Abstract

Fluctuations in image illuminance resulting from various sources of optical noise are studied as a function of the spatial coherence of the illumination. It is shown that noise fluctuations caused by the pupil plane can be reduced considerably by using incoherent, or even partially coherent, rather than coherent illumination. Conversely, noise caused by defects in the object plane is not affected by the degree of coherence, except for phase noise which is suppressed in incoherent light. Expressions for noise fluctuations are developed on the basis of a simplifying Gaussian assumption for the noise sources; the validity of this assumption is justified.

© 1978 Optical Society of America

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References

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  1. P. Chavel and S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,” J. Opt. Soc. Am. 66, 14–23(1976).
    [Crossref]
  2. A. W. Lohmann and W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. (to be published).
  3. G. L. Rogers, “Non-coherent optical processing,” Opt. Las. Technol. 7, 153–162(1975).
    [Crossref]
  4. P. Chavel and S. Lowenthal, “Noise and coherence in optical (and digital) image processing. I. The Callier effect and its influence on image contrast,” J. Opt. Soc. Am. 68, 559–568(1978).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Chap. X.
  6. J. W. Goodman, “Film-grain noise in wavefront-reconstruction imaging,” J. Opt. Soc. Am. 57, 493–502(1967).
    [Crossref] [PubMed]
  7. G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 78–122.
    [Crossref]
  8. H. Fujii and T. Asakura, “Contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–37(1974).
    [Crossref]
  9. H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. d’Optique 6, 5–14(1975).
    [Crossref]
  10. B. Reuter and F. Lanzl, “Ortsfrequenzbeschneidende Filterung bei teilkohärente Beleuchtung,” D.G.a.O. Meeting, Berlin, 1977.
  11. S. Lowenthal and P. Chavel, “Noise problems in optical image processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics, Jerusalem, 1976), paper TuA-1.
  12. D. Tichenor and J. W. Goodman, “Practical Noise Limitations in Holographic Image Deblurring,” in Proceedings of the International Optical Computing Conference (IEEE, Washington, D.C., 1975, IEEE Catalog No. 75, CH 0941-5C), pp. 82–84.
  13. J. W. Goodman, “Noise in Coherent Optical Information Processing,” in Optical Information Processing, edited by Yu. E. Nesterikhin, G. W. Stroke, and W. E. Kock (Plenum, New York, 1976), pp. 85–103.
    [Crossref]
  14. J. D. Armitage and A. W. Lohmann, “Optical character recognition by incoherent matched filtering,” Appl. Opt. 4, 461–467(1965).
    [Crossref]
  15. S. Lowenthal and A. Werts, “Filtrage des fréquences spatiales en lumière incohérente à l’aide d’hologrammes,” C. R. Acad. Sci. (Paris) 2 66B, 542–545 (1968).
  16. T. S. McKechnie, “Measurement of some second-order statistical properties of speckle,” Optik 39, 258–267(1974).

1978 (1)

1976 (1)

1975 (2)

G. L. Rogers, “Non-coherent optical processing,” Opt. Las. Technol. 7, 153–162(1975).
[Crossref]

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. d’Optique 6, 5–14(1975).
[Crossref]

1974 (2)

H. Fujii and T. Asakura, “Contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–37(1974).
[Crossref]

T. S. McKechnie, “Measurement of some second-order statistical properties of speckle,” Optik 39, 258–267(1974).

1968 (1)

S. Lowenthal and A. Werts, “Filtrage des fréquences spatiales en lumière incohérente à l’aide d’hologrammes,” C. R. Acad. Sci. (Paris) 2 66B, 542–545 (1968).

1967 (1)

1965 (1)

Armitage, J. D.

Asakura, T.

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. d’Optique 6, 5–14(1975).
[Crossref]

H. Fujii and T. Asakura, “Contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–37(1974).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Chap. X.

Chavel, P.

P. Chavel and S. Lowenthal, “Noise and coherence in optical (and digital) image processing. I. The Callier effect and its influence on image contrast,” J. Opt. Soc. Am. 68, 559–568(1978).
[Crossref]

P. Chavel and S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,” J. Opt. Soc. Am. 66, 14–23(1976).
[Crossref]

S. Lowenthal and P. Chavel, “Noise problems in optical image processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics, Jerusalem, 1976), paper TuA-1.

Fujii, H.

