Abstract

A nonuniform sampling scheme is described for measuring the mutual intensity of the wave field produced in a plane a distance z from a spatially incoherent, three-dimensional source object. Both uniform and nonuniform sampling are analyzed and discussed in detail, and comparisons of the two schemes are made. It is shown that nonuniform sampling requires fewer measurements than uniform sampling to specify the coherence function. Additionally, the smallest separation required between measurement points is larger for the nonuniform case.

© 2003 Optical Society of America

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References

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    [CrossRef]
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2000 (1)

1999 (1)

1998 (1)

C. E. Shannon, “Communication in the Presence of Noise,” Proc. IEEE 86, 447–457 (1998).
[CrossRef]

1996 (2)

1995 (1)

1990 (1)

1984 (1)

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Brady, D. J.

Brenner, K. H.

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Goodman, J. W.

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

Marks, D. L.

Marks, D. M.

Ojeda-Castaneda, J.

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Rhodes, W. T.

G. Welch, W. T. Rhodes, “Image reconstruction by spatio-temporal coherence transfer ,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds. Proc. SPIE4489, 60–65 (2001).
[CrossRef]

Rosen, J.

Roux, F. S.

Shannon, C. E.

C. E. Shannon, “Communication in the Presence of Noise,” Proc. IEEE 86, 447–457 (1998).
[CrossRef]

Stack, R. A.

Vanderlugt, A.

Welch, G.

G. Welch, W. T. Rhodes, “Image reconstruction by spatio-temporal coherence transfer ,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds. Proc. SPIE4489, 60–65 (2001).
[CrossRef]

Yariv, A.

Appl. Opt. (3)

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

Opt. Acta (1)

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

C. E. Shannon, “Communication in the Presence of Noise,” Proc. IEEE 86, 447–457 (1998).
[CrossRef]

Other (2)

G. Welch, W. T. Rhodes, “Image reconstruction by spatio-temporal coherence transfer ,” in Free-Space Laser Communication and Laser Imaging, D. G. Voelz, J. C. Ricklin, eds. Proc. SPIE4489, 60–65 (2001).
[CrossRef]

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

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Figures (12)

Fig. 1
Fig. 1

Optical coherence function of the wave field generated by the source intensity distribution I ( x ,   z ) sampled at the point ( x 1 ,   x 2 ) in a plane a distance z away from the center of the rectangular area bounding I ( x ,   z ) . Points ( x 1 ,   x 2 ) are centered at x ¯ and separated by Δ x .

Fig. 2
Fig. 2

Relationship between different functions involved in the analysis. J ( x 1 ,   x 2 ) , upper left, is related to J ( x ¯ ,   Δ x ) through a simple geometrical coordinate transformation. The 1D Fourier transformation of J ( x ¯ ,   Δ x ) in the x ¯ direction yields J ^ x ¯ ( Δ ν ,   Δ x ) (upper right), and the 1D transformation in the Δ x direction yields the function J ^ Δ x ( x ¯ ,   ν ) (lower middle). The regions of support of the two 1D transforms are indicated by shading. Dotted lines upper left and upper middle show the relevant domains of x 1 , x 2 , x ¯ , and Δ x .

Fig. 3
Fig. 3

Region of support for J ^ Δ x ( x ¯ ,   ν ) . The bandwidth B Δ x ( x ¯ ) is approximately fixed for all values of x ¯ .

Fig. 4
Fig. 4

Region of support for J ^ x ¯ ( Δ ν ,   Δ x ) . The bandwidth B x ¯ ( Δ x ) increases in proportion to | Δ x | .

Fig. 5
Fig. 5

Contour lines from the two sampling functions. Intersections of these lines indicate the coordinates ( x ¯ ,   Δ x ) of a coherence sample. The dashed lines indicate the measurement-plane boundary. The geometry parameters used to determine these points were λ = 630   nm , W x = 25   μ m , W z = 0.1   mm , z = 1   m , and D = 0.4   m .

Fig. 6
Fig. 6

Sample locations in ( x 1 ,   x 2 ) coordinates for the conditions used for Fig. 5. The line of symmetry for the coherence function is shown, indicating the boundary for coherence samples.

Fig. 7
Fig. 7

Distribution of sample measurement points for the conditions used for Fig. 5. Vertical lines mark the locations of measurement points ( x 1 ,   x 2 ) along the x axis in the measurement plane.

Fig. 8
Fig. 8

Mutual intensity sample distribution produced by numerical simulation of a previous laboratory experiment10 using a reasoned estimate of the experimenters’ original sample locations. Darker regions correspond to greater magnitudes.

Fig. 9
Fig. 9

Reconstructed source intensity associated with the mutual intensity distribution shown in Fig. 8. Darker valued pixels correspond to greater intensity.

Fig. 10
Fig. 10

Central section (about z = 0 ) of the reconstructed source intensity of Fig. 8 showing the two barely-resolved point sources. Darker valued pixels correspond to greater intensity.

Fig. 11
Fig. 11

Nonuniform sampling scheme distribution of mutual intensity measurements for the same source object used to produce the distribution shown in Fig. 8.

