Abstract

Encoding a complex-amplitude diffraction pattern on a photographic film by addirtg a suitably chosen coherent reference beam forms a conventional hologram. A hologram of a spatially noncoherent object, referred to as a Γ hologram in this paper, is formed by encoding the complex-valued spatial-coherence function on a square-law detector such as photographic film. This record is made possible by means of a self-refetencing interferometer. Such a record behaves much as a hologram does; it permits reconstruction of the original object by illuminating it with a spatially noncoherent planar source of uniform (constant) intensity. If a conventional coherent-light setup is used with a Γ hologram, the intensity distribution of the reconstruction equals the square of the intensity of the original object. In the research reported in this paper, optical processing of spatially noncoherent objects is accomplished by using and modifying the spatial-coherence function. The Γ hologram is used to gain access to this function. This procedure opens new possibilities of noncoherent-object information processing. Examples of matched filtering, low-pass filtering, and high-pass filtering are discussed. The underlying theory has its roots in the fundamental Van Cittert–Zernike theorem of the theory of partial coherence.

© 1987 Optical Society of America

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References

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  1. L. Mertz, N. O. Young, “Fresnel transformations of images,” in Proceedings of Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman and Hall, London, 1962); see also G. L. Rogers, “Experiments in diffraction microscopy,” Proc. Phys. Soc. Edinburgh A 63(III), 193–221 (1952).
  2. G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).
  3. L. Mertz, Transformation in Optics (Wiley, New York, 1965).
  4. A. W. Lohmann, “Wavefront reconstruction for incoherent objects,”J. Opt. Soc. Am. 55, 1555–1556 (1965).
    [CrossRef]
  5. H. R. Worthington, “Production of holograms with incoherent illumination,”J. Opt. Soc. Am. 56, 1397–1398 (1966).
    [CrossRef]
  6. G. Cochran, “New method of making Fresnel transforms with incoherent light,”J. Opt. Soc. Am. 56, 1513–1517 (1966).
    [CrossRef]
  7. G. D. Collins, “Achromatic Fourier transform holography,” Appl. Opt. 20, 3109–3119 (1981).
    [CrossRef] [PubMed]
  8. E. N. Leith, G. J. Swanson, “Achromatic interferometers for white light optical processing and holography,” Appl. Opt. 19, 638–644 (1980).
    [CrossRef] [PubMed]
  9. G. M. Morris, N. George, “Space and wavelength dependence of a dispersion-compensated matched filter,” Appl. Opt. 19, 3843–3850 (1980).
    [CrossRef] [PubMed]
  10. N. George, S. Wang, “Cosinusoidal transforms in white light,” Appl. Opt. 23, 787–796 (1984).
    [CrossRef] [PubMed]
  11. A. M. Tai, C. C. Aleksoff, “Grating-based interferometric processor for realtime optical Fourier transformation,” Appl. Opt. 23, 2282–2291 (1984).
    [CrossRef] [PubMed]
  12. G. Indebetouw, C. Varamit, “Spatial filtering with complementary source-pupil masks,” J. Opt. Soc. Am. A 2, 794–798 (1985).
    [CrossRef]
  13. A. W. Lohmann, “Incoherent optical processing of complex data,” Appl. Opt. 16, 261–263 (1977).
    [CrossRef] [PubMed]
  14. A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1141–1151 (1978).
    [CrossRef] [PubMed]
  15. W. T. Rhodes, “Bipolar pointspread function synthesis by phase switching,” Appl. Opt. 16, 265–267 (1977).
    [CrossRef] [PubMed]
  16. W. Stoner, “Edge enhancement with incoherent optics,” Appl. Opt. 16, 1451–1452 (1977).
    [CrossRef] [PubMed]
  17. W. Stoner, “Incoherent optical processing via spatially offset pupil masks,” Appl. Opt. 17, 2454–2467 (1978).
    [CrossRef] [PubMed]
  18. D. K. Angell, “Incoherent spatial filtering with grating interferometers,” Appl. Opt. 24, 2903–2906 (1985).
    [CrossRef] [PubMed]
  19. P. Kellman, S. Leonard, E. Barrett, “Digital hologram reconstruction of radio telescope data,” Appl. Opt. 16, 1113–1114 (1977).
    [CrossRef] [PubMed]
  20. A. S. Marathay, “Hologram and optical processing of spatially noncoherent objects,” J. Opt. Soc. Am. A 3(13), P50 (1986).
  21. K. Itoh, Y. Ohtsuka, “Holographic spectral imaging,” J. Opt. Soc. Am. A 3, 1239–1242 (1986).
    [CrossRef]
  22. A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982).
  23. M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).
  24. For the present discussion we have omitted the quadratic phase factor that occurs in front of the Fourier transform of the intensity of the noncoherent source (see Ref. 22).
  25. M. V. R. K. Murty, “Lateral shearing interferometer,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 4, pp. 105–148.
  26. M. V. R. K. Murty, “Interference between wavefronts rotated or reversed with respect to each other and its relation to spatial coherence,”J. Opt. Soc. Am. 54, 1187–1190 (1964).
    [CrossRef]
  27. J. D. Armitage, A. Lohmann, “Rotary shearing interferometry,” Opt. Acta 12, 185–192 (1965).
    [CrossRef]
  28. J. B. Breckinridge, “Coherence interferometer and astronomical applications,” Appl. Opt. 11, 2996–2998 (1972).
    [CrossRef] [PubMed]
  29. J. J. Burke, J. B. Breckinridge, “Passive imaging through the turbulent atmosphere: fundamental limits on the spatial frequency resolution of a rotational shearing interferometer,”J. Opt. Soc. Am. 68, 67–77 (1978).
    [CrossRef]
  30. J. C. Dainty, R. J. Scaddan, “A coherence interferometer for the direct measurement of the atmospheric transfer function,” Mon. Not. R. Astron. Soc. 167, 69–73 (1974).
  31. The nomenclature is arbitrary to some extent; as used here, the conjugate reconstruction results from the Fourier transform of the function Γ.