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. d’Optique 6, 5–14(1975).
[Crossref]

H. Fujii and T. Asakura, “Contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–37(1974).
[Crossref]

Goodman, J. W.

J. W. Goodman, “Film-grain noise in wavefront-reconstruction imaging,” J. Opt. Soc. Am. 57, 493–502(1967).
[Crossref] [PubMed]

D. Tichenor and J. W. Goodman, “Practical Noise Limitations in Holographic Image Deblurring,” in Proceedings of the International Optical Computing Conference (IEEE, Washington, D.C., 1975, IEEE Catalog No. 75, CH 0941-5C), pp. 82–84.

J. W. Goodman, “Noise in Coherent Optical Information Processing,” in Optical Information Processing, edited by Yu. E. Nesterikhin, G. W. Stroke, and W. E. Kock (Plenum, New York, 1976), pp. 85–103.
[Crossref]

Lanzl, F.

B. Reuter and F. Lanzl, “Ortsfrequenzbeschneidende Filterung bei teilkohärente Beleuchtung,” D.G.a.O. Meeting, Berlin, 1977.

Lohmann, A. W.

J. D. Armitage and A. W. Lohmann, “Optical character recognition by incoherent matched filtering,” Appl. Opt. 4, 461–467(1965).
[Crossref]

A. W. Lohmann and W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. (to be published).

Lowenthal, S.

P. Chavel and S. Lowenthal, “Noise and coherence in optical (and digital) image processing. I. The Callier effect and its influence on image contrast,” J. Opt. Soc. Am. 68, 559–568(1978).
[Crossref]

P. Chavel and S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,” J. Opt. Soc. Am. 66, 14–23(1976).
[Crossref]

S. Lowenthal and A. Werts, “Filtrage des fréquences spatiales en lumière incohérente à l’aide d’hologrammes,” C. R. Acad. Sci. (Paris) 2 66B, 542–545 (1968).

S. Lowenthal and P. Chavel, “Noise problems in optical image processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics, Jerusalem, 1976), paper TuA-1.

McKechnie, T. S.

T. S. McKechnie, “Measurement of some second-order statistical properties of speckle,” Optik 39, 258–267(1974).

Parry, G.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 78–122.
[Crossref]

Reuter, B.

B. Reuter and F. Lanzl, “Ortsfrequenzbeschneidende Filterung bei teilkohärente Beleuchtung,” D.G.a.O. Meeting, Berlin, 1977.

Rhodes, W. T.

A. W. Lohmann and W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. (to be published).

Rogers, G. L.

G. L. Rogers, “Non-coherent optical processing,” Opt. Las. Technol. 7, 153–162(1975).
[Crossref]

Tichenor, D.

D. Tichenor and J. W. Goodman, “Practical Noise Limitations in Holographic Image Deblurring,” in Proceedings of the International Optical Computing Conference (IEEE, Washington, D.C., 1975, IEEE Catalog No. 75, CH 0941-5C), pp. 82–84.

Werts, A.

S. Lowenthal and A. Werts, “Filtrage des fréquences spatiales en lumière incohérente à l’aide d’hologrammes,” C. R. Acad. Sci. (Paris) 2 66B, 542–545 (1968).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Chap. X.

Appl. Opt. (1)

C. R. Acad. Sci. (Paris) (1)

S. Lowenthal and A. Werts, “Filtrage des fréquences spatiales en lumière incohérente à l’aide d’hologrammes,” C. R. Acad. Sci. (Paris) 2 66B, 542–545 (1968).

J. Opt. Soc. Am. (3)

Nouv. Rev. d’Optique (1)

H. Fujii and T. Asakura, “Statistical properties of image speckle patterns in partially coherent light,” Nouv. Rev. d’Optique 6, 5–14(1975).
[Crossref]

Opt. Commun. (1)

H. Fujii and T. Asakura, “Contrast variation of image speckle intensity under illumination of partially coherent light,” Opt. Commun. 12, 32–37(1974).
[Crossref]

Opt. Las. Technol. (1)

G. L. Rogers, “Non-coherent optical processing,” Opt. Las. Technol. 7, 153–162(1975).
[Crossref]

Optik (1)

T. S. McKechnie, “Measurement of some second-order statistical properties of speckle,” Optik 39, 258–267(1974).

Other (7)

B. Reuter and F. Lanzl, “Ortsfrequenzbeschneidende Filterung bei teilkohärente Beleuchtung,” D.G.a.O. Meeting, Berlin, 1977.

