Abstract

Image irradiance distributions from objects illuminated with partially coherent, quasi-monochromatic light, viewed against a spatially uniform background and received with a photosensitive detector are analyzed. A general expression for the SNR at the detector output is obtained as a function of the coherence of the illuminating light, the object surface roughness, the width of the telescope point spread function, and the aperture and integration time of the detector. The expression is evaluated for several limiting cases of coherence of illumination and of object surface roughness.

© 1976 Optical Society of America

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References

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  1. H. H. Hopkins, Proc. R. Soc. London, Ser. A 208, 408 (1953).
  2. Y. Ichioka, J. Opt. Soc. Am. 64, 919 (1974).
    [Crossref]
  3. L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).
  4. S. Lowenthal, H. H. Arsenault, J. Opt. Soc. Am. 60, 1478 (1970).
    [Crossref]
  5. J. C. Dainty, Opt. Acta 18, 327 (1971).
    [Crossref]
  6. M. Elbaum, M. Greenebaum, M. King, Opt. Commun. 5, 171 (1972).
    [Crossref]
  7. M. George, A. Jain, Appl. Phys. 4, 201 (1974).
    [Crossref]
  8. H. Fujii, T. Asakura, Opt. Commun. 11, 351 (1974).
    [Crossref]
  9. J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [Crossref]
  10. E. Wolf, Proc. R. Soc. London Ser. A 230, 246 (1955).
    [Crossref]
  11. L. Mandel, Proc. Phys. Soc. (London)74, 233 (1959).
    [Crossref]
  12. J. Bures, J. Opt. Soc. Am. 64, 1598 (1974).
    [Crossref]
  13. K. W. Pratt, Laser Communication Systems (Wiley, New York, 1969).
  14. he contrast of speckle patterns formed with partially coherent light at the image plane of a diffusing object has been studied by H. Fujii, T. Asakura, Opt. Commun. 12, 32, (1972) and Nouv. Rev. Opt. 6, 5 (1975).
    [Crossref]
  15. K. Miller, SIAM Rev. 11 (October1969).
    [Crossref]
  16. The assumption that the field is a complex Gaussian stochastic process is not necessary to validate Eq. (11). It has been shown by S. H. Chen, P. Tartaglia, Opt. Commun. 6, 119 (1972) that when laser light is scattered from an assembly of N independent particles, 〈I(x1)I(x2)〉 = 〈I〉2 + (1 + N−1)|Γ(x1 − x2)|2. N is large in our case.
    [Crossref]
  17. J. Perina, Coherence of Light (Van Nostrand Reinhold, New York, 1972).
  18. L. Mandel, J. Opt. Soc. Am. 51, 1342 (1961).
    [Crossref]
  19. J. Fujii, T. Asakura, Optik 39, 284 (1974).
  20. G. Parry, Opt. Acta 21, 763 (1974).
    [Crossref]
  21. H. M. Pedersen, Opt. Acta 22, 523 (1975).
    [Crossref]
  22. H. M. Pedersen, C. T. Stansberg, Opt. Commun. 15, 222 (1975).
    [Crossref]
  23. J. Ohtsubo, T. Asakura, Opt. Commun. 14, 30 (1975).
    [Crossref]
  24. J. W. Goodman, Opt. Commun. 14, 324 (1975).
    [Crossref]
  25. N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
    [Crossref]
  26. J. W. Goodman, Stanford Electronics Laboratory Report, Stanford, California (1963).
  27. The dependence of speckle contrast computed without this assumption has been studied in Refs. (7) and (20).
  28. J. C. Dainty, Opt. Acta 17, 761 (1970).
    [Crossref]

1975 (4)

H. M. Pedersen, Opt. Acta 22, 523 (1975).
[Crossref]

H. M. Pedersen, C. T. Stansberg, Opt. Commun. 15, 222 (1975).
[Crossref]

J. Ohtsubo, T. Asakura, Opt. Commun. 14, 30 (1975).
[Crossref]

J. W. Goodman, Opt. Commun. 14, 324 (1975).
[Crossref]

1974 (7)

N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
[Crossref]