1986 (2)

A. S. Marathay, “Hologram and optical processing of spatially noncoherent objects,” J. Opt. Soc. Am. A 3(13), P50 (1986).

K. Itoh, Y. Ohtsuka, “Holographic spectral imaging,” J. Opt. Soc. Am. A 3, 1239–1242 (1986).
[CrossRef]

1985 (2)

1984 (2)

1981 (1)

1980 (2)

1978 (3)

1977 (4)

1974 (1)

J. C. Dainty, R. J. Scaddan, “A coherence interferometer for the direct measurement of the atmospheric transfer function,” Mon. Not. R. Astron. Soc. 167, 69–73 (1974).

1972 (1)

1966 (2)

1965 (2)

J. D. Armitage, A. Lohmann, “Rotary shearing interferometry,” Opt. Acta 12, 185–192 (1965).
[CrossRef]

A. W. Lohmann, “Wavefront reconstruction for incoherent objects,”J. Opt. Soc. Am. 55, 1555–1556 (1965).
[CrossRef]

1964 (1)

Aleksoff, C. C.

Angell, D. K.

Armitage, J. D.

J. D. Armitage, A. Lohmann, “Rotary shearing interferometry,” Opt. Acta 12, 185–192 (1965).
[CrossRef]

Barrett, E.

Beran, M. J.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Breckinridge, J. B.

Burke, J. J.

Cochran, G.

Collins, G. D.

Dainty, J. C.

J. C. Dainty, R. J. Scaddan, “A coherence interferometer for the direct measurement of the atmospheric transfer function,” Mon. Not. R. Astron. Soc. 167, 69–73 (1974).

George, N.

Indebetouw, G.

Itoh, K.

Kellman, P.

Leith, E. N.

Leonard, S.

Lohmann, A.

J. D. Armitage, A. Lohmann, “Rotary shearing interferometry,” Opt. Acta 12, 185–192 (1965).
[CrossRef]

Lohmann, A. W.

Marathay, A. S.

