Abstract

We study the diffraction effect of gratings under partially coherent illumination generated by a source slit. Various fringe distributions from Lau fringes to the Talbot self-image can be observed by continuous adjustment of the slit width. An expression for describing the effect is derived in terms of an ambiguity function, which is a joint correlation among the functions of the source slit and the two gratings. It is concluded that the profile of the fringe pattern changes not only as the degree of spatial coherence is varied but also as one of the gratings is shifted in a lateral direction. The experimental results are in good agreement with the theoretical intensity distributions.

© 1988 Optical Society of America

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  1. E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
    [CrossRef]
  2. J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  3. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
    [CrossRef]
  4. R. Sudol, R. J. Thompson, “Lau effect, theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
    [CrossRef] [PubMed]
  5. G. J. Swanson, E. N. Leith, “Lau effect and grating imaging,”J. Opt. Soc. Am. 72, 552–555 (1982).
    [CrossRef]
  6. K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
    [CrossRef]
  7. K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
    [CrossRef]
  8. L. Liu, “Theory for Lau effect of plane objects,” Acta Opt. Sin. 6, 807–814 (1986).
  9. S. Cartright, J. Lightman, “Coherence analysis of the Lau interferometer,” Appl. Opt. 25, 3141–3148 (1986).
    [CrossRef]
  10. K. Hane, C. P. Grover, “Imaging with rectangular transmission gratings,” J. Opt. Soc. Am. A 4, 706–711 (1987).
    [CrossRef]
  11. F. Talbot, “Facts relating to optical science no. IV,” Phil. Mag. 9, 401–407 (1836).
  12. J. M. Cowley, A. F. Moodie, “Fourier images: I—The point source,” Proc. Phys. Soc. 70, 486–496 (1957).
    [CrossRef]
  13. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,”J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  14. W. D. Montgomery, “Self-imaging objects of infinite aperture,”J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  15. R. Jozwicki, “The Talbot effect as a sequence of quadratic phase correlation of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
    [CrossRef]
  16. J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
    [CrossRef]
  17. J. Ojeda-Castaneda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
    [CrossRef]
  18. L. Liu, “Ambiguity function and general Talbot-Lau effects,” Acta Opt. Sin. 7, 501–510 (1987).
  19. A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
    [CrossRef]
  20. A. W. Lohmann, J. Ojeda-Castaneda, N. Streible, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
    [CrossRef]
  21. G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
    [CrossRef]
  22. R. Sudol, “Lau effect: An interference phenomenon in partially coherent light,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1981).
  23. K. Hane, S. Hattori, C. P. Grover, “Lau effect in reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
    [CrossRef]
  24. A. Papoulis, “Ambiguity function in Fourier optics,”J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  25. J.-P. Gurigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
    [CrossRef]
  26. K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–225 (1984).
    [CrossRef]

1987 (3)

K. Hane, C. P. Grover, “Imaging with rectangular transmission gratings,” J. Opt. Soc. Am. A 4, 706–711 (1987).
[CrossRef]

L. Liu, “Ambiguity function and general Talbot-Lau effects,” Acta Opt. Sin. 7, 501–510 (1987).

K. Hane, S. Hattori, C. P. Grover, “Lau effect in reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
[CrossRef]

1986 (2)

L. Liu, “Theory for Lau effect of plane objects,” Acta Opt. Sin. 6, 807–814 (1986).

S. Cartright, J. Lightman, “Coherence analysis of the Lau interferometer,” Appl. Opt. 25, 3141–3148 (1986).
[CrossRef]

1985 (1)

J. Ojeda-Castaneda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

1984 (3)

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
[CrossRef]

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–225 (1984).
[CrossRef]

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

1983 (5)

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase correlation of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, N. Streible, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[CrossRef]

1982 (1)

1981 (1)

1979 (2)

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

1978 (1)

J.-P. Gurigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

1974 (1)

1967 (1)

1965 (1)

1957 (1)

J. M. Cowley, A. F. Moodie, “Fourier images: I—The point source,” Proc. Phys. Soc. 70, 486–496 (1957).
[CrossRef]

1948 (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

1836 (1)

F. Talbot, “Facts relating to optical science no. IV,” Phil. Mag. 9, 401–407 (1836).

Brenner, K. H.

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–225 (1984).
[CrossRef]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

Cartright, S.

Cowley, J. M.

J. M. Cowley, A. F. Moodie, “Fourier images: I—The point source,” Proc. Phys. Soc. 70, 486–496 (1957).
[CrossRef]

Gori, F.

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Grover, C. P.

K. Hane, C. P. Grover, “Imaging with rectangular transmission gratings,” J. Opt. Soc. Am. A 4, 706–711 (1987).
[CrossRef]

K. Hane, S. Hattori, C. P. Grover, “Lau effect in reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
[CrossRef]

Gurigay, J.-P.

