Abstract

The interference pattern in a cross-grating interferometer is analyzed using first-order analysis. To this order, the fringes are found to be wavelength independent. With white light extended source illumination, the cross-gratinglike structure is localized in a plane disregarding the source position. It is also found that, with some constraint on the separation between cross gratings, the amplitude of a generalized cross grating can be imaged in monochromatic spatially incoherent light. By broadening the spectrum of the illuminating source, each four-beam set of equal paths would form its own cross-gratinglike structure.

© 1986 Optical Society of America

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References

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  1. E. Lau, “Beugungserscheinungen an Deppelrastern,” Ann. Phys. 6, 417 (1948).
    [CrossRef]
  2. J. Jahns, A. W. Lohmann, “The Lau Effect: A Diffraction Experiment with Incoherent Illumination,” Opt. Commun. 28, 263 (1979).
    [CrossRef]
  3. H. O. Bartelt, J. Jahns, “Interferometry Based on the Lau Effect,” Opt. Commun. 30, 268 (1979).
    [CrossRef]
  4. F. Gori, “Lau Effect and Coherent Theory,” Opt. Commun. 31, 4 (1979).
    [CrossRef]
  5. R. Sudal, B. J. Thompson, “An Explanation of the Lau Effect Based on Coherence Theory,” Opt. Commun. 31, 105 (1979).
    [CrossRef]
  6. R. Sudal, B. J. Thompson, “Lau Effect: Theory and Experiment,” Appl. Opt. 20, 1107 (1981).
    [CrossRef]
  7. G. J. Swanson, E. N. Leith, “Lau Effect and Grating Imaging,” J. Opt. Soc. Am. 72, 552 (1982).
    [CrossRef]
  8. K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau Effect: OTF Theory,” Opt. Commun. 46, 14 (1983).
    [CrossRef]
  9. K. Patorski, “Incoherent Superposition of Multiple Self-Imaging Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 745 (1983).
    [CrossRef]
  10. K. Patorski, “Heuristic Explanation of Grating Shearing Interferometry Using Incoherent Illumination,” Opt. Acta 31, 33 (1984).
    [CrossRef]
  11. J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, A Parageometrical Approach,” Opt. Acta 31, 313 (1984).
    [CrossRef]
  12. E. N. Leith, R. Hershey, “Transfer Function and Spatial Filtering in Grating Interferometers,” Appl. Opt. 24, 237 (1985).
    [CrossRef] [PubMed]
  13. S. Jutamulia, T. Asakura, H. Fujii, “Lau Effect and Noncoherent Processing,” Opt. Commun. 53, 77 (1985).
    [CrossRef]
  14. H. O. Bartelt, Y. Li, “Lau Interferometry with Cross Gratings,” Opt. Commun. 48, 1 (1983).
    [CrossRef]
  15. E. N. Leith, B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217 (1977).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 128.
  17. Y. S. Cheng, “Fringe Formation in Incoherent Light with a Two-Grating Interferometer,” Appl. Opt. 23, 3057 (1984).
    [CrossRef] [PubMed]
  18. H. W. Lippincott, H. Stark, “Optical-Digital Detection of Dents and Scratches on Specular Metal Surfaces,” Appl. Opt. 21, 2875 (1982).
    [CrossRef] [PubMed]

1985 (2)

E. N. Leith, R. Hershey, “Transfer Function and Spatial Filtering in Grating Interferometers,” Appl. Opt. 24, 237 (1985).
[CrossRef] [PubMed]

S. Jutamulia, T. Asakura, H. Fujii, “Lau Effect and Noncoherent Processing,” Opt. Commun. 53, 77 (1985).
[CrossRef]

1984 (3)

K. Patorski, “Heuristic Explanation of Grating Shearing Interferometry Using Incoherent Illumination,” Opt. Acta 31, 33 (1984).
[CrossRef]

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, A Parageometrical Approach,” Opt. Acta 31, 313 (1984).
[CrossRef]

Y. S. Cheng, “Fringe Formation in Incoherent Light with a Two-Grating Interferometer,” Appl. Opt. 23, 3057 (1984).
[CrossRef] [PubMed]

1983 (3)

H. O. Bartelt, Y. Li, “Lau Interferometry with Cross Gratings,” Opt. Commun. 48, 1 (1983).
[CrossRef]

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

K. Patorski, “Incoherent Superposition of Multiple Self-Imaging Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 745 (1983).
[CrossRef]

1982 (2)

1981 (1)

1979 (4)

