Abstract

A method to image random three-dimensional source distributions is proposed. We show that, by using a Michelson stellar interferometer in a prescribed fashion, one is able to measure a special form of a three-dimensional degree of coherence. The inverse Fourier transform of this coherence function yields the three-dimensional intensity distribution of the source as seen from the paraxial far zone.

© 1996 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 10, p. 491.
  2. J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 5, p. 157.
  3. W. H. Carter, E. Wolf, Opt. Acta 28, 227 (1981).
    [CrossRef]
  4. A. J. Devaney, J. Math. Phys. 20, 1687 (1979).
    [CrossRef]
  5. I. J. LaHaie, J. Opt. Soc. Am. A 2, 35 (1985).
    [CrossRef]
  6. A. T. Friberg, J. Opt. Soc. Am. A 3, 1219 (1986).
    [CrossRef]

1986 (1)

1985 (1)

1981 (1)

W. H. Carter, E. Wolf, Opt. Acta 28, 227 (1981).
[CrossRef]

1979 (1)

A. J. Devaney, J. Math. Phys. 20, 1687 (1979).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 10, p. 491.

Carter, W. H.

W. H. Carter, E. Wolf, Opt. Acta 28, 227 (1981).
[CrossRef]

Devaney, A. J.

A. J. Devaney, J. Math. Phys. 20, 1687 (1979).
[CrossRef]

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 5, p. 157.

LaHaie, I. J.

Wolf, E.

W. H. Carter, E. Wolf, Opt. Acta 28, 227 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 10, p. 491.

J. Math. Phys. (1)

A. J. Devaney, J. Math. Phys. 20, 1687 (1979).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

W. H. Carter, E. Wolf, Opt. Acta 28, 227 (1981).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 10, p. 491.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 5, p. 157.

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Figures (2)

Fig. 1
Fig. 1

Schematic showing the calculation of the degree of coherence between P1 and P2.

Fig. 2
Fig. 2

(a) 3-D source intensity distribution. (b) Part of the 3-D visibility function calculated from the interference gratings of the simulated MSI (q = 1 is q = 8.5 m). (c) Reconstruction from the complete 3-D complex visibility function on a few planes along the zs axis (zs = 1 is zs = 400 m).

Equations (7)

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μ ( P 1 , P 2 ) = E 1 ( t ) E 2 * ( t ) [ | E 1 ( t ) | 2 | E 2 ( t ) | 2 ] 1 / 2 = m n A m ( t ) A n * ( t ) exp [ j k ( R 1 , m R 2 , n ) ] [ m A m ( t ) A m * ( t ) n A n ( t ) A n * ( t ) ] n A m ( t ) A m * ( t ) exp [ j k ( R 1 , m R 2 , m ) ] m A m ( t ) A m * ( t ) = I 0 1 I s ( r ¯ s ) exp [ j k ( R 1 R 2 ) ] d 3 r s ,
μ ( P 1 , P 2 ) C 0 I s ( r ¯ s ) exp { j 2 π λ [ x s Δ x + y s Δ y R Δ z ( x s 2 + y s 2 ) 2 R 2 + z s ( x ^ Δ x + y ^ Δ y ) R 2 ] } d 3 r s ,
μ ( x , y ) = C 0 I s ( r ¯ s ) × exp { j 2 π λ [ ( x 2 + y 2 ) z s 2 R 2 + x x s + y y s R ] } d 3 r s .
μ ( Δ x , Δ y , q ) = C 0 I s ( r ¯ s ) × exp [ j 2 π λ ( x s Δ x + y s Δ y R + z s q Δ r min R 2 ) ] d 3 r s .
I s ( r ¯ s ) F T 3 D 1 { μ ( Δ x λ R , Δ y λ R , q Δ r min λ R 2 ) } ,
μ ( Δ x λ R , Δ y λ R , q Δ r min λ R 2 ) = μ ( Δ x λ R , Δ y λ R , q Δ r min λ R 2 ) × k = K K m = M M n = N N δ ( Δ x n d x , Δ y m d y , q k d q ) ,
I s ( r ¯ s ) FT 3 D 1 [ μ ( Δ x λ R , Δ y λ R , q Δ r min λ R 2 ) ] = FT 3 D 1 [ μ ( Δ x λ R , Δ y λ R , q Δ r min λ R 2 ) ] * [ sinc ( 2 N d x x s λ R ) sinc ( 2 M d y y s λ R ) sinc ( 2 K Δ r min d q z s λ R 2 ) ] * k = m = n = × δ ( x s n λ R d x , y s m λ R d y , z s k λ R 2 Δ r min d q ) .

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