Abstract

Previous articles have been devoted to the study of optical noise as a function of spatial coherence. The present one completes this study by considering temporal coherence. Noise arising from defects in the pupil plane and affecting the high spatial frequencies of an image is notably reduced by white-light illumination. Temporal coherence has little effect on noise arising from defects in the object plane. However, impulse noise due to small isolated defects is reduced in size. Physical arguments are presented to explain these phenomena and a mathematical study of partially coherent imaging in the presence of random defects is given.

© 1980 Optical Society of America

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References

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  1. P. Chavel and S. Lowenthal, “Noise and coherence in optical image processing. I. The Callier effect and its influence on image contrast,” J. Opt. Soc. Am. 68, 559–568 (1978).
    [CrossRef]
  2. P. Chavel and S. Lowenthal, “Noise and coherence in optical image processing. II. Noise fluctuations,” J. Opt. Soc. Am. 68, 721–732 (1978).
    [CrossRef]
  3. G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).
  4. C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de reponsé impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimental et applications,” Nouv. Rev. Opt. Appl. 4, 183–196 (1973).
    [CrossRef]
  5. M. A. Monahan, K. Bromley, and R. Bocker, “Incoherent correlators,” Proc. IEEE 65, 121–129 (1977).
    [CrossRef]
  6. H. H. Barrett and W. Swindell, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
    [CrossRef]
  7. G. Indebetouw, “Production of color coded equidensities using nonlinear filtering,” Appl. Opt. 16, 1951–1954 (1977).
    [CrossRef] [PubMed]
  8. E. N. Leith and J. Roth, “White light optical processing and holography,” Appl. Opt. 16, 2565–2567 (1977).
    [CrossRef] [PubMed]
  9. F. T. S. Yu, “Restoration of smeared photographic image by incoherent optical processing,” Appl. Opt. 17, 3571–3575 (1978).
    [CrossRef] [PubMed]
  10. J. Bescos and T. C. Strand, “Optical pseudocolor encoding of spatial frequency information,” Appl. Opt. 172524–2531 (1978).
    [PubMed]
  11. T. Sato, K. Sasaki, and P. Yamamoto, “Image processing system using incoherent image feedbacks,” Appl. Opt. 17, 717–720 (1978).
    [CrossRef] [PubMed]
  12. J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 717–720 (1978).
  13. F. Dickey and D. Moore, “White light optical processor for edge enhancement and spectral filtering,” Appl. Opt. 18, 1679–1684 (1979).
    [CrossRef] [PubMed]
  14. H. O. Bartelt, “Image correlation in white light by wavelength multiplexing,” Opt. Commun. 29, 1679–1684 (1979).
    [CrossRef]
  15. D. Görlitz and F. Lanzl, “Colour encoded aperture masks used for incoherent filtering of images,” Opt. Commun. 28, 283–286 (1979).
    [CrossRef]
  16. P. Wiersma, “A three colour channel system to synthesize complex point spread functions,” Opt. Commun. 28, 280–281 (1979).
    [CrossRef]
  17. D. Psaltis, D. Casasent, and M. Carlotto, “Iterative color-multiplexed, electro-optical processor,” Opt. Lett. 4, 348–350 (1979).
    [CrossRef] [PubMed]
  18. B. J. Bartholomew and S. H. Lee, “Nonlinear optical processing with Fabry–Perot interferometers containing phase recording media,” Appl. Opt. 19, 201–206 (1980).
    [CrossRef] [PubMed]
  19. E. N. Leith, “Image deblurring using diffraction gratings,” Opt. Lett. 5, 70–72 (1980).
    [CrossRef] [PubMed]
  20. W. T. Rhodes and A. A. Sawchuk, “Incoherent Optical Processing,” in Optical Information Processing, edited by S. H. Lee (Springer-Verlag, Berlin, 1980), Chap. III.
  21. G. L. Rogers, Proc. Tech. Prog. Electro-Opt. Syst. Design Conf., New York, 1970 (unpublished).
  22. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Chap. X.
  23. P. Chavel, Thesis Institut d’Optique, University of Pari XI, Orsay France, 1979 (unpublished).
  24. M. O. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 231.
  25. P. Chavel and S. Lowenthal, “Film grain noise in partially coherent imaging,” Opt. Eng. 19, 404–410 (1980).
    [CrossRef]
  26. D. Tichenor and J. W. Goodman, “Practical noise limitations in holographic image deblurring,” Proceedings of the International Optical Computing Conference (IEEE, Washington, D.C., 1975, IEEE Catalog, No. 75, CH 0941-5c), pp. 82–84.

