

1959 (1)

D. Kelsall, Proc. Phys. Soc. London 73, 465 (1959).
[CrossRef]

1957 (1)

A. Lohmann, Optik 14, 510 (1957).

1955 (3)

H. H. Hopkins, Opt. Acta 2, 23 (1955).
[CrossRef]

L. R. Baker, Proc. Phys. Soc. London Sect. B: 68, 871 (1955).
[CrossRef]

H. H. Hopkins, Proc. R. Soc. London Ser. A: 231, 98 (1955).

Arecchi, F. T.

Baker, L. R.

L. R. Baker, Proc. Phys. Soc. London Sect. B: 68, 871 (1955).
[CrossRef]

Bassan, M.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 462.

Grover, C. P.

C. P. Grover, Opt. Commun. 24, 113 (1978).
[CrossRef]

S. S. Trivedi, C. P. Grover, IEEE/OSA Conference on Laser Engineering and Applications, Washington, D.C. (1979), paper 17.11.

C. P. Grover, H. M. van Driel (in preparation).

Hariharan, P.

P. Hariharan, D. Sen, Proc. Phys. Soc. London 75, 434 (1960).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, Opt. Acta 2, 23 (1955).
[CrossRef]

H. H. Hopkins, Proc. R. Soc. London Ser. A: 231, 98 (1955).

Jacobs, S. F.

Kelsall, D.

D. Kelsall, Appl. Opt. 12, 1398 (1973).
[CrossRef] [PubMed]

D. Kelsall, Proc. Phys. Soc. London 73, 465 (1959).
[CrossRef]

Lohmann, A.

A. Lohmann, Optik 14, 510 (1957).

Molesini, G.

Montgomery, A. J.

Sen, D.

P. Hariharan, D. Sen, Proc. Phys. Soc. London 75, 434 (1960).
[CrossRef]

Trivedi, S. S.

S. S. Trivedi, C. P. Grover, IEEE/OSA Conference on Laser Engineering and Applications, Washington, D.C. (1979), paper 17.11.

Tsuruta, T.

van Driel, H. M.

C. P. Grover, H. M. van Driel (in preparation).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 462.

Wyant, J. C.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

Opt. Acta (1)

H. H. Hopkins, Opt. Acta 2, 23 (1955).
[CrossRef]

Opt. Commun. (2)

J. C. Wyant, Opt. Commun. 19, 120 (1976).
[CrossRef]

C. P. Grover, Opt. Commun. 24, 113 (1978).
[CrossRef]

Optik (1)

A. Lohmann, Optik 14, 510 (1957).

Proc. Phys. Soc. London (2)

D. Kelsall, Proc. Phys. Soc. London 73, 465 (1959).
[CrossRef]

P. Hariharan, D. Sen, Proc. Phys. Soc. London 75, 434 (1960).
[CrossRef]

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

L. R. Baker, Proc. Phys. Soc. London Sect. B: 68, 871 (1955).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

H. H. Hopkins, Proc. R. Soc. London Ser. A: 231, 98 (1955).

Other (3)

S. S. Trivedi, C. P. Grover, IEEE/OSA Conference on Laser Engineering and Applications, Washington, D.C. (1979), paper 17.11.

C. P. Grover, H. M. van Driel (in preparation).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 462.

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Figures (8)

Fig. 1
Fig. 1

Optical arrangement of the interferometer: S, point source of light; L1, lens under test; D1D2, correlated diffusers; L2, Fourier transform lens; P, observation slit; Dt, detector; R, storage oscilloscope/chart recorder.

Fig. 2
Fig. 2

Illustrating how the identical scattering points A and B contribute rays AE and BF, which interfere to produce modulation at the point of observation.

Fig. 3
Fig. 3

MTF of an f/22 achromatic double. ξ = 2 mm/div on the horizontal axis.

Fig. 4
Fig. 4

MTF of an f/22 system: —, MTF of diffraction-limited system (theoretical); ■, experimental.

Fig. 5
Fig. 5

Typical MTF trace of defocused f/22 system; z = 1.25 mm.

Fig. 6
Fig. 6

MTF of defocused f/22 system: (a) z = 0.125 mm; (b) z = 0.375 mm; (c) z = 1.25 mm. —, –·–·–, - - -, theoretical; ■, ●, ▲, experimental.

Fig. 7
Fig. 7

Typical OTF trace for an arbitrary lens system.

Fig. 8
Fig. 8

Effect of finite size of observation slit: (a) δ-aperture —; (b) slit aperture, w = 15 μm - - -.

Equations (16)

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g ( x , y ) = a - b g ( x , y ) ,
A ( x , y ) = a 2 - a b [ g ( x , y ) + g ( x - ξ , y ) ] ,
g ( x , y ) = - + g ˜ ( u , v ) exp i 2 π λ f 2 ( u x + v y ) d u d v ,
g ˜ eff ( u , v ) = δ ( u - u 0 , v - v 0 ) .
A eff ( x , y ) = a b 0 { exp i 2 π λ f 2 ( u 0 x + v 0 y ) + exp i 2 π λ f 2 [ u 0 ( x - ξ ) + v 0 y ] } ,
ψ ( x , y ) = exp [ i 2 π λ W ( x , y ) ] .
A eff = a b 0 { ψ ( x , y ) exp i 2 π λ f 2 [ u 0 x + v 0 y ] + ψ ( x - ξ , y ) e x p i 2 π λ f 2 [ u 0 ( x - ξ ) + v 0 y ] } = a b 0 exp i 2 π λ f 2 ( u 0 x + v 0 y ) [ ψ ( x , y ) + ψ ( x - ξ , y ) exp i 2 π λ f 2 u 0 ξ } .
Φ ( ξ ) = S A eff ( x , y ) A eff * ( x , y ) d x d y ,
H ( ξ ) = 1 α S ψ ( x , y ) ψ * ( x - ξ , y ) d x d y ,
α = S ψ ( x , y ) 2 d x d y ,
H ( ξ ) = T ( ξ ) exp [ i θ ( ξ ) ] ,
Φ ( ξ ) = 2 a 2 b 0 2 α { 1 + T ( ξ ) cos [ δ ( ξ ) + θ ( ξ ) ] } ,
δ ( ξ ) = 2 π λ f 2 u 0 ξ .
ψ ( x , y ) ~ exp i π λ z 2 f 2 2 ( x 2 + y 2 ) .
g ˜ eff ( u , v ) = C for u 0 - w 2 u u 0 + w 2 = 0 elsewhere } .
g eff ( x , y ) = exp ( i 2 π λ f 2 u 0 x ) sinc ( π w x λ f 2 ) .

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