J.-P. Gurigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

Hane, K.

K. Hane, S. Hattori, C. P. Grover, “Lau effect in reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
[CrossRef]

K. Hane, C. P. Grover, “Imaging with rectangular transmission gratings,” J. Opt. Soc. Am. A 4, 706–711 (1987).
[CrossRef]

Hattori, S.

K. Hane, S. Hattori, C. P. Grover, “Lau effect in reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
[CrossRef]

Indebetouw, G.

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
[CrossRef]

Jahns, J.

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Jozwicki, R.

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase correlation of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

Lau, E.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Leith, E. N.

Lightman, J.

Liu, L.

L. Liu, “Ambiguity function and general Talbot-Lau effects,” Acta Opt. Sin. 7, 501–510 (1987).

L. Liu, “Theory for Lau effect of plane objects,” Acta Opt. Sin. 6, 807–814 (1986).

Lohmann, A. W.

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, N. Streible, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[CrossRef]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Montgomery, W. D.

Moodie, A. F.

J. M. Cowley, A. F. Moodie, “Fourier images: I—The point source,” Proc. Phys. Soc. 70, 486–496 (1957).
[CrossRef]

Ojeda-Castaneda, J.

J. Ojeda-Castaneda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–225 (1984).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, N. Streible, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[CrossRef]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

Papoulis, A.

Patorski, K.

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

Sicre, E. E.

J. Ojeda-Castaneda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

Streible, N.

A. W. Lohmann, J. Ojeda-Castaneda, N. Streible, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[CrossRef]

Sudol, R.

R. Sudol, R. J. Thompson, “Lau effect, theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
[CrossRef] [PubMed]

R. Sudol, “Lau effect: An interference phenomenon in partially coherent light,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1981).

Swanson, G. J.

Talbot, F.

F. Talbot, “Facts relating to optical science no. IV,” Phil. Mag. 9, 401–407 (1836).

Thompson, R. J.

Winthrop, J. T.

Worthington, C. R.

Acta Opt. Sin. (2)

L. Liu, “Theory for Lau effect of plane objects,” Acta Opt. Sin. 6, 807–814 (1986).

L. Liu, “Ambiguity function and general Talbot-Lau effects,” Acta Opt. Sin. 7, 501–510 (1987).

Ann. Phys. (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Appl. Opt. (2)

J. Mod. Opt. (1)

K. Hane, S. Hattori, C. P. Grover, “Lau effect in reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Acta (8)

R. Jozwicki, “The Talbot effect as a sequence of quadratic phase correlation of the object Fourier transform,” Opt. Acta 30, 73–84 (1983).
[CrossRef]

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau effects, a parageometrical approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

J. Ojeda-Castaneda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, “Spatial periodicities in partially coherent fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, N. Streible, “Spatial periodicities in coherent and partially coherent fields,” Opt. Acta 30, 1259–1266 (1983).
[CrossRef]

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
[CrossRef]

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

K. H. Brenner, J. Ojeda-Castaneda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–225 (1984).
[CrossRef]

Opt. Commun. (4)

J.-P. Gurigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136–138 (1978).
[CrossRef]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–17 (1983).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Phil. Mag. (1)

F. Talbot, “Facts relating to optical science no. IV,” Phil. Mag. 9, 401–407 (1836).

Proc. Phys. Soc. (1)

J. M. Cowley, A. F. Moodie, “Fourier images: I—The point source,” Proc. Phys. Soc. 70, 486–496 (1957).
[CrossRef]

Other (1)

R. Sudol, “Lau effect: An interference phenomenon in partially coherent light,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1981).

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of an arrangement for observing the partially coherent diffraction effect occurring between the Lau effect and the Talbot effect.

Fig. 2
Fig. 2

Intensity profiles of self-images as h0 → 0: (a) for the single grating g1 and (b) for the single grating g2.

Fig. 3
Fig. 3

(a), (b) Profiles of the joint self-images with d1 = d2 = 0 and d1 = 0 and d2 = T/2, respectively. (c) Lau fringes as h0 → ∞.

Fig. 4
Fig. 4

Intensity profiles of partially coherent fringes as h0 = T/2: (a) with d1 = d2 = 0, (b) with d1 = T/8 and d2 = 0, (c) with d1 = T/4 and d2 = 0.

Fig. 5
Fig. 5

Photograph of the self-image of the first grating as h0 → 0.

Fig. 6
Fig. 6

Same as Fig. 5 but for the second grating.

Fig. 7
Fig. 7

Photograph of the joint self-image of the two gratings with d1 = d2 = 0 as h0 → 0.

Fig. 8
Fig. 8

Same as Fig. 7 but with d1 = 0 and d2 = T/2.

Fig. 9
Fig. 9

Photograph of the Lau fringes as h0 → ∞.