1980 (3)

1979 (5)

F. Dickey and D. Moore, “White light optical processor for edge enhancement and spectral filtering,” Appl. Opt. 18, 1679–1684 (1979).
[CrossRef] [PubMed]

H. O. Bartelt, “Image correlation in white light by wavelength multiplexing,” Opt. Commun. 29, 1679–1684 (1979).
[CrossRef]

D. Görlitz and F. Lanzl, “Colour encoded aperture masks used for incoherent filtering of images,” Opt. Commun. 28, 283–286 (1979).
[CrossRef]

P. Wiersma, “A three colour channel system to synthesize complex point spread functions,” Opt. Commun. 28, 280–281 (1979).
[CrossRef]

D. Psaltis, D. Casasent, and M. Carlotto, “Iterative color-multiplexed, electro-optical processor,” Opt. Lett. 4, 348–350 (1979).
[CrossRef] [PubMed]

1978 (6)

1977 (4)

M. A. Monahan, K. Bromley, and R. Bocker, “Incoherent correlators,” Proc. IEEE 65, 121–129 (1977).
[CrossRef]

H. H. Barrett and W. Swindell, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

G. Indebetouw, “Production of color coded equidensities using nonlinear filtering,” Appl. Opt. 16, 1951–1954 (1977).
[CrossRef] [PubMed]

E. N. Leith and J. Roth, “White light optical processing and holography,” Appl. Opt. 16, 2565–2567 (1977).
[CrossRef] [PubMed]

1973 (1)

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de reponsé impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimental et applications,” Nouv. Rev. Opt. Appl. 4, 183–196 (1973).
[CrossRef]

Abramowitz, M. O.

M. O. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 231.

Barrett, H. H.

H. H. Barrett and W. Swindell, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, “Image correlation in white light by wavelength multiplexing,” Opt. Commun. 29, 1679–1684 (1979).
[CrossRef]

Bartholomew, B. J.

Bescos, J.

Bocker, R.

M. A. Monahan, K. Bromley, and R. Bocker, “Incoherent correlators,” Proc. IEEE 65, 121–129 (1977).
[CrossRef]

Born, M.

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

Bromley, K.

M. A. Monahan, K. Bromley, and R. Bocker, “Incoherent correlators,” Proc. IEEE 65, 121–129 (1977).
[CrossRef]

Carlotto, M.

Casasent, D.

Chavel, P.

Dickey, F.

Froehly, C.

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de reponsé impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimental et applications,” Nouv. Rev. Opt. Appl. 4, 183–196 (1973).
[CrossRef]

Goodman, J. W.

D. Tichenor and J. W. Goodman, “Practical noise limitations in holographic image deblurring,” Proceedings of the International Optical Computing Conference (IEEE, Washington, D.C., 1975, IEEE Catalog, No. 75, CH 0941-5c), pp. 82–84.

Görlitz, D.

D. Görlitz and F. Lanzl, “Colour encoded aperture masks used for incoherent filtering of images,” Opt. Commun. 28, 283–286 (1979).
[CrossRef]

Indebetouw, G.

Jahns, J.

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 717–720 (1978).

Lacourt, A.

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de reponsé impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimental et applications,” Nouv. Rev. Opt. Appl. 4, 183–196 (1973).
[CrossRef]

Lanzl, F.