S. Lowenthal and P. Chavel, “Noise problems in optical image processing,” in Proceedings of the International Conference on Applications of Holography and Optical Data Processing (International Commission of Optics, Jerusalem, 1976), paper TuA-1.

D. Tichenor and J. W. Goodman, “Practical Noise Limitations in Holographic Image Deblurring,” in Proceedings of the International Optical Computing Conference (IEEE, Washington, D.C., 1975, IEEE Catalog No. 75, CH 0941-5C), pp. 82–84.

J. W. Goodman, “Noise in Coherent Optical Information Processing,” in Optical Information Processing, edited by Yu. E. Nesterikhin, G. W. Stroke, and W. E. Kock (Plenum, New York, 1976), pp. 85–103.
[Crossref]

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, edited by J. C. Dainty (Springer-Verlag, Berlin, 1975), pp. 78–122.
[Crossref]

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Chap. X.

A. W. Lohmann and W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. (to be published).

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Figures (7)

FIG. 1
FIG. 1

Schematic diagram of the setup. The object plane Π0 is imaged on the image plane Π by the lens L located in the pupil plane P. The condenser C placed against the object images the plane ∑ of the source of luminance s ˜(Ω) on the pupil plane.

FIG. 2
FIG. 2

Pupil noise.

FIG. 3
FIG. 3

Behavior of the SNR corresponding to pupil noise, as a function of the area SS of the source. SP = pupil area; G = ratio of partially coherent SNR to coherent SNR; N = number of degrees of freedom in the image. SS/SP is equal to the ratio AR/AC between resolution area and coherence area. The coherent case corresponds to SS = 0.

FIG. 4
FIG. 4

Effective number of degrees of freedom that contribute to the noise reduction between the partially coherent and the coherent cases is given by S0 Smin2d2, where Smin is the smaller of the pupil area and the source area.

FIG. 5
FIG. 5

Input grain noise.

FIG. 6
FIG. 6

Two kinds of input impulse noises: object transmittance in the presence of a dust particle P and of the negative photographic image of another dust particle P′, both superimposed on a constant background of amplitude transmittance τc.

FIG. 7
FIG. 7

Input impulse noise: effect of an opaque dust particle P(left) and of the negative image of a dust particle P′ (right), both of size 1/Nd resolution cell, on the illuminance in the image plane of a constant background of intensity τ c 2. (a) τc = 0.3, Nd = 1.1; (b) τc = 0.9, Nd = 1.1; (c) τc = 0.3, Nd = 3; (d) τc = 0.9, Nd = 3.

Tables (1)

Tables Icon

TABLE I Evolution of the signal terms (I0, IN, and Im) and of the noise terms (P1, P2, Q1, and Q2) as a function of the coherence; Ac = coherence area; AR = resolution area; AN = correlation area of the input noise. The crosses X indicate a rigorous limit of validity between the expression to the left and the expression to the right; ellipses between columns mean that no simple expression can be given.

Equations (107)

