Abstract

The role of a sharp autocorrelation phase mask, called the bleached uniformly redundant array, for improving the spatial coherence in the far-field of partially coherent light sources is studied. It is shown both theoretically and experimentally that the input source correlation plays an important role in determining the amount of enhancement introduced by the phase mask.

© 1993 Optical Society of America

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References

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  1. V. Boopathi, R. M. Vasu, “Coherent optical processing with noncoherent light after source masking,” Appl. Opt. 31, 186–191 (1992).
    [CrossRef] [PubMed]
  2. R. Silva, G. L. Rogers, “Tomographical possibilities in coded aperture imaging optical simulations,” Opt. Acta 29, 257–264 (1982); “Coded aperture imaging: a noncoherent approach,” Opt. Acta 28, 1125–1134 (1981).
    [CrossRef]
  3. G. L. Rogers, Noncoherent Optical Processing (Wiley, 1977), Chap. 3, pp. 18–25.
  4. E. E. Fenimore, T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17, 337–347 (1978).
    [CrossRef] [PubMed]
  5. H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steninle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
    [CrossRef]
  6. H. P. Baltes, K.M. Jauch, “Multiplex version of Van Cittert–Zernike theorem,” J. Opt. Soc. Am. 71, 1434–1439 (1981).
    [CrossRef]
  7. K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser: experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
    [CrossRef]
  8. A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
    [CrossRef]
  9. D. Newman, J. C. Dainty, “Detection of gratings hidden by diffusers using intensity interferometry,” J. Opt. Soc. Am. A 1, 403–411 (1984).
    [CrossRef]
  10. D. Calabro, J. K. Wolf, “On the synthesis of two-dimensional arrays with desirable correlation properties,” Inf. Control 11, 537–560 (1968).
    [CrossRef]
  11. F. J. MacWilliams, N. J. A. Sloane, “Pseudo-random sequences and arrays,” Proc. IEEE 64, 1715–1729 (1976).
    [CrossRef]
  12. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 292–296.
  13. The γ(Δr) plots are one-dimensional cross sections of the two-dimensional circular symmetric mutual intensity functions from the URA, which has a circular symmetric autocorrelation function. Since the URA is two dimensional, we replaced it with a 2 mm × 2 mm clear aperture, and the plot 2k is a one-dimensional cross section through the center of the corresponding two-dimensional mutual intensity, which has no circular symmetry.
  14. The surface roughness of the diffuser, or a related parameter, the surface height decorrelation length, is an important parameter that controls the way spatial coherence of light is varied by this method. We prepared the diffuser used by spray painting colorless lacquer onto Perspex disks. Such diffusers are relatively smooth as compared with ground-glass screens that are obtained with the finest emery available.
  15. The bleached URA is prepared such that the phase difference between the two regions present is π. Phase steps were created on a test plate by recording and bleaching an intensity step wedge with different exposure times. The bleached plate was tested in a Twyman–Green interferometer to determine the right exposure time for a π phase difference.
  16. Young’s fringes for a number of double slits are recorded on the same type of emulsion and processed under identical conditions. The processed film is illuminated by an unexpanded laser beam. The illumination intensity and the + 1-order diffracted intensity are measured with a photomultiplier tube. The ratio of the photocurrents corresponding to +1-order diffraction and the illumination intensity is proportional to the fringe visibility in the record.

1992 (1)

1984 (1)

1982 (2)

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

R. Silva, G. L. Rogers, “Tomographical possibilities in coded aperture imaging optical simulations,” Opt. Acta 29, 257–264 (1982); “Coded aperture imaging: a noncoherent approach,” Opt. Acta 28, 1125–1134 (1981).
[CrossRef]

1981 (3)

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steninle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

H. P. Baltes, K.M. Jauch, “Multiplex version of Van Cittert–Zernike theorem,” J. Opt. Soc. Am. 71, 1434–1439 (1981).
[CrossRef]

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser: experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

1978 (1)

1976 (1)

F. J. MacWilliams, N. J. A. Sloane, “Pseudo-random sequences and arrays,” Proc. IEEE 64, 1715–1729 (1976).
[CrossRef]

1968 (1)

D. Calabro, J. K. Wolf, “On the synthesis of two-dimensional arrays with desirable correlation properties,” Inf. Control 11, 537–560 (1968).
[CrossRef]

Baltes, H. P.