D. Görlitz and F. Lanzl, “Colour encoded aperture masks used for incoherent filtering of images,” Opt. Commun. 28, 283–286 (1979).
[CrossRef]

Lee, S. H.

Leith, E. N.

Lohmann, A. W.

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 717–720 (1978).

Lowenthal, S.

Monahan, M. A.

M. A. Monahan, K. Bromley, and R. Bocker, “Incoherent correlators,” Proc. IEEE 65, 121–129 (1977).
[CrossRef]

Moore, D.

Psaltis, D.

Rhodes, W. T.

W. T. Rhodes and A. A. Sawchuk, “Incoherent Optical Processing,” in Optical Information Processing, edited by S. H. Lee (Springer-Verlag, Berlin, 1980), Chap. III.

Rogers, G. L.

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

G. L. Rogers, Proc. Tech. Prog. Electro-Opt. Syst. Design Conf., New York, 1970 (unpublished).

Roth, J.

Sasaki, K.

Sato, T.

Sawchuk, A. A.

W. T. Rhodes and A. A. Sawchuk, “Incoherent Optical Processing,” in Optical Information Processing, edited by S. H. Lee (Springer-Verlag, Berlin, 1980), Chap. III.

Stegun, I. A.

M. O. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 231.

Strand, T. C.

Swindell, W.

H. H. Barrett and W. Swindell, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

Tichenor, D.

D. Tichenor and J. W. Goodman, “Practical noise limitations in holographic image deblurring,” Proceedings of the International Optical Computing Conference (IEEE, Washington, D.C., 1975, IEEE Catalog, No. 75, CH 0941-5c), pp. 82–84.

Vienot, J. C.

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de reponsé impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimental et applications,” Nouv. Rev. Opt. Appl. 4, 183–196 (1973).
[CrossRef]

Wiersma, P.

P. Wiersma, “A three colour channel system to synthesize complex point spread functions,” Opt. Commun. 28, 280–281 (1979).
[CrossRef]

Wolf, E.

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

Yamamoto, P.

Yu, F. T. S.

Appl. Opt. (7)

J. Opt. Soc. Am. (2)

Nouv. Rev. Opt. Appl. (1)

C. Froehly, A. Lacourt, and J. C. Vienot, “Notions de reponsé impulsionnelle et de fonction de transfert temporelles des pupilles optiques, justifications expérimental et applications,” Nouv. Rev. Opt. Appl. 4, 183–196 (1973).
[CrossRef]

Opt. Commun. (4)

J. Jahns and A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 717–720 (1978).

H. O. Bartelt, “Image correlation in white light by wavelength multiplexing,” Opt. Commun. 29, 1679–1684 (1979).
[CrossRef]

D. Görlitz and F. Lanzl, “Colour encoded aperture masks used for incoherent filtering of images,” Opt. Commun. 28, 283–286 (1979).
[CrossRef]

P. Wiersma, “A three colour channel system to synthesize complex point spread functions,” Opt. Commun. 28, 280–281 (1979).
[CrossRef]

Opt. Eng. (1)

P. Chavel and S. Lowenthal, “Film grain noise in partially coherent imaging,” Opt. Eng. 19, 404–410 (1980).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (2)

M. A. Monahan, K. Bromley, and R. Bocker, “Incoherent correlators,” Proc. IEEE 65, 121–129 (1977).
[CrossRef]

H. H. Barrett and W. Swindell, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

Other (7)

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

W. T. Rhodes and A. A. Sawchuk, “Incoherent Optical Processing,” in Optical Information Processing, edited by S. H. Lee (Springer-Verlag, Berlin, 1980), Chap. III.