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I ( r ) = s ˜ ( Ω ) i ( r , Ω ) d Ω
i ( r , Ω ) = τ ( r 1 ) p ( r - r 1 ) exp 2 i π Ω · r 1 d r 1 2 = a ( r , Ω ) 2 ,
a ( r , Ω ) = a ( r , Ω ) + g ( r , Ω ) ,
I ( r ) = I 0 ( r ) + I N ( r ) ,
σ I 2 ( r ) = s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) [ A ( r , Ω 1 , Ω 2 ) + B ( r , Ω 1 , Ω 2 ) ] d Ω 1 d Ω 2
A ( r , Ω 1 , Ω 2 ) = g ( r , Ω 1 ) g * ( r , Ω 2 ) 2 + 2 Re [ a ( r , Ω 1 ) a * ( r , Ω 2 ) × g * ( r , Ω 1 ) g ( r , Ω 2 ) ]
B ( r , Ω 1 , Ω 2 ) = g ( r , Ω 1 ) g ( r , Ω 2 ) 2 + 2 Re [ a ( r , Ω 1 ) a ( r , Ω 2 ) × g * ( r , Ω 1 ) g * ( r , Ω 2 ) ] ,
τ ( r ) = [ τ c + m ( r ) ] F ( r ) ,             max m τ c
p ˜ ( Ω ) = p ˜ ( Ω ) + ñ ( Ω ) ,
p ˜ ( Ω ) = K P ˜ ( Ω ) ,             K < 1
ñ ( Ω ) = P ˜ ( Ω ) Ñ ( Ω ) ,
a ( r , Ω ) = K τ ( r ) P ˜ ( Ω ) exp ( 2 π i Ω · r )
I ( r ) = K 2 τ 2 ( r ) s ˜ ( Ω ) P ˜ ( Ω ) d Ω ,
σ I 2 ( r ) = 2 τ 2 ( r ) Re s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) K 2 P ˜ ( Ω 1 ) P ˜ ( Ω 2 ) × { g * ( r , Ω 1 ) g ( r , Ω 2 ) × exp [ - 2 i π r · ( Ω 2 - Ω 1 ) ] + g ( r , Ω 1 ) g ( r , Ω 2 ) × exp [ - 2 i π r · ( Ω 1 + Ω 2 ) ] } d Ω 1 d Ω 2 ,
g ( r , Ω ) = τ ( r 1 ) n ( r - r 1 ) exp ( 2 i π Ω · r 1 ) d r 1 .
n ( r 1 ) n * ( r 2 ) = ϕ ( r 2 ) P ( r 2 - r 1 )
n ( r 1 ) n ( r 2 ) = ψ ( - r 2 ) P ( r 1 + r 2 ) ,
ϕ ˜ ( Ω ) = Ñ ( Ω 1 ) Ñ * ( Ω 1 + Ω )
ψ ˜ ( Ω ) = Ñ ( Ω 1 ) Ñ ( Ω 1 + Ω ) .
g ( r , Ω 1 ) g * ( r , Ω 2 ) = ϕ P ˜ ( Ω 2 ) τ c 2 F ˜ ( Ω 1 - Ω 2 )
g ( r , Ω 1 ) g ( r , Ω 2 ) = ψ P ˜ ( Ω 2 ) τ c 2 F ˜ ( Ω 2 - Ω 1 , 2 r ) exp ( 4 i π Ω 2 · r ) ,
F ( r 1 , r ) = F ( r 1 ) F ( 2 r - r 1 ) .
I ( r ) σ I ( r ) = τ ( r ) K s ˜ ( Ω ) P ( Ω ) d Ω ( A 1 + A 2 ) 1 / 2
A 1 = 2 ϕ τ c 2 s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) P ˜ ( Ω 1 ) P ˜ ( Ω 2 ) F ˜ ( Ω 1 - Ω 2 ) × exp [ - 2 i π r · ( Ω 2 - Ω 1 ) ] d Ω 1 d Ω 2
A 2 = 2 Re { ψ τ c 2 s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) P ˜ ( Ω 1 ) P ˜ ( Ω 2 ) F ˜ ( Ω 1 - Ω 2 , r ) × exp [ - 2 i π r · ( Ω 2 - Ω 1 ) ] d Ω 1 d Ω 2 } .
I / σ I = K / { 2 ϕ S 0 + 2 Re ψ S 0 } 1 / 2 ,
I σ I = K [ 2 ( ϕ + Re ψ ) ] 1 / 2 ( S min λ 2 d 2 ) 1 / 2 ,
G = C ( S 0 S min / λ 2 d 2 ) 1 / 2 ,
C = ( 1 - Re ψ ( 1 - S 0 / S 0 ) ϕ + 2 Re ψ ) 1 / 2 .
G = ( S 0 / A c ) 1 / 2 if A c > A R G = ( S 0 / A R ) 1 / 2 if A c < A R