Fig. 12
Fig. 12

Reconstructed source intensity associated with the mutual intensity distribution shown in Fig. 11. Darker valued pixels correspond to greater intensity.

Equations (37)

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J ( x 1 ,   x 2 ) = - + - + 1 λ ( z - ζ )   I ( ξ ,   ζ ) × exp j   π λ ( z - ζ )   [ ( x 1 - ξ ) 2 - ( x 2 - ξ ) 2 ] d ξ d ζ .
x ¯ = ( x 1 + x 2 ) / 2 ,
Δ x = ( x 1 - x 2 ) .
J ( x ¯ ,   Δ x ) = - + - + 1 λ ( z - ζ )   I ( ξ ,   ζ ) × exp j   2 π Δ x λ ( z - ζ )   ( x ¯ - ξ ) d ξ d ζ .
J ^ Δ x ( x ¯ ,   ν ) = - + - + 1 λ ( z - ζ )   I ( ξ ,   ζ ) × δ ν - ( x ¯ - ξ ) λ ( z - ζ ) d ξ d ζ .
ν = 1 λ z ± W z 2 - 1 x ¯ ± W x 2 ,
ν = 1 λ z ± W z 2 - 1 x ¯ W x 2 .
B Δ x ( x ¯ ) = W x λ ( z - W z / 2 ) , | x | W x / 2 W z λ ( z 2 - W z 2 / 4 ) x ¯ + W x z W z , | x ¯ | > W x / 2 .
B Δ x W x λ z .
J ^ x ¯ ( Δ ν ,   Δ x ) = - + - + 1 λ ( z - ζ )   I ( ξ ,   ζ ) × exp - j   2 π Δ x ξ λ ( z - ζ ) × δ Δ ν - Δ x λ ( z - ζ ) d ξ d ζ .
Δ ν = Δ x λ z ± W z 2 - 1 .
B x ¯ ( Δ x ) = W z λ ( z 2 - W z 2 / 4 )   Δ x W z λ z 2   Δ x ,
f ( t ) = m f m δ ( t - mT )     *     sinc t T = m f m sinc t T - m ,
Δ ν max = D λ z - W z 2 - 1 D λ z ,
ν max = W x + D 2 λ z - W z 2 - 1 W x + D 2 λ z ,
J meas ( x ¯ ,   Δ x ) = m = - m max m max n = 1 n max J m , n × sinc x ¯ T x ¯ - m ,   Δ x T Δ x - n .
x ¯ m = mT x ¯ = m λ z 2 D ,
Δ x n = nT Δ x = n λ z D + W x .
S uniform = λ z 2 ( D + W x ) .
N Nyquist 2 D 2 ν max Δ ν max = D 4 λ 2 z 2   ( 1 + W x / D ) .
f ( t ) = m f m δ ( t - mT )     *     exp ( - j 2 π ν c t ) sinc t T = m f m exp [ - j 2 π ν c ( t - mT ) ] sinc t T - m ,
N Shannon 1 2   D 2 B Δ x B x ¯ ( D ) = W x W z D 3 2 λ 2 z 3 .
T x ¯ ( Δ x ) = λ z 2 W z Δ x ,
T Δ x = λ z W x .
x ¯ m , n = mW x z nW z ,
Δ x n = n λ z W x ,
S nonuni = λ z 2 W x .
J meas ( x ¯ , Δ x ) = ( [ exp ( - j 2 π x ¯ Δ x λ z ) m , n J m , n × δ ( x ¯ - x ¯ m , n , Δ x - Δ x n ) ] * x ¯ { sinc [ x ¯ T x ¯ ( Δ x ) ] } ) * Δ x [ sinc ( Δ x T Δ x ) ] .
| m | m max < D 2 W z 8 λ z 2 = m ˜ max ,
DW x 2 λ z 1 - 1 - | m | m ˜ max 1 / 2
< n < DW x 2 λ z 1 + 1 - | m | m ˜ max 1 / 2 .
N nonuni < DW x λ z m = - m max m max 1 - | m | m ˜ max 1 / 2 DW x λ z   ( 1 + 4 / 3 m ˜ max ) = DW x λ z 1 + D 2 W z 6 λ z 2 .
N Nyquist N nonuni 1 + D W x λ z D 2 + W z 6 z - 1 .
S nonuni S Nyquist 1 + D W x .
J ( x ¯ ,   Δ x ) 1 λ z - + - + I ( ξ ,   ζ ) × exp j   2 π Δ x λ z       x ¯ - ξ + x ¯ ζ z - ξ ζ z d ξ d ζ .
J ( x ¯ ,   Δ x ) 1 λ z exp j   2 π x ¯ Δ x λ z × F I ( ξ ,   ζ ) exp - j   2 π Δ x λ z 2   ξ ζ w = - x ¯ Δ x / λ z 2 u = Δ x / λ z ,
I ( x ,   z )     | F - 1 { J ( u ,   w ) exp ( - j 2 π zw ) } | .

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