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steninle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

H. P. Baltes, K.M. Jauch, “Multiplex version of Van Cittert–Zernike theorem,” J. Opt. Soc. Am. 71, 1434–1439 (1981).
[CrossRef]

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser: experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

Boopathi, V.

Calabro, D.

D. Calabro, J. K. Wolf, “On the synthesis of two-dimensional arrays with desirable correlation properties,” Inf. Control 11, 537–560 (1968).
[CrossRef]

Cannon, T. M.

Dainty, J. C.

Fenimore, E. E.

Ferwerda, H. A.

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steninle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Glass, A. S.

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steninle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 292–296.

Jauch, K. M.

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser: experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

Jauch, K.M.

MacWilliams, F. J.

F. J. MacWilliams, N. J. A. Sloane, “Pseudo-random sequences and arrays,” Proc. IEEE 64, 1715–1729 (1976).
[CrossRef]

Newman, D.

Rogers, G. L.

R. Silva, G. L. Rogers, “Tomographical possibilities in coded aperture imaging optical simulations,” Opt. Acta 29, 257–264 (1982); “Coded aperture imaging: a noncoherent approach,” Opt. Acta 28, 1125–1134 (1981).
[CrossRef]

G. L. Rogers, Noncoherent Optical Processing (Wiley, 1977), Chap. 3, pp. 18–25.

Silva, R.

R. Silva, G. L. Rogers, “Tomographical possibilities in coded aperture imaging optical simulations,” Opt. Acta 29, 257–264 (1982); “Coded aperture imaging: a noncoherent approach,” Opt. Acta 28, 1125–1134 (1981).
[CrossRef]

Sloane, N. J. A.

F. J. MacWilliams, N. J. A. Sloane, “Pseudo-random sequences and arrays,” Proc. IEEE 64, 1715–1729 (1976).
[CrossRef]

Steninle, B.

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steninle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

Vasu, R. M.

Wolf, J. K.

D. Calabro, J. K. Wolf, “On the synthesis of two-dimensional arrays with desirable correlation properties,” Inf. Control 11, 537–560 (1968).
[CrossRef]

Appl. Opt. (2)

Inf. Control (1)

D. Calabro, J. K. Wolf, “On the synthesis of two-dimensional arrays with desirable correlation properties,” Inf. Control 11, 537–560 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (4)

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser: experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

A. S. Glass, H. P. Baltes, “The significance of far-zone coherence for sources or scatterers with hidden periodicity,” Opt. Acta 29, 169–185 (1982).
[CrossRef]

H. P. Baltes, H. A. Ferwerda, A. S. Glass, B. Steninle, “Retrieval of structural information from the far-zone intensity and coherence of scattered radiation,” Opt. Acta 28, 11–28 (1981).
[CrossRef]

R. Silva, G. L. Rogers, “Tomographical possibilities in coded aperture imaging optical simulations,” Opt. Acta 29, 257–264 (1982); “Coded aperture imaging: a noncoherent approach,” Opt. Acta 28, 1125–1134 (1981).
[CrossRef]

Proc. IEEE (1)

F. J. MacWilliams, N. J. A. Sloane, “Pseudo-random sequences and arrays,” Proc. IEEE 64, 1715–1729 (1976).
[CrossRef]

Other (6)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 292–296.

The γ(Δr) plots are one-dimensional cross sections of the two-dimensional circular symmetric mutual intensity functions from the URA, which has a circular symmetric autocorrelation function. Since the URA is two dimensional, we replaced it with a 2 mm × 2 mm clear aperture, and the plot 2k is a one-dimensional cross section through the center of the corresponding two-dimensional mutual intensity, which has no circular symmetry.

The surface roughness of the diffuser, or a related parameter, the surface height decorrelation length, is an important parameter that controls the way spatial coherence of light is varied by this method. We prepared the diffuser used by spray painting colorless lacquer onto Perspex disks. Such diffusers are relatively smooth as compared with ground-glass screens that are obtained with the finest emery available.