N = S 0 S P / λ 2 d 2 ,
N = S 0 S s / λ 2 d 2 .
G { = N , N < N = N , N > N
τ ( r ) = τ ( r ) + X ( r ) ,
τ ( r ) = τ c + m ( r ) + X ( r ) ,
I 0 ( r ) = τ c + m ( r ) 2 s ˜ ( Ω ) p ˜ ( Ω ) d Ω
R = I / σ I ,
R m = I m / σ I ,
I τ c 2 + I m + I N ,
I m 2 τ c m ( r ) s ˜ ( Ω ) p ˜ ( Ω ) d Ω
g ( r , Ω ) = X ( r 1 ) p ( r - r 1 ) exp ( 2 i π Ω · r 1 ) d r 1 .
ϕ X ( r ) = X ( r 1 ) X * ( r 1 - r ) , ψ X ( r ) = X ( r 1 ) X ( r 1 - r ) .
g ( r , Ω 1 ) g * ( r , Ω 2 ) = ϕ ˜ X ( - Ω 1 ) ϕ P ˜ ( Ω 1 - Ω 2 ) × exp [ 2 i π r · ( Ω 1 - Ω 2 ) ] , g ( r , Ω 1 ) g ( r , Ω 2 ) = ψ ˜ X ( - Ω 1 ) ϕ P ˜ ( Ω 1 + Ω 2 ) × exp [ 2 i π r · ( Ω 1 + Ω 2 ) ] ,
I N ( r ) = ( 1 / A R ) s ˜ ( Ω ) ϕ ˜ X ( - Ω ) d Ω ,
σ I ( r ) = P 1 + P 2 + 2 Q 1 + 2 Re Q 2
P 1 = s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) ϕ ˜ x ( - Ω 1 ) 2 ϕ P ˜ 2 ( Ω 1 - Ω 2 ) d Ω 1 d Ω 2 , P 2 = s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) ψ ˜ x ( - Ω 1 ) 2 ϕ P ˜ 2 ( Ω 1 + Ω 2 ) d Ω 1 d Ω 2 , Q 1 = s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) τ c 2 P ˜ ( Ω 1 ) P ˜ ( Ω 2 ) × ϕ ˜ x ( Ω 1 ) ϕ P ˜ ( Ω 1 - Ω 2 ) d Ω 1 d Ω 2 , Q 2 = s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) τ c 2 P ˜ ( Ω 1 ) P ˜ ( Ω 2 ) × ψ ˜ x ( Ω 1 ) ϕ P ˜ ( Ω 1 + Ω 2 ) d Ω 1 d Ω 2 .
ϕ ˜ x ( 0 ) = ϕ x ( r ) d r = A N σ x 2 .
A c = λ 2 d 2 / S s
A R = λ 2 d 2 / S P
N 0 = A R A N = λ 2 d 2 S P A N = resolution area noise correlation area
ϕ ˜ P ( Ω ) 2 d Ω = K 0 / A R 3 ,
ϕ ˜ x ( Ω ) 2 d Ω = ϕ x ( r ) 2 d r = K 1 σ x 4 A N
ψ ˜ x ( 0 ) = ψ x ( r ) d r = k σ x 2 A N = ( k r + i k i ) σ x 2 A N
ψ ˜ x ( Ω ) 2 d Ω = ψ x ( r ) 2 d r = K 2 σ x 4 A N .
σ I ( r ) = P 1 + P 2 + 2 Q 1 + 2 Re Q 2 .
G = [ I / σ I ] incoherent [ I / σ I ] coherent ,             G m = [ I m / σ I ] incoherent [ I m / σ I ] coherent .
G = 1 + α 2 1 + α 2 / N 0 G m             and             G m [ 1 + α 2 / 2 N 0 1 + α 2 / 2 ] 1 / 2 ,
τ ( r ) = τ R exp [ i ϕ ( r ) ] .
X ( r ) = τ R exp [ i ϕ ( r ) ] - τ ( r ) .
I ( r ) = τ R exp [ i ϕ ( r 1 ) ] p ( r - r 1 ) exp ( 2 i π Ω · r 1 ) d r 1 2 ,
I ( r ) = τ R 2 exp [ i ϕ ( r 1 ) ] 2 p ( r - r 1 ) 2 d r 1 = τ R 2 Σ P ,
τ ( r ) = τ ( r ) + X ( r ) ,
τ ( r ) 2 = τ ( r ) + X ( r ) 2 .
τ ( r ) = τ c + X ( r ) ,