Fig. 10
Fig. 10

Photograph of the partially coherent fringes with d1 = d2 = 0 as h0 = T/2.

Fig. 11
Fig. 11

Same as Fig. 10 but with d1 = T/8 and d2 = 0.

Fig. 12
Fig. 12

Same as Fig. 10 but with d1 = T/4 and d2 = 0.

Equations (42)

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I ( x 1 , x 2 ) = υ ( x 1 ) υ * ( x 2 ) ,
J ( x , Δ x ) = I ( x + Δ x / 2 , x Δ x / 2 ) .
A ( Δ ν , Δ x ) = J ( x , Δ x ) exp ( 2 π i Δ ν x ) d x ,
I s ( x ) = I 0 rect ( x / h 0 ) ,
J s ( x , Δ x ) = I 0 rect ( x / h 0 ) δ ( Δ x ) .
A s ( Δ ν , Δ x ) = I 0 h 0 sinc ( h 0 Δ ν ) δ ( Δ x ) ,
[ 1 0 λ z 1 ] .
A 1 ( ) ( Δ ν , Δ x ) = A s ( Δ ν , Δ x λ z 1 Δ ν ) = I 0 h 0 λ z 1 sinc ( h 0 Δ x λ z 1 ) δ ( Δ ν Δ x λ z 1 ) .
J 1 ( ) ( x , Δ x ) = F Δ ν , Δ x 1 A 1 ( ) ( Δ ν , Δ x ) = I 0 h 0 λ z 1 sinc ( h 0 Δ x λ z 1 ) exp ( 2 π i x Δ x λ z 1 ) .
A 1 ( + ) ( Δ ν , Δ x ) = A 1 ( ) ( Δ ν Δ ν , Δ x ) A g 1 ( Δ ν , Δ x ) d Δ ν .
A 1 ( + ) ( Δ ν , Δ x ) = I 0 h 0 λ z 1 sinc ( h 0 Δ x λ z 1 ) A g 1 ( Δ ν Δ x λ z 1 , Δ x ) ,
g ( x ) = n rect ( x n T h ) .
J g ( x , Δ x ) = n m ( rect [ x n T d h Λ ( Δ x 2 m T h ) ] + rect { x ( n + 1 2 ) T d h Λ ( Δ x ( 2 m + 1 ) T h ) } ) .
A g ( Δ ν , Δ x ) = n m ( h T Λ ( Δ x 2 m T h ) × sinc [ Δ ν h Λ ( Δ x 2 m T h ) ] + ( 1 ) n h T × Λ [ Δ x ( 2 m + 1 ) T h ] × sinc { Δ ν h Λ [ Δ x ( 2 m + 1 ) T h ] } ) × exp ( 2 π i d Δ ν ) δ ( Δ ν n T ) ,
A 2 ( ) ( Δ ν , Δ x ) = A 1 ( + ) ( Δ ν , Δ x λ z 2 Δ ν ) = I 0 h 0 λ z 1 sinc [ h 0 λ z 1 ( Δ x λ z 2 Δ ν ) ] × A g 1 ( Δ ν Δ x λ z 2 Δ ν λ z 1 , Δ x λ z 2 Δ ν ) .
A 2 ( + ) ( Δ ν , Δ x ) = I 0 h 0 λ z 1 sinc [ h 0 λ z 1 ( Δ x λ z 2 Δ ν ) ] × A g 1 ( M Δ ν Δ x λ z 1 , Δ x λ z 2 Δ ν ) × A g 2 ( Δ ν Δ ν , Δ x ) d Δ ν ,
[ 1 0 λ ( z 3 f ) 1 ]
[ 0 1 / λ f λ f 0 ]
A f ( Δ ν , Δ x ) = A 2 ( + ) [ Δ x λ f , λ f Δ ν ( z 3 f ) Δ x f ] = I 0 h 0 λ z 1 sinc { h 0 [ f z 1 Δ ν + z 2 z 1 Δ ν + ( z 3 f ) Δ x λ z 1 f ] } × A g 1 [ M Δ ν + f z 1 Δ ν + ( z 3 f ) Δ x λ z 1 f , λ f Δ ν λ z 2 Δ ν ( z 3 f ) Δ x f ] × A g 2 [ Δ x λ f Δ ν , λ f Δ ν ( z 3 f ) Δ x f ] d Δ ν ,
I ( x ) = A ( Δ ν , 0 ) exp ( 2 π i Δ ν x ) d Δ ν .
I ( x ) = I 0 h 0 λ z 1 sinc [ h 0 ( f z 1 Δ ν + z 2 z 1 Δ ν ) ] × A g 1 ( M Δ ν + f z 1 Δ ν , λ f Δ ν λ z 2 Δ ν ) × A g 2 ( Δ ν , λ f Δ ν ) exp ( 2 π i Δ ν Δ x ) d Δ ν d Δ ν .
Δ ν = n / T
Δ ν M + f z 1 Δ ν = k / T ,