G. L. Rogers, Proc. Tech. Prog. Electro-Opt. Syst. Design Conf., New York, 1970 (unpublished).

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

P. Chavel, Thesis Institut d’Optique, University of Pari XI, Orsay France, 1979 (unpublished).

M. O. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 231.

D. Tichenor and J. W. Goodman, “Practical noise limitations in holographic image deblurring,” Proceedings of the International Optical Computing Conference (IEEE, Washington, D.C., 1975, IEEE Catalog, No. 75, CH 0941-5c), pp. 82–84.

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Figures (5)

FIG. 1
FIG. 1

System considered. S is a pointlike source emitting a spectrum B(λ). The noise in the image is mainly due to defects in the pupil and to defects in the object.

FIG. 2
FIG. 2

Pupil noise. A given spatial frequency corresponds to one spot in the pupil plane for each wavelength. (a) Monochromatic case; (b) two wavelengths.

FIG. 3
FIG. 3

Reduction of pupil noise by the use of temporally partially coherent illumination. Plots of the SNR improvement when compared with the coherent case, in the case of a uniform spectrum between λ1, and λ1(1 + ) and of an object of side a containing only spatial frequency μ; (a) SNR improvement versus the number of periods in the object μa for several values of ; (b) SNR improvement versus the relative spectral width for several values of μa.

FIG. 4
FIG. 4

Illustration of amplitude impulse noise due to a defect in the object under violet, red, and white illumination. (a) Dark dust particle; (b) small hole in a constant background.

FIG. 5
FIG. 5

Illustration of phase impulse noise due to a defect in the object under violet, red, and white illumination. (a) Path delay δ = 0.4 μm, invisible in violet light, maximal in red light; (b) δ = 5.1 μm.

Equations (51)