I ( r ) = s ˜ ( Ω ) τ 0 p ˜ ( Ω ) exp ( 2 i π Ω · r ) + p ( r ) X ˜ ( - Ω ) 2 d Ω .
I c ( r ) = ( S P / λ 2 d 2 ) [ τ 0 + p ( r ) X ˜ ( 0 ) ]
I i ( r ) = { τ 0 S P / λ 2 d 2 + 2 p 2 ( r ) τ 0 X ˜ ( 0 ) + p 2 ( r ) X ˜ 2 ( Ω ) d Ω } ,
p ( r ) = p ( r ) / p ( 0 ) = [ p ( r ) / S P ] λ 2 d 2
N d = resolution area dust particle area = λ 2 d 2 S P = A R .
I c ( r ) = S S λ 2 d 2 [ τ 0 + p ( r ) X ( 0 ) N d ] 2 ,
I i ( r ) = S P λ 2 d 2 [ τ 0 2 + p 2 ( r ) ( 2 τ 0 X ( 0 ) N d + X ( 0 ) 2 N d ) ] .
C = peak intensity variation due to the noise intensity due to the signal alone .
C c = ( 2 τ 0 X ( 0 ) N d + X ( 0 ) 2 N d 2 ) / τ 0 2 ,
C i = ( 2 τ 0 X ( 0 ) N d + X ( 0 ) 2 N d ) / τ 0 2 ,
C i C c = ( 1 + X ( 0 ) / 2 τ 0 1 + X ( 0 ) / 2 N d τ 0 ) .
C i / C c = N d / ( 2 N d - 1 ) .
C i C c = 1 + ( 1 - τ 0 ) / 2 τ 0 1 + ( 1 - τ 0 ) / 2 τ 0 N d
C = I max - I min I max + I min = 2 ρ 1 / 2 1 + ρ ,
σ I 2 ( r ) = S 2 + S 3 + S 4 = s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) [ T 2 ( r , Ω 1 , Ω 2 ) + T 3 ( r , Ω 1 , Ω 2 ) + T 4 ( r , Ω 1 , Ω 2 ) ] d r d Ω 1 d Ω 2 .
T 2 ( r , Ω 1 , Ω 2 ) = 2 Re [ a ( r , Ω 1 ) a * ( r , Ω 2 ) g * ( r , Ω 1 ) g ( r , Ω 2 ) + a ( r , Ω 1 ) a ( r , Ω 2 ) g * ( r , Ω 1 ) g * ( r , Ω 2 ) ] ,
T 3 ( r , Ω 1 , Ω 2 ) = 4 Re [ a ( r , Ω 1 ) g * ( r , Ω 1 ) g ( r , Ω 2 ) 2 ]
T 4 ( r , Ω 1 , Ω 2 ) = { g ( r , Ω 1 ) 2 - g ( r , Ω 1 ) 2 } × { g ( r , Ω 2 ) 2 - g ( r , Ω 2 ) 2 } .
g ( r , Ω ) = τ ( r 1 ) n ( r - r 1 ) exp ( 2 i π Ω · r 1 ) d r 1 ,
a ( r , Ω ) = K τ ( r ) P ( Ω ) exp ( 2 i π Ω · r ) .
g ( r , Ω ) = X ( r 1 ) p ( r - r 1 ) exp ( 2 i π Ω · r 1 ) d r 1 .
T 3 T 2 , T 4 - g ( r , Ω 1 ) g * ( r , Ω 2 ) 2 T 2
T 2 ( r , 0 , 0 ) = 2 Re [ τ 0 ( r ) 2 X * ( r 1 ) X ( r 2 ) × p ( r - r 1 ) p ( r - r 2 ) d r 1 d r 2 + τ 0 2 ( r ) X * ( r 1 ) X * ( r 2 ) × p ( r - r 1 ) p ( r - r 2 ) d r 1 d r 2 ]
T 3 ( r , 0 , 0 ) = 4 Re [ τ 0 ( r ) X * ( r 1 ) X ( r 2 ) X * ( r 3 ) × p ( r - r 1 ) p ( r - r 2 ) × p ( r - r 3 ) d r 1 d r 2 d r 3 ] .
Γ X ( r 2 - r 1 , r 3 - r 1 ) = X * ( r 1 ) X ( r 2 ) X * ( r 3 )
T 2 = τ 0 2 ( r ) ϕ 0 A N p 2 ( r ) d r = τ 0 2 ( r ) ϕ 0 A N S P / λ 2 d 2 ,
T 3 = τ 0 ( r ) ϕ 0 A N 2 p 3 ( r ) d r τ 0 ( r ) ϕ 0 A N 2 S p 2 / λ 4 d 4 .