J. Jahns, A. W. Lohmann, “The Lau Effect: A Diffraction Experiment with Incoherent Illumination,” Opt. Commun. 28, 263 (1979).
[CrossRef]

H. O. Bartelt, J. Jahns, “Interferometry Based on the Lau Effect,” Opt. Commun. 30, 268 (1979).
[CrossRef]

F. Gori, “Lau Effect and Coherent Theory,” Opt. Commun. 31, 4 (1979).
[CrossRef]

R. Sudal, B. J. Thompson, “An Explanation of the Lau Effect Based on Coherence Theory,” Opt. Commun. 31, 105 (1979).
[CrossRef]

1977 (1)

E. N. Leith, B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217 (1977).
[CrossRef]

1948 (1)

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

Asakura, T.

S. Jutamulia, T. Asakura, H. Fujii, “Lau Effect and Noncoherent Processing,” Opt. Commun. 53, 77 (1985).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, Y. Li, “Lau Interferometry with Cross Gratings,” Opt. Commun. 48, 1 (1983).
[CrossRef]

H. O. Bartelt, J. Jahns, “Interferometry Based on the Lau Effect,” Opt. Commun. 30, 268 (1979).
[CrossRef]

Brenner, K. H.

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

Chang, B. J.

E. N. Leith, B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217 (1977).
[CrossRef]

Cheng, Y. S.

Fujii, H.

S. Jutamulia, T. Asakura, H. Fujii, “Lau Effect and Noncoherent Processing,” Opt. Commun. 53, 77 (1985).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 128.

Gori, F.

F. Gori, “Lau Effect and Coherent Theory,” Opt. Commun. 31, 4 (1979).
[CrossRef]

Hershey, R.

Jahns, J.

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, A Parageometrical Approach,” Opt. Acta 31, 313 (1984).
[CrossRef]

H. O. Bartelt, J. Jahns, “Interferometry Based on the Lau Effect,” Opt. Commun. 30, 268 (1979).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau Effect: A Diffraction Experiment with Incoherent Illumination,” Opt. Commun. 28, 263 (1979).
[CrossRef]

Jutamulia, S.

S. Jutamulia, T. Asakura, H. Fujii, “Lau Effect and Noncoherent Processing,” Opt. Commun. 53, 77 (1985).
[CrossRef]

Lau, E.

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

Leith, E. N.

Li, Y.

H. O. Bartelt, Y. Li, “Lau Interferometry with Cross Gratings,” Opt. Commun. 48, 1 (1983).
[CrossRef]

Lippincott, H. W.

Lohmann, A. W.

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, A Parageometrical Approach,” Opt. Acta 31, 313 (1984).
[CrossRef]

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

J. Jahns, A. W. Lohmann, “The Lau Effect: A Diffraction Experiment with Incoherent Illumination,” Opt. Commun. 28, 263 (1979).
[CrossRef]

Ojeda-Castaneda, J.

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, A Parageometrical Approach,” Opt. Acta 31, 313 (1984).
[CrossRef]

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

Patorski, K.

K. Patorski, “Heuristic Explanation of Grating Shearing Interferometry Using Incoherent Illumination,” Opt. Acta 31, 33 (1984).
[CrossRef]

K. Patorski, “Incoherent Superposition of Multiple Self-Imaging Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 745 (1983).
[CrossRef]

Stark, H.

Sudal, R.

R. Sudal, B. J. Thompson, “Lau Effect: Theory and Experiment,” Appl. Opt. 20, 1107 (1981).
[CrossRef]

R. Sudal, B. J. Thompson, “An Explanation of the Lau Effect Based on Coherence Theory,” Opt. Commun. 31, 105 (1979).
[CrossRef]

Swanson, G. J.

Thompson, B. J.

R. Sudal, B. J. Thompson, “Lau Effect: Theory and Experiment,” Appl. Opt. 20, 1107 (1981).
[CrossRef]

R. Sudal, B. J. Thompson, “An Explanation of the Lau Effect Based on Coherence Theory,” Opt. Commun. 31, 105 (1979).
[CrossRef]

Ann. Phys. (1)

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

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

Opt. Acta (3)

K. Patorski, “Incoherent Superposition of Multiple Self-Imaging Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 745 (1983).
[CrossRef]

K. Patorski, “Heuristic Explanation of Grating Shearing Interferometry Using Incoherent Illumination,” Opt. Acta 31, 33 (1984).
[CrossRef]

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, A Parageometrical Approach,” Opt. Acta 31, 313 (1984).
[CrossRef]