A. S. Marathay, “Hologram and optical processing of spatially noncoherent objects,” J. Opt. Soc. Am. A 3(13), P50 (1986).

A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982).

Mertz, L.

L. Mertz, N. O. Young, “Fresnel transformations of images,” in Proceedings of Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman and Hall, London, 1962); see also G. L. Rogers, “Experiments in diffraction microscopy,” Proc. Phys. Soc. Edinburgh A 63(III), 193–221 (1952).

L. Mertz, Transformation in Optics (Wiley, New York, 1965).

Morris, G. M.

Murty, M. V. R. K.

M. V. R. K. Murty, “Interference between wavefronts rotated or reversed with respect to each other and its relation to spatial coherence,”J. Opt. Soc. Am. 54, 1187–1190 (1964).
[CrossRef]

M. V. R. K. Murty, “Lateral shearing interferometer,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 4, pp. 105–148.

Ohtsuka, Y.

Parrent, G. B.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Rhodes, W. T.

Rogers, G. L.

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).

Scaddan, R. J.

J. C. Dainty, R. J. Scaddan, “A coherence interferometer for the direct measurement of the atmospheric transfer function,” Mon. Not. R. Astron. Soc. 167, 69–73 (1974).

Stoner, W.

Swanson, G. J.

Tai, A. M.

Varamit, C.

Wang, S.

Worthington, H. R.

Young, N. O.

L. Mertz, N. O. Young, “Fresnel transformations of images,” in Proceedings of Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman and Hall, London, 1962); see also G. L. Rogers, “Experiments in diffraction microscopy,” Proc. Phys. Soc. Edinburgh A 63(III), 193–221 (1952).

Appl. Opt. (13)

A. W. Lohmann, “Incoherent optical processing of complex data,” Appl. Opt. 16, 261–263 (1977).
[CrossRef] [PubMed]

A. W. Lohmann, W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1141–1151 (1978).
[CrossRef] [PubMed]

W. Stoner, “Incoherent optical processing via spatially offset pupil masks,” Appl. Opt. 17, 2454–2467 (1978).
[CrossRef] [PubMed]

E. N. Leith, G. J. Swanson, “Achromatic interferometers for white light optical processing and holography,” Appl. Opt. 19, 638–644 (1980).
[CrossRef] [PubMed]

G. M. Morris, N. George, “Space and wavelength dependence of a dispersion-compensated matched filter,” Appl. Opt. 19, 3843–3850 (1980).
[CrossRef] [PubMed]

G. D. Collins, “Achromatic Fourier transform holography,” Appl. Opt. 20, 3109–3119 (1981).
[CrossRef] [PubMed]

N. George, S. Wang, “Cosinusoidal transforms in white light,” Appl. Opt. 23, 787–796 (1984).
[CrossRef] [PubMed]

A. M. Tai, C. C. Aleksoff, “Grating-based interferometric processor for realtime optical Fourier transformation,” Appl. Opt. 23, 2282–2291 (1984).
[CrossRef] [PubMed]

D. K. Angell, “Incoherent spatial filtering with grating interferometers,” Appl. Opt. 24, 2903–2906 (1985).
[CrossRef] [PubMed]

W. Stoner, “Edge enhancement with incoherent optics,” Appl. Opt. 16, 1451–1452 (1977).
[CrossRef] [PubMed]

W. T. Rhodes, “Bipolar pointspread function synthesis by phase switching,” Appl. Opt. 16, 265–267 (1977).
[CrossRef] [PubMed]

P. Kellman, S. Leonard, E. Barrett, “Digital hologram reconstruction of radio telescope data,” Appl. Opt. 16, 1113–1114 (1977).
[CrossRef] [PubMed]

J. B. Breckinridge, “Coherence interferometer and astronomical applications,” Appl. Opt. 11, 2996–2998 (1972).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

Mon. Not. R. Astron. Soc. (1)

J. C. Dainty, R. J. Scaddan, “A coherence interferometer for the direct measurement of the atmospheric transfer function,” Mon. Not. R. Astron. Soc. 167, 69–73 (1974).