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N = ( λ 2 λ 1 ) d μ λ d / a + 1 = Δ λ λ μ a + 1 = Δ λ δ λ + 1
I ( r ) = 0 B ( λ ) | b ( r , λ ) | 2 d λ ,
b ( r , λ ) = e i ψ r λ 2 d 2 p τ ( ρ λ d , λ ) p ( ρ ) exp ( 2 i π ρ · r λ d ) d ρ ,
τ ( r , λ ) = τ ( r , λ ) + X ( r , λ ) = τ 0 ( r , λ ) + X ( r , λ )
p ( ρ ) = p ( ρ ) + P ( ρ ) N ( ρ ) = P ( ρ ) [ K + N ( ρ ) ] .
I ( r ) = i = 1 4 I i ( r ) ,
I 1 ( r ) = B ( λ ) λ 4 d 4 τ 0 ( ρ 1 λ d , λ ) τ 0 * ( ρ 2 λ d , λ ) × K 2 P ( ρ 1 ) P ( ρ 2 ) ϕ d ρ 1 d ρ 2 d λ ,
I 2 ( r ) = B ( λ ) λ 4 d 4 τ 0 ( ρ 1 λ d , λ ) τ 0 * ( ρ 2 λ d , λ ) K P ( ρ 1 ) P ( ρ 2 ) × [ N ( ρ 1 ) + N * ( ρ 2 ) ] ϕ d ρ 1 d ρ 2 d λ ,
I 3 ( r ) = B ( λ ) λ 4 d 4 [ X ( ρ 1 λ d , λ ) τ 0 * ( ρ 2 λ d , λ ) + τ 0 * ( ρ 1 λ d , λ ) X * ( ρ 2 λ d , λ ) ] K 2 P ( ρ 1 ) P ( ρ 2 ) ϕ d ρ 1 d ρ 2 d λ ,
I 4 ( r ) = B ( λ ) λ 4 d 4 X ( ρ 1 λ d , λ ) X * ( ρ 2 λ d , λ ) × K 2 P ( ρ 1 ) P ( ρ 2 ) ϕ d ρ 1 d ρ 2 d λ ,
ϕ = exp ( 2 i π ( ρ 1 ρ 2 ) · r λ d ) .
τ 0 ( ρ λ d , λ ) P ( ρ ) = τ 0 ( ρ λ d , λ ) ,
τ 0 ( ρ λ d , λ ) P ( ρ ) exp ( 2 i π ρ · r λ d ) d ρ = λ 2 d 2 τ 0 ( r ) ,
I 1 ( r ) = K 2 | τ 0 ( r ) | 2 B ( λ ) d λ .
σ I 2 ( r ) = [ I ( r ) I ( r ) ] 2 = I 2 ( r ) 2 .
ρ = I / σ I
σ I 2 ( r ) = 2 K 2 { B ( λ ) B ( λ ) λ 2 d 2 λ 2 d 2 × | τ 0 * ( r ) | 2 τ 0 ( ρ 1 λ d , λ ) τ 0 * ( ρ 2 λ d , λ ) × ϕ N ( ρ 1 , ρ 2 ) exp [ 2 i π r · ( ρ 1 λ d ρ 1 λ d ) ] d ρ 1 d ρ 2 d λ d λ + Re B ( λ ) B ( λ ) λ 2 d 2 λ 2 d 2 τ 0 * ( r ) 2 τ 0 ( ρ 1 λ d , λ ) τ 0 ( ρ 1 λ d , λ ) × ψ N ( ρ 1 , ρ 1 ) exp [ 2 i π r · ( ρ 1 λ d ρ 1 λ d ) ] d ρ 1 d ρ 1 d λ d λ } .
ϕ N ( ρ 1 , ρ 2 ) = N ( ρ 1 ) N * ( ρ 2 ) ,
ψ N ( ρ 1 , ρ 1 ) = N ( ρ 1 ) N ( ρ 2 ) .
σ I 2 ( r ) = 2 K 2 | τ 0 ( r ) | 2 ϕ N ( 0 ) B ( λ ) B ( λ ) λ 2 d 2 λ 2 d 2 τ 0 ( ρ 1 λ d , λ ) × τ 0 * ( ρ 1 λ d , λ ) exp [ 2 i π r · ρ 1 d ( 1 λ 1 λ ) ] d ρ 1 d λ d λ .
τ 0 ( r ) = rect ( x / a ) rect ( y / f ) exp ( 2 i π μ x ) .
σ I 2 ( 0 ) = 2 K 2 ϕ N ( 0 ) s N S 0 × [ 0 0 λ B ( λ ) λ 2 d 2 B ( λ ) sinc μ ( 1 λ λ ) a d λ d λ + 0 λ B ( λ ) B ( λ ) λ 2 d 2 sinc μ ( 1 λ λ ) a d λ d λ ] .
B ( λ ) = { B 0 if λ 1 < λ < λ 1 ( 1 + ) , 0 otherwise .
ρ c = K 2 ϕ N ( 0 ) λ 2 d 2 s N S 0 .