Y. Ichioka, J. Opt. Soc. Am. 64, 919 (1974).
[Crossref]

M. George, A. Jain, Appl. Phys. 4, 201 (1974).
[Crossref]

H. Fujii, T. Asakura, Opt. Commun. 11, 351 (1974).
[Crossref]

J. Bures, J. Opt. Soc. Am. 64, 1598 (1974).
[Crossref]

J. Fujii, T. Asakura, Optik 39, 284 (1974).

G. Parry, Opt. Acta 21, 763 (1974).
[Crossref]

1972 (3)

The assumption that the field is a complex Gaussian stochastic process is not necessary to validate Eq. (11). It has been shown by S. H. Chen, P. Tartaglia, Opt. Commun. 6, 119 (1972) that when laser light is scattered from an assembly of N independent particles, 〈I(x1)I(x2)〉 = 〈I〉2 + (1 + N−1)|Γ(x1 − x2)|2. N is large in our case.
[Crossref]

he contrast of speckle patterns formed with partially coherent light at the image plane of a diffusing object has been studied by H. Fujii, T. Asakura, Opt. Commun. 12, 32, (1972) and Nouv. Rev. Opt. 6, 5 (1975).
[Crossref]

M. Elbaum, M. Greenebaum, M. King, Opt. Commun. 5, 171 (1972).
[Crossref]

1971 (1)

J. C. Dainty, Opt. Acta 18, 327 (1971).
[Crossref]

1970 (2)

1969 (1)

K. Miller, SIAM Rev. 11 (October1969).
[Crossref]

1967 (1)

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

1965 (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

1961 (1)

1955 (1)

E. Wolf, Proc. R. Soc. London Ser. A 230, 246 (1955).
[Crossref]

1953 (1)

H. H. Hopkins, Proc. R. Soc. London, Ser. A 208, 408 (1953).

Arsenault, H. H.

Asakura, T.

J. Ohtsubo, T. Asakura, Opt. Commun. 14, 30 (1975).
[Crossref]

J. Fujii, T. Asakura, Optik 39, 284 (1974).

H. Fujii, T. Asakura, Opt. Commun. 11, 351 (1974).
[Crossref]

he contrast of speckle patterns formed with partially coherent light at the image plane of a diffusing object has been studied by H. Fujii, T. Asakura, Opt. Commun. 12, 32, (1972) and Nouv. Rev. Opt. 6, 5 (1975).
[Crossref]

Bures, J.

Chen, S. H.

The assumption that the field is a complex Gaussian stochastic process is not necessary to validate Eq. (11). It has been shown by S. H. Chen, P. Tartaglia, Opt. Commun. 6, 119 (1972) that when laser light is scattered from an assembly of N independent particles, 〈I(x1)I(x2)〉 = 〈I〉2 + (1 + N−1)|Γ(x1 − x2)|2. N is large in our case.
[Crossref]

Dainty, J. C.

J. C. Dainty, Opt. Acta 18, 327 (1971).
[Crossref]

J. C. Dainty, Opt. Acta 17, 761 (1970).
[Crossref]

Elbaum, M.

M. Elbaum, M. Greenebaum, M. King, Opt. Commun. 5, 171 (1972).
[Crossref]

Enloe, L. H.

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

Fujii, H.

H. Fujii, T. Asakura, Opt. Commun. 11, 351 (1974).
[Crossref]

he contrast of speckle patterns formed with partially coherent light at the image plane of a diffusing object has been studied by H. Fujii, T. Asakura, Opt. Commun. 12, 32, (1972) and Nouv. Rev. Opt. 6, 5 (1975).
[Crossref]

Fujii, J.

J. Fujii, T. Asakura, Optik 39, 284 (1974).

George, M.

M. George, A. Jain, Appl. Phys. 4, 201 (1974).
[Crossref]

Goodman, J. W.

J. W. Goodman, Opt. Commun. 14, 324 (1975).
[Crossref]

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

J. W. Goodman, Stanford Electronics Laboratory Report, Stanford, California (1963).

Greenebaum, M.

