Abstract

Three-dimensional imaging of incoherent light sources by the Michelson stellar interferometer is considered. When the interferometer’s pinholes are arranged properly, its output result is equivalent to a two-dimensional Fourier hologram that stores information about the source object’s three-dimensional intensity distribution.

© 1996 Optical Society of America

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References

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  1. W. J. Tango, R. Q. Twiss, “Michelson stellar interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1980), Vol. 17, pp. 239–277.
    [CrossRef]
  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.
  3. W. H. Carter, E. Wolf, “Correlation theory of wavefieldsgenerated by fluctuating, three-dimensional, primary, scalar sources,” Opt. Acta 28, 227–244 (1981).
    [CrossRef]
  4. C. W. McCutchen, “Generalized source and the Van Cittert–Zernike theorem: a study of the spatial coherence required for interferometry,” J. Opt. Soc. Am. 56, 727–733 (1966); J. Rosen, A. Yariv, “Longitudinal partial coherence of optical radiation,” Opt. Commun. 117, 8–12 (1995).
    [CrossRef]
  5. A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Ca., 1991), App. E, p. 705.
  6. A. W. Lohmann, “Wavefront reconstruction for incoherent objects,” J. Opt. Soc. Am. 55, 1555–1556 (1965); G. Sirat, D. Psaltis, “Conoscopic holography,” Opt. Lett. 10, 4–6 (1985).
    [CrossRef] [PubMed]

1981

W. H. Carter, E. Wolf, “Correlation theory of wavefieldsgenerated by fluctuating, three-dimensional, primary, scalar sources,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

1966

1965

Born, M.

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.

Carter, W. H.

W. H. Carter, E. Wolf, “Correlation theory of wavefieldsgenerated by fluctuating, three-dimensional, primary, scalar sources,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

Lohmann, A. W.

McCutchen, C. W.

Tango, W. J.

W. J. Tango, R. Q. Twiss, “Michelson stellar interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1980), Vol. 17, pp. 239–277.
[CrossRef]

Twiss, R. Q.

W. J. Tango, R. Q. Twiss, “Michelson stellar interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1980), Vol. 17, pp. 239–277.
[CrossRef]

Wolf, E.

W. H. Carter, E. Wolf, “Correlation theory of wavefieldsgenerated by fluctuating, three-dimensional, primary, scalar sources,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

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.

Yariv, A.

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Ca., 1991), App. E, p. 705.

J. Opt. Soc. Am.

Opt. Acta

W. H. Carter, E. Wolf, “Correlation theory of wavefieldsgenerated by fluctuating, three-dimensional, primary, scalar sources,” Opt. Acta 28, 227–244 (1981).
[CrossRef]

Other

A. Yariv, Optical Electronics, 4th ed. (Saunders, Philadelphia, Ca., 1991), App. E, p. 705.

W. J. Tango, R. Q. Twiss, “Michelson stellar interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1980), Vol. 17, pp. 239–277.
[CrossRef]

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 system of the MSI for 3-D imaging.

Fig. 2
Fig. 2

Computer simulation of the MSI. The magnitude (a) and the phase (b) of the 2-D complex visibility function were calculated from the interference MSI’s gratings. (c)–(e) The reconstruction of the hologram shown in (a) and (b) at three different planes along the z axis. In each plane only one abbreviation is in focus, indicating its original location in the object space.

Equations (10)

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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 ) ] 1 / 2 m 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 ,
R 1 - R 2 = [ ( R + z 1 - z s ) 2 + ( x 1 - x s ) 2 + ( y 1 - y s ) 2 ] 1 / 2 - [ ( R + z 2 - z s ) 2 + ( x 2 - x s ) 2 + ( y 2 - y s ) 2 ] 1 / 2 Δ z + x ^ Δ x + y ^ Δ y R - 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 ,
μ ( r ¯ 1 , r ¯ 2 ) = exp ( j k [ Δ z + ( x ^ Δ x + y ^ Δ y ) / R ] ) I 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 .
I ( y 0 ) = I s ( r ¯ s ) | 1 R 1 exp [ j k ( R 1 + y 0 sin φ ) ] + 1 R 2 exp [ j k ( R 2 - y 0 sin φ ) ] | 2 d 3 r s ( 1 / R ) 2 2 I 0 [ 1 + μ ( r ¯ 1 , r ¯ 2 ) cos ( 2 k y 0 sin φ + arg { μ } ) ] ,
μ ( Δ x , Δ y , Δ z ) = exp ( j k Δ z ) I 0 × I ^ s ( x s , y s ) exp { - j 2 π λ R [ Δ x x s + Δ y y s - ( x s 2 + y s 2 ) Δ z 2 R ] } d x s d y s ,
I ^ s ( x s , y s ) = I s ( x s , y s , z s ) d z s .
μ ( x , y ) = C 0 I s ( r ¯ s ) exp { - j 2 π λ R [ x x s + y y s + ( x 2 + y 2 ) z s 2 R ] } d 3 r ¯ s ,
u r ( x 0 , y 0 , z 0 ) = μ ( x , y ) exp { j 2 π λ [ x x 0 + y y 0 L 1 + z 0 2 L 2 2 ( x 2 + y 2 ) ] } d x d y ,
μ ( x , y ) = C 0 n Δ z s , min I s ( x s , y s ; z s , n ) × exp { - j 2 π λ R [ x x s + y y s + ( x 2 + y 2 ) z s , m 2 R ] } d x s d y s .
u r ( x 0 , y 0 ; z ¯ 0 ) = Δ z s , min I s ( - x 0 R L 1 , - y 0 R L 1 ; - z ¯ 0 R 2 L 2 2 ) + z s , n R 2 z ¯ 0 / L 2 2 Δ z s , min I s ( x s , y s ; z s , n ) × λ R 2 L 2 2 z s , n L 2 2 - R 2 z ¯ 0 × exp { - j π R 2 L 2 2 λ ( z s , n L 2 2 - R 2 z ¯ 0 ) [ ( x 0 L 1 + x s R ) 2 + ( y 0 L 1 + y s R ) 2 ] } d x s d y s .

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