S 0 s N / λ 2 d 2 .
ρ ( ρ c , , μ a ) = ρ c ( λ 1 ) / I ( μ a , ) 1 / 2 ,
I ( n , ) = 0 1 π n ( 1 + ξ ) [ Si ( π n ξ 1 + ξ ) + Si ( π n ξ 1 + ) ] d ξ ,
ρ ( ρ c , , 0 ) = ρ c ( λ 1 ) 2 log ( 1 + ) 2 / ( 1 + ) ,
ρ ( ρ c , , μ a ) μ a large ρ c ( λ 1 ) log ( 1 + ) μ a .
ϕ x ( r 1 r 2 ) = X ( r 1 ) X * ( r 2 ) ,
I ( r ) = I 1 ( r ) + I 4 ( r )
I ( r ) = | τ 0 ( r ) | 2 B ( λ ) d λ + B ( λ ) ϕ x ( ρ 1 λ d ) P 2 ( ρ 1 ) d ρ 1 λ 2 d 2 d λ ,
I ( r ) = L ( | τ 0 ( r ) | 2 + ϕ x ( Ω ) P 2 ( λ d Ω ) d Ω ) ,
σ I 2 ( r ) = [ I 3 ( r ) + I 4 ( r ) I 4 ( r ) ] 2 .
σ I 2 ( r ) = B ( λ ) B ( λ ) [ 2 | τ 0 ( r ) | 2 T ( λ , λ ) + T 2 ( λ , λ ) ] d λ d λ
T ( λ , λ ) = ϕ x ( Ω ) P ( λ d Ω ) P ( λ d Ω ) d Ω .
σ I 2 ( r ) = L 2 [ 2 | τ 0 ( r ) | 2 T ( λ , λ ) + T 2 ( λ , λ ) ]
σ I 2 ( r ) = 0 B ( λ ) 0 λ B ( λ ) [ 2 | τ 0 ( r ) | 2 T ( λ , λ ) + T 2 ( λ , λ ) ] d λ d λ + 0 B ( λ ) λ B ( λ ) × [ 2 | τ 0 ( r ) | 2 T ( λ , λ ) + T 2 ( λ , λ ) ] d λ d λ ,
τ ( r ) = τ 0 + X ( r , λ ) ,
I ( r ) = B ( λ ) λ 4 d 4 | [ τ 0 δ ( ρ λ d ) + X ( ρ λ d , λ ) ] P ( ρ ) × exp ( 2 i π ρ · r λ d d ρ ) | 2 d λ .
I ( r ) = B ( λ ) | τ 0 + X ( 0 , λ ) λ 2 d 2 P ( r λ d ) | 2 d λ .
I ( x , 0 ) = τ 0 2 B ( λ ) d λ + 2 τ 0 X 0 s B ( λ ) a 2 λ 2 d 2 sinc x a λ d d λ + X 0 2 s 2 B ( λ ) λ 4 d 4 a 4 sinc 2 x a λ d d λ .
I ( x , 0 ) = B 0 λ 1 τ 0 2 + 2 B 0 τ 0 X 0 β π ξ ( Si π ξ λ 1 Si π ξ λ 1 ( 1 + ) ) + B 0 X 0 2 β 2 2 π 3 ξ 3 ( π ξ λ 1 π ξ λ 1 ( 1 + ) 1 2 sin 2 π ξ λ 1 + 1 2 sin 2 π ξ λ 1 ( 1 + ) ) ,
β = s a 2 / d 2
ξ = a x / d
X 0 = τ 0
X 0 = 1 τ 0 .
X ( r ) = τ 0 ( exp 2 i π δ ( r ) λ 1 ) .
X ( 0 , λ ) = τ 0 s ( exp 2 i π δ λ 1 ) .
I ( x , 0 ) = τ 0 2 B 0 { λ 1 + β π ξ [ Si π ( ξ + 2 δ ) λ 1 Si π ( ξ + 2 δ ) λ 1 ( 1 + ) 2 Si π ξ λ 1 + 2 Si π ξ λ 1 ( 1 + ) Si π ( 2 δ ξ ) λ 1 + Si π ( 2 δ ξ ) λ 1 ( 1 + ) ] + β 2 π 2 ξ 2 λ 1 1 + β 2 2 π 2 ξ 2 [ 2 sin 2 π ξ / λ 1 sin 2 π ξ / λ 1 ( 1 + ) 2 π ξ + 2 sin 2 π δ / λ 1 sin 2 π δ / λ 1 ( 1 + ) 2 π δ sin 2 π ( ξ + δ ) / λ 1 sin 2 π ( ξ + δ ) / λ 1 ( 1 + ) 2 π ( ξ + δ ) sin 2 π ( ξ δ ) / λ 1 sin 2 π ( ξ δ ) / λ 1 ( 1 + ) 2 π ( ξ δ ) ] } .
I ( x , 0 ) = B 0 τ 0 2 [ 1 2 β λ 2 sinc ξ λ ( 1 cos 2 π δ λ ) + 2 β 2 λ 4 sinc 2 ξ λ ( 1 cos 2 π δ λ ) ] .