M. Elbaum, M. Greenebaum, M. King, Opt. Commun. 5, 171 (1972).
[Crossref]

Hopkins, H. H.

H. H. Hopkins, Proc. R. Soc. London, Ser. A 208, 408 (1953).

Ichioka, Y.

Jain, A.

M. George, A. Jain, Appl. Phys. 4, 201 (1974).
[Crossref]

King, M.

M. Elbaum, M. Greenebaum, M. King, Opt. Commun. 5, 171 (1972).
[Crossref]

Lowenthal, S.

Mandel, L.

L. Mandel, J. Opt. Soc. Am. 51, 1342 (1961).
[Crossref]

L. Mandel, Proc. Phys. Soc. (London)74, 233 (1959).
[Crossref]

Miller, K.

K. Miller, SIAM Rev. 11 (October1969).
[Crossref]

Ohtsubo, J.

J. Ohtsubo, T. Asakura, Opt. Commun. 14, 30 (1975).
[Crossref]

Parry, G.

G. Parry, Opt. Acta 21, 763 (1974).
[Crossref]

Pedersen, H. M.

H. M. Pedersen, Opt. Acta 22, 523 (1975).
[Crossref]

H. M. Pedersen, C. T. Stansberg, Opt. Commun. 15, 222 (1975).
[Crossref]

Perina, J.

J. Perina, Coherence of Light (Van Nostrand Reinhold, New York, 1972).

Pratt, K. W.

K. W. Pratt, Laser Communication Systems (Wiley, New York, 1969).

Stansberg, C. T.

H. M. Pedersen, C. T. Stansberg, Opt. Commun. 15, 222 (1975).
[Crossref]

Takai, N.

N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
[Crossref]

Tartaglia, P.

The assumption that the field is a complex Gaussian stochastic process is not necessary to validate Eq. (11). It has been shown by S. H. Chen, P. Tartaglia, Opt. Commun. 6, 119 (1972) that when laser light is scattered from an assembly of N independent particles, 〈I(x1)I(x2)〉 = 〈I〉2 + (1 + N−1)|Γ(x1 − x2)|2. N is large in our case.
[Crossref]

Wolf, E.

E. Wolf, Proc. R. Soc. London Ser. A 230, 246 (1955).
[Crossref]

Appl. Phys. (1)

M. George, A. Jain, Appl. Phys. 4, 201 (1974).
[Crossref]

Bell Syst. Tech. J. (1)

L. H. Enloe, Bell Syst. Tech. J. 46, 1479 (1967).

J. Opt. Soc. Am. (4)

Jpn. J. Appl. Phys. (1)

N. Takai, Jpn. J. Appl. Phys. 13, 2025 (1974).
[Crossref]

Opt. Acta (4)

G. Parry, Opt. Acta 21, 763 (1974).
[Crossref]

H. M. Pedersen, Opt. Acta 22, 523 (1975).
[Crossref]

J. C. Dainty, Opt. Acta 17, 761 (1970).
[Crossref]

J. C. Dainty, Opt. Acta 18, 327 (1971).
[Crossref]

Opt. Commun. (7)

M. Elbaum, M. Greenebaum, M. King, Opt. Commun. 5, 171 (1972).
[Crossref]

H. Fujii, T. Asakura, Opt. Commun. 11, 351 (1974).
[Crossref]

he contrast of speckle patterns formed with partially coherent light at the image plane of a diffusing object has been studied by H. Fujii, T. Asakura, Opt. Commun. 12, 32, (1972) and Nouv. Rev. Opt. 6, 5 (1975).
[Crossref]

The assumption that the field is a complex Gaussian stochastic process is not necessary to validate Eq. (11). It has been shown by S. H. Chen, P. Tartaglia, Opt. Commun. 6, 119 (1972) that when laser light is scattered from an assembly of N independent particles, 〈I(x1)I(x2)〉 = 〈I〉2 + (1 + N−1)|Γ(x1 − x2)|2. N is large in our case.
[Crossref]