Opt. Acta (1)

J. D. Armitage, A. Lohmann, “Rotary shearing interferometry,” Opt. Acta 12, 185–192 (1965).
[CrossRef]

Other (8)

The nomenclature is arbitrary to some extent; as used here, the conjugate reconstruction results from the Fourier transform of the function Γ.

L. Mertz, N. O. Young, “Fresnel transformations of images,” in Proceedings of Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman and Hall, London, 1962); see also G. L. Rogers, “Experiments in diffraction microscopy,” Proc. Phys. Soc. Edinburgh A 63(III), 193–221 (1952).

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).

L. Mertz, Transformation in Optics (Wiley, New York, 1965).

A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982).

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

For the present discussion we have omitted the quadratic phase factor that occurs in front of the Fourier transform of the intensity of the noncoherent source (see Ref. 22).

M. V. R. K. Murty, “Lateral shearing interferometer,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), Chap. 4, pp. 105–148.

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Figures (5)

Fig. 1
Fig. 1

(a) The object is in the (xs, ys) plane and placed on the axis of an image-forming lens L; the wave-front-folding interferometer is shown in image space, BD is the beam divider, and 1 and 2 are roof prisms. The observer’s eye is focused on the pupil plane (α, β). The eye is shown next to the image plane. The distance s1 between the lens L and the image plane is not shown. (b) Same as (a) except that the observer’s eye is replaced by a lens L1 to image the exit pupil onto a receiving plane to record the intensity distribution Ip of Eq. (2), which is the Γ hologram.

Fig. 2
Fig. 2

Reconstruction setup. Planes P1 and P2 are in proximity. Plane P1 contains a large noncoherent source of uniform intensity I0, and the plane P2 contains the Γ hologram whose intensity transmittance is proportional to Ip. The rest of the diagram is the same as in Fig. 1(a). To photograph the reconstruction, the lens L1 of Fig. 1(b) may be used.

Fig. 3
Fig. 3

(a) Original object. (b) Reconstruction in the pupil plane (α, β).

Fig. 4
Fig. 4

Reconstructions of the two noncoherent objects (I1, I2), their convolution (CON), and their cross correlation (XCOR). Pluses indicate the locations of the objects CON and XCOR in the plane of the exit pupil (α, β).

Fig. 5
Fig. 5

Spatial-frequency filtering: (a) low-pass filter and (b) high-pass filter. The label as indicates the extent of the source in the xs direction.

Equations (17)