Opt. Commun. (8)

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

J. Jahns, A. W. Lohmann, “The Lau Effect: A Diffraction Experiment with Incoherent Illumination,” Opt. Commun. 28, 263 (1979).
[CrossRef]

H. O. Bartelt, J. Jahns, “Interferometry Based on the Lau Effect,” Opt. Commun. 30, 268 (1979).
[CrossRef]

F. Gori, “Lau Effect and Coherent Theory,” Opt. Commun. 31, 4 (1979).
[CrossRef]

R. Sudal, B. J. Thompson, “An Explanation of the Lau Effect Based on Coherence Theory,” Opt. Commun. 31, 105 (1979).
[CrossRef]

S. Jutamulia, T. Asakura, H. Fujii, “Lau Effect and Noncoherent Processing,” Opt. Commun. 53, 77 (1985).
[CrossRef]

H. O. Bartelt, Y. Li, “Lau Interferometry with Cross Gratings,” Opt. Commun. 48, 1 (1983).
[CrossRef]

E. N. Leith, B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217 (1977).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 128.

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Figures (4)

Fig. 1
Fig. 1

Cross-grating interferometer illuminated with a plane wave.

Fig. 2
Fig. 2

Cross-grating interferometer illuminated with a diverging wave.

Fig. 3
Fig. 3

Cross-grating interferometer illuminated with a converging wave.

Fig. 4
Fig. 4

Localized cross-gratinglike pattern (2 × 105 dots/mm2).

Equations (30)