H. M. Pedersen, C. T. Stansberg, Opt. Commun. 15, 222 (1975).
[Crossref]

J. Ohtsubo, T. Asakura, Opt. Commun. 14, 30 (1975).
[Crossref]

J. W. Goodman, Opt. Commun. 14, 324 (1975).
[Crossref]

Optik (1)

J. Fujii, T. Asakura, Optik 39, 284 (1974).

Proc. IEEE (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[Crossref]

Proc. R. Soc. London Ser. A (1)

E. Wolf, Proc. R. Soc. London Ser. A 230, 246 (1955).
[Crossref]

Proc. R. Soc. London, Ser. A (1)

H. H. Hopkins, Proc. R. Soc. London, Ser. A 208, 408 (1953).

SIAM Rev. (1)

K. Miller, SIAM Rev. 11 (October1969).
[Crossref]

Other (5)

J. Perina, Coherence of Light (Van Nostrand Reinhold, New York, 1972).

K. W. Pratt, Laser Communication Systems (Wiley, New York, 1969).

L. Mandel, Proc. Phys. Soc. (London)74, 233 (1959).
[Crossref]

J. W. Goodman, Stanford Electronics Laboratory Report, Stanford, California (1963).

The dependence of speckle contrast computed without this assumption has been studied in Refs. (7) and (20).

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Figures (4)

Fig. 1
Fig. 1

Geometry of the imaging system: O = object; T = imaging lens; I = image; A = array of detectors.

Fig. 2
Fig. 2

Imaging of rough target, showing various object resolution cells: 1 = specular; 2 = optically rough but nearly normal to the line-of-sight; 3 = optically rough but skewed to the line-of-sight.

Fig. 3
Fig. 3

Fading-limited SNR for discrete frequencies (solid lines) and for a continuous spectrum (dashed lines) vs the total illumination bandwidth for different scales of roughness assumed greater than the illumination wavelength.

Fig. 4
Fig. 4

Typical plot of SNR vs signal level 〈ns〉 with its asymptotic approximations. The steepest asymptote is for background-noise-limited operation: SNR = 〈nS〉/〈nN1/2, The middle asymptote is for signal-shot-noise-limited operation: SNR = 〈nS1/2. The horizontal asymptote represents target-fading limitation: SNR = CW−1. Noise level 〈nN〉 = 9, fading contrast CW = 0.1.

Equations (47)

