Abstract

A new imaging system for incoherent objects is constructed. In this system, the coherence function of the diffracted field is derived from the signals of three scanning intensity detectors by using computational manipulations. The concrete optical and electronic systems, the details of the signal processings for the derivation of the coherence function, and calculations for image reconstruction are shown. The reconstructed images of asymmetric objects show the usefulness of the system.

© 1978 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), Chap. 10.
  2. J. C. Dainty, Ed., Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, 1975).
  3. Special issue on speckle in optics, J. Opt. Soc. Am. 66, 1145 (1976).
  4. R. H. Brown, The Intensity Interferometer (Taylor & Francis, London, 1974).
  5. A. Labeyrie, in Progress in Optics Vol. 14, E. Wolf, Ed. (North-Holland, Amsterdam, 1976), pp. 49–87.
  6. H. Gamo, J. Opt. Soc. Am. 34, 875 (1963).
  7. D. R. Brillinger, Time Series Analysis (Holt, Rinehart and Winston, New York, 1975), pp. 19–21.
  8. W. Martienssen, E. Spiller, Am. J. Phys. 32, 919 (1964).
    [CrossRef]
  9. M. Rousseau, J. Opt. Soc. Am. 61, 1307 (1971).
    [CrossRef]
  10. L. E. Estes, L. M. Narducci, R. A. Tuft, J. Opt. Soc. Am. 61, 1301 (1971).
    [CrossRef]

1976 (1)

1971 (2)

1964 (1)

W. Martienssen, E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

1963 (1)

H. Gamo, J. Opt. Soc. Am. 34, 875 (1963).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), Chap. 10.

Brillinger, D. R.

D. R. Brillinger, Time Series Analysis (Holt, Rinehart and Winston, New York, 1975), pp. 19–21.

Brown, R. H.

R. H. Brown, The Intensity Interferometer (Taylor & Francis, London, 1974).

Estes, L. E.

Gamo, H.

H. Gamo, J. Opt. Soc. Am. 34, 875 (1963).

Labeyrie, A.

A. Labeyrie, in Progress in Optics Vol. 14, E. Wolf, Ed. (North-Holland, Amsterdam, 1976), pp. 49–87.

Martienssen, W.

W. Martienssen, E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Narducci, L. M.

Rousseau, M.

Spiller, E.

W. Martienssen, E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

Tuft, R. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), Chap. 10.

Am. J. Phys. (1)

W. Martienssen, E. Spiller, Am. J. Phys. 32, 919 (1964).
[CrossRef]

J. Opt. Soc. Am. (4)

Other (5)

D. R. Brillinger, Time Series Analysis (Holt, Rinehart and Winston, New York, 1975), pp. 19–21.

R. H. Brown, The Intensity Interferometer (Taylor & Francis, London, 1974).

A. Labeyrie, in Progress in Optics Vol. 14, E. Wolf, Ed. (North-Holland, Amsterdam, 1976), pp. 49–87.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), Chap. 10.

J. C. Dainty, Ed., Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, 1975).

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Figures (9)

Fig. 1
Fig. 1

The relation between the average intensity distribution of an incoherent object and the coherence function in the far field. (u,υ) and (x,y), rectangular coordinates on the object and the observation planes, respectively; (α,β), α = (x1x2)/R, β = (y1y2)/R, angular coordinates at the two points P1,P2 on the observation plane.

Fig. 2
Fig. 2

Arrangement of three intensity detectors PD1, PD2, and PD3 for imaging of a 1-D object. PD1 is scanned stepwise by the pitch of a small distance Δx. PD2 and PD3 are fixed and separated by Δx. I1(x0 + nΔx,t) (n = 1,2,…), I2(x0,t), and I3(x0 − Δx,t) are the intensity signals detected by PD1, PD2, and PD3, respectively.

Fig. 3
Fig. 3

Construction of the system.

Fig. 4
Fig. 4

Preprocessing of detected signals in analog correlators.

Fig. 5
Fig. 5

Signal processing for calculation of the coherence function and image reconstruction in a minicomputer.

Fig. 6
Fig. 6

Scanning of detectors and signal processing for the case of 2-D objects.

Fig. 7
Fig. 7

Preprocessed signals recorded on a chart recorder for the case of an 1-D object: (a) original object; (b) the second order cumulant function; (c) the third order cumulant function. The abscissa in (b) and (c) corresponds to the spatial coordinate on the detection plane.

Fig. 8
Fig. 8

Experimental results for a 1-D object: (a) original object; (b) reconstructed image; (c) calculated and normalized modulus of coherence function; (d) calculated phase of coherence function.

Fig. 9
Fig. 9

Experimental results for a 2-D object: (a) original object; (b) reconstructed image displayed by contour lines.

Equations (8)

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Γ ( α , β ) = S ( u , υ ) exp [ j k ( α u + β υ ) ] dud υ ,
K 2 ( I 1 , I 2 ) = | Γ ( n Δ x ) | 2 ( n = 1 , 2 , . . . ) ,
K 2 ( I 1 , I 3 ) = | Γ [ ( n + 1 ) Δ x ] | 2 ( n = 1 , 2 , . . . ) ,
K 2 ( I 2 , I 3 ) = | Γ ( Δ x ) | 2 ,
K 3 ( I 1 , I 2 , I 3 ) = 2 | Γ ( n Δ x ) | | Γ [ ( n + 1 ) Δ x ] | | Γ ( Δ x ) | × cos { ϕ [ ( n + 1 ) Δ x ] ϕ ( n Δ x ) ϕ ( Δ x ) } ( n = 1 , 2 , . . . ) ,
G ( n Δ x ) ϕ [ ( n + 1 ) Δ x ] ϕ ( n Δ x ) ϕ ( Δ x ) = ± cos 1 { K 3 ( I 1 , I 2 , I 3 ) / 2 [ K 2 ( I 1 , I 2 ) K 2 ( I 2 , I 3 ) K 3 ( I 1 , I 3 ) ] 1 / 2 } ( n = 1 , 2 , . . . ) .
ϕ ( n Δ x ) = k = 1 n 1 G ( k Δ x ) + n ϕ ( Δ x ) ( n = 1 , 2 , . . . ) .
G ( n Δ x ) = ϕ [ ( n + 2 ) Δ x ] ϕ ( n Δ x ) ϕ ( 2 Δ x ) ( n = 1 , 2 , . . . ) .

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