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u = exp [ i 2 π ( f a x + f b y ) ] ,
g 1 ( x , y ) = ¼ exp [ i π ( f 1 x + f 2 y ) ] + ¼ exp [ - i π ( f 1 x + f 2 y ) ] + ¼ exp [ i π ( f 1 x - f 2 y ) ] + ¼ exp [ - i π ( f 1 x - f 2 y ) ] .
u 1 2 = exp i [ 2 π ( f a x + f b y ) ± π ( f 1 x + f 2 y ) ] , u 3 4 = exp i [ 2 π ( f a x + f b y ) ± π ( f 1 x - f 2 y ) ] ,
u 1 2 = exp i { 2 π ( f a x + f b y ) ± π ( f 1 x + f 2 y ) - π λ d 1 [ ( f a ± f 1 / 2 ) 2 + ( f b ± f 2 / 2 ) 2 ] } , u 3 4 = exp i { 2 π ( f a x + f b y ) ± π ( f 1 x - f 2 y ) - π λ d 1 [ ( f a ± f 1 / 2 ) 2 + ( f b f 2 / 2 ) 2 ] } ,
u 1 2 = exp ± i { π ( f 1 - f 3 ) x + ( f 2 - f 4 ) y ] - π λ f a [ d 1 f 1 + d 2 ( f 1 - f 3 ) ] - π λ f b [ d 1 f 2 + d 2 ( f 2 - f 4 ) ] } , u 3 4 = exp ± i { π ( f 1 - f 3 ) x - ( f 2 - f 4 ) y ] - π λ f a [ d 1 f 1 + d 2 ( f 1 - f 3 ) ] + π λ f b [ d 1 f 2 + d 2 ( f 2 - f 4 ) ] } .
I ( θ , ϕ , λ ; x , y , z ) = u 1 + u 2 + u 3 + u 4 2 = ( ½ + ½ cos { 2 π ( f 1 - f 3 ) x - 2 π sin θ sin ϕ [ d 1 f 1 + d 2 ( f 1 - f 3 ) ] } ) · ( ½ + ½ cos { 2 π ( f 2 - f 4 ) y - 2 π sin θ sin ϕ [ d 1 f 2 + d 2 ( f 2 - f 4 ) ] } ) .
I = { ½ + ½ cos [ 2 π ( f 3 - f 1 ) x ] } { ½ + ½ cos [ 2 π ( f 3 f 2 / f 1 - f 2 ) ] } .
I ( θ , ϕ , λ ; x , y , z ) = { ½ + ½ cos [ 2 π ( f 3 - f 1 ) ( x - z sin θ cos ϕ ) ] } · { ½ + ½ cos [ 2 π ( f 4 - f 2 ) ( y - z sin θ sin ϕ ) ] } .
cos ( π f 1 x ) cos ( π f 2 y ) = 1 8 m = - n = - [ sin c ( m - ½ ) + sinc ( m + ½ ) ] [ sinc ( n - ½ ) + sinc ( n + ½ ) ] exp ( i 2 π m f 1 x ) × exp ( i 2 π n f 2 y ) ,
u 1 2 = exp i { 2 π ( f a x + f b y ) ± 2 π ( f 1 - f 3 ) x - π λ d 1 [ ( f a ± f 1 ) 2 + f b 2 ] - π λ d 2 [ ( f a ± f 1 f 3 ) 2 + f b 2 ] } , u 3 4 = exp i { 2 π ( f a x + f b y ) ± 2 π ( f 2 - f 4 ) y - π λ d 1 [ f a 2 + ( f b ± f 2 ) 2 ] - π λ d 2 [ f a 2 + ( f b ± f 2 f 4 ) 2 ] } .
u 1 + u 2 = cos [ 2 π ( f 3 - f 1 ) ( x - z sin θ cos ϕ ) ] × exp { - i π λ [ d 1 f 1 f 3 + z ( f 3 - f 1 ) 2 ] } , u 3 + u 4 = cos [ 2 π ( f 4 - f 2 ) ( y - z sin θ sin ϕ ) ] × exp { - i π λ [ d 1 f 2 f 4 + z ( f 4 - f 2 ) 2 ] } .
I ( x , y , 0 ) = { ½ + ½ cos [ 2 π ( f 3 - f 1 ) ( x + y ) ] } × { ½ + ½ cos [ 2 π ( f 3 - f 1 ) ( x - y ) ] } .
I ( x , y , 0 ) = { ½ + ½ cos [ 2 2 π ( f 3 - f 1 ) x ] } × { ½ + ½ cos [ 2 2 π ( f 3 - f 1 ) y ] } .
I ( θ , ϕ , λ ; x , y , z ) = { ½ + ½ cos [ 4 π ( f 3 - f 1 ) ( x - z sin θ cos ϕ ) ] } · { ½ + ½ cos [ 4 π ( f 4 - f 2 ) ( y - z sin θ cos ϕ ) ] } .
u = exp i π λ z 0 [ ( x - x 0 ) 2 + ( y - y 0 ) 2 ] ,
u 1 2 = exp i ( π λ z 0 { [ x - ( x 0 ½ λ f 1 z 0 ) ] 2 + [ y - ( y 0 ½ λ f 2 z 0 ) ] 2 } ± π f 1 x 0 ± π f 2 y 0 - ¼ π λ z 0 ( f 1 2 + f 2 2 ) ) , u 3 4 = exp i ( π λ z 0 { [ x - ( x 0 ± ½ λ f 1 z 0 ) ] 2 + [ y - ( y 0 ½ λ f 2 z 0 ) ] 2 } π f 1 x 0 ± π f 2 y 0 - ¼ π λ z 0 ( f 1 2 + f 2 2 ) ) .
u 1 2 = exp i ( π λ ( z 0 + d 1 ) { [ x - ( x 0 ½ λ f 1 z 0 ) ] 2 + [ y - ( y 0 ½ λ f 2 z 0 ) ] 2 } ± π f 1 x 0 ± π f 2 y 0 ) ; u 3 4 = exp i ( π λ ( z 0 + d 1 ) { [ x - ( x 0 ± ½ λ f 1 z 0 ) ] 2 + [ y - ( y 0 ½ λ f 2 z 0 ) ] 2 } π f 1 x 0 ± π f 2 y 0 ) .