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n T = n S + n B + n D .
N = σ T = ( σ S 2 + σ B 2 + σ D 2 ) 1 / 2 .
n = α W
σ n 2 = n + α 2 σ W 2 .
W = T d t A d 2 x i I ( x i , t ) ,
SNR = n S σ T = n S ( n S + α 2 σ W 2 + n N ) 1 / 2 ,
SNR = ( 1 n S + σ W 2 W 2 + n N n S 2 ) - 1 / 2 .
SNR = n S 1 / 2 [ 1 + C W 2 n S + ( n N / n S ) ] 1 / 2 ,
C W 2 = d t 1 d t 2 d 2 x i 1 d 2 x i 2 cov { I 1 , I 2 } d t 1 d t 2 d 2 x i 1 d 2 x i 2 I 1 I 2 .
cov { I 1 , I 2 } = I 1 I 2 - I 1 I 2 .
cov { I 1 , I 2 } = γ 12 ( t 1 , t 2 ) 2 I 1 I 2 ,
γ 12 ( t 1 , t 2 ) = V ( 3 ) ( x i 1 , t 1 ) V ( 3 ) * ( x i 2 , t 2 ) / ( I 1 I 2 ) 1 / 2 .
C W 2 = A - 2 T - 2 d t 1 d t 2 d t 2 x i 1 d 2 x i 2 γ 12 ( Δ t ) 2 ,
γ 12 ( Δ t ) = F 12 * γ 12 ( 2 ) ( Δ t + R / c ) ,
γ 12 ( 2 ) ( Δ t + R / c ) = γ 12 ( 2 ) ( Δ t ) exp ( j 2 π ν ¯ R / c ) .
Z ( θ ) = [ z ( x 01 ) - z ( x 02 ) ] ( 1 + cos θ ) ,
exp ( j 2 π ν ¯ Z / c ) = ϕ ( k , - k ) ,
γ 12 ( Δ t ) = d 2 x 01 d 2 x 02 H 12 ( x 01 , x 02 ) × ϕ ( k , - k ) ρ η ( Δ x 0 ) γ 12 ( 1 ) ( Δ t ) ,
γ 12 ( 0 ) = d 2 x 01 d 2 x 02 H 12 ( x 01 , x 02 ) ϕ ( k , - k ) × ρ η ( Δ x 0 ) γ 12 ( 1 ) ( 0 ) ,
C a 2 = [ T - 2 d t 1 d t 2 γ 11 ( 1 ) ( Δ t ) 2 ] ( A - 2 d 2 x i 1 d 2 x i 2 γ 12 2 ) .
V ( 1 ) ( x 01 , t ) = U ( x 0 ) exp ( - j 2 π ν 0 t ) .
γ 12 ( 1 ) ( Δ t ) = ρ U ( Δ x 0 ) exp ( - j 2 π ν 0 Δ t ) ,
C b 2 = A - 2 d 2 x i 1 d 2 x i 2 γ 12 2 .
V M ( 1 ) = U ( x 0 ) m = 1 M exp ( - j 2 π ν m t ) .
V M ( 3 ) ( x i , t ) = m = 1 M V m ( 3 ) ( x i , t ) .
W M = T d 2 x i m = 1 M V m ( 3 ) ( x i ) 2 .
C M 2 = M - 2 A - 2 d 2 x i 1 d 2 x i 2 m , n = 1 M γ 12 , m n 2 ,
γ 12 , m n = d 2 x 01 d 2 x 02 H 12 , min ( x 01 , x 02 ) ϕ ( k m , - k n ) × ρ η ( Δ x 0 ) γ 12 ( 1 ) ( 0 ) .
W c = d 2 x i d ν S ( ν ) I ( x i , ν ) ,
C c 2 = B - 2 A - 2 d 2 x i 2 ν 1 ν 2 d ν m ν 1 ν 2 d ν n C 12 , m n 2 ,
ϕ ( k 1 , k 2 ) = exp [ - 1 2 σ z 2 ( k 1 2 + k 2 2 + 2 k 1 k 2 ρ ) ] ,
ρ ( Δ x 0 ) = z ( x 01 ) z ( x 02 ) / σ z 2 .
ϕ ( k , - k ) exp [ - ( k σ z ) 2 Δ x 0 2 / a 2 ] .
ϕ ( k , - k ) π [ a / ( k σ z ) ] 2 δ ( Δ x 0 ) ,
ϕ ( k m , - k n ) = exp [ - 1 2 σ z 2 ( k m - k n ) 2 ] π a 2 k m k n σ z 2 δ ( Δ x 0 ) .
γ * 12 ( 0 ) = d 2 x 0 H 12 ( x 0 , x 0 )
C * a 2 = { 1 for A 3 A , T 3 T , A 3 A T 3 T for A 3 A , T 3 T ,
A 3 = d 2 x γ 12 ( x ) 2 ,
T 3 = d t γ 11 ( t ) 2 .
γ 0 12 = d 2 x 01 d 2 x 02 H 12 ( x 01 , x 02 ) ρ η ( Δ x 0 ) γ 12 ( 1 ) .
C 0 a 2 = { 1 for T 3 T , T 3 T for T 3 T .
C * b 2 = { 1 A 3 / A for A 3 A , A 3 A .
C 0 b = 0.
C * M 2 = C * b 2 M [ 1 + 2 M n = 1 M - 1 ( M - n ) exp ( - b 2 n 2 ) ] ,
C * M 2 = { 1 Δ ν = 0 and A 3 A , 1 / M Δ ν > c / σ z and A 3 A , ( A 3 / A ) / M Δ ν > c / σ z and A A 3 .
C * c 2 = C * b 2 B 2 ν 1 ν 2 d ν m ν 1 ν 2 d ν n exp [ - ( k m - k n ) 2 σ z 2 ]
C * c 2 = C * b 2 [ π 1 / 2 β erf β - 1 - exp ( - β 2 ) β 2 ] ,

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