The bleached URA is prepared such that the phase difference between the two regions present is π. Phase steps were created on a test plate by recording and bleaching an intensity step wedge with different exposure times. The bleached plate was tested in a Twyman–Green interferometer to determine the right exposure time for a π phase difference.

Young’s fringes for a number of double slits are recorded on the same type of emulsion and processed under identical conditions. The processed film is illuminated by an unexpanded laser beam. The illumination intensity and the + 1-order diffracted intensity are measured with a photomultiplier tube. The ratio of the photocurrents corresponding to +1-order diffraction and the illumination intensity is proportional to the fringe visibility in the record.

G. L. Rogers, Noncoherent Optical Processing (Wiley, 1977), Chap. 3, pp. 18–25.

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Figures (6)

Fig. 1
Fig. 1

Far field of the illuminated code SC is available at plane F, which is the back focal plane of lens L.

Fig. 2
Fig. 2

Theoretical curves of the moduli of normalized mutual intensities at the far field of the source code (i.e., plane F of Fig. 4) for the following correlation lengths of the illumination source l (μm): (a) 100, (b) 200, (c) 300, (d) 400, (e) 500, (f) 600, (g) 700, and (h) 1000. Curve k is |γ(Δr)| for a 2 mm × 2 mm aperture without the mask (l = 100 μm) (the focal length of lens L3 is 380 mm, and the wavelength of light is 0.6328 μm). The URA used in this calculation is the one described in Subsection 3.B.

Fig. 3
Fig. 3

Variation of |γ(Δr)| for Δr = 5 mm with l, the correlation length of the source.

Fig. 4
Fig. 4

Schematic layout of the experimental setup. The laser, lens L1, and rotating diffuser RD combination, along with lens L2, provides partially coherent illumination at input plane IP containing the source code SC. At F, the back focal plane of lens L3, a double slit (DS) is used to measure the spatial coherence of light by forming Young’s fringes at the observation plane O (f3 is the focal length of lens L3 and is 380 mm).

Fig. 5
Fig. 5

Experimental results of normalized mutual intensity measurement at plane F (Fig. 4): (curve a) poor source correlation; Young’s fringes were not visible for the 50- μm double slit at the source (plane SC of Fig. 4); (curve b) improved source correlation; the 100- μm double slit at the source produced Young’s fringes that were clearly seen.

Fig. 6
Fig. 6

Variation of |γ(Δr)| for Δr = 5 mm with the spatial coherence of the source. On the x axis is d, the distance by which RD is short of the focal distance from lens L1 (Fig. 4). d = 0 represents the best possible source coherence with diffuser RD. In parentheses are given the laser beam spot sizes on the diffuser in micrometers, obtained by different positions of lens L1.

Equations (9)

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A c ( I , J ) = 0 , I = 0 = 1 , J = 0 , I 0 = 1 , C r ( I ) C s ( J ) = 1 = 0 , otherwise ,
A d ( i , j ) = 1 , A c ( i , j ) = 1 = - 1 , A c ( i , j ) = 0.
μ ( α 1 , α 2 ) = exp [ - ( α 1 - α 2 ) 2 / 2 l 2 ]
Γ ( r 1 , r 2 ) = - - Γ ( α 1 , α 2 ) exp [ - j k f ( α 2 r 2 - α 1 r 1 ) ] d α 2 d α 1 .
Γ ( α 1 , α 2 ) = t ( α 1 ) t * ( α 2 ) I [ ( α 1 + α 2 ) / 2 ] μ ( α 1 - α 2 ) ,
Γ ( r , - r ) = - - μ ( q ) I ( q ) t ( p + q 2 ) × t * ( p - q 2 ) exp [ - j k f ( 2 r p ) ] d q d p ,
Γ ( Δ r ) = p μ ( p ) I ( p ) exp [ - j k f ( Δ r p ) ] d p ,
μ ( p ) = q μ ( q ) t ( p + q 2 ) t * ( p - q 2 ) d q .
I ( r ) = - - μ ( q ) I ( q ) t ( p + q 2 ) × t * ( p - q 2 ) exp [ - j k f ( r q ) ] d q d q .

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