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Γ p s ( α 12 , β 12 ) = c 0 I s ( x s , y s ) × exp [ - i 2 π ( x s α 12 + y s β 12 ) λ s 0 ] d x s d y s ,
I p ( α , β ) = 2 Γ p s ( 0 , 0 ) + Γ p s ( - 2 α , 2 β ) × exp [ - i 2 π ( a 12 α + b 12 β ) λ s 1 ] + Γ p s * ( - 2 α , 2 β ) × exp [ i 2 π ( a 12 α + b 12 β ) λ s 1 ] .
Γ p s ( - 2 α , 2 β ) = E [ V p s ( - α , β ) V p s * ( α , - β ) ] .
f = ( a 12 2 + b 12 2 ) 1 / 2 λ s 1 ,
θ = arctan ( b 12 a 12 ) .
T ( α , β ) = c I p ( α , β ) ,
I p r ( α , β ) = 2 Γ p r ( 0 , 0 ) + Γ p r ( - 2 α , 2 β ) exp [ - i 2 π ( a 34 α + b 34 β ) λ s 3 ] + Γ p r * ( - 2 α , 2 β ) exp [ i 2 π ( a 34 α + b 34 β ) λ s 3 ] .
Γ p r ( α 12 , β 12 ) = c 2 ( c I 0 ) A p I p ( m 1 x s , m 1 y s ) × exp [ - i 2 π ( x s α 12 + y s β 12 ) λ s 2 ] d x s d y s .
A ˜ p ( α 12 , β 12 ) = A p ( x s , y s ) exp [ - i 2 π ( x s α 12 + y s β 12 ) λ s 2 ] d x s d y s ,
A ˜ p ( α 12 , β 12 ) ( λ s 2 ) 2 δ ( α 12 ) δ ( β 12 ) .
Γ p r ( - 2 α , 2 β ) = c I 0 [ 2 c 2 ( λ s 2 ) 2 Γ p s ( 0 , 0 ) δ ( - 2 α ) δ ( 2 β ) - ( λ 2 π s 2 m 1 ) 2 × I s ( - α s 0 m 1 s 2 + a 12 s 0 2 s 1 , - β s 0 m 1 s 2 - b 12 s 0 2 s 1 ) - ( λ 2 π s 2 m 1 ) 2 × I s ( α s 0 m 1 s 2 + a 12 s 0 2 s 1 , β s 0 m 1 s 2 - b 12 s 0 2 s 1 ) ] .
I p r ( α , β ) = 2 Γ p r ( 0 , 0 ) + 2 Γ p r ( - 2 α , 2 β ) × cos [ 2 π ( a 34 α + b 34 β ) λ s 3 - ϕ ( - 2 α , 2 β ) ] ,
Γ p r ( 0 , 0 ) = 2 c 2 ( λ s 2 ) 2 Γ p s ( 0 , 0 ) δ ( - 2 α ) δ ( 2 β ) .
I p 1 ( x s , y s ) I p 2 ( x s , y s ) = { 2 Γ 1 ( 0 , 0 ) + Γ 1 ( - 2 x s , 2 y s ) × exp [ - i 2 π ( a 12 x s + b 12 y s ) λ s 1 ] + Γ 1 * ( - 2 x s , 2 y s ) × exp [ i 2 π ( a 12 x s + b 12 y s ) λ s 1 ] } { 2 Γ 2 ( 0 , 0 ) + Γ 2 ( - 2 x s , 2 y s ) × exp [ - i 2 π ( a 12 x s + b 12 y s ) λ s 1 ] + Γ 2 * ( - 2 x s , 2 y s ) × exp [ i 2 π ( a 12 x s + b 12 y s ) λ s 1 ] } ,
Γ p r ( - 2 α , 2 β ) = C { Γ 1 ( 0 , 0 ) Γ 2 ( 0 , 0 ) δ ( - 2 α ) δ ( 2 β ) - ( c 0 8 ) [ Γ 1 ( 0 , 0 ) I 2 ( - α + μ 12 , - β - ν 12 ) + Γ 1 ( 0 , 0 ) I 2 ( α + μ 12 , β - ν 12 ) + Γ 2 ( 0 , 0 ) I 1 ( - α + μ 12 , - β - ν 12 ) + Γ 2 ( 0 , 0 ) I 1 ( α + μ 12 , β - ν 12 ) ] + ( c 0 2 16 ) d x s d y s I 1 ( x s y s ) I 2 ( - x s - α + μ 12 + μ 12 , - y s - β - ν 12 - ν 12 ) + ( c 0 2 16 ) d x s d y s I 1 ( x s , y s ) I 2 ( - x s + α + μ 12 + μ 12 , - y s + β - ν 12 - ν 12 ) + ( c 0 2 16 ) d x s d y s I 1 ( x s , y s ) I 2 ( x s + α - μ 12 + μ 12 , y s + β + ν 12 - ν 12 ) + ( c 0 2 16 ) d x s d y s I 1 ( x s , y s ) I 2 ( x s - α - μ 12 + μ 12 , y s - β + ν 12 - ν 12 ) } ,
μ 12 ( a 12 s 0 2 s 1 ) , ν 12 ( b 12 s 0 2 s 1 ) , μ 12 ( a 12 s 0 2 s 1 ) , ν 12 ( b 12 s 0 2 s 1 ) .
Γ p f ( α 12 , β 12 ) = 4 a 0 b 0 ( λ s 2 ) 2 Γ p r ( α 12 , β 12 ) × sinc { 2 π a 0 ( α 12 - α 12 ) λ s 2 } × sinc { 2 π b 0 ( β 12 - β 12 ) λ s 2 } d α 12 d β 12 ,

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