S 2 = s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) T 2 ( r , Ω 1 , Ω 2 ) d Ω 1 d Ω 2 = 2 s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) p ˜ ( Ω 1 ) p ˜ ( Ω 2 ) p ( r - r 1 ) p ( r - r 2 ) × Re [ τ 0 ( r ) 2 ϕ X ( r 2 - r 1 ) exp 2 i π { Ω 1 · ( r - r 1 ) - Ω 2 · ( r - r 2 ) } + τ 0 2 ( r ) ψ X * ( r 1 - r 2 ) exp 2 i π { Ω 1 · ( r - r 1 ) + Ω 2 · ( r - r 2 ) } ] d Ω 1 d Ω 2 d r 1 d r 2
S 3 = s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) T 3 ( r , Ω 2 , Ω 3 ) d Ω 1 d Ω 2 = 4 Re [ τ 0 ( r ) s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) p ˜ ( Ω 1 ) Γ X ( r 2 - r 1 , r 3 - r 1 ) p ( r - r 1 ) p ( r - r 2 ) p ( r - r 3 ) × exp 2 i π { Ω 1 · ( r - r 1 ) + Ω 2 · ( r 2 - r 3 ) } d Ω 1 d Ω 2 d r 1 d r 2 d r 3 ] .
S 2 = 2 p 2 ( r - r 1 ) p 2 ( r - r 2 ) × Re [ τ 0 ( r ) 2 ϕ X ( r 2 - r 1 ) + τ 0 2 ( r ) ψ X * ( r 1 - r 2 ) ] d r 1 d r 2 ,
S 3 = 4 Re s ( r 2 - r 3 ) p 2 ( r - r 1 ) p ( r - r 2 ) p ( r - r 3 ) × τ 0 ( r ) Γ X ( r 2 - r 1 , r 3 - r 1 ) d r 1 d r 2 d r 3 .
S 3 = 4 Re p 2 ( r - r 1 ) p 2 ( r - r 2 ) τ 0 ( r ) × Γ X ( r 2 - r 1 , r 2 - r 1 ) d r 1 d r 2 .
S 4 = s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) T 4 ( r , Ω 1 , Ω 2 ) d Ω 1 d Ω 2 = s ˜ ( Ω 1 ) s ˜ ( Ω 2 ) p ( r - r 1 ) p ( r - r 2 ) p ( r - r 1 ) p ( r - r 2 ) × { X ( r 1 ) X * ( r 1 ) X ( r 2 ) X * ( r 2 ) - X ( r 1 ) X * ( r 1 ) X ( r 2 ) X * ( r 2 ) } × exp 2 i π [ Ω 1 · ( r 1 - r 1 ) + Ω 2 · ( r 2 - r 2 ) ] d Ω 1 d Ω 2 d r 1 d r 2 d r 1 d r 2 = p 2 ( r - r 1 ) p 2 ( r - r 2 ) Δ X ( r 1 - r 2 ) d r 1 d r 2 ,
Δ X ( r 1 - r 2 ) = X ( r 1 ) 2 X ( r 2 ) 2 - X ( r 1 ) 2 X ( r 2 ) 2 .
H 1 ( r 2 - r 1 ) = exp i [ ϕ ( r 2 ) - ϕ ( r 1 ) ] ,
H 2 ( r 2 - r 1 ) = exp i [ ϕ ( r 2 ) + ϕ ( r 1 ) ] ,
ϕ X ( r 2 - r 1 ) = X * ( r 1 ) X ( r 2 ) = τ R 2 H 1 ( r - r 1 ) - τ 0 2 ,
ψ X * ( r 1 - r 2 ) = X * ( r 1 ) X * ( r 2 ) = τ R 2 H 2 * ( r 2 - r 1 ) - τ 0 * 2 ,
Γ X ( r 2 - r 1 , r 2 - r 1 ) = X * ( r 1 ) X ( r 2 ) 2 = 2 τ 0 2 τ 0 * - τ R 2 τ 0 * H 1 ( r 2 - r 1 ) - τ R 0 τ 0 H 2 * ( r 2 - r 1 ) ,
Δ X ( r 1 - r 2 ) = 2 Re [ τ R 2 τ 0 2 H 1 ( r 2 - r 1 ) + τ R τ 0 2 H 2 * ( r 2 - r 1 ) ] - 4 τ 0 4 .
σ I 2 = S 2 + S 3 + S 4 = p 2 ( r - r 1 ) p 2 ( r - r 2 ) S ( r 1 , r 2 ) d r 1 d r 2 ,
S ( r 1 , r 2 ) = 2 Re { τ 0 2 ϕ 1 ( r 2 - r 1 ) + τ 0 2 ψ X * ( r 1 - r 2 ) } + 4 Re { τ 0 Γ X ( r 2 - r 1 , r 2 - r 1 ) } + Δ X ( r 1 - r 2 ) .
S 3 = - 2 S 2 = - 2 S 4 .