I ( x ) = I 0 h 0 λ f n k sinc [ h 0 ( k n ) T ] A g 1 [ k T , λ z 1 ( k n T ) ] × A g 2 [ n T , λ z 2 T ( k n z 2 z 1 n ) ] × exp [ 2 π i z 1 x f T ( k M n ) ] .
z 1 = α 1 β 1 T 2 λ , z 2 = α 2 β 2 T 2 λ ,
z 12 = α 12 β 12 T 2 λ ,
( 1 ) α 1 β 1 ( n + p ) p ( 1 ) n ( α 1 β 1 p + α 2 β 2 n ) = ( 1 ) α 1 β 1 p 2 ( 1 ) α 2 β 2 n 2 .
exp ( ± 2 π i α 1 2 β 1 p 2 ) exp ( ± 2 π i α 2 2 β 2 n 2 ) .
I ( x ) = I 0 h 0 λ f p n sinc ( h 0 p T ) h 1 T sinc [ h 1 ( n + p ) T ] × exp [ 2 π i d 1 T ( n + p ) ] h 2 T sinc ( h 2 n T ) exp ( 2 π i d 2 T n ) × exp ( ± 2 π i α 2 2 β 2 n 2 ) exp ( ± 2 π i α 1 2 β 1 p 2 ) × exp ( 2 π i z 1 p z 2 n f T x ) .
n h 1 T sinc [ h 1 ( n + p ) T ] exp ( 2 π i z 2 x f n T ) = n rect ( z 2 f x n T h 1 ) exp ( 2 π i z 2 p x f T ) .
n h 2 T sinc ( h 2 n T ) exp [ 2 π i ( d 2 d 1 ) n T ] exp ( ± 2 π i α 2 2 β 2 n 2 ) × exp ( 2 π i z 2 x f n T ) = n h 2 T sinc ( h 2 β 2 n T ) exp { 2 π i [ ( d 2 d 1 ) ± α 2 T 2 ] n β 2 T } × exp ( 2 π i z 2 x f n β 2 T ) = 1 β 2 m rect [ z 2 f x m T β 2 + ( d 2 d 1 ) ± α 2 T 2 h 2 ] .
I ( x ) = I 0 h 0 λ f p 1 T C T / 2 C + T / 2 n rect ( z 2 f α n T h 1 ) × exp ( 2 π i z 2 α f p T ) × 1 β 2 m rect [ z 2 f x + z 2 f α m T β 2 + ( d 2 d 1 ) ± α 2 T 2 h 2 ] × d ( z 2 f α ) sinc ( h 0 p T ) exp ( ± 2 π i α 1 2 β 1 p 2 ) × exp ( 2 π i d 1 T p ) exp ( 2 π i z 1 x p f T ) ,
I ( x ) = I 0 λ f 1 β 1 k = 0 ( β 1 1 ) rect ( z 1 f x + α d 1 ± α 1 T 2 k T β 1 h 0 ) × n rect ( α n T h 1 ) × 1 β 2 m rect ( z 2 f x + α + ( d 2 d 1 ) ± α 2 T 2 m T β 2 h 2 ) d α ,
I ( x ) = I 0 λ f n rect ( α n T h 1 ) × 1 β 2 m rect ( α z 2 f x + d 2 d 1 ± α 1 T 2 m T β 2 h 2 ) d α .
T 1 = f z 2 T .
lim h 0 0 [ 1 h 0 rect ( x h 0 ) ] = δ ( x ) ,
I ( x ) = I 0 h 0 λ f β 1 β 2 k = 0 ( β 1 1 ) n rect ( z 1 f x d 1 ± α 1 T 2 k T β 1 n T h 1 ) × m rect ( z 1 + z 2 f x d 2 ± α 1 T 2 ± α 2 T 2 m T β 2 h 2 ) = I 0 h 0 λ f β 1 β 2 n rect ( z 1 f x d 1 ± α 1 T 2 n T β 1 h 1 ) × m rect ( z 1 + z 2 f x d 2 ± α 1 T 2 ± α 2 T 2 m T β 2 h 2 ) .
I ( x ) = I 0 h 0 λ f β 1 n rect ( z 1 f x d 1 n T β 1 ± α 1 T 2 h 1 ) .
T t = f T z 1 β 1 = z 2 z 1 β 1 T 1 .
I ( x ) = I 0 h 0 λ f β 12 m rect ( z 12 f x d 2 ± α 12 T 2 m T β 12 h 2 ) .
T t = f z 12 β 12 T = z 12 z 2 β 12 T 1 .
I ( x ) = I 0 h 0 λ f rect ( α h 0 ) × 1 β 1 n rect ( z 1 f x + α d 1 ± α 1 T 2 n T β 1 h 1 ) d α .

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