u 1 2 = exp i ( π λ ( z 0 + d 1 ) { x - [ x 0 ½ λ f 1 z 0 ± ½ λ f 3 ( z 0 + d 1 ) ] } 2 + π λ ( z 0 + d 1 ) { y - [ y 0 ½ λ f 2 z 0 ± ½ λ f 4 ( z 0 + d 1 ) ] } 2 π ( f 3 - f 1 ) x 0 π ( f 4 - f 2 ) y 0 + ½ π λ z 0 ( f 1 f 3 + f 2 f 4 ) - ¼ π λ ( z 0 + d 1 ) ( f 3 2 + f 4 2 ) ) , u 3 4 = exp i ( π λ ( z 0 + d 1 ) { x - [ x 0 ± ½ λ f 1 z 0 ½ λ f 3 ( z 0 + d 1 ) ] } 2 + π λ ( z 0 + d 1 ) { y - [ y 0 ½ λ f 2 z 0 ± ½ λ f 4 ( z 0 + d 1 ) ] } 2 ± π ( f 3 - f 1 ) x 0 π ( f 4 - f 2 ) y 0 + ½ π λ z 0 ( f 1 f 3 + f 2 f 4 ) - ¼ π λ ( z 0 + d 1 ) ( f 3 2 + f 4 2 ) ) .
u 1 2 = exp i ( π λ ( z 0 + d 1 + d 2 ) { ( x - x 0 ) ± ½ λ [ z 0 f 1 - ( z 0 + d 1 ) f 3 ] 2 + π λ ( z 0 + d 1 + d 2 ) { ( y - y 0 ) ± ½ λ [ z 0 f 2 - ( z 0 + d 1 ) f 4 ] } 2 π ( f 3 - f 1 ) x 0 π ( f 4 - f 2 ) y 0 ) ; u 3 4 = exp i ( π λ ( z 0 + d 1 + d 2 ) { ( x - x 0 ) ½ λ [ z 0 f 1 - ( z 0 + d 1 ) f 3 ] } 2 + π λ ( z 0 + d 1 + d 2 ) { ( y - y 0 ) ± ½ λ [ z 0 f 2 - ( z 0 + d 1 ) f 4 ] } 2 ± π ( f 3 - f 1 ) x 0 π ( f 4 - f 2 ) y 0 ) .
I ( x 0 , y 0 , λ ; x , y ) = { ½ + ½ cos [ 2 π z 0 f 1 - ( z 0 + d 1 ) f 3 z 0 + d 1 + d 2 ( x - x 0 ) - 2 π ( f 3 - f 1 ) x 0 ] } · { ½ + ½ cos [ 2 π z 0 f 2 - ( z 0 + d 1 ) f 4 z 0 + d 1 + d 2 ( y - y 0 ) - 2 π ( f 4 - f 2 ) y 0 ] } .
I ( x 0 , y 0 , λ ; x , y ) = { ½ + ½ cos [ 2 π ( f 3 - f 1 ) x ] } × { ½ + ½ cos [ 2 π ( f 4 - f 2 ) y ] } ,
u = exp { - i π λ z 0 [ ( x - x 0 ) 2 + ( y - y 0 ) 2 ] } .
g 1 ( x , y ) = m n a m b n exp i 2 π ( m f 1 x + n f 2 y )
g 2 ( x , y ) = m n A m B n exp i 2 π ( m f 3 x + n f 4 y ) .
u = m n m n a m b n A m B n × exp [ i 2 π ( f a + m f 1 + m f 3 ) x ] exp [ i 2 π ( f b + n f 2 + n f 4 ) y ] · exp { - i π λ d 1 [ ( f a + m f 1 ) 2 + ( f b + n f 2 ) 2 ] } × exp { - i π λ d 2 [ ( f a + m f 1 + m f 3 ) 2 + ( f b + n f 2 + n f 4 ) 2 ] } .
u = m n a m b n A - m B - n exp [ - i 2 π m ( f 3 - f 1 ) x ] × exp [ - i 2 π n ( f 4 - f 2 ) y ] · exp { - i 2 π λ f a m [ d 1 f 1 - d 2 ( f 3 - f 1 ) ] } · exp { - i 2 π λ f b n [ d 1 f 2 - d 2 ( f 4 - f 2 ) ] } · exp ( - i π λ { d 1 ( m 2 f 1 2 + n 2 f 2 2 ) + d 2 [ m 2 ( f 3 - f 1 ) 2 + n 2 ( f 4 - f 2 ) 2 ] } ) .
u = m n a m b n A - m B - n exp [ - i 2 π m ( f 3 - f 1 ) x ] × exp [ - i 2 π n ( f 4 - f 2 ) y ] · exp [ - i π λ d 1 ( m 2 f 1 f 3 + n 2 f 2 f 4 ) ] .
u = m n a m b n A - m B - n exp [ i 2 π m ( f 3 - f 1 ) x ] × exp [ i 2 π n ( f 4 - f 2 ) y ] · exp [ - i π λ d 1 ( m 2 f 1 f 3 + n 2 f 2 f 4 ) ] .
u = m n a m b n ( A - m B - n ) exp [ i 2 π m ( f 3 - f 1 ) x ] × exp [ i 2 π n ( f 4 - f 2 ) y ] .
I = m n m n a m b n ( A - m B - n ) a m * b n * ( A - m * B - n * ) × exp [ i 2 π ( m - m ) ( f 3 - f 1 ) x ] · exp [ i 2 π ( n - n ) ( f 4 - f 2 ) y ] × exp { - i π λ d 1 [ ( m 2 - m 2 ) f 1 f 3 + ( n 2 - n 2 ) f 2 f 4 ] } .

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